From fc2287d3bb8bef54fc9a30f1846e31f7e307480b Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 05:42:39 +0000 Subject: [PATCH 1/7] Soft rejected pairs (d = 2): the 1/kappa tail of the rejected cusp kernel, the fold-scale rejection function, and a conjectured l^(2/3) term New packet frontiers/soft_rejected_pairs_20261002 (CL-SOFT-REJECTED-20261002-v1), author-side proof candidate and conjectures, scientific effect NONE: - Theorem 1: kappa A^rej(b, kappa, u) = F0(b, u) + O(1/kappa), F0 = (5/24) pi0 pA(0|b) E gamma^6; the rejected weight sits on soft transverse curvatures lambda in [gamma^2/(24 kappa), gamma^2/(8 kappa)]. - The fold-scale soft model G_k (new jets B = d_u A, C3 = d_Theta^3 f); Lemma 2 (the elder decision by a 1-D scan, including a case where the saddle is a slice minimum); Lemma 3 (an a priori elder region). - Proposition 4 (Gaussian kernel): the fold-scale rejection rate is F0(b) H(k), H = 1 + (12/25) k^2 + O(k^3). - Lemma 5: the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)). - Conjectures 6 (fold-scale limit) and 7 (rho_rej + nu_eld^far = B_{2,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 for the Gaussian kernel, not certified). Standard-library checker soft_check.py (controls S1-S10, mutants M1-M10), RESULTS.json, SOURCES.json, and the workflow soft-rejected-pairs.yml. A same-family referee pass (MAJOR REVISION, then ACCEPT WITH MINOR FIXES on the delta) is applied; nonauthor review is required. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .github/workflows/soft-rejected-pairs.yml | 157 ++++++ .../soft_rejected_pairs_20261002/PROOF.md | 515 +++++++++++++++++ .../soft_rejected_pairs_20261002/README.md | 89 +++ .../soft_rejected_pairs_20261002/RESULTS.json | 239 ++++++++ .../soft_rejected_pairs_20261002/SOURCES.json | 200 +++++++ .../soft_check.py | 516 ++++++++++++++++++ 6 files changed, 1716 insertions(+) create mode 100644 .github/workflows/soft-rejected-pairs.yml create mode 100644 frontiers/soft_rejected_pairs_20261002/PROOF.md create mode 100644 frontiers/soft_rejected_pairs_20261002/README.md create mode 100644 frontiers/soft_rejected_pairs_20261002/RESULTS.json create mode 100644 frontiers/soft_rejected_pairs_20261002/SOURCES.json create mode 100644 frontiers/soft_rejected_pairs_20261002/soft_check.py diff --git a/.github/workflows/soft-rejected-pairs.yml b/.github/workflows/soft-rejected-pairs.yml new file mode 100644 index 000000000..223457911 --- /dev/null +++ b/.github/workflows/soft-rejected-pairs.yml @@ -0,0 +1,157 @@ +name: Soft rejected pairs exact controls + +'on': + pull_request: + paths: + - 'frontiers/soft_rejected_pairs_20261002/**' + - '.github/workflows/soft-rejected-pairs.yml' + # bound merged sources (consumed and cited in SOURCES.json); the cited unmerged sources are bound through the API + - 'frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md' + - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md' + - 'reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md' + - 'frontiers/cusp_second_order_20261001/PROOF.md' + - 'frontiers/candidate_third_order_20261001/PROOF.md' + - 'frontiers/elder_third_order_20261001/PROOF.md' + - 'frontiers/third_order_rate_20261001/PROOF.md' + - 'frontiers/c7_total_bounded_20260929/PROOF.md' + - 'frontiers/far_elder_rate_20260930/PROOF.md' + workflow_dispatch: + +permissions: + contents: read + pull-requests: read + +jobs: + exact-controls: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Verify sources, replay both modes, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + GH_TOKEN: ${{ github.token }} + GH_REPOSITORY: ${{ github.repository }} + run: | + set -euo pipefail + python -B -S - <<'PY' + import base64, hashlib, json, os, pathlib, subprocess, sys, urllib.parse, urllib.request + root = pathlib.Path('frontiers/soft_rejected_pairs_20261002').resolve() + manifest = json.loads((root / 'SOURCES.json').read_text()) + present = sorted(p.name for p in root.iterdir()) + expected_names = sorted([e['path'] for e in manifest['files']] + ['SOURCES.json']) + if present != expected_names: + raise ValueError('packet tree differs from manifest: ' + repr(present)) + for entry in manifest['files']: + path = root / entry['path'] + if path.is_symlink(): + raise ValueError('symlink: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('packet file identity mismatch: ' + entry['path']) + def same(data, entry): + framed = b'blob ' + str(len(data)).encode() + b'\0' + data + return len(data) == entry['bytes'] and hashlib.sha256(data).hexdigest() == entry['sha256'] and hashlib.sha1(framed).hexdigest() == entry['git_blob'] + def tree_entry(path): + # the checked-out commit's tree entry for path: (mode, type, blob) or None; symlinks have mode 120000 + out = subprocess.run(['git', 'ls-tree', '-z', 'HEAD', '--', path], check=True, capture_output=True).stdout + if not out: + return None + meta, name = out.rstrip(b'\0').split(b'\t', 1) + if name.decode() != path: + raise ValueError('unexpected tree entry for ' + path) + mode, kind, sha = meta.decode().split() + return mode, kind, sha + def tree_bytes(entry): + # a source read from the checkout must be a regular file whose tree blob is the recorded one + te = tree_entry(entry['path']) + if te is None: + return None + mode, kind, sha = te + if mode != '100644' or kind != 'blob' or sha != entry['git_blob']: + raise ValueError('tree entry is not the recorded regular-file blob (mode/type/blob ' + ' '.join(te) + '): ' + entry['path']) + path = pathlib.Path(entry['path']) + if path.is_symlink() or not path.is_file(): + raise ValueError('checkout path is not a regular file: ' + entry['path']) + return path.read_bytes() + for entry in manifest['consumed'] + manifest['cited_only']: + data = tree_bytes(entry) + if data is None or not same(data, entry): + raise ValueError('merged source drifted or absent: ' + entry['path']) + # unmerged sources are bound to their recorded commit, path and blob through the repository API: the head must + # resolve, the path in that head's tree must be a file with the recorded blob id and size, and the bytes must have + # the recorded sha256; a consumed dependency that has landed is instead read from the tree and must be identical + # to the pinned version (else this packet needs rebinding) + api = 'https://api.github.com/repos/' + os.environ['GH_REPOSITORY'] + def get_json(url): + req = urllib.request.Request(url, headers={'Authorization': 'Bearer ' + os.environ['GH_TOKEN'], + 'Accept': 'application/vnd.github+json'}) + with urllib.request.urlopen(req, timeout=60) as resp: + return json.load(resp) + def bound(entry): + commit = get_json(api + '/commits/' + entry['head']) + if commit.get('sha') != entry['head']: + raise ValueError('recorded head does not resolve: ' + entry['head']) + node = get_json(api + '/contents/' + urllib.parse.quote(entry['path']) + '?ref=' + entry['head']) + if (node.get('type') != 'file' or node.get('path') != entry['path'] or node.get('sha') != entry['git_blob'] + or node.get('size') != entry['bytes']): + raise ValueError('recorded head/path does not map to the recorded blob: ' + entry['path']) + blob = get_json(api + '/git/blobs/' + entry['git_blob']) + if blob.get('encoding') != 'base64': + raise ValueError('unexpected blob encoding for ' + entry['path']) + return base64.b64decode(blob['content']) + def current(entry): + # drift gate (as on #220, after Copilot's review there): the recorded head keeps resolving after a dependency PR advances, so + # also resolve the dependency PR now. Open: the file at its current head must be the pinned blob. Merged: + # the file at its merge commit must be the pinned blob. Closed unmerged: rebinding is required. + pr = get_json(api + '/pulls/' + entry['pr'].split('#', 1)[1]) + if pr.get('merged'): + ref = pr.get('merge_commit_sha') + elif pr.get('state') == 'open': + ref = pr['head']['sha'] + else: + raise ValueError('dependency ' + entry['pr'] + ' closed without merge: rebind required') + node = get_json(api + '/contents/' + urllib.parse.quote(entry['path']) + '?ref=' + ref) + if node.get('type') != 'file' or node.get('sha') != entry['git_blob'] or node.get('size') != entry['bytes']: + raise ValueError('dependency ' + entry['pr'] + ' advanced: ' + entry['path'] + ' at ' + ref + + ' is not the pinned blob (rebind required)') + return ref + for kind in ('consumed_unmerged', 'cited_unmerged'): + for entry in manifest.get(kind, []): + # a landed consumed dependency must be the recorded regular-file blob in this tree (symlinks and any + # other mode are rejected by tree_bytes); otherwise the source is bound through the API + landed = tree_bytes(entry) if kind == 'consumed_unmerged' else None + if landed is not None: + data, where = landed, 'tree' + else: + if os.path.lexists(entry['path']) and kind == 'consumed_unmerged': + raise ValueError('path present in the checkout but not as a tracked blob: ' + entry['path']) + data, where = bound(entry), 'head/path/blob api' + if not same(data, entry): + raise ValueError(kind + ' source differs from its pinned blob (rebind required): ' + entry['path'] + ' via ' + where) + now = current(entry) if kind == 'consumed_unmerged' else None + print(json.dumps({kind + '_bound': entry['path'], 'git_blob': entry['git_blob'], 'head': entry['head'], + 'via': where, 'current_ref_checked': now})) + script = str(root / 'soft_check.py') + expected = (root / 'RESULTS.json').read_bytes() + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script], check=True, capture_output=True, timeout=600).stdout + if out != expected: + raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags)) + for m in manifest['mutants']: + rc = subprocess.run([sys.executable, *flags, script, '--mutant', m], capture_output=True, timeout=600).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + rc = subprocess.run([sys.executable, *flags, script, '--mutant', 'M99'], capture_output=True, timeout=600).returncode + if rc != 2: + raise RuntimeError('unknown mutant label not rejected with exit 2') + print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False})) + PY + git diff --exit-code diff --git a/frontiers/soft_rejected_pairs_20261002/PROOF.md b/frontiers/soft_rejected_pairs_20261002/PROOF.md new file mode 100644 index 000000000..d33a5ba1e --- /dev/null +++ b/frontiers/soft_rejected_pairs_20261002/PROOF.md @@ -0,0 +1,515 @@ +# Soft rejected pairs in `d = 2`: the `1/κ` tail of the rejected cusp kernel, the fold-scale rejection function, and a conjectured `ℓ^{2/3}` term + +Object: CL-SOFT-REJECTED-20261002-v1. +Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 2 October 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem 1, Lemmas 2, 3 and 5, Proposition 4); CONJECTURES 6 and 7, with numerical +evidence. Nonauthor review required. Scientific effect: NONE — no register, graph, STATUS, PROOF_INDEX, prize or Boolean +change; no numerical constant is certified. Same GitHub account as every lane; zero organizational independence. + +**Why.** Math- #229 (merged) proves +`ρ_rej(ℓ) = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} − ν_eld^{far,r_0^*}(ℓ) + O(ℓ^{3/7})` ((R⁺.1)). Math- #240 (open) proves that each +fixed cusp window `κ₀ ≤ ℓ/r⁴ ≤ κ₁` contributes a multiple of `ℓ^{1/4}` with error `O(ℓ^{3/4})`, and its Remark 1 leaves the +ends `κ → ∞` and `κ → 0` open. This note studies the end `κ → ∞` for the rejected pairs in `d = 2`. Formally it is the only +source of the next term of `ρ_rej` (§4). + +**What is new.** +- **Theorem 1** (§1). In `d = 2`, `κ𝒜^{rej}(b, κ, u) = F₀(b, u) + O(κ^{−1})` as `κ → ∞`, uniformly with Gaussian decay in + `b`, where `F₀ = (5/24)π₀(u; v₀(b, 0))p_A(0 | b)E[γ⁶] > 0`. The rejected weight sits on *soft* transverse curvatures + `λ = −A ∈ [γ²/(24κ), γ²/(8κ)]`, up to relative errors `O(1/κ)`. +- **The fold-scale soft model** (§2). At fixed `k = ℓ/r³` and `λ = λ̃r/k`, the window field divided by `κ = k/r` tends to an + explicit polynomial `G_k` in which two more jets enter: `B = ∂_uA` and `C₃ = ∂_Θ³f`. **Lemma 2** decides the elder mark in + `G_k` exactly by a one-dimensional scan, including a case in which the saddle is not on the ridge. **Lemma 3** gives an a + priori elder region. The typed weight is `(γ⁴/16)(φ^{−2} − (1 − 12kB/γ²)²)₊`. +- **Proposition 4** (§3). For the Gaussian kernel, the fold-scale rejection rate is `F(k; b) = F₀(b)H(k)` with `H` free of `b`, + and `H(k) = 1 + (12/25)k² + O(k³)`. The elder edge moves to `φ_e = 1/3 + t/3 + 10t²/27 + O(t³)` (`t = 12kB/γ²`). The + resulting `+312/25` nearly cancels the `−12` of the pin density `e^{−12k²}`. +- **Lemma 5 and Conjecture 7** (§4). The two-scale composite density has the expansion + `(I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`, with `R_{2/3}` given by `F` and `F₀`. Conjecture 7 says that + `ρ_rej + ν_eld^{far,r_0^*} − B_{2,L}` has the same expansion. For the Gaussian kernel, numerically `R_{2/3} ≈ −0.049` + (§5). +- **Evidence** (§5, exploration): + - the tail of Theorem 1 by quadrature; + - Lemma 2 against a two-dimensional flood fill; + - `H` by quadrature; + - #216's Monte Carlo of the rejected adjacent pairs, binned by lifetime and by `s = r/ℓ^{1/4}`. It shows the predicted + suppression at small `s`: for `s < 0.8`, 37 pairs are observed and 35.3 predicted, against 4,322 from the cusp kernel + alone. + +**Dependencies.** +- Consumed (all merged): #220 (`frontiers/elder_third_order_20261001/PROOF.md`, blob `c8767dde`): §0 ((0.1), (0.2)) and §1 + (window coordinates, (1.1), Lemma Q, (M1)–(M6)). #218 (`frontiers/candidate_third_order_20261001/PROOF.md`, blob + `70ca57ef`): §0 (`𝒜^{cand}`, `I^{cand}`, `B_{d,L}`). #229 (`frontiers/third_order_rate_20261001/PROOF.md`, blob + `110ed33a`): (R⁺.1) and §0 (`Y_r`). #207 (`frontiers/cusp_second_order_20261001/PROOF.md`, blob `f6df5a73`): §§0, 4, 6 and + Theorem CU.2(c). [R] (`frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, blob `247b3ecf`): (R5). [P] §2, through #220. +- Cited only: [C7-K] (`frontiers/c7_total_bounded_20260929/PROOF.md`, blob `28748b08`): (K2). #240 (open): Lemma Q′, Step N6, + Corollary N′, Remarks 1 and 4. #237 (open): Theorem P, Remark 4. #216 (open): the Monte Carlo. #187 (merged) and #188 (open): the far + elder density. + +## 0. Setting + +Setting and notation are those of #220 §0 and #218 §0 (through #207 §§0, 4, 6), specialized to `d = 2`, `m = 1`: +- the [P] field on `X = R²/(LZ²)`; near pins `M = −ru/2`, `S = ru/2` at heights `b` and `b − kr³`; `κ = k/r`, so the lifetime is + `ℓ = kr³ = κr⁴`; `Θ` is the unit vector orthogonal to `u`; +- the jets at `0` under the contact law at `v₀(b, k)` (#220 §2; expectation `E_{v₀(b,k)}`, and `E₀[· | b]` at `k = 0`): + `A = ∂_Θ²f` (a scalar), `f₄ = ∂_u⁴f`, `γ = ∂_u²∂_Θf`, `B = ∂_u∂_Θ²f = ∂_uA`, `C₃ = ∂_Θ³f`. On `{A < 0}` put `λ := −A`, so + `Δ = A = −λ`, `adj A = 1`, and + + Y = (f₄/12)Δ − γ²/4 = −(λ/12) z, z := f₄ + 3γ²/λ = f₄ − 3γᵀA^{−1}γ, φ := z/(72κ); (0.1) + +- the cusp kernels (#220 (0.1), #218 §0): `𝒜^*(b, κ, u) = 12π₀(u; v₀(b, 0))E₀[w_κ 1_* | b]` with + `w_κ = 36κ²Δ² − Y² = (λ²/144)(5184κ² − z²)`, `1_cand = 1{A < 0, |z| < 72κ}` and `1_eld = 1{A < 0, |z| < 24κ}`. So + + 𝒜^{rej}(b, κ, u) := (𝒜^{cand} − 𝒜^{eld})(b, κ, u) = 12π₀(u; v₀(b, 0)) E₀[(λ²/144)(5184κ² − z²) 1{A < 0, 24κ ≤ |z| < 72κ} | b]. (0.2) + + The contact terms cancel, so by #220 (0.2) and #218 §0, `I^{cand} − c₁ = ∫_{S¹}∫_R∫_0^∞𝒜^{rej}(b, s^{−4}, u) ds db dσ(u)`. + +Two facts about the contact law are used. The pin set splits into even pins `(f, ∂_u²f, ∂_u∂_Θf)` and odd pins +`(∂_uf, ∂_u³f, ∂_Θf)`, and even- and odd-order derivatives of a stationary field at a point are uncorrelated; the +nondegeneracy statements come from the finite-jet rank of [P] §2. +- (G1) Under `v₀(b, 0)`, `γ` is centred Gaussian with variance `σ_γ²(u) > 0` and independent of `(A, f₄)`. +- (G2) Given `b`, `(A, f₄)` is a nondegenerate Gaussian vector whose mean is affine in `b`. Write `p_A(a | b)` for the density + of `A`, and `p_A(a | f₄, b)` for its conditional density given `f₄`. + +**The Gaussian kernel.** For `C(x) = e^{−|x|²/2}` on `R²`, an exact computation (control S3) gives the following. Given +`∇f(0) = 0`, the third-order jets `(∂_u³f, γ, B, C₃)` are independent, with variances `(6, 2, 2, 6)`, and uncorrelated with +every even jet. Hence under `v₀(b, k)` the jets `(γ, B, C₃)` are centred and independent of `(A, f₄)`, and the gap pin +`∂_u³f = 12k` contributes the factor `e^{−144k²/(2·6)} = e^{−12k²}` to the pin density. (0.3) +The periodized kernel of the torus field agrees with `C` up to `O(e^{−L²/8})`. Statements labelled *Gaussian kernel* are +about the model case `C` on `R²`, as in #207 §8. + +## 1. The soft tail of the rejected cusp kernel + +**Theorem 1.** There are `C, c, N` such that for `κ ≥ 1`, `b ∈ R` and `u ∈ S¹`, + + |κ 𝒜^{rej}(b, κ, u) − F₀(b, u)| ≤ C κ^{−1} (1 + |b|)^N e^{−cb²}, F₀(b, u) := (5/24) π₀(u; v₀(b, 0)) p_A(0 | b) E[γ⁶]. (1.0) + +For the Gaussian kernel `E[γ⁶] = 15σ_γ⁶ = 120`, so `F₀ = 25π₀(v₀(b, 0))p_A(0 | b)`. + +*Proof.* Fix `b` and `u`. The constants below are uniform in `u` and polynomial in `b` before the factor `π₀`, which carries +`e^{−c₀b²}` ([R] (R5)). Split the expectation in (0.2) on `{A < 0}`. + +*Step 1 (negative `z` and large `|f₄|`).* By (0.1), `z ≥ f₄` on `{A < 0}`. So `z ≤ −24κ` forces `f₄ ≤ −24κ`, and the weight is +at most `36κ²λ²`. On `{|f₄| ≥ κ}`, (G2) gives `E₀[λ²1{|f₄| ≥ κ} | b] ≤ C(1 + |b|)^N e^{−c(κ − C|b|)₊²}`. Multiplied by `π₀`, +this is `≤ C(1 + |b|)^N e^{−cb²}e^{−cκ²}` (if `|b| ≥ κ/(2C)`, use `e^{−c₀b²}`). Both parts are `O(κ^{−2})` after +multiplying by `κ`. + +*Step 2 (the soft window).* On `{|f₄| < κ, 24κ ≤ z < 72κ}` we have `3γ²/λ = z − f₄ ∈ (23κ, 73κ)`, so the event is +`{λ ∈ (3γ²/(72κ − f₄), 3γ²/(24κ − f₄)]}`. For fixed `(γ, f₄)`, `λ ↦ φ = (f₄ + 3γ²/λ)/(72κ)` is a decreasing bijection onto +`[1/3, 1)`, with `λ = 3γ²/(72κφ − f₄)` and `|dλ/dφ| = 216κγ²/(72κφ − f₄)²`. Using `5184κ² − z² = 5184κ²(1 − φ²)`, + + ∫ p_A(−λ | f₄, b) (λ²/144)(5184κ² − z²) 1{…} dλ = ∫_{1/3}^{1} p_A(−λ(φ) | f₄, b) · 69984 κ³ γ⁶ (1 − φ²)/(72κφ − f₄)⁴ dφ. (1.1) + +Since `69984/72⁴ = 1/384`, the factor `69984κ³/(72κφ − f₄)⁴` equals `(384κφ⁴)^{−1}(1 − f₄/(72κφ))^{−4}`, and +`|f₄/(72κφ)| ≤ 1/24` on this event. So + + |(1 − f₄/(72κφ))^{−4} − 1| ≤ C|f₄|/κ, |p_A(−λ | f₄, b) − p_A(0 | f₄, b)| ≤ C_b λ ≤ C_b γ²/κ, (1.2) + +where `C_b` is a polynomial in `|b|` and `|f₄|` (the conditional density is Gaussian, with variance bounded below and mean +affine in `(f₄, b)`). Hence (1.1) equals `(γ⁶/(384κ))p_A(0 | f₄, b)∫_{1/3}^1(φ^{−4} − φ^{−2})dφ + κ^{−2}O(γ⁸ + γ⁶|f₄|)·poly`, +and `∫_{1/3}^1(φ^{−4} − φ^{−2})dφ = 20/3`. + +*Step 3 (integration).* Integrate over `f₄` given `b`; restoring `|f₄| ≥ κ` costs Step 1's bound. Use +`E[p_A(0 | f₄, b) | b] = p_A(0 | b)`, and integrate over `γ` by (G1), with `E[γ⁶], E[γ⁸] < ∞`. Multiplying by `12π₀` and +by `κ` gives `κ𝒜^{rej} = 12π₀p_A(0 | b)E[γ⁶](20/3)/384 + O(κ^{−1})·poly(b)π₀`, and `12·(20/3)/384 = 5/24`. ∎ + +*Remarks on Theorem 1.* +1. Theorem 1 is about the cusp kernel `𝒜^{rej}` itself, not about the non-uniformity of #240's Theorem N as `κ → ∞`. The + rejected weight sits on `λ ∈ [γ²/(24κ), γ²/(8κ)]` (Step 2), so the rejected pairs at large `κ` are soft in the transverse + direction. In #220's elder decision they are the pairs with `|φ| ∈ (1/3, 1)` coming from `3γ²/λ ≈ z`, not from `f₄`. +2. Expanding (1.2) to first order gives `κ𝒜^{rej}/F₀ = 1 + c(b)/κ + O(κ^{−2})` with + `c(b) = (12/5)[E(f₄ | A = 0, b)/18 − ∂_a log p_A(0 | b)·E[γ⁸]/(24E[γ⁶])]`, where `12/5` is the mean of `1/φ` under the + weight `φ^{−4} − φ^{−2}` on `[1/3, 1]`. For the Gaussian kernel, `A ~ N(−b, 2)` and `f₄ ~ N(−3b, 24)` are independent + given `b`, and `c(b) = 0.3b`. +3. With `s = κ^{−1/4}`, for an isotropic field `∫_{S¹}𝒜^{rej}(b, s^{−4}, u) dσ(u) = 2πF₀(b)s⁴ + O(s⁸(1 + |b|)^N e^{−cb²})`. + So the `s`-integral defining `I^{cand} − c₁` converges at `s = 0` like `s⁵`, and `s ≤ s₀` contributes + `2π∫F₀ db · s₀⁵/5 + O(s₀⁹)`. +4. *Numerically* (Gaussian kernel; §5): + - `F₀(0) = 0.008207`, `F₀(1) = 0.003019` and `∫F₀ db = 0.0145472`, so the `s⁴`-coefficient is `2π∫F₀ db = 0.091403`. + - `κ𝒜^{rej}(0, κ)/F₀(0) = 0.509, 0.751, 0.917, 0.977, 0.9985, 0.9999` at `κ = 1/2, 1, 2, 4, 16, 64` (`c(0) = 0`). + - `κ(κ𝒜^{rej}/F₀ − 1) = 0.280, 0.295` at `b = 1` and `−0.318, −0.305` at `b = −1`, at `κ = 16, 64`, as `c(±1) = ±0.3` + predicts. + +## 2. The fold-scale soft model + +Theorem 1 shows where the rejected weight sits as `κ → ∞` along the cusp scale. The *fold scale* is the regime of fixed +`k = ℓ/r³` and `r → 0`, so again `κ = k/r → ∞`. There the pair is rejected only if the transverse curvature is of order `r` +([C7-K] (K2) bounds the rejected weight by `O(r³/k)`), and two more jets enter. This section is formal for the field (the +limit is Conjecture 6) and exact for the model. + +*The limit window field.* In the window coordinates of #220 §1 (`Φ(X, Ξ) = rXu + r²ΞΘ`, `𝔉 = r^{−4}(f∘Φ − b)`), put `λ = λ̃/κ` +and `Ξ = κζ`, so that `x = rX` and `y = rkζ`. Every monomial `x^iy^j` of the Taylor expansion at `0` contributes +`r^{i+j−3}k^{j−1}X^iζ^j` to `𝔉/κ = (f(rX, rkζ) − b)/(kr³)`. After the pin corrections of #220 (1.1), + + 𝔉(X, κζ)/κ = G_k(X, ζ) + r G₁(X, ζ) + O_k(r²(1 + |ζ|)⁵) (|X| ≤ 3, rk|ζ| ≤ 1), + G_k(X, ζ) := 2(X + ½)²(X − 1) + ½(X² − ¼)γζ − ½λ̃ζ² + ½kBXζ² + (k²/6)C₃ζ³, (2.1) + G₁ = (f₄/(24k))(X² − ¼)² + (k/4)∂_u²∂_Θ²f·X²ζ² + (k²/6)∂_u∂_Θ³f·Xζ³ + (k³/24)∂_Θ⁴f·ζ⁴ + (1/6)∂_u³∂_Θf·X(X² − ¼)ζ. + +Control S10 checks (2.1), including `G₁`, on exactly pinned polynomial fields. The pins are exact: `G_k(−½, 0) = 0`, +`G_k(½, 0) = −1`, and `∇G_k = 0` at both. + +*Normalization.* For `γ > 0` set `ζ = (γ/λ̃)z` and + + φ := γ²/(24λ̃), t := 12kB/γ², β := kB/λ̃ = 2tφ, χ₀ := 576k²C₃/γ³, χ := χ₀φ². (2.2) + +(For `γ < 0`, replace `(γ, C₃, ζ)` by `(−γ, −C₃, −ζ)`.) Then `G_k = (γ²/λ̃) G` with + + G(X, z) = A₀(X) + ½(X² − ¼)z − ½(1 − βX)z² + (χ/6)z³, A₀(X) := 2(X + ½)²(X − 1)/(24φ), (2.3) + +with `M = (−½, 0)` at level `0` and `S = (½, 0)` at level `L_S := −1/(24φ)`. At `t = χ₀ = 0`, the ridge of (2.3) is the cusp +ridge `g/(24φκ)` of #220 §1 with `f₄ = 0`, and `φ` is #220's `φ` in the limit `κ → ∞` (Theorem 1, Step 2). + +*Typing and weight.* The Hessians of `G_k` at the pins have determinants `6λ̃ + Y` (at `M`) and `−6λ̃ + Y` (at `S`), with +`Y := 3kB − γ²/4`; this is `Y_r = Y′ + 3kΔ_B` of #229 §0 in the limit. So `M` is a nondegenerate maximum and `S` a saddle iff +`|Y| < 6λ̃`, that is + + 0 < φ < 1/|c′|, c′ := 1 − t. (2.4) + +The scaled Kac–Rice weight is `36λ̃² − Y² = (γ⁴/16)(φ^{−2} − c′²)`. With `dλ̃ = (γ²/(24φ²))dφ`, the weight per unit `φ` is +`(γ⁶/384)φ^{−2}(φ^{−2} − c′²)`. By (2.4), `1 + β/2 − φ = 1 − φc′ > 0`, so the `z`-curvature `−(1 + β/2)` at `M` is negative. + +*Slices.* For fixed `X`, `z ↦ G(X, z)` is a polynomial of degree at most three. Write `p := X² − ¼`, `a := 1 − βX` and +`D := a² − χp`. Call `X` *open* if the slice has no local maximum (`χ = 0` and `a ≤ 0`, or `χ ≠ 0` and `D ≤ 0`). Otherwise +`R(X)` is the value at the local maximum and `V(X)` the value at the local minimum (`V := −∞` if `χ = 0`). For `χ ≠ 0` the +critical points are `z_r = (a − √D)/χ` (maximum) and `z_v = (a + √D)/χ` (minimum); for `χ = 0`, `z_r = p/(2a)`. At a critical +point `z_*`, `G = A₀ + (p/3)z_* − (a/6)z_*²`. + +**Lemma 2 (the elder decision in the model).** Let `(φ, β, χ)` satisfy (2.4) with `β ≠ 2`. Let `I_M` be the connected component +of `{X not open: R(X) > L_S}` containing `−½`. Suppose that `M` is the only critical point of `G` at level `0`, that `S` is the only one at level `L_S`, and that `R` has no +finite limit along an unbounded `I_M`. This excludes a null set of parameters. +Then `S` kills `M` in the superlevel filtration of `G` (the elder rule) iff one of the following holds. + +- (D) `1 − β/2 > 0`, the right end of `I_M` is `½`, its left end `X_L` is finite, and on `(X_L, ½)` we have `V < L_S` and `R < 0` + for `X ≠ −½`. +- (D′) `1 − β/2 < 0` and `χ ≠ 0`, `½ ∈ I_M`, `I_M = (X_L, X_R)` is bounded, and on `I_M` we have `R < 0` for `X ≠ −½` and + `V < L_S` for `X ≠ ½`. + +If `1 − β/2 < 0` and `χ = 0`, then `e = 0`. + +*Proof.* Fix a level `c`. On a non-open slice, `{z : G(X, z) > c}` has a bounded *ridge piece* around `z_r` when `R(X) > c`. +When `χ ≠ 0` it also has an unbounded *far piece* beyond `z_v`, and the two are one interval iff `V(X) > c`. On an open slice +the set is unbounded (one or two half-lines, or `R`), and `G → +∞` along it, as along every far piece. +1. *Above `L_S`.* Let `c ↓ L_S`. Suppose no `X ∈ I_M` has `V(X) > c`. Then the component of `{G > c}` containing `M` is the union of + the ridge pieces over `I_M`. These vary continuously and shrink to points at finite ends of `I_M`, and the supremum of the + union is `max_{I_M} R`. So `M` is still the oldest point of its component iff `I_M` is bounded and `R < 0` on + `I_M ∖ {−½}`. + - If `I_M` is unbounded, then as `|X| → ∞` along it, either some limit point of `I_M` is open or `R → ±∞` (a finite limit is + excluded by hypothesis). The value `−∞` contradicts `R > L_S` on `I_M`, so the component reaches above `0`. + - If some `X ∈ I_M` has `V(X) > c`, the component meets a far piece. + In all of these cases `M` has died at a level above `L_S`, and `e = 0`. +2. *`1 − β/2 > 0`.* At `X = ½`, where `p = 0`, the slice has its local maximum at `z = 0`, with value `L_S`. So `S` lies on the + ridge, and the right end of `I_M` is at most `½`. + - If it is smaller, `R` drops below `L_S` between `M` and `S`, so `S` is not in the closure of `M`'s component above `L_S`, + and `e = 0`. + - Otherwise, for `c < L_S` close to `L_S`, the component of `M` contains `S` and the segment `{(X, 0) : X > ½}`. Along it + `G = A₀(X)` increases from `L_S` to `+∞`. So `S` merges `M`'s component with one containing points above `0`, and `M` dies + at `S`. +3. *`1 − β/2 < 0`.* Now `a(½) < 0`. + - If `χ = 0`, `a` vanishes at `X = 1/β ∈ (0, ½)`, and `R = A₀ + p²/(8a) → +∞` as `X ↑ 1/β`. Meanwhile + `R ≥ G(X, 0) = A₀ > L_S` on `(−½, 1/β)`. So `R > 0` somewhere on `I_M`, and `e = 0`. + - If `χ ≠ 0`, the slice at `X = ½` has its local minimum at `z = 0` (value `L_S`) and its local maximum at `z = 2a(½)/χ` + (value `L_S + (2/3)|a(½)|³/χ² > L_S`). So `S` is the valley point of this slice. `V` has a strict local maximum `L_S` at + `X = ½`, because the Schur complement of `∂_z²G(S) > 0` in the indefinite Hessian is negative. + - If `½ ∉ I_M`, `S` is not in the closure of `M`'s component above `L_S`, and `e = 0`. + - If `½ ∈ I_M` and Step 1 leaves `M` alive, then for `c < L_S` close to `L_S` the slice at `X = ½` is one interval. So `M`'s + component contains the far piece at `X = ½`, along which `G → +∞`, and `S` kills `M`. ∎ + +Case (D′) is not empty. Numerically (control S8), `(φ, β, χ) = (0.2401, 2.1425, 0.5536)` is elder, with `I_M ≈ (−1.032, 0.515)`. +The margins vanish quadratically at the special points `X = −½` (for `R`) and `X = ½` (for `V`), as the lemma requires. A +flood fill of `{G > L_S ± ε}` on a grid agrees with Lemma 2 at 900 random typed parameter points, once one near-degenerate +point is rechecked with a smaller `ε` (§5). + +Let `𝓡(t, χ₀) ⊂ (0, 1/|1 − t|)` be the set of `φ` that are rejected at `(φ, 2tφ, χ₀φ²)`, and set + + I(t, χ₀) := ∫_{𝓡(t, χ₀)} φ^{−2}(φ^{−2} − (1 − t)²) dφ. (2.5) + +**Lemma 3 (an a priori elder region).** If `φ ≤ (1/20)·min(1, |t|^{−1}, |χ₀|^{−2/3})`, read with `0^{−1} = ∞`, the configuration +is elder. Hence `I(t, χ₀) ≤ (8000/3)·max(1, |t|³, χ₀²)`. + +*Proof.* Then `φ ≤ 1/20`, `|β| ≤ 1/10` and `|χ| ≤ 1/400`, with `χ² ≤ φ/8000`. We check (D) of Lemma 2 with `X_L ∈ (−2, −½)`. +- *Critical values.* On `[−2, ½]`, `a ∈ [0.8, 1.2]`, `|p| ≤ 15/4` and `|χp| ≤ 1/100`. So `D > 0`, and `s := √D` satisfies + `s/a ∈ [0.99, 1.01]`. Exactly, `R − A₀ = p²(a + 2s)/(6(a + s)²)` and `V − A₀ = (a + s)²(a − 2s)/(6χ²)`. Hence + `R − A₀ ≤ 0.128p²/a` and `V − A₀ ≤ −0.33/χ²`. +- *Middle.* On `(−½, ½)`, `z = 0` lies on the descending side of the local maximum, so `R(X) ≥ G(X, 0) = A₀(X) > L_S`. Also + `1 − β/2 > 0`. So the right end of `I_M` is `½`. +- *Left dip.* At `X = −2`, `R − A₀ ≤ 0.128·(225/16)/0.8 < 2.3`, while `A₀(−2) = 13.5L_S`. Since `12.5/(24φ) > 2.3`, + `R(−2) < L_S`. Hence `X_L > −2`. +- *No trigger on `(X_L, ½)`.* On `[−2, −½)`, `R ≤ (X + ½)²[2(X − 1)/(24φ) + 0.128(X − ½)²/a] < 0`, since + `2(X − 1)/(24φ) ≤ −1/(8φ) ≤ −5/2` and `0.128(X − ½)²/a ≤ 1`. On `(−½, ½)` the bracket is at most `−1/(24φ) + 0.135 < 0`. + And `V ≤ A₀ − 0.33/χ² < L_S`, since `A₀ ≤ 0` on `[−2, ½]` and `χ² ≤ φ/8000`. +Finally `φ^{−2}(φ^{−2} − c′²) ≤ φ^{−4}`, so `I ≤ ∫_{φ_*}^∞ φ^{−4} dφ = φ_*^{−3}/3`. ∎ + +The constants of Lemma 3 are crude. Numerically: +- the elder edge `φ_e(t, χ₀)` satisfies `φ_e|t| → 1/2` as `t → −∞`, and `φ_e|χ₀|^{2/3} → 16^{1/3}` as `|χ₀| → ∞`. In the second + case the valley at `M`, `V(−½) = −(2/3)/χ²`, reaches `L_S` when `χ² = 16φ`; +- for `t ≥ t*`, with `t* ∈ (0.65, 0.68)`, the elder edge at `χ₀ = 0` is `β = 2`, that is `φ_e = 1/t`; +- correspondingly `I(t, 0) = (2 − t)(2t − 1)²/3` on `[t*, 1]`, `I(t, 0) = t − 2/3` for `t ≥ 1`, `I(t, 0) ~ (4/3)|t|³` as + `t → −∞`, and `I(0, χ₀) ~ χ₀²/48`. + +## 3. The fold-scale rejection function + +Formally, at fixed `k` the rejected kernel is the expectation of the rejected weight of §2 over the jets. With +`λ = λ̃r/k`, the law of `λ` near `0` contributes `p_A(0 | b)·(r/k)dλ̃`, and the contact pin density at gap `k` is +`π₀(u; v₀(b, k))`. This motivates + + F(k; b, u) := 12 π₀(u; v₀(b, k)) p_A(0 | b, k) E_{v₀(b,k)}[(γ⁶/384) I(12kB/γ², 576k²C₃/γ³) | A = 0], (3.1) + +with the expectation under the contact law at `v₀(b, k)`, conditioned on `A = 0`. Formally, +`r^{−2}A_r^{rej}(b, k, u) = (r/k)F(k; b, u) + o(r)` (Conjecture 6 below). As `k → 0`, `I → I(0, 0) = 20/3`, and (3.1) tends +to the `F₀` of Theorem 1. + +**Proposition 4 (Gaussian kernel).** Let `C(x) = e^{−|x|²/2}`. Then: +1. `F(k; b) = F₀(b)H(k)`, where `H(k) := e^{−12k²}E[γ⁶I(t, χ₀)]/(E[γ⁶]·20/3)` does not depend on `b`; here `(γ, B, C₃)` are + independent `N(0, 2)`, `N(0, 2)` and `N(0, 6)`, and `(t, χ₀) = (12kB/γ², 576k²C₃/γ³)`. +2. `H(k) = 1 + (12/25)k² + O(k³)` as `k → 0`. +3. `H(k) ≤ C(1 + k⁴)e^{−12k²}`; in particular `|H(k) − 1| ≤ C min(1, k²)`. + +*Proof.* (1) By (0.3), the gap pin contributes `e^{−12k²}`, so `π₀(v₀(b, k)) = π₀(v₀(b, 0))e^{−12k²}`. This is the factor +`e^{−a′k²}`, `a′ = 12`, of #240 Step N6. `p_A(0 | b, k) = p_A(0 | b)`, and the law of `(γ, B, C₃)` is as stated, centred and +independent of `(A, f₄, b)`. + +(2) *The edge at `χ₀ = 0`.* For `χ = 0` and `a > 0` the ridge is `R = A₀ + p²/(8a)`. At `t = 0` it is +`[2(X + ½)²(X − 1) + 3φ(X² − ¼)²]/(24φ)`, and + + R − L_S = (X − ½)²[2(X + 1) + 3φ(X + ½)²]/(24φ) (t = 0), (3.2) + +whose bracket has discriminant `4 − 12φ` in `X + ½`. At `φ = 1/3`, (3.2) is `((X − ½)(X + 3/2))²/(24φ)`, with a nondegenerate +double zero at `X = −3/2`. The elder edge solves `R(X) = L_S`, `R′(X) = 0` in `(X, φ)` at `β = 2tφ`. The Jacobian at +`(−3/2, 1/3, t = 0)` is `−96 ≠ 0`, so the implicit function theorem gives analytic `X_e(t)` and `φ_e(t)`. Their series +(control S4) are + + X_e(t) = −3/2 − t/3 − t²/3 + O(t³), φ_e(t) = 1/3 + t/3 + (10/27)t² + O(t³). (3.3) + +*The rejected set near `(0, 0)`.* For `|t| ≤ t₀` and `|χ₀| ≤ δ₀` (small constants), +`𝓡(t, χ₀) = (φ_e(t, χ₀), 1/|1 − t|)`, where `φ_e(t, χ₀)` is the same tangency for the ridge of the cubic slices. +- *Elder side.* For `φ_* ≤ φ < φ_e`, Lemma 2's condition (D) involves only `X ∈ [X_L, ½] ⊂ [−2, ½]`. At `t = χ₀ = 0` it holds + with strict margins for `φ ≤ 1/3 − δ`; this is the exact cusp-ridge decision of #207 Theorem CU.2(c) and #220 (M1)–(M6). + Strict inequalities on a compact set persist for small `|t|, |χ₀|`. Near the edge, the implicit function theorem applies. + Lemma 3 covers `φ < φ_*`. +- *Rejected side.* For `φ ∈ [1/3 + δ, 2]` at `t = χ₀ = 0`, the bracket in (3.2) is positive, so `R − L_S ≥ c(δ) > 0` on `[−3, −½)`. + Also `R(−3) > 0`, since `24φR(−3) = −50 + (3675/16)φ > 0`. Again these are strict inequalities on a compact set. On it + `a ∈ [1 − 3|β|, 1 + 3|β|]` and `D > 0` for small `|t|` and `|χ₀|`. So the left part of `I_M` reaches a point with `R > 0` before + closing, and `e = 0`, up to the typed edge `1/|1 − t| ≤ 2`. +- *`χ₀`-direction.* For `|χ₀| ≤ δ₀`, the valley satisfies `V ≤ L_S − c/χ²` on `[−3, ½]`, as in Lemma 3, so it never triggers. The + tangency `φ_e(t, χ₀)` is smooth in `(t, χ₀)`, and so is `I`. + +With `∫_a^bφ^{−2}(φ^{−2} − c′²)dφ = (a^{−3} − b^{−3})/3 − c′²(a^{−1} − b^{−1})` and `b = 1/c′`, + + I(t, 0) = φ_e^{−3}/3 − (1 − t)²/φ_e + (2/3)(1 − t)³ = 20/3 − 20t + (52/3)t² + O(t³). (3.4) + +*Averaging.* Split the expectation on `𝒢 := {|t| ≤ t₀, |χ₀| ≤ δ₀}`. +- On `𝒢`, Taylor's formula gives `I = 20/3 − 20t + ∂_{χ₀}I(0, 0)χ₀ + (52/3)t² + O(|t|³ + |t||χ₀| + χ₀²)`. The linear terms have mean + zero: `B` and `C₃` are symmetric and independent of `γ`, and `𝒢` is symmetric in each. +- The moments are finite: `γ⁶t² = 144k²B²γ²`, `γ⁶|t|³ = 1728|k|³|B|³`, `γ⁶|t||χ₀| = 6912|k|³|B||C₃||γ|` and + `γ⁶χ₀² = 331776k⁴C₃²`. Moreover `E[γ⁶t²] = 576k²`. +- Off `𝒢`, Lemma 3 bounds `γ⁶I` by `C(γ⁶ + k³|B|³ + k⁴C₃²)`. For `k ≤ 1`, `𝒢^c` forces `γ² ≤ ε := Ck(|B| + |C₃|^{2/3})`, and + `E[γ^{2j}1{γ² ≤ ε} | B, C₃] ≤ Cε^{j+1/2}`. So `𝒢^c` costs `O(k^{7/2})`, and so does the mean of `γ⁶t² = 144k²B²γ²` over `𝒢^c`. +Hence + + E[γ⁶I(t, χ₀)] = 120·(20/3) + (52/3)·576k² + O(k³), H(k) = e^{−12k²}(1 + (312/25)k²) + O(k³) = 1 + (12/25)k² + O(k³). + +(The true next correction is of order `k^{7/2}`, since `E[γ⁶t⁴] = ∞`.) + +(3) Lemma 3 gives `γ⁶I ≤ C(γ⁶ + 1728k³|B|³ + 331776k⁴C₃²)`. ∎ + +So for the Gaussian kernel the rejected-pair rate at the fold scale starts at its cusp-scale value `F₀`. The pin density +lowers it by `12k²`, and the motion of the elder edge raises it by `(312/25)k²`, so the two nearly cancel (`12/25` +remains). Numerically `H₀ − 1` (with `C₃` ignored) already changes sign near `k ≈ 0.08`. + +For another stationary kernel with `B` and `γ` independent of each other and of `∂_u³f` under the contact law, the two +coefficients are `−144/(2σ₃²)` and `(1872/75)σ_B²/σ_γ⁴`, with `σ₃² := Var(∂_u³f | ∇f(0) = 0)`. For the Gaussian kernel, +`σ_B² = σ_γ² = 2` and `σ₃² = 6`. + +## 4. The composite density and the conjectured `ℓ^{2/3}` term + +The cusp kernel is the `k → 0` limit, and `(r/k)F(k; b, u)` the `κ → ∞` limit, of the same rejected kernel. Theorem 1 says +that they agree in the overlap: `F(k)/F₀ → 1`. Put `H(k; b, u) := F(k; b, u)/F₀(b, u)`; for the Gaussian kernel this is the `H(k)` +of Proposition 4. The simplest density built from both limits is the *composite* + + ν^c(ℓ) := ∫_0^∞∫_R∫_{S¹} 𝒜^{rej}(b, ℓ/r⁴, u) H(ℓ/r³; b, u) dσ(u) db dr. (4.1) + +**Lemma 5 (expansion of the composite).** Let `H ≥ 0` be measurable with `|H(k; b, u) − 1| ≤ C_H min(1, k²)` uniformly +(Proposition 4(3) for the Gaussian kernel). Then, as `ℓ ↓ 0`, + + ν^c(ℓ) = (I^{cand} − c₁) ℓ^{1/4} + R_{2/3} ℓ^{2/3} + O(ℓ^{3/4}), + R_{2/3} := ∫_{S¹}∫_R∫_0^∞ v⁴[F(v^{−3}; b, u) − F₀(b, u)] dv db dσ(u), F := F₀H. (4.2) + +For the Gaussian kernel, `R_{2/3} = 2π∫F₀ db · Ĩ` with `Ĩ := ∫_0^∞ v⁴[H(v^{−3}) − 1] dv`. + +*Proof.* The inner integral converges: its integrand is `O(v⁴)` at `0` and `O(v^{−2})` at `∞`, with Gaussian decay in `b`. +Write `H = 1 + (H − 1)`. The term with `1` is `ℓ^{1/4}(I^{cand} − c₁)` after `r = sℓ^{1/4}` (§0). For the term with `H − 1`, let +`P(b) := (1 + |b|)^N e^{−cb²}`. +- *`r > ℓ^{1/4}` (`κ < 1`).* `0 ≤ 𝒜^{rej} ≤ 𝒜^{con} ≤ Cκ²P(b)` and `|H − 1| ≤ Cℓ²r^{−6}`, so this part is + `≤ Cℓ⁴∫_{ℓ^{1/4}}^∞ r^{−14} dr = O(ℓ^{3/4})`. +- *`r ≤ ℓ^{1/4}` (`κ ≥ 1`).* By Theorem 1, `𝒜^{rej} = F₀r⁴/ℓ + O(P(b)r⁸/ℓ²)`. With `r = vℓ^{1/3}`, the main part is + `ℓ^{2/3}∫_0^{ℓ^{−1/12}}v⁴(F(v^{−3}) − F₀)dv`, which equals `ℓ^{2/3}` times the full `v`-integral up to `O(ℓ^{1/12})`. The + error is `≤ CP(b)ℓ∫_0^{ℓ^{−1/12}}v⁸min(1, v^{−6})dv = O(P(b)ℓ^{3/4})`. +Integrating over `b` and `u` gives (4.2). ∎ + +**Conjecture 6 (the fold-scale limit).** In `d = 2`, for every `k > 0`, `b` and `u`, +`lim_{r↓0} (k/r)·r^{−2}A_r^{rej}(b, k, u) = F(k; b, u)`, locally uniformly in `k`. + +**Conjecture 7 (the next term of the rejected density).** In `d = 2`, + + ρ_rej(ℓ) + ν_eld^{far,r_0^*}(ℓ) = B_{2,L} + (I^{cand} − c₁) ℓ^{1/4} + R_{2/3} ℓ^{2/3} + O(ℓ^{3/4}), (4.3) + +with `R_{2/3}` as in (4.2); equivalently, `ρ_rej + ν_eld^{far,r_0^*} − B_{2,L} − ν^c = O(ℓ^{3/4})`. This is (R⁺.1) of #229 with +its remainder resolved. If `ν_eld^{far,r_0^*}(ℓ) = O(ℓ^{3/4})` (#188, open, claims `O(ℓ^N)`), then +`ρ_rej = B_{2,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`. For the Gaussian kernel, numerically +`R_{2/3} ≈ −0.049` (§5). + +*What a proof needs.* Inputs (i) and (ii) are missing; (iii) is needed only to drop the far term. +- (i) *The soft layer.* A two-scale version of Conjecture 6, uniform from the cusp scale to the fold scale (`κ ≥ κ₁`), with + errors integrating to `O(ℓ^{3/4})`. In fixed cusp windows this is #240's Theorem N (`O(r²)`). The new part is the window + field's elder decision in the soft scaling `λ ≍ r`: Lemma 2 with margins for the field, as #240 Lemma Q′ gives them with + `ε̃` for `λ ≍ 1`. +- (ii) *The end `κ → 0`.* On the intermediate separations `ℓ^{1/4} ≪ r ≤ r_0^*`, the rejected kernel must match `B_{2,L}`'s part + plus the composite's own part, up to errors integrating to `O(ℓ^{3/4})`. (The composite is not negligible there: + `𝒜^{rej} ≍ κ³` as `κ → 0`, so `r ≥ Sℓ^{1/4}` contributes about `ℓ^{1/4}S^{−11}`.) The best available bounds on these + separations are #229's `O(ℓ^{4/9}log(1/ℓ))` (elder) and #237's `O(ℓ^{3/5})` (candidate), both larger than `ℓ^{2/3}`. +- (iii) *The far elder density.* #187 (merged) gives `ν_eld^{far,r_0^*}(ℓ) ≤ Cℓ^{2/3}`, which is the same order as the + conjectured term. Conjecture 7 is stated with `ν_eld^{far,r_0^*}` on the left for that reason; #188's `O(ℓ^N)` would remove + it. + +Formally, a matched-asymptotics count gives cusp-side powers `ℓ^{(m+1)/4}` and fold-side powers `ℓ^{(m+1)/3}`. In `ρ_rej` the +fold-scale kernel is `O(r³/k)` ([C7-K] (K2)), with leading coefficient `F` (formally, Theorem 1 and Proposition 4). This +makes `ℓ^{2/3}` the first fold-side power and `ℓ^{3/4}` the next cusp-side one (#240 Corollary N′). + +*Remarks.* +1. `ν_cand` has no term from this mechanism. In `ν_eld = ν_cand − ρ_rej` the soft layer contributes `−R_{2/3}ℓ^{2/3}`, in + addition to any `ℓ^{2/3}` term of `ν_cand`: the typed boundary `λ ≍ r` also produces an `|r|³` term in the candidate kernel. + #237's remainder `ℓ^{3/5}` allows both. Only `ρ_rej`, or the rejected Monte Carlo row binned in `s`, tests `R_{2/3}` alone, + and only subject to (ii)–(iii). +2. For the Gaussian kernel, `Ĩ` is dominated by `v ∈ [1, 1.5]`, that is `k ∈ [0.3, 1]`, where `H` falls from `≈ 1` to `≈ 0`. The + contributions are `−0.200` (`v < 1`), `−0.563` (`1 ≤ v ≤ 1.5`), `+0.003` (`1.5 ≤ v ≤ 2`) and `+0.223` (`v ≥ 2`). +3. `R_{2/3}/(I^{cand} − c₁) ≈ −0.53`. Since `ℓ^{5/12} = 0.147` at `ℓ = 10^{−2}`, the `ℓ^{2/3}` term is about 8% of the `ℓ^{1/4}` + term there. + +## 5. Numerical evidence (exploration; not part of the proofs) + +Gaussian kernel `e^{−|z|²/2}`, `d = 2`. The scripts are in the project archive +`V2_2/frontiers_soft_rejected_pairs_20261002/exploration/` (numpy, scipy, mpmath, sympy); none is part of the repository. + +1. *Theorem 1.* Quadrature of (0.2) under the contact law (`fastk.py`, `soft_d2.py`) gives the numbers of Remark 4 of §1. The + `b`-integrated tail is `κ∫𝒜^{rej}(b, κ) db → 0.0145472 = ∫F₀ db`. Also `∫∫𝒜^{rej}(b, s^{−4}) ds db = 0.014662`, and its `2π` + multiple is `I^{cand} − c₁ = 0.092124`. +2. *Lemma 2.* The one-dimensional decision (`dec1d.py`) was compared with a flood fill of `{G > L_S ± ε}` on an `(X, z)` grid + (`model2d.py`, `validate_D2.py`). + - 600 parameter points drawn as in the computation of `H`: no disagreement. + - 300 points with `β > 2`: one disagreement. There the valley exceeds `L_S` by `2·10^{−4}`, below the flood fill's + default `ε`; with `ε_rel = 5·10^{−4}` the flood fill agrees. + - The referee's independent comparison (500 points, 40 of them (D′)-elder) agrees after resolving two numerical + artifacts. +3. *The function `I`.* + - A `101 × 121` table of `I(t, χ₀)` on `|t| ≤ 30`, `|χ₀| ≤ 2000` (`Itab2d_v2.py`), and an exact evaluator for other points + (`Ifast.py`). + - `I(t, 0)/|t|³ = 1.95, 1.51, 1.39` at `t = −10², −10³, −10⁴`, tending to `4/3`. + - `I(0, χ₀)/χ₀² = 0.0228, 0.0212, 0.0209` at `χ₀ = 10³, 10⁴, 10⁵`, tending to `1/48`. +4. *The function `H`.* `H = H₀ + Δ_χ`. + - `H₀`, the value with `C₃` ignored, comes from a 639-point table of `I(t, 0)` on `|t| ≤ 400`, its asymptotic forms beyond + (`|t|³(4/3 + 5.74|t|^{−1/2})` and `t − 2/3`), and adaptive quadrature in `t` and `γ²` down to `γ → 0` (`Hk0b.py`). Its + small-`k` values reproduce `(H₀ − 1)/k² → 12/25`, and it agrees with the referee's independent `H₀` to `3·10^{−5}`. + - `Δ_χ` is the difference of two direct quadratures over `(|γ|, B, C₃)`, with and without `C₃` (`Hdirect4.py`). These use + 32 Gauss–Legendre nodes in `|γ|` on `[0, 7.5]` and 24 Gauss–Hermite nodes in `B` and in `C₃`, with the table inside its + range and the exact evaluator outside it. + - Accuracy checks: without the table, `Δ_χ(0.2)` changes by `2·10^{−5}`; with 48 nodes in `|γ|`, `H` changes by + `5·10^{−5}`. For small `k`, `Δ_χ ≈ 62k⁴`. The quadrature `H` agrees with a table-free direct quadrature at `k = 0.2` to + `1.3·10^{−4}`. + - Values: + + | `k` | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 1.0 | + |---|---|---|---|---|---|---|---|---|---| + | `H₀` | 0.9976 | 0.9334 | 0.7403 | 0.4663 | 0.2268 | 0.0845 | 0.0241 | 0.0053 | 0.0001 | + | `H` | 1.0039 | 1.0041 | 0.9345 | 0.7271 | 0.4397 | 0.2007 | 0.0687 | 0.0177 | 0.0005 | + | `e^{−12k²}` | 0.887 | 0.619 | 0.340 | 0.147 | 0.050 | 0.013 | 0.003 | 0.0005 | 0.000006 | + +5. *`R_{2/3}`.* + - `Ĩ = −0.536` (`assemble2.py`; a dense trapezoid rule gives `−0.537`), so `R_{2/3} = 0.0914028·Ĩ = −0.049`. + - Sensitivity: a uniform shift of `H` by `±0.002` on `k ≥ 0.1` moves `Ĩ` by `±0.018`. Halving the small-`k` term `62k⁴` + moves it by `−0.021`, and multiplying it by `1.5` by `+0.021`. So `Ĩ = −0.54 ± 0.03` and `R_{2/3} = −0.049 ± 0.003`. + - With `C₃` ignored (`H = H₀`), `Ĩ = −1.133` and `R_{2/3} = −0.104`. With the pin density alone (`H = e^{−12k²}`), + `Ĩ = 12^{5/6}Γ(−5/6)/6 = −8.829` and `R_{2/3} = −0.807`. So the soft model's extra jets matter: `B` nearly cancels the pin + density at small `k`, and `C₃` raises `H` at `k ≈ 0.2–0.7`. +6. *#216's Monte Carlo.* The raw `d = 2` records (`batchA2.npz`: 4,000 samples, 271,272 rejected adjacent pairs with `ℓ < 0.3`) + were compared with the composite (`composite.py`, `comp2d.py`). #216's row counts *adjacent* rejected pairs, which carry no + `B_{2,L}` (#218 Remark 3). By §4 (inputs (ii)–(iii), and the `ℓ^{3/4}` order), these totals do not test the global + `ℓ^{2/3}` coefficient. What they do test, bin by bin in `s = r/ℓ^{1/4}`, is the fold-side suppression described by `H`. + - *Small `s` (the fold side).* For `s < 0.8` and `ℓ ∈ [10^{−4}, 0.1)`, 37 pairs are observed, 35.3 predicted by the + composite, and 4,322 by the cusp kernel alone. In the bin `ℓ ∈ [3·10^{−3}, 10^{−2})`, `s ∈ [0.6, 0.8)` the three numbers + are 12, 13.1 and 143. For `s ∈ [0.8, 1)` they are 2,248, 1,904 and 7,766. + - *`s ≥ 1` (the cusp side).* The composite and the cusp kernel agree. The data exceed both by up to 25% at `ℓ ≥ 10^{−2}`, + which is not modelled here (finite-`r` cusp corrections, the `ℓ^{3/4}` order). + - *Totals.* On `[10^{−4}, 10^{−2}]` the ratio of the counts to `(I^{cand} − c₁)ℓ^{1/4}` alone is `0.959 ± 0.017`; to the + composite, `1.040`; to the two-term law with `R_{2/3} = −0.049`, `1.018`. + +## 6. Scope and relation to other packets + +- **#229 (R⁺.1).** It proves `ρ_rej = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} − ν_eld^{far,r_0^*} + O(ℓ^{3/7})`. Conjecture 7 would resolve + its remainder in `d = 2`, given inputs (i)–(ii) of §4 (and (iii) to drop the far term). Theorem 1, Lemma 5 and Proposition 4 do not change any statement + of #229. +- **#240 Remark 1.** It leaves the ends `κ → ∞` and `κ → 0` of Theorem N open. Theorem 1 describes the `κ → ∞` end of the + *rejected* cusp kernel in `d = 2`: it decays like `1/κ`, while `𝒜^{cand}` and `𝒜^{eld}` grow like `κ²`. Lemma Q′'s margins do + not reach the soft window `λ ≍ γ²/κ` where the rejected weight sits. +- **#237 Remark 4 and #240 Remark 4.** They found residuals of #216's Monte Carlo about the three-term laws that two-power + fits could not attribute (`ℓ^{1/2}` against `ℓ^{2/3} + ℓ^{3/4}`). Conjecture 7 predicts a contribution `−R_{2/3}ℓ^{2/3}` to + `ν_eld`, in addition to any `ℓ^{2/3}` term of `ν_cand`, with a coefficient computed independently of the Monte Carlo. +- **[C7-K] (K2)** bounds the fold-scale rejected weight by `O(r³/k)`. Formally (Conjecture 6) the bound is attained: the + leading term of `r^{−2}A_r^{rej}` is `(r/k)F(k; b, u)`, and `F(k) → F₀ ≠ 0` as `k → 0` (Theorem 1). +- **Not claimed.** + - Conjectures 6 and 7 are not proved. The values of `H` and `R_{2/3}` are numerical and not certified. + - Nothing for `d ≥ 3`. There the soft direction is an eigenvector of a `(d − 1) × (d − 1)` matrix, and `F₀` needs the density + of its smallest eigenvalue at `0`; the `1/κ` form of Theorem 1 is expected to persist. + - Nothing about the candidate density's own `ℓ^{2/3}` term. + - Lemma 2, Lemma 3 and Proposition 4 are statements about the limit model (2.1). Their transfer to the field is + Conjecture 6. + - The Gaussian-kernel statements are about the model case on `R²`. + +## 7. Controls + +`soft_check.py` uses the standard library and exact rationals, except S8 (fixed floating-point cases). Its output is +`RESULTS.json`. + +| Control | Checks | +|---|---| +| S1 | Theorem 1: `69984/72⁴ = 1/384`; `∫_{1/3}^1(φ^{−4} − φ^{−2})dφ = 20/3`; `12·(20/3)/384 = 5/24`; `F₀ = 25π₀p_A(0 \| b)`; the soft window at `f₄ = 0` | +| S2 | the algebra of `G_k` at 60 random rational points: its derivatives (exact four-point stencils), pins, `det H_M = 6λ̃ + Y`, `det H_S = −6λ̃ + Y`, the normalization (both signs of `γ`), weight, measure and typed window | +| S3 | the Gaussian kernel's jet covariances from Hermite numbers: (0.3), and `a′ = 144/(2·6) = 12` | +| S4 | (3.3) as exact truncated series; the double zero at `t = 0`; the Jacobian `−96` | +| S5 | (3.4), and the antiderivative of the weight | +| S6 | `E[γ⁶] = 120`, `E[γ⁶t²] = 576k²`, `312/25`, `12/25`, and the moment identities of Proposition 4's proof | +| S7 | every numerical inequality of Lemma 3's proof, including the critical-value formulas as identities | +| S8 | Lemma 2 on 16 fixed parameter points with known answers, including case (D′), in floating point | +| S9 | the error exponents of Lemma 5 | +| S10 | the limit (2.1), including `G₁`, on three exactly pinned degree-6 fields with `A = −λ̃r/k`, at `r = 10^{−3}` and `10^{−4}` | + +Mutants `M1`–`M10` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, `M4` S5, `M5` S7, `M6` S8, `M7` S9, `M8` S2, +`M9` S6, `M10` S10); an unknown label exits 2. + +## 8. Sources (exact identities in `SOURCES.json`) + +| Tag | Path | Role | +|---|---|---| +| [R] | `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (blob `247b3ecf`) | (R5): Gaussian decay of the pin density — consumed | +| [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d`) | §2 finite-jet rank, through #220 — consumed | +| [E1], [E2], [REC] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`, `reviews/d1_section9_borel_repair_20260925/REPAIR.md`, `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` | reading rules and the Borel elder mark, through #220 — consumed | +| #207 | `frontiers/cusp_second_order_20261001/PROOF.md` (blob `f6df5a73`) | §§0, 4, 6 (the cusp objects); Theorem CU.2(c) — consumed | +| #218 | `frontiers/candidate_third_order_20261001/PROOF.md` (blob `70ca57ef`) | §0: `𝒜^{cand}`, `I^{cand}`, `B_{d,L}`; Remark 3 — consumed | +| #220 | `frontiers/elder_third_order_20261001/PROOF.md` (blob `c8767dde`) | §0 ((0.1), (0.2)); §1 (window coordinates, (1.1), Lemma Q, (M1)–(M6)) — consumed | +| #229 | `frontiers/third_order_rate_20261001/PROOF.md` (blob `110ed33a`) | (R⁺.1); §0 (`Y_r`); §5 — consumed | +| [C7-K] | `frontiers/c7_total_bounded_20260929/PROOF.md` (blob `28748b08`) | (K2) — cited | +| #187 | `frontiers/far_elder_rate_20260930/PROOF.md` (blob `07260114`) | (0.2): `ν_eld^{far} ≤ Cℓ^{2/3}` — cited | +| #240 | `frontiers/elder_cusp_parity_20261002/PROOF.md` (open PR; blob `16a1db06`) | Lemma Q′, Step N6, Corollary N′, Remarks 1 and 4 — cited | +| #237 | `frontiers/candidate_parity_rate_20261001/PROOF.md` (open PR) | Theorem P's remainder, Remark 4 — cited | +| #216 | `frontiers/third_order_coefficient_20261001/NOTE.md` (open PR) | the full-field Monte Carlo (§5) — cited | +| #188 | far elder density `O(ℓ^N)` (open PR) | §4 (iii) — cited | + +## 9. Review slices + +- **A** (§1): Theorem 1, its proof and Remarks 1–3. +- **B** (§2): the limit field (2.1) and its monomial bookkeeping, the normalization, typing and weight, Lemma 2 (including case + (D′)) and Lemma 3. +- **C** (§3): Proposition 4, namely the edge (3.3), the identification of the rejected set near `(0, 0)`, the series (3.4), the + averaging and the bound off `𝒢`. +- **D** (§4): Lemma 5, and whether Conjectures 6 and 7 are stated precisely, with inputs (i)–(iii) complete and consistent with + #187, #229 and #240. +- **E** (§§5–7): the evidence is described accurately and kept separate from the proofs, and the controls match their + claims. diff --git a/frontiers/soft_rejected_pairs_20261002/README.md b/frontiers/soft_rejected_pairs_20261002/README.md new file mode 100644 index 000000000..f0c726471 --- /dev/null +++ b/frontiers/soft_rejected_pairs_20261002/README.md @@ -0,0 +1,89 @@ +# Soft rejected pairs in `d = 2` + +**Author-side proof candidate and conjectures (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** + +Object `CL-SOFT-REJECTED-20261002-v1`. Full text: [`PROOF.md`](PROOF.md). + +## Result + +Math- #229 proves `ρ_rej(ℓ) = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} − ν_eld^{far,r_0^*}(ℓ) + O(ℓ^{3/7})`. Math- #240 Remark 1 leaves +the cusp end `κ = ℓ/r⁴ → ∞` open. In `d = 2` this note describes that end. + +| | Statement | Status | +|---|---|---| +| **Theorem 1** | `κ𝒜^{rej}(b, κ, u) = F₀(b, u) + O(κ^{−1})`, with `F₀ = (5/24)π₀p_A(0 \| b)E[γ⁶]` (Gaussian kernel: `25π₀p_A(0 \| b)`). The rejected weight sits on `λ = −A ∈ [γ²/(24κ), γ²/(8κ)]`. | proof | +| **Lemma 2** | The elder decision in the fold-scale soft model `G_k` (new jets `B = ∂_uA`, `C₃ = ∂_Θ³f`), by a one-dimensional scan, including a case where the saddle is a slice minimum | proof (model) | +| **Lemma 3** | Elder whenever `φ ≤ (1/20)min(1, \|t\|^{−1}, \|χ₀\|^{−2/3})`; so `I(t, χ₀) ≤ (8000/3)max(1, \|t\|³, χ₀²)` | proof (model) | +| **Proposition 4** | Gaussian kernel: the fold-scale rejection rate is `F₀(b)H(k)` with `H(k) = 1 + (12/25)k² + O(k³)`. The elder edge `φ_e = 1/3 + t/3 + 10t²/27` contributes `+312/25`, and the pin density `e^{−12k²}` contributes `−12`. | proof (given the model) | +| **Lemma 5** | The composite `∫∫∫𝒜^{rej}(b, ℓ/r⁴, u)H(ℓ/r³)` equals `(I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`, where `R_{2/3} = ∫∫∫u⁴[F(u^{−3}) − F₀]` | proof | +| **Conjecture 6** | `(k/r)r^{−2}A_r^{rej}(b, k, u) → F(k; b, u)` as `r → 0` at fixed `k` | conjecture | +| **Conjecture 7** | `ρ_rej + ν_eld^{far,r_0^*} = B_{2,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`; Gaussian kernel: `R_{2/3} ≈ −0.049 ± 0.003` | conjecture, with evidence | + +Conjecture 7 needs two inputs that are not yet available (PROOF §4): +- (i) the soft layer `λ ≍ r`; +- (ii) the intermediate separations, where the best bounds are `ℓ^{4/9}log(1/ℓ)` (#229) and `ℓ^{3/5}` (#237). + +A third input, (iii) the far elder density, is needed only to remove `ν_eld^{far,r_0^*}` from the left side. #187 bounds it by +`Cℓ^{2/3}`, and #188 (open) claims `O(ℓ^N)`. + +## Evidence (exploration; outside the repository) + +| | Result | +|---|---| +| Theorem 1 | `κ𝒜^{rej}(0, κ)/F₀(0) = 0.751, 0.977, 0.9985, 0.9999` at `κ = 1, 4, 16, 64`; the `1/κ` coefficient `c(b) = 0.3b` is confirmed at `b = ±1` | +| Lemma 2 | agrees with a two-dimensional flood fill on 900 random typed parameter points (one near-degenerate point needs a smaller level offset) | +| `H` | `H(0.2) = 1.004`, `H(0.4) = 0.727`, `H(0.6) = 0.201`, `H(0.8) = 0.018`. With `C₃` ignored, `R_{2/3}` would be `−0.104`; with the pin density alone, `−0.807`. | +| #216's Monte Carlo | rejected adjacent pairs with `s = r/ℓ^{1/4} < 0.8`: 37 observed, 35.3 from the composite, 4,322 from the cusp kernel alone | + +## Controls + +`soft_check.py` uses the standard library and exact rationals; control S8 is a fixed floating-point implementation of +Lemma 2. Its output is `RESULTS.json`, byte-identical under `-O`. + +| Control | Checks | +|---|---| +| S1 | Theorem 1: the change of variables (`1/384`), `∫_{1/3}^1(φ^{−4} − φ^{−2}) = 20/3`, `5/24`, the factor `25` | +| S2 | the algebra of `G_k`: derivatives, pins, `det H_M = 6λ̃ + Y`, `det H_S = −6λ̃ + Y`, the normalization, weight and measure, the typed window | +| S3 | Gaussian kernel: given `∇f = 0`, the third-order jets are independent with variances `(6, 2, 2, 6)` and uncorrelated with even jets; `a′ = 12` | +| S4 | the elder edge (3.3) by exact truncated series; the double zero at `t = 0`; the Jacobian | +| S5 | `I(t, 0) = 20/3 − 20t + (52/3)t² + O(t³)` | +| S6 | `E[γ⁶t²] = 576k²`, `h₂ = 312/25`, `312/25 − 12 = 12/25`, the moment identities | +| S7 | every inequality in the proof of Lemma 3, with exact rational bounds | +| S8 | Lemma 2 on 16 fixed parameter points with known answers, including case (D′) | +| S9 | Lemma 5's error exponents (all `3/4`) | +| S10 | the limit field (2.1), including its `r`-correction, on exactly pinned degree-6 fields | + +Mutants `M1`–`M10` each fail only their own control, and an unknown label exits 2: + + python3 -B -S soft_check.py # exit 0, output = RESULTS.json + python3 -B -S soft_check.py --mutant M1 # exit 1 + +## Review record + +- **Same-family referee.** A clean-context referee (Anthropic Claude) read the note against its sources at the declared blobs. + Its verdict was **MAJOR REVISION**, with no BLOCKING finding: + - one MAJOR finding: Conjecture 7 must keep the far elder density and state the missing inputs (ii)–(iii); + - eight MINOR findings and nineteen NITs. + All of them are applied. Among them: + - the definition of `F` and `R_{2/3}` for the torus field; + - Lemma 2's genericity hypothesis; + - a false closed form for `I(t, 0)` on `[t*, 1)`; + - one number in Theorem 1's remarks; + - a new control S10 for the limit field (2.1). +- **The referee's independent checks.** A sympy derivation of (2.1); the contact law of the Gaussian kernel; an independent + quadrature of Theorem 1; a 500-point two-dimensional test of Lemma 2; the `I(t, 0)` table; `H₀`. +- **Delta check of the revision** (the same referee): **ACCEPT WITH MINOR FIXES**, with no BLOCKING or MAJOR finding. Its three + new MINOR findings and seven NITs are applied: + - Lemma 2's genericity hypothesis, in its final form; + - input (ii) now accounts for the composite's own part at small `κ`; + - `H₀` had truncated the small-`γ` end. Correcting it moves `R_{2/3}` from `−0.050` to `−0.049`, inside the quoted + uncertainty. +- **Independence.** The referee is the same provider and the same GitHub account as the author, so it carries zero + organizational independence. Nonauthor review is required; the review slices are in PROOF §9. + +## Not claimed + +- Conjectures 6 and 7 are not proved, and `R_{2/3}` is not certified. +- Nothing for `d ≥ 3`. +- Nothing about the candidate density's own `ℓ^{2/3}` term. +- No change to #229, #240 or any other packet. diff --git a/frontiers/soft_rejected_pairs_20261002/RESULTS.json b/frontiers/soft_rejected_pairs_20261002/RESULTS.json new file mode 100644 index 000000000..7d696ca24 --- /dev/null +++ b/frontiers/soft_rejected_pairs_20261002/RESULTS.json @@ -0,0 +1,239 @@ +{ + "S1": { + "F0_over_pi0_pA0": "25", + "change_of_variables_constant": "1/384", + "phi_integral": "20/3" + }, + "S10": { + "fields": 3, + "max_error_after_r1_term_at_r_1e-4": "0.0004916683775860822" + }, + "S2_instances": 60, + "S3": { + "conditional_covariance_third_jets": [ + [ + "6", + "0", + "0", + "0" + ], + [ + "0", + "2", + "0", + "0" + ], + [ + "0", + "0", + "2", + "0" + ], + [ + "0", + "0", + "0", + "6" + ] + ], + "pin_exponent_a_prime": "12" + }, + "S4": { + "X_e": [ + "-3/2", + "-1/3", + "-1/3" + ], + "jacobian": "-96", + "phi_e": [ + "1/3", + "1/3", + "10/27" + ] + }, + "S5": { + "I_t_series": [ + "20/3", + "-20", + "52/3" + ] + }, + "S6": { + "H_k2_coefficient": "12/25", + "h2": "312/25" + }, + "S7": { + "I_bound_constant": "8000/3", + "c_R": "15100/118803", + "phi_star_factor": "1/20" + }, + "S8": { + "cases": [ + [ + [ + 0.3, + 0.0, + 0.0 + ], + 1, + 1 + ], + [ + [ + 0.36, + 0.0, + 0.0 + ], + 0, + 0 + ], + [ + [ + 0.3333, + 0.0, + 0.0 + ], + 1, + 1 + ], + [ + [ + 0.3334, + 0.0, + 0.0 + ], + 0, + 0 + ], + [ + [ + 0.95, + 0.0, + 0.0 + ], + 0, + 0 + ], + [ + [ + 0.5, + 2.2, + 0.0 + ], + 0, + 0 + ], + [ + [ + 0.04, + 0.09, + 0.002 + ], + 1, + 1 + ], + [ + [ + 0.3, + 0.0, + 3.0 + ], + 0, + 0 + ], + [ + [ + 0.2, + 0.0, + 1.0 + ], + 0, + 0 + ], + [ + [ + 0.2, + 0.0, + 0.5 + ], + 1, + 1 + ], + [ + [ + 0.05, + 0.0, + 0.5 + ], + 1, + 1 + ], + [ + [ + 0.05, + 0.0, + 0.85 + ], + 0, + 0 + ], + [ + [ + 0.03, + -0.1, + 0.0 + ], + 1, + 1 + ], + [ + [ + 0.2401, + 2.1425, + 0.5536 + ], + 1, + 1 + ], + [ + [ + 0.6267, + 2.3645, + 2.8796 + ], + 1, + 1 + ], + [ + [ + 0.5, + 2.2, + 0.01 + ], + 0, + 0 + ] + ] + }, + "S9": { + "error_exponents": [ + "3/4", + "3/4", + "3/4" + ] + }, + "controls": { + "S10_limit_field": true, + "S1_theorem_S": true, + "S2_model_algebra": true, + "S3_gaussian_kernel_jets": true, + "S4_elder_edge": true, + "S5_I_series": true, + "S6_proposition_H": true, + "S7_lemma_B": true, + "S8_lemma_D_cases": true, + "S9_lemma_M": true + }, + "object": "CL-SOFT-REJECTED-20261002-v1", + "scientific_effect": "NONE" +} diff --git a/frontiers/soft_rejected_pairs_20261002/SOURCES.json b/frontiers/soft_rejected_pairs_20261002/SOURCES.json new file mode 100644 index 000000000..79af497cc --- /dev/null +++ b/frontiers/soft_rejected_pairs_20261002/SOURCES.json @@ -0,0 +1,200 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "object": "CL-SOFT-REJECTED-20261002-v1", + "kind": "author-side proof candidate and conjectures, d = 2: Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 pA(0|b) E gamma^6, with the weight on soft transverse curvatures lambda in [gamma^2/(24 kappa), gamma^2/(8 kappa)]); 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not a nonauthor review", + "delta": { + "verdict": "ACCEPT WITH MINOR FIXES, no BLOCKING or MAJOR finding: three new MINOR findings (Lemma 2 hypothesis; input (ii); H0 truncated the small-gamma end, R_{2/3} -0.050 -> -0.049) and seven NITs, all applied", + "report": "project archive V2_2/frontiers_soft_rejected_pairs_20261002/REFEREE_A_DELTA.md", + "sha256": "5ccfb644e1a58abf4a022a1fe83a9c3969fdca448724de315a9eace5dc60ca01", + "bytes": 13940 + } + }, + "exploration": "project archive V2_2/frontiers_soft_rejected_pairs_20261002/exploration/ (numpy, scipy, mpmath, sympy; not part of the repository)", + "delivered_under": "explicit owner instructions of Dylan Roy in claude.ai session session_01NMeKEismAyeqgdB4sy2NJU, given after OWNER_STOP (2026-09-27): 2026-09-30 \"Continue to Work on closing as many mathematical items as possible while working with and supporting the other models in the GitHub. Also advance the mathematics in every place that's justified based on what we currently know\", followed by \"Continue\"; standing: \"Any request or permissions are preapproved in advance by me Dylan Roy\", pull requests only, never self-merge, no register/GRAPH/STATUS/lemma_closed/prize/premise edits, OWNER_STOP.md untouched. Nothing is merged by its author.", + "files": [ + { + "path": "PROOF.md", + "bytes": 43001, + "sha256": "19e911e99298daf9e7879a88cc2d727278472c68ed80f4fd6701f38596c98fe7" + }, + { + "path": "README.md", + "bytes": 6239, + "sha256": "8a98e9039bdde282ab6c4e23717d7ec5b096d9e1368e4d922bb1e0376df9f589" + }, + { + "path": "soft_check.py", + "bytes": 25199, + "sha256": "feb656ce94401a63f994dca47d09416cf130f520dc53d8ece68e18a044b4a9ab" + }, + { + "path": "RESULTS.json", + "bytes": 2395, + "sha256": "e1e0e8885d6e5a6d49865026d6c5d0a27ecc14f7fac2d61cb55f082a9d791d38" + } + ] +} diff --git a/frontiers/soft_rejected_pairs_20261002/soft_check.py b/frontiers/soft_rejected_pairs_20261002/soft_check.py new file mode 100644 index 000000000..9e6d38d79 --- /dev/null +++ b/frontiers/soft_rejected_pairs_20261002/soft_check.py @@ -0,0 +1,516 @@ +#!/usr/bin/env python3 +"""Exact controls for CL-SOFT-REJECTED-20261002-v1 (Theorem 1, Lemmas 2, 3 and 5, Proposition 4; PROOF.md). + +Standard library only; exact rationals except control S8 (a deterministic floating-point implementation of Lemma 2 on +fixed cases). Output: RESULTS.json (sorted keys), byte-identical under -O. Usage: + python3 -B -S soft_check.py # exit 0, prints RESULTS.json + python3 -B -S soft_check.py --mutant M3 # exit 1 (only its control fails) + unknown mutant label -> exit 2 + +Controls + S1 Theorem 1: the change of variables in (1.1) (69984 kappa^3 / 72^4 = 1/384 per kappa), int_{1/3}^1 (phi^-4 - phi^-2) + = 20/3, 12 (20/3)/384 = 5/24, and F0 = 25 pi0 pA(0|b) for E gamma^6 = 15 sigma^6, sigma^2 = 2; the soft window + lambda in [gamma^2/(24 kappa), gamma^2/(8 kappa)] at f4 = 0 + S2 the algebra of the model (2.1)-(2.4): partial derivatives of G_k (exact 4-point stencils), the pins, + det H_M = 6 lt + Y, det H_S = -6 lt + Y with Y = 3kB - gamma^2/4, the normalization G_k(X, (gamma/lt) z) = + (gamma^2/lt) G(X, z), the weight 36 lt^2 - Y^2 = (gamma^4/16)(phi^-2 - c'^2), the measure, and the typed window (2.4), + on random rationals + S3 the Gaussian kernel exp(-|x|^2/2): covariances of (f, grad, Hessian, third and fourth jets) from Hermite numbers; + given grad f = 0 the third-order jets (f_uuu, gamma, B, C3) are independent with variances (6, 2, 2, 6) and + uncorrelated with every even jet; the pin factor exp(-144 k^2 / (2 * 6)) = exp(-12 k^2) [(0.3)] + S4 the elder edge (3.3): truncated power series in t (exact) for the tangency R = L_S, R' = 0 of the ridge + 2(X + 1/2)^2 (X - 1) + 3 phi (X^2 - 1/4)^2 / (1 - 2 t phi X) at X = -3/2 - t/3 - t^2/3, phi = 1/3 + t/3 + 10 t^2/27; + the factorization at t = 0 and its nondegeneracy (Jacobian -96) + S5 (3.4): I(t, 0) = 20/3 - 20 t + 52 t^2/3 + O(t^3) from (3.3) and the typed edge 1/(1 - t) + S6 Proposition 4: E gamma^6 = 120, E[gamma^6 t^2] = 576 k^2, (52/3) 576 / 800 = 312/25, 312/25 - 12 = 12/25; the moment + identities gamma^6 |t|^3 = 1728 |k|^3 |B|^3, gamma^6 chi0^2 = 331776 k^4 C3^2, gamma^6 |t chi0| = 6912 |k|^3 |B C3 gamma| + S7 Lemma 3: every numerical inequality of its proof, with exact rational bounds (including s/a in [0.99, 1.01] and the + critical-value formulas h(z_r) = p^2 (a + 2s)/(6 (a + s)^2), h(z_v) = (a + s)^2 (a - 2s)/(6 chi^2), checked as + identities at rational points with s rational) + S8 Lemma 2 (floating point, fixed cases): the decision on 16 parameter points with known answers (cusp edge 1/3, + beta > 2 with and without C3, the valley at M, chi^2 = 16 phi, the open slice, the a priori region of Lemma 3) + S9 Lemma 5: the exponent bookkeeping of its three error terms (all 3/4) and the convergence exponents of I~ + S10 the limit (2.1): on three exactly pinned degree-6 fields with A = -lt r/k, (f(rX, rk zeta) - b)/(k r^3) equals + G_k + r G1 + O(r^2) at four points (G1 explicit), checked at r = 1/1000 and 1/10000 in exact arithmetic +""" +import json +import math +import sys +from fractions import Fraction as Fr + +MUTANTS = ('M1', 'M2', 'M3', 'M4', 'M5', 'M6', 'M7', 'M8', 'M9', 'M10') +MUT = None + + +class LCG: + def __init__(self, seed): + self.s = seed % (2 ** 61 - 1) + + def nxt(self): + self.s = (self.s * 437799614237992725 + 1) % (2 ** 61 - 1) + return self.s + + def rat(self, num, den, nonzero=False): + while True: + x = Fr(self.nxt() % (2 * num + 1) - num, 1 + self.nxt() % den) + if x != 0 or not nonzero: + return x + + +# ------------------------------------------------------------------------------------- truncated power series in t +class TS: + """Power series in t truncated after degree N (exact rationals).""" + N = 3 + + def __init__(self, c): + c = [Fr(x) for x in c][:TS.N + 1] + self.c = c + [Fr(0)] * (TS.N + 1 - len(c)) + + def __add__(self, o): + o = o if isinstance(o, TS) else TS([o]) + return TS([a + b for a, b in zip(self.c, o.c)]) + + __radd__ = __add__ + + def __neg__(self): + return TS([-a for a in self.c]) + + def __sub__(self, o): + return self + (-(o if isinstance(o, TS) else TS([o]))) + + def __rsub__(self, o): + return TS([o]) - self + + def __mul__(self, o): + o = o if isinstance(o, TS) else TS([o]) + out = [Fr(0)] * (TS.N + 1) + for i, a in enumerate(self.c): + for j, b in enumerate(o.c): + if i + j <= TS.N: + out[i + j] += a * b + return TS(out) + + __rmul__ = __mul__ + + def inv(self): + a0 = self.c[0] + out = [Fr(1) / a0] + [Fr(0)] * TS.N + for n in range(1, TS.N + 1): + out[n] = -sum(self.c[j] * out[n - j] for j in range(1, n + 1)) / a0 + return TS(out) + + def __pow__(self, n): + r = TS([1]) + for _ in range(n): + r = r * self + return r + + +# ------------------------------------------------------------------------------------- S1: Theorem 1 algebra +def control_S1(): + ok = True + const = Fr(9 * 5184 * 216, 144) # (9 gamma^4/144) * 5184 kappa^2 * 216 kappa gamma^2 / ... + ok &= const == 69984 + c384 = Fr(69984, 72 ** 4) if MUT != 'M1' else Fr(69984, 72 ** 4) * Fr(385, 384) + ok &= c384 == Fr(1, 384) + # int_{1/3}^1 (phi^-4 - phi^-2) dphi with the antiderivative -phi^-3/3 + phi^-1 + F = lambda p: -Fr(1, 3) / p ** 3 + 1 / p + I0 = F(Fr(1)) - F(Fr(1, 3)) + ok &= I0 == Fr(20, 3) + ok &= 12 * I0 / 384 == Fr(5, 24) + Eg6 = 15 * Fr(2) ** 3 + ok &= Eg6 == 120 and Fr(5, 24) * Eg6 == 25 + # the soft window at f4 = 0: lambda = 3 gamma^2 / (72 kappa phi), phi in [1/3, 1) + for g2, kap in ((Fr(2), Fr(7)), (Fr(5, 3), Fr(100))): + lam = lambda phi: 3 * g2 / (72 * kap * phi) + ok &= lam(Fr(1, 3)) == g2 / (8 * kap) and lam(Fr(1)) == g2 / (24 * kap) + # the bound |f4/(72 kappa phi)| <= 1/24 on |f4| < kappa, phi >= 1/3 + ok &= Fr(1) / (72 * Fr(1, 3)) == Fr(1, 24) + return ok, {'change_of_variables_constant': str(c384), 'phi_integral': str(I0), 'F0_over_pi0_pA0': str(Fr(5, 24) * Eg6)} + + +# ------------------------------------------------------------------------------------- S2: the soft model +def Gk(X, ze, g, lt, k, B, C3): + return (2 * (X + Fr(1, 2)) ** 2 * (X - 1) + Fr(1, 2) * (X * X - Fr(1, 4)) * g * ze - Fr(1, 2) * lt * ze * ze + + Fr(1, 2) * k * B * X * ze * ze + k * k * C3 * ze ** 3 / 6) + + +def Gn(X, z, phi, beta, chi): + return 2 * (X + Fr(1, 2)) ** 2 * (X - 1) / (24 * phi) + Fr(1, 2) * (X * X - Fr(1, 4)) * z - Fr(1, 2) * (1 - beta * X) * z * z + chi * z ** 3 / 6 + + +def d1(f, x, h): + return (-f(x + 2 * h) + 8 * f(x + h) - 8 * f(x - h) + f(x - 2 * h)) / (12 * h) + + +def control_S2(): + ok = True + rng = LCG(20261002) + n = 0 + for _ in range(60): + g = rng.rat(9, 7, True); lt = abs(rng.rat(9, 7, True)); k = rng.rat(5, 7); B = rng.rat(9, 5); C3 = rng.rat(9, 5) + X = rng.rat(9, 4); ze = rng.rat(9, 4); h = Fr(1, 3) + f = lambda x, y: Gk(x, y, g, lt, k, B, C3) + GX = d1(lambda x: f(x, ze), X, h) + GZ = d1(lambda y: f(X, y), ze, h) + GXX = d1(lambda x: d1(lambda xx: f(xx, ze), x, h), X, h) + GZZ = d1(lambda y: d1(lambda yy: f(X, yy), y, h), ze, h) + GXZ = d1(lambda y: d1(lambda x: f(x, y), X, h), ze, h) + ok &= GX == 6 * (X * X - Fr(1, 4)) + X * g * ze + k * B * ze * ze / 2 + ok &= GZ == (X * X - Fr(1, 4)) * g / 2 - lt * ze + k * B * X * ze + k * k * C3 * ze * ze / 2 + ok &= GXX == 12 * X + g * ze and GZZ == -lt + k * B * X + k * k * C3 * ze and GXZ == X * g + k * B * ze + Y = 3 * k * B - g * g / 4 + detM = (-6) * (-lt - k * B / 2) - (g / 2) ** 2 + detS = 6 * (-lt + k * B / 2) - (g / 2) ** 2 + ok &= detM == 6 * lt + Y and detS == -6 * lt + Y + ok &= f(Fr(-1, 2), Fr(0)) == 0 and f(Fr(1, 2), Fr(0)) == -1 + # normalization (2.2)-(2.3), gamma > 0 branch and the reflection for gamma < 0 + gg, CC, sgn = (g, C3, 1) if g > 0 else (-g, -C3, -1) + phi = gg * gg / (24 * lt); beta = k * B / lt; chi0 = 576 * (k * k) * CC / gg ** 3 + if MUT == 'M8': + chi0 = chi0 / 2 + chi = chi0 * phi * phi + t = 12 * k * B / (gg * gg) + ok &= beta == 2 * t * phi + z = rng.rat(9, 4) + ok &= f(X, sgn * gg / lt * z) == gg * gg / lt * Gn(X, z, phi, beta, chi) + # weight and measure + cp = 1 - t + ok &= 36 * lt * lt - Y * Y == gg ** 4 / 16 * (1 / phi ** 2 - cp * cp) + ok &= gg ** 2 / (24 * phi ** 2) * gg ** 4 / 16 == gg ** 6 / 384 / phi ** 2 + # typed window (2.4): |Y| < 6 lt <=> phi < 1/|c'| + ok &= (abs(Y) < 6 * lt) == (cp == 0 or phi < 1 / abs(cp)) + n += 1 + return ok, n + + +# ------------------------------------------------------------------------------------- S3: Gaussian kernel covariances +def dfac(n): + out = 1 + for j in range(n - 1, 0, -2): + out *= j + return out + + +def d0(n): + return 0 if n % 2 else (-1) ** (n // 2) * dfac(n) + + +def cov(a, b): + return Fr((-1) ** (b[0] + b[1]) * d0(a[0] + b[0]) * d0(a[1] + b[1])) + + +def control_S3(): + ok = True + first = [(1, 0), (0, 1)] + third = [(3, 0), (2, 1), (1, 2), (0, 3)] # f_uuu, gamma = f_uuT, B = f_uTT, C3 = f_TTT + S11 = [[cov(a, b) for b in first] for a in first] + ok &= S11 == [[1, 0], [0, 1]] + C = [[cov(a, b) - sum(cov(a, first[i]) * cov(first[i], b) / S11[i][i] for i in range(2)) for b in third] for a in third] + target = [[6, 0, 0, 0], [0, 2, 0, 0], [0, 0, 2, 0], [0, 0, 0, 6]] + if MUT == 'M2': + target[2][2] = 3 + ok &= C == [[Fr(x) for x in row] for row in target] + even = [(0, 0), (2, 0), (1, 1), (0, 2), (4, 0), (2, 2), (0, 4)] + ok &= all(cov(e, o) == 0 for e in even for o in third + first) + var3 = C[0][0] + ok &= Fr(144) / (2 * var3) == 12 # exp(-(12k)^2/(2*6)) = exp(-12 k^2) + return ok, {'conditional_covariance_third_jets': [[str(x) for x in row] for row in C], 'pin_exponent_a_prime': str(Fr(144) / (2 * var3))} + + +# ------------------------------------------------------------------------------------- S4, S5: the edge (3.3) and I(t, 0) (3.4) +def ridge_series(Xs, Ps, t): + beta = 2 * t * Ps + p = Xs * Xs - Fr(1, 4) + a = 1 - beta * Xs + g = 2 * (Xs + Fr(1, 2)) ** 2 * (Xs - 1) + 3 * Ps * p * p * a.inv() + # derivative in X of the same expression at fixed phi, beta: d/dX[2(X+1/2)^2(X-1)] = 6(X^2 - 1/4), + # d/dX[p^2/a] = (4 X p a + beta p^2)/a^2 + gx = 6 * p + 3 * Ps * (4 * Xs * p * a + beta * p * p) * (a * a).inv() + return g, gx + + +def control_S4(): + ok = True + t = TS([0, 1]) + p2 = Fr(10, 27) if MUT != 'M3' else Fr(1, 3) + Xs = TS([Fr(-3, 2), Fr(-1, 3), Fr(-1, 3)]) + Ps = TS([Fr(1, 3), Fr(1, 3), p2]) + g, gx = ridge_series(Xs, Ps, t) + ok &= (g + 1).c[:3] == [0, 0, 0] and gx.c[:3] == [0, 0, 0] + # t = 0: g + 1 = ((X - 1/2)(X + 3/2))^2 at phi = 1/3 (identity at rational points) + for X in (Fr(-7, 3), Fr(-3, 2), Fr(1, 5), Fr(9, 4)): + ok &= 2 * (X + Fr(1, 2)) ** 2 * (X - 1) + (X * X - Fr(1, 4)) ** 2 + 1 == ((X - Fr(1, 2)) * (X + Fr(3, 2))) ** 2 + # Jacobian of (g + 1, g_X) in (X, phi) at (-3/2, 1/3, t = 0): [[g_X, g_phi], [g_XX, g_Xphi]] = [[0, 12], [g_XX, -36]] + X0 = Fr(-3, 2) + gphi = 3 * (X0 * X0 - Fr(1, 4)) ** 2 + gXX = 12 * X0 + Fr(1, 3) * 3 * (12 * X0 * X0 - 1) # d2/dX2 [2(X+1/2)^2(X-1) + 3 phi (X^2-1/4)^2] at phi = 1/3 + gXphi = 3 * 4 * X0 * (X0 * X0 - Fr(1, 4)) + jac = 0 * gXphi - gphi * gXX + ok &= gphi == 12 and gXX == 8 and jac != 0 + return ok, {'X_e': ['-3/2', '-1/3', '-1/3'], 'phi_e': ['1/3', '1/3', str(p2)], 'jacobian': str(jac)} + + +def control_S5(): + ok = True + t = TS([0, 1]) + pe = TS([Fr(1, 3), Fr(1, 3), Fr(10, 27)]) + cp = 1 - t if MUT != 'M4' else 1 + t + I = Fr(1, 3) * (pe.inv() ** 3) - cp * cp * pe.inv() + Fr(2, 3) * cp ** 3 + ok &= I.c[:3] == [Fr(20, 3), Fr(-20), Fr(52, 3)] + # the general antiderivative: int_a^b phi^-2 (phi^-2 - c^2) = (a^-3 - b^-3)/3 - c^2 (a^-1 - b^-1), b = 1/c + for a, c in ((Fr(1, 3), Fr(1)), (Fr(2, 7), Fr(5, 4))): + b = 1 / c + lhs = (a ** -3 - b ** -3) / 3 - c * c * (1 / a - 1 / b) + ok &= lhs == a ** -3 / 3 - c * c / a + Fr(2, 3) * c ** 3 + return ok, {'I_t_series': [str(x) for x in I.c[:3]]} + + +# ------------------------------------------------------------------------------------- S6: Proposition 4 constants +def control_S6(): + ok = True + m = lambda n, s2: dfac(n) * Fr(s2) ** (n // 2) if n % 2 == 0 else Fr(0) # E X^n, X ~ N(0, s2) + EB2, Eg2, Eg6 = m(2, 2), m(2, 2), m(6, 2) + ok &= Eg6 == 120 and EB2 == 2 and Eg2 == 2 + Eg6t2 = 144 * EB2 * Eg2 if MUT != 'M9' else 144 * EB2 * Eg2 * Fr(3, 2) # gamma^6 t^2 = 144 k^2 B^2 gamma^2 + ok &= Eg6t2 == 576 + h2 = Fr(52, 3) * Eg6t2 / (Eg6 * Fr(20, 3)) + ok &= h2 == Fr(312, 25) and h2 - 12 == Fr(12, 25) + # pointwise identities at rational (k, B, C3, gamma) + rng = LCG(7) + for _ in range(20): + k, B, C3, g = rng.rat(5, 3), rng.rat(5, 3), rng.rat(5, 3), rng.rat(5, 3, True) + t = 12 * k * B / (g * g); c0 = 576 * k * k * C3 / g ** 3 + ok &= g ** 6 * abs(t) ** 3 == 1728 * abs(k) ** 3 * abs(B) ** 3 and g ** 6 * c0 ** 2 == 331776 * k ** 4 * C3 ** 2 + ok &= g ** 6 * abs(t) * abs(c0) == 6912 * abs(k) ** 3 * abs(B) * abs(C3) * abs(g) + return ok, {'h2': str(h2), 'H_k2_coefficient': str(h2 - 12)} + + +# ------------------------------------------------------------------------------------- S7: Lemma 3 +def control_S7(): + ok = True + phimax = Fr(1, 20) if MUT != 'M5' else Fr(1, 5) + betamax, chimax = Fr(1, 10), Fr(1, 400) + # |beta| = 2|t| phi <= 1/10 when phi <= |t|^-1/20; |chi| = |chi0| phi^2 <= 1/400; chi^2 <= phi/8000 + ok &= 2 * Fr(1, 20) == betamax and Fr(1, 20) ** 2 == chimax + ok &= Fr(1, 20) ** 3 * 8000 == 1 # phi^3 <= |chi0|^-2/8000 => chi^2 = chi0^2 phi^4 <= phi/8000 + # a = 1 - beta X on [-2, 1/2] lies in [4/5, 6/5]; on [-1/2, 1/2] in [19/20, 21/20] + ok &= 1 - betamax * 2 == Fr(4, 5) and 1 + betamax * 2 == Fr(6, 5) and 1 - betamax / 2 == Fr(19, 20) + # D > 0 on [-2, 1/2]: a^2 - |chi| |p| >= 16/25 - (1/400)(15/4) > 0 + ok &= Fr(16, 25) - chimax * Fr(15, 4) > 0 + # s/a in [0.99, 1.01]: |chi p| / a^2 <= (1/400)(15/4)/(16/25) = 375/25600 < 1/64 + ok &= chimax * Fr(15, 4) / Fr(16, 25) <= Fr(1, 64) + ok &= Fr(99, 100) ** 2 <= 1 - Fr(1, 64) and Fr(101, 100) ** 2 >= 1 + Fr(1, 64) + # critical values of h(z) = p z/2 - a z^2/2 + chi z^3/6 at z_r = p/(a+s), z_v = (a+s)/chi, s^2 = a^2 - chi p + rng = LCG(11) + for _ in range(30): + a, s, chi = rng.rat(9, 5, True), rng.rat(9, 5, True), rng.rat(9, 5, True) + p = (a * a - s * s) / chi + h = lambda z: p * z / 2 - a * z * z / 2 + chi * z ** 3 / 6 + ok &= h(p / (a + s)) == p * p * (a + 2 * s) / (6 * (a + s) ** 2) if a + s != 0 else True + ok &= h((a + s) / chi) == (a + s) ** 2 * (a - 2 * s) / (6 * chi * chi) + # R - A0 <= p^2 (a + 2s)/(6 (a + s)^2) <= (a + 2.02 a)/(6 (1.99 a)^2) p^2 = c_R p^2 / a with c_R <= 0.128 + cR = (1 + Fr(202, 100)) / (6 * Fr(199, 100) ** 2) + ok &= cR <= Fr(128, 1000) + # left dip at X = -2: p = 15/4, a >= 4/5: R - A0 <= cR (225/16)/(4/5) <= 2.3; A0(-2) = 13.5 L_S; 12.5/(24 phi) > 2.3 + ok &= cR * Fr(225, 16) / Fr(4, 5) <= Fr(23, 10) + ok &= 2 * Fr(9, 4) * (-3) == Fr(-27, 2) + ok &= Fr(25, 2) / (24 * phimax) > Fr(23, 10) + # on [-2, -1/2): bracket 2(X - 1)/(24 phi) + (cR/a)(X - 1/2)^2 <= -1/(8 phi) + (cR/(4/5)) (25/4) < 0 + ok &= -1 / (8 * phimax) + cR / Fr(4, 5) * Fr(25, 4) < 0 and -1 / (8 * Fr(1, 20)) == Fr(-5, 2) + # middle (-1/2, 1/2): 2(X - 1) <= -1 and (X - 1/2)^2 <= 1: bracket <= -1/(24 phi) + cR/(19/20) < 0 + ok &= -1 / (24 * phimax) + cR / Fr(19, 20) < 0 + # valley: (a + s)^2 (2s - a) >= (1.99 a)^2 (0.98 a) >= 3.88 * 0.512 = 1.98 -> V - A0 <= -0.33/chi^2; vs L_S: chi^2 < 7.92 phi + low = Fr(199, 100) ** 2 * Fr(98, 100) * Fr(4, 5) ** 3 + ok &= low >= Fr(198, 100) and low / 6 >= Fr(33, 100) + ok &= Fr(33, 100) * 24 > Fr(1, 8000) # 0.33/chi^2 > 1/(24 phi) when chi^2 <= phi/8000 + # the resulting bound: I <= int_{phi*}^inf phi^-4 = phi*^-3/3 = (20^3/3) max(1, |t|^3, chi0^2) + ok &= Fr(20) ** 3 / 3 == Fr(8000, 3) + return ok, {'phi_star_factor': str(phimax), 'c_R': str(cR), 'I_bound_constant': '8000/3'} + + +# ------------------------------------------------------------------------------------- S8: Lemma 2 (floating point) +def slice_vals(X, phi, beta, chi): + """(open, R, V) of z -> G(X, z) for the normalized model (2.3).""" + p = X * X - 0.25 + a = 1.0 - beta * X + A0 = 2 * (X + 0.5) ** 2 * (X - 1) / (24 * phi) + if chi == 0.0: + if a <= 0: + return True, None, None + return False, A0 + p * p / (8 * a), -math.inf + D = a * a - chi * p + if D <= 0: + return True, None, None + s = math.sqrt(D) + # stable roots of (chi/2) z^2 - a z + p/2 = 0 (product p/chi): no cancellation in either branch + if a >= 0: + zr = p / (a + s); zv = (a + s) / chi + else: + zr = (a - s) / chi; zv = p / (a - s) + crit = lambda z: A0 + p * z / 3 - a * z * z / 6 # the value of G at a critical point of the slice + R = crit(zr) + V = crit(zv) if MUT != 'M6' else -math.inf + return False, R, V + + +def decide(phi, beta, chi, n_mid=2001, n_side=6000, Xside=40.0): + """Lemma 2: 1 = elder, 0 = rejected.""" + LS = -1.0 / (24 * phi) + aS = 1 - beta / 2 + if aS <= 0 and chi == 0.0: + return 0 # the ridge blows up at X = 1/beta < 1/2 + q = (Xside - 0.5) / 1e-6 + side = [1e-6 * q ** (i / (n_side - 1)) for i in range(n_side)] + mid = [-0.5 + i / (n_mid - 1) for i in range(1, n_mid - 1)] + + def walk(xs, at_half_is_S): + """Follow M's ridge interval along xs: 'closed' at the first X with R <= L_S, 'trigger' or 'open-ended'.""" + for X in xs: + op, R, V = slice_vals(X, phi, beta, chi) + if op: + return 'trigger', X + if R <= LS: + return 'closed', X + if R > 0: + return 'trigger', X + if V > LS and not (at_half_is_S and abs(X - 0.5) < 2e-3): + return 'trigger', X + return 'open-ended', None + + if aS > 0: # S on the ridge: M's interval must reach S + st, X = walk(mid, False) + if st == 'trigger' or (st == 'closed' and X < 0.5 - 1e-3): + return 0 + else: # S is the valley point at X = 1/2: the interval must contain 1/2 + st, X = walk(mid + [0.5] + [0.5 + d for d in side], True) + if st != 'closed' or X <= 0.5: + return 0 + st, X = walk([-0.5 - d for d in side], False) + return 1 if st == 'closed' else 0 + + +def control_S8(): + cases = [((0.30, 0.0, 0.0), 1), ((0.36, 0.0, 0.0), 0), ((0.3333, 0.0, 0.0), 1), ((0.3334, 0.0, 0.0), 0), + ((0.95, 0.0, 0.0), 0), ((0.5, 2.2, 0.0), 0), ((0.04, 0.09, 0.002), 1), + ((0.30, 0.0, 3.0), 0), ((0.20, 0.0, 1.0), 0), ((0.20, 0.0, 0.5), 1), ((0.05, 0.0, 0.5), 1), + ((0.05, 0.0, 0.85), 0), ((0.03, -0.1, 0.0), 1), + ((0.2401, 2.1425, 0.5536), 1), ((0.6267, 2.3645, 2.8796), 1), ((0.5, 2.2, 0.01), 0)] + # (0.05, 0, chi): the valley at M reaches L_S when chi^2 = 16 phi = 0.8, chi = 0.894, and slightly left of M earlier; + # (0.2, 0, 1): the slice opens (D = 0) at p = 1 before the left ridge falls below L_S + res = [decide(*c) for c, _ in cases] + ok = all(r == e for r, (_, e) in zip(res, cases)) + return ok, {'cases': [[list(c), e, r] for (c, e), r in zip(cases, res)]} + + +# ------------------------------------------------------------------------------------- S9: Lemma 5 bookkeeping +def control_S9(): + ok = True + e1 = 4 + Fr(-13, 4) if MUT != 'M7' else 4 + Fr(-12, 4) # l^4 * l^(-13/4): r > l^(1/4), kappa^2 * k^2 + e2 = Fr(2, 3) + Fr(1, 12) # l^(2/3) * l^(1/12): the tail of I~ beyond u = l^(-1/12) + e3 = Fr(8, 3) - 2 + Fr(1, 3) - Fr(3, 12) # l^(8/3 - 2 + 1/3) * l^(-3/12): the O(kappa^-2) part + ok &= e1 == e2 == e3 == Fr(3, 4) + # int_{l^(1/4)}^inf r^-14 dr = l^(-13/4)/13 ; the integrand of I~: u^4 * u^-6 = u^-2 at infinity, u^4 at 0 + ok &= Fr(-14) + 1 == -13 and 4 - 6 == -2 + ok &= Fr(3, 4) > Fr(2, 3) > Fr(1, 4) + return ok, {'error_exponents': [str(e1), str(e2), str(e3)]} + + + +# ------------------------------------------------------------------------------------- S10: the limit (2.1) +FACT = [1, 1, 2, 6, 24, 120, 720, 5040] + + +def solve(M, v): + """Exact Gaussian elimination.""" + n = len(v) + A = [row[:] + [v[i]] for i, row in enumerate(M)] + for c in range(n): + piv = next(i for i in range(c, n) if A[i][c] != 0) + A[c], A[piv] = A[piv], A[c] + for i in range(n): + if i != c and A[i][c] != 0: + f = A[i][c] / A[c][c] + A[i] = [a - f * b for a, b in zip(A[i], A[c])] + return [A[i][n] / A[i][i] for i in range(n)] + + +def control_S10(): + """Exactly pinned degree-6 fields with A = -lt r/k: (f(rX, rk zeta) - b)/(k r^3) = G_k + r G1 + O(r^2), with + G1 = (f4/(24k))(X^2 - 1/4)^2 + (k/4) c22 X^2 zeta^2 + (k^2/6) c13 X zeta^3 + (k^3/24) c04 zeta^4 + (1/6) c31 X (X^2 - 1/4) zeta.""" + ok = True + rng = LCG(240) + pinned = [(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (1, 1)] + mono = [(i, n - i) for n in range(7) for i in range(n, -1, -1)] + worst = Fr(0) + for trial in range(3): + k = Fr(1 + rng.nxt() % 4, 1 + rng.nxt() % 3); lt = Fr(1 + rng.nxt() % 5, 1 + rng.nxt() % 4); b = rng.rat(3, 2) + free = {m: rng.rat(5, 3) for m in mono if m not in pinned} + errs = [] + for r in (Fr(1, 10 ** 3), Fr(1, 10 ** 4)): + c = dict(free); c[(0, 2)] = -lt * r / k + def val(cc, x, y, dx=0, dy=0): + out = Fr(0) + for (i, j), v in cc.items(): + if i >= dx and j >= dy: + out += v * x ** (i - dx) * y ** (j - dy) / (FACT[i - dx] * FACT[j - dy]) + return out + rows, rhs = [], [] + for (x, target, kind) in ((-r / 2, b, 'f'), (r / 2, b - k * r ** 3, 'f'), (-r / 2, 0, 'fx'), (r / 2, 0, 'fx'), + (-r / 2, 0, 'fy'), (r / 2, 0, 'fy')): + dx, dy = {'f': (0, 0), 'fx': (1, 0), 'fy': (0, 1)}[kind] + rows.append([val({m: Fr(1)}, x, Fr(0), dx, dy) for m in pinned]) + rhs.append(target - val(c, x, Fr(0), dx, dy)) + sol = solve(rows, rhs) + for m, v in zip(pinned, sol): + c[m] = v + g, B, C3, f4 = c[(2, 1)], c[(1, 2)], c[(0, 3)], c[(4, 0)] + e = [] + for X, ze in ((Fr(-3, 2), Fr(1, 3)), (Fr(-1, 2), Fr(2)), (Fr(1, 3), Fr(-1)), (Fr(2), Fr(1, 2))): + F = (val(c, r * X, r * k * ze) - b) / (k * r ** 3) + C3m = C3 if MUT != 'M10' else 0 + G = Gk(X, ze, g, lt, k, B, C3m) + p4 = (X * X - Fr(1, 4)) + G1 = (f4 / (24 * k) * p4 ** 2 + k / 4 * c[(2, 2)] * X * X * ze * ze + k * k / 6 * c[(1, 3)] * X * ze ** 3 + + k ** 3 / 24 * c[(0, 4)] * ze ** 4 + c[(3, 1)] / 6 * X * p4 * ze) + e.append(abs((F - G) / r - G1)) + errs.append(max(e)) + # O(r^2) remainder: the r^1-corrected error must scale like r (factor 10 between r = 1e-3 and 1e-4) + ok &= errs[1] * 5 <= errs[0] and errs[1] <= Fr(1, 10) + worst = max(worst, errs[1]) + return ok, {'fields': 3, 'max_error_after_r1_term_at_r_1e-4': str(float(worst))} + + +def main(argv): + global MUT + if len(argv) == 2 and argv[0] == '--mutant': + if argv[1] not in MUTANTS: + print('unknown mutant label', file=sys.stderr) + return 2 + MUT = argv[1] + elif argv: + print('usage: soft_check.py [--mutant M1..M10]', file=sys.stderr) + return 2 + s1, s1d = control_S1() + s2, n2 = control_S2() + s3, s3d = control_S3() + s4, s4d = control_S4() + s5, s5d = control_S5() + s6, s6d = control_S6() + s7, s7d = control_S7() + s8, s8d = control_S8() + s9, s9d = control_S9() + s10, s10d = control_S10() + res = {'object': 'CL-SOFT-REJECTED-20261002-v1', 'scientific_effect': 'NONE', + 'controls': {'S1_theorem_S': s1, 'S2_model_algebra': s2, 'S3_gaussian_kernel_jets': s3, 'S4_elder_edge': s4, + 'S5_I_series': s5, 'S6_proposition_H': s6, 'S7_lemma_B': s7, 'S8_lemma_D_cases': s8, + 'S9_lemma_M': s9, 'S10_limit_field': s10}, + 'S1': s1d, 'S2_instances': n2, 'S3': s3d, 'S4': s4d, 'S5': s5d, 'S6': s6d, 'S7': s7d, 'S8': s8d, 'S9': s9d, 'S10': s10d} + ok = all(res['controls'].values()) + if MUT is None: + print(json.dumps(res, indent=1, sort_keys=True)) + return 0 if ok else 1 + + +if __name__ == '__main__': + sys.exit(main(sys.argv[1:])) From b744f03f913e415de7f7139844a01c7711083ee8 Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 07:48:46 +0000 Subject: [PATCH 2/7] Soft rejected pairs v1.1: Theorem 1 in every d >= 2, the d = 3 constants, the stiff directions, and a d = 3 Monte Carlo test Made before any nonauthor review of v1 (6bae0b3); see the v1.1 notice on the PR. - Theorem 1 holds in every d >= 2: kappa A^rej(b, kappa, u) = F0 + O(1/kappa), with F0 = (5/24) pi0 D, where D is the density at 0 of the smallest eigenvalue lambda_1 of -A, weighted by gamma_1^6 (lambda_2...lambda_m)^2. The proof uses Weyl spectral coordinates, a double-soft bound and the Step 2 substitution with f4~ = f4 + 3 sum_{i>=2} gamma_i^2/lambda_i. - Corollary 1' (Gaussian kernel): int int F0 = 25 sqrt3/(48 pi^2) in d = 2 and 125 sqrt30/(192 pi^3) in d = 3. - Proposition 2' (d >= 3): the stiff directions live at scale r^(3/2) and decouple, so the elder decision is Lemma 2's. For the Gaussian kernel H is the same in every d. Lemma 5 holds, and Conjectures 6-7 are stated, in every d. R_{2/3} is about -0.062 in d = 3. - Section 5 adds the d = 3 quadrature and #216's raw d = 3 records. For s < 0.8 there are 11 observed pairs, against 10.7 from the composite and 1,329 from the cusp kernel alone. - Fixes A-1: v1's "-0.537" for Itilde was not reproducible from the recorded scripts. - New exact controls S11-S13 and mutants M11-M13. In mutant mode the checker names the failing control on stderr. - A second clean-context same-family referee reviewed the delta: ACCEPT WITH MINOR FIXES (4 MINOR, 14 NIT), all applied. Scientific effect: NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .../soft_rejected_pairs_20261002/PROOF.md | 548 +++++++++++++----- .../soft_rejected_pairs_20261002/README.md | 73 +-- .../soft_rejected_pairs_20261002/RESULTS.json | 106 +++- .../soft_rejected_pairs_20261002/SOURCES.json | 36 +- .../soft_check.py | 313 +++++++++- 5 files changed, 864 insertions(+), 212 deletions(-) diff --git a/frontiers/soft_rejected_pairs_20261002/PROOF.md b/frontiers/soft_rejected_pairs_20261002/PROOF.md index d33a5ba1e..8b2f5fae2 100644 --- a/frontiers/soft_rejected_pairs_20261002/PROOF.md +++ b/frontiers/soft_rejected_pairs_20261002/PROOF.md @@ -1,140 +1,283 @@ -# Soft rejected pairs in `d = 2`: the `1/κ` tail of the rejected cusp kernel, the fold-scale rejection function, and a conjectured `ℓ^{2/3}` term +# Soft rejected pairs: the `1/κ` tail of the rejected cusp kernel in every dimension, the fold-scale rejection function, and a conjectured `ℓ^{2/3}` term + +Object: CL-SOFT-REJECTED-20261002-v1.1. Version history: +- v1 → v1.1, made before any nonauthor review: + - Theorem 1 holds in every `d ≥ 2`; the soft direction is the eigenvector of `−A` with the smallest eigenvalue. + - Corollary 1′ gives the Gaussian-kernel constants in `d = 2, 3`. + - Proposition 2′ treats the stiff directions for `d ≥ 3`. So Proposition 4 and Lemma 5 hold, and Conjectures 6–7 are + stated, in every `d`. + - §5 adds the `d = 3` quadrature and #216's `d = 3` records. + - Finding A-1 is fixed. The author found it while archiving v1 and recorded it in the PR's disposition: v1 quoted a + value of `Ĩ` that was not reproducible from the recorded scripts. + - Controls S11–S13 are new. + - A same-family referee checked the delta (`REFEREE_B.md`: ACCEPT WITH MINOR FIXES; 4 MINOR, 14 NIT, all applied). +- §2's Lemmas 2 and 3 and the `d = 2` proof of Proposition 4 are unchanged. -Object: CL-SOFT-REJECTED-20261002-v1. Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 2 October 2026. -Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem 1, Lemmas 2, 3 and 5, Proposition 4); CONJECTURES 6 and 7, with numerical -evidence. Nonauthor review required. Scientific effect: NONE — no register, graph, STATUS, PROOF_INDEX, prize or Boolean -change; no numerical constant is certified. Same GitHub account as every lane; zero organizational independence. +Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem 1, Corollary 1′, Lemmas 2, 3 and 5, Propositions 2′ and 4); CONJECTURES 6 +and 7, with numerical evidence. Nonauthor review required. Scientific effect: NONE — no register, graph, STATUS, +PROOF_INDEX, prize or Boolean change; no numerical constant is certified. Same GitHub account as every lane; zero +organizational independence. **Why.** Math- #229 (merged) proves `ρ_rej(ℓ) = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} − ν_eld^{far,r_0^*}(ℓ) + O(ℓ^{3/7})` ((R⁺.1)). Math- #240 (open) proves that each fixed cusp window `κ₀ ≤ ℓ/r⁴ ≤ κ₁` contributes a multiple of `ℓ^{1/4}` with error `O(ℓ^{3/4})`, and its Remark 1 leaves the -ends `κ → ∞` and `κ → 0` open. This note studies the end `κ → ∞` for the rejected pairs in `d = 2`. Formally it is the only -source of the next term of `ρ_rej` (§4). +ends `κ → ∞` and `κ → 0` open. This note studies the end `κ → ∞` for the rejected pairs. Formally it is the only source of +the next term of `ρ_rej` (§4). **What is new.** -- **Theorem 1** (§1). In `d = 2`, `κ𝒜^{rej}(b, κ, u) = F₀(b, u) + O(κ^{−1})` as `κ → ∞`, uniformly with Gaussian decay in - `b`, where `F₀ = (5/24)π₀(u; v₀(b, 0))p_A(0 | b)E[γ⁶] > 0`. The rejected weight sits on *soft* transverse curvatures - `λ = −A ∈ [γ²/(24κ), γ²/(8κ)]`, up to relative errors `O(1/κ)`. -- **The fold-scale soft model** (§2). At fixed `k = ℓ/r³` and `λ = λ̃r/k`, the window field divided by `κ = k/r` tends to an - explicit polynomial `G_k` in which two more jets enter: `B = ∂_uA` and `C₃ = ∂_Θ³f`. **Lemma 2** decides the elder mark in - `G_k` exactly by a one-dimensional scan, including a case in which the saddle is not on the ridge. **Lemma 3** gives an a - priori elder region. The typed weight is `(γ⁴/16)(φ^{−2} − (1 − 12kB/γ²)²)₊`. -- **Proposition 4** (§3). For the Gaussian kernel, the fold-scale rejection rate is `F(k; b) = F₀(b)H(k)` with `H` free of `b`, - and `H(k) = 1 + (12/25)k² + O(k³)`. The elder edge moves to `φ_e = 1/3 + t/3 + 10t²/27 + O(t³)` (`t = 12kB/γ²`). The - resulting `+312/25` nearly cancels the `−12` of the pin density `e^{−12k²}`. -- **Lemma 5 and Conjecture 7** (§4). The two-scale composite density has the expansion - `(I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`, with `R_{2/3}` given by `F` and `F₀`. Conjecture 7 says that - `ρ_rej + ν_eld^{far,r_0^*} − B_{2,L}` has the same expansion. For the Gaussian kernel, numerically `R_{2/3} ≈ −0.049` - (§5). +- **Theorem 1** (§1). In every `d ≥ 2`, `κ𝒜^{rej}(b, κ, u) = F₀(b, u) + O(κ^{−1})` as `κ → ∞`, uniformly with Gaussian decay + in `b`. + - The constant is `F₀ = (5/24)π₀(u; v₀(b, 0))𝔇(b, u) > 0`. Here `𝔇` is the density at `0` of the smallest eigenvalue `λ₁` + of `−A`, weighted by `γ₁⁶(λ₂⋯λ_m)²`, where `γ₁` is the component of `γ` along the soft eigenvector. + - In `d = 2`, `𝔇 = p_A(0 | b)E[γ⁶]`. + - The rejected weight sits on *soft* transverse curvatures `λ₁ ∈ [γ₁²/(24κ), γ₁²/(8κ)]`, up to relative errors + `O(1/κ)`. +- **Corollary 1′** (Gaussian kernel). `∫∫F₀ db dσ = 25√3/(48π²) ≈ 0.09140` in `d = 2`, and `125√30/(192π³) ≈ 0.11501` in + `d = 3`. +- **The fold-scale soft model** (§2). At fixed `k = ℓ/r³` and `λ₁ = λ̃r/k`, the window field divided by `κ = k/r` tends to an + explicit polynomial `G_k`, in which two more jets enter: `B = ∂_uA` and `C₃ = ∂_Θ³f`, taken along the soft direction. + - **Lemma 2** decides the elder mark in `G_k` exactly by a one-dimensional scan, including a case in which the saddle is + not on the ridge. + - **Lemma 3** gives an a priori elder region. + - The typed weight is `(γ⁴/16)(φ^{−2} − (1 − 12kB/γ²)²)₊`. + - **Proposition 2′** (`d ≥ 3`). The stiff transverse directions live at the scale `r^{3/2}` and decouple. The limit is + `G_k` plus a negative definite quadratic form in them, with the same elder decision. +- **Proposition 4** (§3). For the Gaussian kernel in every `d ≥ 2` (through Proposition 2′ for `d ≥ 3`), the fold-scale + rejection rate is `F(k; b) = F₀(b)H(k)`, with `H` independent of `b` and of `d`, and `H(k) = 1 + (12/25)k² + O(k³)`. + - The elder edge moves to `φ_e = 1/3 + t/3 + 10t²/27 + O(t³)`, where `t = 12kB/γ²`. + - The resulting `+312/25` nearly cancels the `−12` of the pin density `e^{−12k²}`. +- **Lemma 5 and Conjecture 7** (§4). + - The two-scale composite density has the expansion `(I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`, with `R_{2/3}` + given by `F` and `F₀`. + - Conjecture 7 says that `ρ_rej + ν_eld^{far,r_0^*} − B_{d,L}` has the same expansion. + - For the Gaussian kernel, numerically `R_{2/3} ≈ −0.049` in `d = 2` and `≈ −0.062` in `d = 3` (§5). - **Evidence** (§5, exploration): - - the tail of Theorem 1 by quadrature; + - the tail of Theorem 1, by quadrature in `d = 2` and `d = 3`; - Lemma 2 against a two-dimensional flood fill; - - `H` by quadrature; - - #216's Monte Carlo of the rejected adjacent pairs, binned by lifetime and by `s = r/ℓ^{1/4}`. It shows the predicted - suppression at small `s`: for `s < 0.8`, 37 pairs are observed and 35.3 predicted, against 4,322 from the cusp kernel - alone. + - `H`, by quadrature; + - #216's Monte Carlo of the rejected adjacent pairs in `d = 2` and `d = 3`, binned by lifetime and by `s = r/ℓ^{1/4}`. It + shows the predicted suppression at small `s`: + + | | observed (`s < 0.8`) | composite | cusp kernel alone | + |---|---|---|---| + | `d = 2` | 37 | 35.3 | 4,322 | + | `d = 3` | 11 | 10.7 | 1,329 | **Dependencies.** -- Consumed (all merged): #220 (`frontiers/elder_third_order_20261001/PROOF.md`, blob `c8767dde`): §0 ((0.1), (0.2)) and §1 - (window coordinates, (1.1), Lemma Q, (M1)–(M6)). #218 (`frontiers/candidate_third_order_20261001/PROOF.md`, blob - `70ca57ef`): §0 (`𝒜^{cand}`, `I^{cand}`, `B_{d,L}`). #229 (`frontiers/third_order_rate_20261001/PROOF.md`, blob - `110ed33a`): (R⁺.1) and §0 (`Y_r`). #207 (`frontiers/cusp_second_order_20261001/PROOF.md`, blob `f6df5a73`): §§0, 4, 6 and - Theorem CU.2(c). [R] (`frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, blob `247b3ecf`): (R5). [P] §2, through #220. -- Cited only: [C7-K] (`frontiers/c7_total_bounded_20260929/PROOF.md`, blob `28748b08`): (K2). #240 (open): Lemma Q′, Step N6, - Corollary N′, Remarks 1 and 4. #237 (open): Theorem P, Remark 4. #216 (open): the Monte Carlo. #187 (merged) and #188 (open): the far - elder density. +- Consumed (all merged): + - #220 (`frontiers/elder_third_order_20261001/PROOF.md`, blob `c8767dde`): §0 ((0.1), (0.2)) and §1 (window coordinates, + (1.1), Lemma Q, (M1)–(M6)). + - #218 (`frontiers/candidate_third_order_20261001/PROOF.md`, blob `70ca57ef`): §0 (`𝒜^{cand}`, `I^{cand}`, `B_{d,L}`). + - #229 (`frontiers/third_order_rate_20261001/PROOF.md`, blob `110ed33a`): (R⁺.1) and §0 (`Y_r`). + - #207 (`frontiers/cusp_second_order_20261001/PROOF.md`, blob `f6df5a73`): §§0, 4, 6; the parity factorization of + [P] §15, as stated after (CU.2); Theorem CU.1, for the transverse pins; and Theorem CU.2(c). + - [R] (`frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, blob `247b3ecf`): (R5). + - [P] §2, through #220. +- Cited only: + - [C7-K] (`frontiers/c7_total_bounded_20260929/PROOF.md`, blob `28748b08`): (K2). + - #240 (open): Lemma Q′, Step N6, Corollary N′, Remarks 1 and 4. + - #237 (open): Theorem P, Remark 4. + - #216 (open): the Monte Carlo in `d = 2, 3`. + - #187 (merged) and #188 (open): the far elder density. ## 0. Setting -Setting and notation are those of #220 §0 and #218 §0 (through #207 §§0, 4, 6), specialized to `d = 2`, `m = 1`: -- the [P] field on `X = R²/(LZ²)`; near pins `M = −ru/2`, `S = ru/2` at heights `b` and `b − kr³`; `κ = k/r`, so the lifetime is - `ℓ = kr³ = κr⁴`; `Θ` is the unit vector orthogonal to `u`; -- the jets at `0` under the contact law at `v₀(b, k)` (#220 §2; expectation `E_{v₀(b,k)}`, and `E₀[· | b]` at `k = 0`): - `A = ∂_Θ²f` (a scalar), `f₄ = ∂_u⁴f`, `γ = ∂_u²∂_Θf`, `B = ∂_u∂_Θ²f = ∂_uA`, `C₃ = ∂_Θ³f`. On `{A < 0}` put `λ := −A`, so - `Δ = A = −λ`, `adj A = 1`, and +Setting and notation are those of #220 §0 and #218 §0 (through #207 §§0, 4, 6). Fix `d ≥ 2` and put `m := d − 1`. +- **The field and the pair.** The [P] field lives on `X = R^d/(LZ^d)`. The near pins are `M = −ru/2` and `S = ru/2`, at + heights `b` and `b − kr³`. Put `κ = k/r`, so the lifetime is `ℓ = kr³ = κr⁴`; `Θ` ranges over `u^⊥`. +- **The jets** at `0`, under the contact law at `v₀(b, k)` (#220 §2; expectation `E_{v₀(b,k)}`, and `E₀[· | b]` at `k = 0`): + - `A = D_Θ²f ∈ Sym(m)`, `f₄ = ∂_u⁴f` and `γ = ∇_Θ∂_u²f ∈ R^m`; + - `Δ = det A` and `Y = (f₄/12)Δ − γᵀadj(A)γ/4`; + - `φ = Y/(6κΔ) = (f₄ − 3γᵀA^{−1}γ)/(72κ)`. +- **The eigenframe.** On `{A < 0}` write `Λ := −A`. Its eigenvalues are `0 < λ₁ ≤ λ₂ ≤ ⋯ ≤ λ_m`, with an orthonormal + eigenbasis `e₁, …, e_m`, and `γ_i := γ·e_i`. Then - Y = (f₄/12)Δ − γ²/4 = −(λ/12) z, z := f₄ + 3γ²/λ = f₄ − 3γᵀA^{−1}γ, φ := z/(72κ); (0.1) + z := 72κφ = f₄ + 3Σ_i γ_i²/λ_i, Y = (Δ/12)z, Δ² = (λ₁⋯λ_m)², w_κ := 36κ²Δ² − Y² = 36κ²Δ²(1 − φ²). (0.1) -- the cusp kernels (#220 (0.1), #218 §0): `𝒜^*(b, κ, u) = 12π₀(u; v₀(b, 0))E₀[w_κ 1_* | b]` with - `w_κ = 36κ²Δ² − Y² = (λ²/144)(5184κ² − z²)`, `1_cand = 1{A < 0, |z| < 72κ}` and `1_eld = 1{A < 0, |z| < 24κ}`. So + In `d = 2` (`m = 1`), `λ := λ₁ = −A` and `adj A = 1`, so `z = f₄ + 3γ²/λ` and `w_κ = (λ²/144)(5184κ² − z²)`. +- **The jets along a transverse direction.** For a unit vector `e ⊥ u` put `γ_e := ∂_u²∂_ef`, `B_e := ∂_u∂_e²f` and + `C_e := ∂_e³f`. In `d = 2`, write `B := B_Θ = ∂_uA` and `C₃ := C_Θ = ∂_Θ³f`. +- **The cusp kernels** (#220 (0.1), #218 §0) are `𝒜^*(b, κ, u) = 12π₀(u; v₀(b, 0))E₀[w_κ1_* | b]`, with weight `w_κ`, + `1_cand = 1{A < 0, |φ| < 1}` and `1_eld = 1{A < 0, |φ| < 1/3}`. So - 𝒜^{rej}(b, κ, u) := (𝒜^{cand} − 𝒜^{eld})(b, κ, u) = 12π₀(u; v₀(b, 0)) E₀[(λ²/144)(5184κ² − z²) 1{A < 0, 24κ ≤ |z| < 72κ} | b]. (0.2) + 𝒜^{rej}(b, κ, u) := (𝒜^{cand} − 𝒜^{eld})(b, κ, u) = 12π₀(u; v₀(b, 0)) E₀[36κ²Δ²(1 − φ²) 1{A < 0, 1/3 ≤ |φ| < 1} | b]. (0.2) - The contact terms cancel, so by #220 (0.2) and #218 §0, `I^{cand} − c₁ = ∫_{S¹}∫_R∫_0^∞𝒜^{rej}(b, s^{−4}, u) ds db dσ(u)`. + The contact terms cancel. So, by #220 (0.2) and #218 §0, `I^{cand} − c₁ = ∫_{S^{d−1}}∫_R∫_0^∞𝒜^{rej}(b, s^{−4}, u) ds db dσ(u)`. -Two facts about the contact law are used. The pin set splits into even pins `(f, ∂_u²f, ∂_u∂_Θf)` and odd pins -`(∂_uf, ∂_u³f, ∂_Θf)`, and even- and odd-order derivatives of a stationary field at a point are uncorrelated; the +**Two facts about the contact law.** The pin set splits into even pins `(f, ∂_u²f, ∇_Θ∂_uf)` and odd pins +`(∂_uf, ∂_u³f, ∇_Θf)`. Even-order and odd-order derivatives of a stationary field at a point are uncorrelated. The nondegeneracy statements come from the finite-jet rank of [P] §2. -- (G1) Under `v₀(b, 0)`, `γ` is centred Gaussian with variance `σ_γ²(u) > 0` and independent of `(A, f₄)`. -- (G2) Given `b`, `(A, f₄)` is a nondegenerate Gaussian vector whose mean is affine in `b`. Write `p_A(a | b)` for the density - of `A`, and `p_A(a | f₄, b)` for its conditional density given `f₄`. - -**The Gaussian kernel.** For `C(x) = e^{−|x|²/2}` on `R²`, an exact computation (control S3) gives the following. Given -`∇f(0) = 0`, the third-order jets `(∂_u³f, γ, B, C₃)` are independent, with variances `(6, 2, 2, 6)`, and uncorrelated with -every even jet. Hence under `v₀(b, k)` the jets `(γ, B, C₃)` are centred and independent of `(A, f₄)`, and the gap pin -`∂_u³f = 12k` contributes the factor `e^{−144k²/(2·6)} = e^{−12k²}` to the pin density. (0.3) -The periodized kernel of the torus field agrees with `C` up to `O(e^{−L²/8})`. Statements labelled *Gaussian kernel* are -about the model case `C` on `R²`, as in #207 §8. +- (G1) Under `v₀(b, 0)`, `γ` is a centred Gaussian vector with nondegenerate covariance `Σ_γ(u)`, independent of `(A, f₄)`. + In `d = 2`, `Σ_γ = σ_γ²`. +- (G2) Given `b`, `(A, f₄)` is a nondegenerate Gaussian vector on `Sym(m) × R` whose mean is affine in `b`. + - `p(A, f₄ | b)` is its density with respect to Lebesgue measure on `Sym(m) × R`. + - In `d = 2`, `p_A(a | b)` is the density of `A`, and `p_A(a | f₄, b)` its conditional density given `f₄`. + +**The Gaussian kernel.** For `C(x) = e^{−|x|²/2}` on `R^d`, exact computations (controls S3 and S12) give the following. +1. *Odd jets.* Given `∇f(0) = 0`, and for every unit vector `e ⊥ u`, the jets `(∂_u³f, γ_e, B_e, C_e)` are independent, with + variances `(6, 2, 2, 6)`, and uncorrelated with every even jet. Hence: + - under `v₀(b, k)` the jets `(γ_e, B_e, C_e)` are centred and independent of `(A, f₄)`; + - `γ ~ N(0, 2I_m)`; + - the gap pin `∂_u³f = 12k` contributes the factor `e^{−144k²/(2·6)} = e^{−12k²}` to the pin density. +2. *Even jets.* Given the even pins `(f, ∂_u²f, ∇_Θ∂_uf) = (b, 0, 0)`: + - `A = −bI_m + G`, where `G` has independent entries with `Var G_ii = 2` and `Var G_ij = 1` for `i ≠ j` (the Gaussian + orthogonal ensemble); + - `f₄ ~ N(−3b, 24)` is independent of `A`. + +This is (0.3). The periodized kernel of the torus field agrees with `C` up to `O(e^{−L²/8})`. Statements labelled +*Gaussian kernel* are about the model case `C` on `R^d`, as in #207 §8. ## 1. The soft tail of the rejected cusp kernel -**Theorem 1.** There are `C, c, N` such that for `κ ≥ 1`, `b ∈ R` and `u ∈ S¹`, - - |κ 𝒜^{rej}(b, κ, u) − F₀(b, u)| ≤ C κ^{−1} (1 + |b|)^N e^{−cb²}, F₀(b, u) := (5/24) π₀(u; v₀(b, 0)) p_A(0 | b) E[γ⁶]. (1.0) - -For the Gaussian kernel `E[γ⁶] = 15σ_γ⁶ = 120`, so `F₀ = 25π₀(v₀(b, 0))p_A(0 | b)`. - -*Proof.* Fix `b` and `u`. The constants below are uniform in `u` and polynomial in `b` before the factor `π₀`, which carries -`e^{−c₀b²}` ([R] (R5)). Split the expectation in (0.2) on `{A < 0}`. - -*Step 1 (negative `z` and large `|f₄|`).* By (0.1), `z ≥ f₄` on `{A < 0}`. So `z ≤ −24κ` forces `f₄ ≤ −24κ`, and the weight is -at most `36κ²λ²`. On `{|f₄| ≥ κ}`, (G2) gives `E₀[λ²1{|f₄| ≥ κ} | b] ≤ C(1 + |b|)^N e^{−c(κ − C|b|)₊²}`. Multiplied by `π₀`, -this is `≤ C(1 + |b|)^N e^{−cb²}e^{−cκ²}` (if `|b| ≥ κ/(2C)`, use `e^{−c₀b²}`). Both parts are `O(κ^{−2})` after -multiplying by `κ`. - -*Step 2 (the soft window).* On `{|f₄| < κ, 24κ ≤ z < 72κ}` we have `3γ²/λ = z − f₄ ∈ (23κ, 73κ)`, so the event is -`{λ ∈ (3γ²/(72κ − f₄), 3γ²/(24κ − f₄)]}`. For fixed `(γ, f₄)`, `λ ↦ φ = (f₄ + 3γ²/λ)/(72κ)` is a decreasing bijection onto -`[1/3, 1)`, with `λ = 3γ²/(72κφ − f₄)` and `|dλ/dφ| = 216κγ²/(72κφ − f₄)²`. Using `5184κ² − z² = 5184κ²(1 − φ²)`, +**Theorem 1.** Let `d ≥ 2`. The limit - ∫ p_A(−λ | f₄, b) (λ²/144)(5184κ² − z²) 1{…} dλ = ∫_{1/3}^{1} p_A(−λ(φ) | f₄, b) · 69984 κ³ γ⁶ (1 − φ²)/(72κφ − f₄)⁴ dφ. (1.1) + 𝔇(b, u) := lim_{ε↓0} ε^{−1} E₀[γ₁⁶ (λ₂⋯λ_m)² 1{A < 0, λ₁ < ε} | b] ∈ (0, ∞) (1.0a) -Since `69984/72⁴ = 1/384`, the factor `69984κ³/(72κφ − f₄)⁴` equals `(384κφ⁴)^{−1}(1 − f₄/(72κφ))^{−4}`, and -`|f₄/(72κφ)| ≤ 1/24` on this event. So +exists, and there are `C, c, N` such that for `κ ≥ 1`, `b ∈ R` and `u ∈ S^{d−1}`, - |(1 − f₄/(72κφ))^{−4} − 1| ≤ C|f₄|/κ, |p_A(−λ | f₄, b) − p_A(0 | f₄, b)| ≤ C_b λ ≤ C_b γ²/κ, (1.2) + |κ 𝒜^{rej}(b, κ, u) − F₀(b, u)| ≤ C κ^{−1} (1 + |b|)^N e^{−cb²}, F₀(b, u) := (5/24) π₀(u; v₀(b, 0)) 𝔇(b, u). (1.0) -where `C_b` is a polynomial in `|b|` and `|f₄|` (the conditional density is Gaussian, with variance bounded below and mean -affine in `(f₄, b)`). Hence (1.1) equals `(γ⁶/(384κ))p_A(0 | f₄, b)∫_{1/3}^1(φ^{−4} − φ^{−2})dφ + κ^{−2}O(γ⁸ + γ⁶|f₄|)·poly`, -and `∫_{1/3}^1(φ^{−4} − φ^{−2})dφ = 20/3`. +The empty product `λ₂⋯λ_m` equals `1` when `m = 1`. +- *Special forms of `𝔇`.* + - By (G1), `E[γ₁⁶ | A] = 15(e₁ᵀΣ_γe₁)³`. + - In `d = 2`, `𝔇 = p_A(0 | b)E[γ⁶]`, so `F₀ = (5/24)π₀p_A(0 | b)E[γ⁶]`. + - If `Σ_γ = σ_γ²I_m`, then `𝔇 = 15σ_γ⁶𝔡`, where `𝔡` is the density at `0` of `λ₁`, weighted by `(λ₂⋯λ_m)²`. +- *Gaussian kernel* (`σ_γ² = 2`). `𝔇 = 120𝔡` and `F₀ = 25π₀𝔡`. -*Step 3 (integration).* Integrate over `f₄` given `b`; restoring `|f₄| ≥ κ` costs Step 1's bound. Use -`E[p_A(0 | f₄, b) | b] = p_A(0 | b)`, and integrate over `γ` by (G1), with `E[γ⁶], E[γ⁸] < ∞`. Multiplying by `12π₀` and -by `κ` gives `κ𝒜^{rej} = 12π₀p_A(0 | b)E[γ⁶](20/3)/384 + O(κ^{−1})·poly(b)π₀`, and `12·(20/3)/384 = 5/24`. ∎ +*Proof.* Fix `b` and `u`. The constants below are uniform in `u` and polynomial in `b` before the factor `π₀`, which carries +`e^{−c₀b²}` ([R] (R5)). For `m = 1` put `λ₁ := λ`. For `m ≥ 2`, matrices with a repeated eigenvalue form a null set. By Weyl's +integration formula, Lebesgue measure on the negative definite cone pulls back, under +`(λ, O) ↦ −O diag(λ)Oᵀ` (`0 < λ₁ < ⋯ < λ_m`, `O ∈ O(m)`, `e_i = Oε_i`), to `c_m V(λ) dλ dO` with +`V(λ) := Π_{i 0` and consider the event `{λ₁ < λ₂ ≤ δ}` (only for `m ≥ 2`). + - There `w_κ ≤ 36κ²λ₁²λ₂²(λ₃⋯λ_m)²` and `V ≤ λ₂·Π_{j≥3}λ_j^{j−1}`. The density `p` is bounded and has Gaussian decay in + `(λ₃, …, λ_m, f₄)`. + - Hence `E₀[w_κ1{λ₁ < λ₂ ≤ δ} | b, γ] ≤ Cκ²∫_0^δ∫_0^{λ₂}λ₁²λ₂³ dλ₁dλ₂ ≤ Cκ²δ⁷`, polynomially in `b`. + - There are two uses. First, if `|f₄| < κ/2` and `f̃₄ ≥ κ`, then `3|γ|²/λ₂ ≥ 3Σ_{i≥2}γ_i²/λ_i ≥ κ/2`, which forces + `λ₂ ≤ δ := 6|γ|²/κ`. + Second, the event of Step 2 with `λ₂ ≤ γ₁²/(7κ)` has `δ := γ₁²/(7κ)`. + - In both cases `E_γ` of `κ·κ²δ⁷` is `O(κ^{−4})`. + +All these parts are `O(κ^{−2})` after multiplying by `κ`. + +*Step 2 (the soft window).* Work on `{|f̃₄| < κ, 24κ ≤ z < 72κ, λ₂ > γ₁²/(7κ), |γ|² ≤ κ}`. The complement of the last +condition costs `O(e^{−cκ})` by (G1), since the weight is at most `36κ²Δ²`. +- *The window.* `3γ₁²/λ₁ = z − f̃₄ ∈ (23κ, 73κ)`, so `λ₁ ∈ (3γ₁²/(72κ − f̃₄), 3γ₁²/(24κ − f̃₄)]`, and in particular + `λ₁ < 3γ₁²/(23κ) < min(λ₂, 1)`. +- *The substitution.* Fix `(λ₂, …, λ_m, O, f₄, γ)`. Then `f̃₄` is fixed, and `λ₁ ↦ φ = (f̃₄ + 3γ₁²/λ₁)/(72κ)` is a decreasing + bijection of the window onto `[1/3, 1)`, with `λ₁ = 3γ₁²/(72κφ − f̃₄)` and `|dλ₁/dφ| = 216κγ₁²/(72κφ − f̃₄)²`. +- *The integral.* Write `Π′ := (λ₂⋯λ_m)²`, so `Δ² = λ₁²Π′`. Then + + ∫ ψ 36κ²Δ²(1 − φ²) 1{…} dλ₁ = ∫_{1/3}^{1} ψ|_{λ₁=λ₁(φ)} · 69984 κ³ γ₁⁶ Π′ (1 − φ²)/(72κφ − f̃₄)⁴ dφ. (1.1) + +Since `69984/72⁴ = 1/384`, the factor `69984κ³/(72κφ − f̃₄)⁴` equals `(384κφ⁴)^{−1}(1 − f̃₄/(72κφ))^{−4}`, and +`|f̃₄/(72κφ)| ≤ 1/24` on this event. So + + |(1 − f̃₄/(72κφ))^{−4} − 1| ≤ C|f̃₄|/κ, |ψ − ψ₀| ≤ λ₁K ≤ γ₁²K/(7κ) (by (1.1b)). (1.2) + +Hence (1.1) equals `(γ₁⁶Π′/(384κ))ψ₀∫_{1/3}^1(φ^{−4} − φ^{−2})dφ + κ^{−2}O((γ₁⁸ + γ₁⁶|f̃₄|)Π′(ψ₀ + K))`, and +`∫_{1/3}^1(φ^{−4} − φ^{−2})dφ = 20/3`. + +*Step 3 (integration).* Integrate over `(λ₂, …, λ_m, O, f₄)` and `γ`, and restore the sets excluded in Step 2 at the costs of +Step 1. +- *The errors are finite.* `Π′|f̃₄| ≤ Π′|f₄| + 3Σ_{i≥2}γ_i²λ_iΠ_{j≥2, j≠i}λ_j²` is a polynomial: each `1/λ_i` is cancelled by + `λ_i²`. So the error terms are finite, with Gaussian decay. +- *The excluded sets, for the main term.* The main-term integrand `(20/3)/(384κ)·E_γ[γ₁⁶]Π′ψ₀` must also be bounded on the + sets excluded in Step 2. + - Since `V(0, λ₂, …, λ_m) ≤ λ₂Π_{j≥3}λ_j^{j−1}`, we have `Π′ψ₀ ≤ Cλ₂³·poly(λ₃, …, λ_m)·p₀`. So on `{λ₂ ≤ δ}` the + integrand integrates to `O(δ⁴/κ)`, which is `O(κ^{−5})` for both choices of `δ` in Step 1. + - On `{|f₄| ≥ κ/2}` and `{|γ|² > κ}` the Gaussian tails of `p₀` and of `γ` apply. Since `f̃₄ ≥ f₄`, this covers + `{|f̃₄| ≥ κ}`, as in Step 1. +- *The main term.* It is `(20/3)/(384κ)` times `c_m∫∫∫E_γ[γ₁⁶]Π′ψ₀ dλ₂⋯dλ_m dO df₄`. This equals `𝔇`: by (1.1a), + `ε^{−1}E₀[γ₁⁶Π′1{A < 0, λ₁ < ε} | b]` is the average over `λ₁ ∈ [0, ε)` of the same integral with `ψ` in place of `ψ₀`, + and (1.1b) with dominated convergence gives the limit. This also proves that the limit (1.0a) exists. +- *Positivity.* `𝔇 > 0` because `ψ₀ > 0` and `γ₁ ≠ 0` on sets of positive measure. + +Multiplying by `12π₀` and by `κ` gives (1.0), since `12·(20/3)/384 = 5/24`. In `d = 2`, `ψ = p(−λ, f₄ | b)` and `𝔇 = p_A(0 | b)E[γ⁶]` +(by (G1)). ∎ *Remarks on Theorem 1.* -1. Theorem 1 is about the cusp kernel `𝒜^{rej}` itself, not about the non-uniformity of #240's Theorem N as `κ → ∞`. The - rejected weight sits on `λ ∈ [γ²/(24κ), γ²/(8κ)]` (Step 2), so the rejected pairs at large `κ` are soft in the transverse - direction. In #220's elder decision they are the pairs with `|φ| ∈ (1/3, 1)` coming from `3γ²/λ ≈ z`, not from `f₄`. -2. Expanding (1.2) to first order gives `κ𝒜^{rej}/F₀ = 1 + c(b)/κ + O(κ^{−2})` with - `c(b) = (12/5)[E(f₄ | A = 0, b)/18 − ∂_a log p_A(0 | b)·E[γ⁸]/(24E[γ⁶])]`, where `12/5` is the mean of `1/φ` under the - weight `φ^{−4} − φ^{−2}` on `[1/3, 1]`. For the Gaussian kernel, `A ~ N(−b, 2)` and `f₄ ~ N(−3b, 24)` are independent - given `b`, and `c(b) = 0.3b`. -3. With `s = κ^{−1/4}`, for an isotropic field `∫_{S¹}𝒜^{rej}(b, s^{−4}, u) dσ(u) = 2πF₀(b)s⁴ + O(s⁸(1 + |b|)^N e^{−cb²})`. - So the `s`-integral defining `I^{cand} − c₁` converges at `s = 0` like `s⁵`, and `s ≤ s₀` contributes - `2π∫F₀ db · s₀⁵/5 + O(s₀⁹)`. +1. Theorem 1 is about the cusp kernel `𝒜^{rej}` itself, not about the non-uniformity of #240's Theorem N as `κ → ∞`. + - The rejected weight sits on `λ₁ ∈ [γ₁²/(24κ), γ₁²/(8κ)]` (Step 2), with the other eigenvalues of order one. So the + rejected pairs at large `κ` are soft in exactly one transverse direction. + - In #220's elder decision they are the pairs with `|φ| ∈ (1/3, 1)` coming from `3γ₁²/λ₁ ≈ z`, not from `f₄`. +2. In `d = 2`, expanding (1.2) to first order gives `κ𝒜^{rej}/F₀ = 1 + c(b)/κ + O(κ^{−2})`, with + `c(b) = (12/5)[E(f₄ | A = 0, b)/18 − ∂_a log p_A(0 | b)·E[γ⁸]/(24E[γ⁶])]`. Here `12/5` is the mean of `1/φ` under the + weight `φ^{−4} − φ^{−2}` on `[1/3, 1]`. For the Gaussian kernel, `A ~ N(−b, 2)` and `f₄ ~ N(−3b, 24)` are independent given + `b`, and `c(b) = 0.3b`. + In `d ≥ 3` the coefficient also involves the positive shift `f̃₄ − f₄ = 3Σ_{i≥2}γ_i²/λ_i` and the `λ₁`-derivative of the + eigenvalue density, and it is not odd in `b`. For the Gaussian kernel in `d = 3`, numerically + `κ(κ𝒜^{rej}/F₀ − 1) → −0.266` at `b = 0` and `+0.083` at `b = 1` (`perb3.py`, at `κ` up to `10⁴`). +3. With `s = κ^{−1/4}`, `∫_{S^{d−1}}𝒜^{rej}(b, s^{−4}, u) dσ(u) = (∫F₀ dσ)s⁴ + O(s⁸(1 + |b|)^N e^{−cb²})`. So the `s`-integral + defining `I^{cand} − c₁` converges at `s = 0` like `s⁵`, and `s ≤ s₀` contributes `(∫∫F₀ db dσ)s₀⁵/5 + O(s₀⁹)`. 4. *Numerically* (Gaussian kernel; §5): - - `F₀(0) = 0.008207`, `F₀(1) = 0.003019` and `∫F₀ db = 0.0145472`, so the `s⁴`-coefficient is `2π∫F₀ db = 0.091403`. - - `κ𝒜^{rej}(0, κ)/F₀(0) = 0.509, 0.751, 0.917, 0.977, 0.9985, 0.9999` at `κ = 1/2, 1, 2, 4, 16, 64` (`c(0) = 0`). - - `κ(κ𝒜^{rej}/F₀ − 1) = 0.280, 0.295` at `b = 1` and `−0.318, −0.305` at `b = −1`, at `κ = 16, 64`, as `c(±1) = ±0.3` - predicts. + - *`d = 2`.* `F₀(0) = 0.008207`, `F₀(1) = 0.003019` and `∫F₀ db = 0.0145472`, so `∫∫F₀ db dσ = 0.091403`. + - `κ𝒜^{rej}(0, κ)/F₀(0) = 0.509, 0.751, 0.917, 0.977, 0.9985, 0.9999` at `κ = 1/2, 1, 2, 4, 16, 64`; here `c(0) = 0`. + - `κ(κ𝒜^{rej}/F₀ − 1) = 0.280, 0.295` at `b = 1` and `−0.318, −0.305` at `b = −1`, at `κ = 16, 64`, as `c(±1) = ±0.3` + predicts. + - *`d = 3`*, integrated over `b` and `u`. `κ∫∫𝒜^{rej} db dσ / ∫∫F₀ db dσ = 0.6697, 0.9899, 0.99936, 0.99994` at + `κ = 1, 10, 100, 1000`. + - *`d = 3`*, per `b`. `κ𝒜^{rej}(b, κ)/F₀(b) = 0.99730, 0.99973, 0.999973` at `b = 0` and `1.00079, 1.000082, 1.0000083` at + `b = 1`, at `κ = 10², 10³, 10⁴`. + +**Corollary 1′ (Gaussian kernel).** Let `C(x) = e^{−|x|²/2}` on `R^d`. +- *`d = 2`.* `F₀(b) = 25π₀(b)p_A(0 | b)`, where `A ~ N(−b, 2)` given `b`, and + + ∫_{S¹}∫_R F₀ db dσ = 25√3/(48π²) ≈ 0.0914028. + +- *`d = 3`.* Given `b`, the eigenvalues of `Λ = −A` are `λ_{1,2} = −t ∓ ρ`, with `t ~ N(−b, 1)` and `ρ ~ Rayleigh(1)` + independent. Hence, with `ϕ` the standard normal density, + + 𝔡(b) = ∫_0^∞ (2ρ)² ρe^{−ρ²/2} ϕ(ρ − b) dρ, F₀ = 25π₀𝔡, ∫_{S²}∫_R F₀ db dσ = 125√30/(192π³) ≈ 0.1150058. + +*Proof.* +- *The law of `Λ`.* By (0.3), `A = −bI + G`. For `m = 2`, write `G = τI + G₀` with `G₀` traceless. Then + `τ = (G₁₁ + G₂₂)/2 ~ N(0, 1)`, and `G₀` has the independent `N(0, 1)` coordinates `(G₁₁ − G₂₂)/2` and `G₁₂`. So the + eigenvalues of `G₀` are `±ρ`, with `ρ ~ Rayleigh(1)` independent of `τ`, and `t := τ − b`. At `λ₁ = 0` we have `t = −ρ` and + `λ₂ = 2ρ`. The density of `λ₁` at `0` given `ρ` is that of `t` at `−ρ`, namely `ϕ(ρ − b)`. This gives `𝔡`. With + `E[γ₁⁶] = 15·2³ = 120`, it follows that `F₀ = (5/24)·120π₀𝔡`. +- *The integrals.* We use the parity factorization of #207 (stated after (CU.2)): + `∫_R π₀(u; v₀(b, 0))E₀[g | b] db = p_G(0)p_{V_u}(0)(2π)^{−1/2}τ_u^{−1}E[g | V_u = 0, G = 0, t_u = 0]`, where `f(0)` is + free in the law on the right. + - For the Gaussian kernel the prefactor is `(2π)^{−d}/(6√π)`. Indeed `G ~ N(0, I_d)`; `V_u` has independent coordinates with + variances `3` and `1` (`m` times); and `τ_u² = 6`. + - Under the law on the right (control S12), `A ~ N(0, 8/3)` in `d = 2`. In `d = 3`, `t ~ N(0, 5/3)`, with the traceless part + as before. + - So `𝔡` integrates to `(16π/3)^{−1/2}` in `d = 2`. In `d = 3` it integrates to + `∫_0^∞4ρ³e^{−ρ²/2}(10π/3)^{−1/2}e^{−3ρ²/10} dρ = (25/8)(10π/3)^{−1/2}`. + - The factorization is applied to `𝔡`, which is an `ε`-limit. The exchange of `lim_ε` and `∫db` is justified by dominated + convergence: by (1.1a)–(1.1b), `ε^{−1}E₀[Π′1{A < 0, λ₁ < ε} | b] ≤ C(1 + |b|)^N` uniformly in `ε ≤ 1`. + - Multiplying by `25`, `|S^{d−1}|` and `(2π)^{−d}/(6√π)` gives the two values (control S12). ∎ + +In the cusp variable the integrated tail coefficient is thus larger in `d = 3` (`0.1150`) than in `d = 2` (`0.0914`). The +rejected coefficient itself is smaller: `I^{cand} − c₁ = 0.07244` against `0.09212` (#207 §8). ## 2. The fold-scale soft model Theorem 1 shows where the rejected weight sits as `κ → ∞` along the cusp scale. The *fold scale* is the regime of fixed `k = ℓ/r³` and `r → 0`, so again `κ = k/r → ∞`. There the pair is rejected only if the transverse curvature is of order `r` ([C7-K] (K2) bounds the rejected weight by `O(r³/k)`), and two more jets enter. This section is formal for the field (the -limit is Conjecture 6) and exact for the model. +limit is Conjecture 6) and exact for the model. It is written for `d = 2`; Proposition 2′ at its end reduces `d ≥ 3` to it. *The limit window field.* In the window coordinates of #220 §1 (`Φ(X, Ξ) = rXu + r²ΞΘ`, `𝔉 = r^{−4}(f∘Φ − b)`), put `λ = λ̃/κ` and `Ξ = κζ`, so that `x = rX` and `y = rkζ`. Every monomial `x^iy^j` of the Taylor expansion at `0` contributes @@ -245,27 +388,84 @@ The constants of Lemma 3 are crude. Numerically: - correspondingly `I(t, 0) = (2 − t)(2t − 1)²/3` on `[t*, 1]`, `I(t, 0) = t − 2/3` for `t ≥ 1`, `I(t, 0) ~ (4/3)|t|³` as `t → −∞`, and `I(0, χ₀) ~ χ₀²/48`. +*The stiff directions (`d ≥ 3`).* Let `m ≥ 2`, and work in the eigenframe of `A` at `0`. There `A = −diag(λ₁, …, λ_m)`, with +soft direction `e₁`, `λ₁ = λ̃r/k`, and stiff eigenvalues `λ₂, …, λ_m` fixed. Put `x = rX` and +`y = rkζe₁ + r^{3/2}Σ_{i≥2}η_ie_i`. Let `G_k` be (2.1) with the jets along `e₁`: `γ = γ_{e₁}`, `B = B_{e₁}`, `C₃ = C_{e₁}`. + +**Proposition 2′.** On `{|X| ≤ 3, |ζ| + |η| ≤ R}`, + + (f(rXu + y) − b)/(kr³) = G_k(X, ζ) − (1/(2k))Σ_{i≥2}λ_iη_i² + r^{1/2}G_{1/2}(X, ζ, η) + O_{k,R}(r), + G_{1/2} := Σ_{i≥2} [(γ_{e_i}/(2k))(X² − ¼) + (∂_u∂_{e₁}∂_{e_i}f) Xζ + (k/2)(∂_{e₁}²∂_{e_i}f) ζ²] η_i. (2.6) + +The limit `G_k^{(d)} := G_k(X, ζ) − (1/(2k))Σ_{i≥2}λ_iη_i²` has the elder decision of `G_k`: +- (a) the critical points of `G_k^{(d)}` are those of `G_k` with `η = 0`, at the same levels, with `m − 1` additional negative + Hessian eigenvalues `−λ_i/k`. So `M` is a nondegenerate maximum, and `S` a nondegenerate critical point of index `m`, + exactly when (2.4) holds; +- (b) for every level `c`, the projection `(X, ζ, η) ↦ (X, ζ)` maps the connected components of `{G_k^{(d)} > c}` + bijectively onto those of `{G_k > c}`, and the supremum of a component equals that of its image; +- (c) hence `S` kills `M` for `G_k^{(d)}` iff it does for `G_k`, and Lemma 2 decides the elder mark. In the original + units, the Hessians of the window field at the pins are block diagonal up to `O(r)`, with stiff block + `−diag(λ₂, …, λ_m)`. So the typed weight gains the factor `(λ₂⋯λ_m)²`, as in Theorem 1's `Π′`. + +*Proof.* The monomial `x^i y₁^j Π_{i′≥2} y_{i′}^{l_{i′}}` of the Taylor expansion at `0` contributes `r^{i+j+3|l|/2−3}k^{j−1}` +times a monomial in `(X, ζ, η)`. Exponents below `1` occur only for three groups: +- the monomials of (2.1) (`|l| = 0`, `i + j ≤ 3`); +- the stiff quadratic (`|l| = 2`, `i = j = 0`, exponent `0`); +- `|l| = 1` with `i + j ≤ 2` (exponent `i + j − 3/2`). Within this group: + - `y₁y_{i′}` (exponent `−1/2`) has coefficient `∂_{e₁}∂_{e_{i′}}f(0) = 0` in the eigenframe; + - `y_{i′}` (exponent `−3/2`) and `xy_{i′}` (exponent `−1/2`) have the pinned coefficients + `∂_{e_{i′}}f(0) = −(r²/8)γ_{e_{i′}} + O(r⁴)` and `∂_u∂_{e_{i′}}f(0) = O(r²)` (the transverse pins, as in #207 Theorem + CU.1). They contribute `−(γ_{e_{i′}}/(8k))r^{1/2}η_{i′}` and `O(r^{3/2})`; + - `x²y_{i′}`, `xy₁y_{i′}` and `y₁²y_{i′}` have exponent `1/2` and give `G_{1/2}`. + +The other monomials, and the Taylor remainder, are `O(r)` on the window. This proves (2.6). +- For (a), `∇_ηG_k^{(d)} = −(λ_iη_i/k)_i` vanishes only at `η = 0`. +- For (b), the fiber of `{G_k^{(d)} > c}` over `(X, ζ)` is the open ellipsoid `{Σλ_iη_i² < 2k(G_k(X, ζ) − c)}`, which is + nonempty iff `G_k(X, ζ) > c`. The projection is continuous and open, its fibers are connected, and its image is + `{G_k > c}`, so it induces a bijection of components. The supremum over a fiber is attained at `η = 0`. +- (c) follows from (a) and (b), and from the block form of the Hessians. ∎ + +Control S13 checks (2.6), including `G_{1/2}`, on exactly pinned degree-6 fields in `d = 3`, at `r = 10^{−6}` and `10^{−8}` +(so that `r^{1/2}` is rational). Like (2.1) and #207 Theorem CU.1, (2.6) is a deterministic Taylor statement for every `C⁵` +field with these pins. What is formal is the use of the limit model's elder decision for the field (Conjecture 6). + +*Consistency with Theorem 1.* Maximize the `γ`-part of `r^{1/2}G_{1/2}` over `η_i` against the stiff quadratic +`−λ_iη_i²/(2k)`. The maximum is at `η_i = r^{1/2}γ_{e_i}(X² − ¼)/(2λ_i)`, and its value is +`rγ_{e_i}²(X² − ¼)²/(8kλ_i) = r(3γ_{e_i}²/λ_i)(X² − ¼)²/(24k)`. Added to the term `r(f₄/(24k))(X² − ¼)²` of `G₁`, it turns `f₄` +into `f̃₄ = f₄ + 3Σ_{i≥2}γ_{e_i}²/λ_i`. This is the fold-scale image of Step 2 of Theorem 1. + ## 3. The fold-scale rejection function -Formally, at fixed `k` the rejected kernel is the expectation of the rejected weight of §2 over the jets. With -`λ = λ̃r/k`, the law of `λ` near `0` contributes `p_A(0 | b)·(r/k)dλ̃`, and the contact pin density at gap `k` is -`π₀(u; v₀(b, k))`. This motivates +Formally, at fixed `k` the rejected kernel is the expectation of the rejected weight of §2 over the jets. +- With `λ₁ = λ̃r/k`, the law of `λ₁` near `0` contributes its density at `0` times `(r/k)dλ̃`. +- The stiff directions contribute the weight factor `(λ₂⋯λ_m)²` (Proposition 2′(c)). +- The contact pin density at gap `k` is `π₀(u; v₀(b, k))`. + +This motivates - F(k; b, u) := 12 π₀(u; v₀(b, k)) p_A(0 | b, k) E_{v₀(b,k)}[(γ⁶/384) I(12kB/γ², 576k²C₃/γ³) | A = 0], (3.1) + F(k; b, u) := 12 π₀(u; v₀(b, k)) lim_{ε↓0} ε^{−1} E_{v₀(b,k)}[(γ₁⁶/384)(λ₂⋯λ_m)² I(12kB₁/γ₁², 576k²C₁/γ₁³) 1{A < 0, λ₁ < ε}], (3.1) -with the expectation under the contact law at `v₀(b, k)`, conditioned on `A = 0`. Formally, +with `(γ₁, B₁, C₁) := (γ_{e₁}, B_{e₁}, C_{e₁})` and the expectation under the contact law at `v₀(b, k)`. In `d = 2` this is +`12π₀(u; v₀(b, k))p_A(0 | b, k)E_{v₀(b,k)}[(γ⁶/384)I(12kB/γ², 576k²C₃/γ³) | A = 0]`. Formally, `r^{−2}A_r^{rej}(b, k, u) = (r/k)F(k; b, u) + o(r)` (Conjecture 6 below). As `k → 0`, `I → I(0, 0) = 20/3`, and (3.1) tends to the `F₀` of Theorem 1. -**Proposition 4 (Gaussian kernel).** Let `C(x) = e^{−|x|²/2}`. Then: -1. `F(k; b) = F₀(b)H(k)`, where `H(k) := e^{−12k²}E[γ⁶I(t, χ₀)]/(E[γ⁶]·20/3)` does not depend on `b`; here `(γ, B, C₃)` are - independent `N(0, 2)`, `N(0, 2)` and `N(0, 6)`, and `(t, χ₀) = (12kB/γ², 576k²C₃/γ³)`. +**Proposition 4 (Gaussian kernel).** Let `C(x) = e^{−|x|²/2}` on `R^d`, `d ≥ 2`. Then: +1. `F(k; b) = F₀(b)H(k)`, where `H(k) := e^{−12k²}E[γ⁶I(t, χ₀)]/(E[γ⁶]·20/3)` depends on neither `b` nor `d`. Here + `(γ, B, C₃)` are independent `N(0, 2)`, `N(0, 2)` and `N(0, 6)`, and `(t, χ₀) = (12kB/γ², 576k²C₃/γ³)`. 2. `H(k) = 1 + (12/25)k² + O(k³)` as `k → 0`. 3. `H(k) ≤ C(1 + k⁴)e^{−12k²}`; in particular `|H(k) − 1| ≤ C min(1, k²)`. *Proof.* (1) By (0.3), the gap pin contributes `e^{−12k²}`, so `π₀(v₀(b, k)) = π₀(v₀(b, 0))e^{−12k²}`. This is the factor -`e^{−a′k²}`, `a′ = 12`, of #240 Step N6. `p_A(0 | b, k) = p_A(0 | b)`, and the law of `(γ, B, C₃)` is as stated, centred and -independent of `(A, f₄, b)`. +`e^{−a′k²}`, `a′ = 12`, of #240 Step N6. +- The law of the even jets does not depend on `k`. So `p_A(0 | b, k) = p_A(0 | b)`, and in general the weighted density of + `λ₁` at `0` is the `𝔡(b)` of Theorem 1. +- The law of `(γ, B, C₃)` is as stated, centred and independent of `(A, f₄, b)`. +- In `d ≥ 3` the soft direction `e₁` is a function of `A`. By (0.3) the odd jets are independent of `A`, and for every + fixed unit `e ⊥ u` the triple `(γ_e, B_e, C_e)` has the stated law. So `(γ₁, B₁, C₁)` has that law and is independent of + `(A, f₄)`. + +Hence the expectation in (3.1) factorizes as `𝔡(b)` times the soft-jet average, and `F = F₀H` with the same `H` in every `d`. (2) *The edge at `χ₀ = 0`.* For `χ = 0` and `a > 0` the ridge is `R = A₀ + p²/(8a)`. At `t = 0` it is `[2(X + ½)²(X − 1) + 3φ(X² − ¼)²]/(24φ)`, and @@ -311,9 +511,11 @@ Hence (3) Lemma 3 gives `γ⁶I ≤ C(γ⁶ + 1728k³|B|³ + 331776k⁴C₃²)`. ∎ -So for the Gaussian kernel the rejected-pair rate at the fold scale starts at its cusp-scale value `F₀`. The pin density -lowers it by `12k²`, and the motion of the elder edge raises it by `(312/25)k²`, so the two nearly cancel (`12/25` -remains). Numerically `H₀ − 1` (with `C₃` ignored) already changes sign near `k ≈ 0.08`. +So for the Gaussian kernel the rejected-pair rate at the fold scale starts at its cusp-scale value `F₀`, in every +dimension. +- The pin density lowers it by `12k²`, and the motion of the elder edge raises it by `(312/25)k²`. The two nearly cancel, + and `12/25` remains. +- Numerically, `H₀ − 1` (with `C₃` ignored) already changes sign near `k ≈ 0.08`. For another stationary kernel with `B` and `γ` independent of each other and of `∂_u³f` under the contact law, the two coefficients are `−144/(2σ₃²)` and `(1872/75)σ_B²/σ_γ⁴`, with `σ₃² := Var(∂_u³f | ∇f(0) = 0)`. For the Gaussian kernel, @@ -325,15 +527,16 @@ The cusp kernel is the `k → 0` limit, and `(r/k)F(k; b, u)` the `κ → ∞` l that they agree in the overlap: `F(k)/F₀ → 1`. Put `H(k; b, u) := F(k; b, u)/F₀(b, u)`; for the Gaussian kernel this is the `H(k)` of Proposition 4. The simplest density built from both limits is the *composite* - ν^c(ℓ) := ∫_0^∞∫_R∫_{S¹} 𝒜^{rej}(b, ℓ/r⁴, u) H(ℓ/r³; b, u) dσ(u) db dr. (4.1) + ν^c(ℓ) := ∫_0^∞∫_R∫_{S^{d−1}} 𝒜^{rej}(b, ℓ/r⁴, u) H(ℓ/r³; b, u) dσ(u) db dr. (4.1) **Lemma 5 (expansion of the composite).** Let `H ≥ 0` be measurable with `|H(k; b, u) − 1| ≤ C_H min(1, k²)` uniformly (Proposition 4(3) for the Gaussian kernel). Then, as `ℓ ↓ 0`, ν^c(ℓ) = (I^{cand} − c₁) ℓ^{1/4} + R_{2/3} ℓ^{2/3} + O(ℓ^{3/4}), - R_{2/3} := ∫_{S¹}∫_R∫_0^∞ v⁴[F(v^{−3}; b, u) − F₀(b, u)] dv db dσ(u), F := F₀H. (4.2) + R_{2/3} := ∫_{S^{d−1}}∫_R∫_0^∞ v⁴[F(v^{−3}; b, u) − F₀(b, u)] dv db dσ(u), F := F₀H. (4.2) -For the Gaussian kernel, `R_{2/3} = 2π∫F₀ db · Ĩ` with `Ĩ := ∫_0^∞ v⁴[H(v^{−3}) − 1] dv`. +For the Gaussian kernel, `R_{2/3} = (∫∫F₀ db dσ)·Ĩ`, with `Ĩ := ∫_0^∞ v⁴[H(v^{−3}) − 1] dv` the same in every `d`, and +`∫∫F₀ db dσ` given by Corollary 1′. *Proof.* The inner integral converges: its integrand is `O(v⁴)` at `0` and `O(v^{−2})` at `∞`, with Gaussian decay in `b`. Write `H = 1 + (H − 1)`. The term with `1` is `ℓ^{1/4}(I^{cand} − c₁)` after `r = sℓ^{1/4}` (§0). For the term with `H − 1`, let @@ -345,23 +548,24 @@ Write `H = 1 + (H − 1)`. The term with `1` is `ℓ^{1/4}(I^{cand} − c₁)` a error is `≤ CP(b)ℓ∫_0^{ℓ^{−1/12}}v⁸min(1, v^{−6})dv = O(P(b)ℓ^{3/4})`. Integrating over `b` and `u` gives (4.2). ∎ -**Conjecture 6 (the fold-scale limit).** In `d = 2`, for every `k > 0`, `b` and `u`, +**Conjecture 6 (the fold-scale limit).** In every `d ≥ 2`, with `F` as in (3.1), for every `k > 0`, `b` and `u`, `lim_{r↓0} (k/r)·r^{−2}A_r^{rej}(b, k, u) = F(k; b, u)`, locally uniformly in `k`. -**Conjecture 7 (the next term of the rejected density).** In `d = 2`, +**Conjecture 7 (the next term of the rejected density).** In every `d ≥ 2`, - ρ_rej(ℓ) + ν_eld^{far,r_0^*}(ℓ) = B_{2,L} + (I^{cand} − c₁) ℓ^{1/4} + R_{2/3} ℓ^{2/3} + O(ℓ^{3/4}), (4.3) + ρ_rej(ℓ) + ν_eld^{far,r_0^*}(ℓ) = B_{d,L} + (I^{cand} − c₁) ℓ^{1/4} + R_{2/3} ℓ^{2/3} + O(ℓ^{3/4}), (4.3) -with `R_{2/3}` as in (4.2); equivalently, `ρ_rej + ν_eld^{far,r_0^*} − B_{2,L} − ν^c = O(ℓ^{3/4})`. This is (R⁺.1) of #229 with -its remainder resolved. If `ν_eld^{far,r_0^*}(ℓ) = O(ℓ^{3/4})` (#188, open, claims `O(ℓ^N)`), then -`ρ_rej = B_{2,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`. For the Gaussian kernel, numerically -`R_{2/3} ≈ −0.049` (§5). +with `R_{2/3}` as in (4.2). +- Equivalently, `ρ_rej + ν_eld^{far,r_0^*} − B_{d,L} − ν^c = O(ℓ^{3/4})`. This is (R⁺.1) of #229 with its remainder resolved. +- If `ν_eld^{far,r_0^*}(ℓ) = O(ℓ^{3/4})` (#188, open, claims `O(ℓ^N)`), then + `ρ_rej = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`. +- For the Gaussian kernel, numerically `R_{2/3} ≈ −0.049` in `d = 2` and `R_{2/3} ≈ −0.062` in `d = 3` (§5). *What a proof needs.* Inputs (i) and (ii) are missing; (iii) is needed only to drop the far term. - (i) *The soft layer.* A two-scale version of Conjecture 6, uniform from the cusp scale to the fold scale (`κ ≥ κ₁`), with errors integrating to `O(ℓ^{3/4})`. In fixed cusp windows this is #240's Theorem N (`O(r²)`). The new part is the window field's elder decision in the soft scaling `λ ≍ r`: Lemma 2 with margins for the field, as #240 Lemma Q′ gives them with - `ε̃` for `λ ≍ 1`. + `ε̃` for `λ ≍ 1`. In `d ≥ 3` it also needs Proposition 2′ for the field, uniformly in the stiff eigenvalues. - (ii) *The end `κ → 0`.* On the intermediate separations `ℓ^{1/4} ≪ r ≤ r_0^*`, the rejected kernel must match `B_{2,L}`'s part plus the composite's own part, up to errors integrating to `O(ℓ^{3/4})`. (The composite is not negligible there: `𝒜^{rej} ≍ κ³` as `κ → 0`, so `r ≥ Sℓ^{1/4}` contributes about `ℓ^{1/4}S^{−11}`.) The best available bounds on these @@ -370,7 +574,8 @@ its remainder resolved. If `ν_eld^{far,r_0^*}(ℓ) = O(ℓ^{3/4})` (#188, open, conjectured term. Conjecture 7 is stated with `ν_eld^{far,r_0^*}` on the left for that reason; #188's `O(ℓ^N)` would remove it. -Formally, a matched-asymptotics count gives cusp-side powers `ℓ^{(m+1)/4}` and fold-side powers `ℓ^{(m+1)/3}`. In `ρ_rej` the +Formally, a matched-asymptotics count gives cusp-side powers `ℓ^{(j+1)/4}` and fold-side powers `ℓ^{(j+1)/3}` +(`j = 0, 1, 2, …`). In `ρ_rej` the fold-scale kernel is `O(r³/k)` ([C7-K] (K2)), with leading coefficient `F` (formally, Theorem 1 and Proposition 4). This makes `ℓ^{2/3}` the first fold-side power and `ℓ^{3/4}` the next cusp-side one (#240 Corollary N′). @@ -381,17 +586,27 @@ makes `ℓ^{2/3}` the first fold-side power and `ℓ^{3/4}` the next cusp-side o and only subject to (ii)–(iii). 2. For the Gaussian kernel, `Ĩ` is dominated by `v ∈ [1, 1.5]`, that is `k ∈ [0.3, 1]`, where `H` falls from `≈ 1` to `≈ 0`. The contributions are `−0.200` (`v < 1`), `−0.563` (`1 ≤ v ≤ 1.5`), `+0.003` (`1.5 ≤ v ≤ 2`) and `+0.223` (`v ≥ 2`). -3. `R_{2/3}/(I^{cand} − c₁) ≈ −0.53`. Since `ℓ^{5/12} = 0.147` at `ℓ = 10^{−2}`, the `ℓ^{2/3}` term is about 8% of the `ℓ^{1/4}` - term there. +3. `R_{2/3}/(I^{cand} − c₁) ≈ −0.53` in `d = 2` and `≈ −0.85` in `d = 3`. Since `ℓ^{5/12} = 0.147` at `ℓ = 10^{−2}`, the + `ℓ^{2/3}` term there is about 8% (`d = 2`) and 12% (`d = 3`) of the `ℓ^{1/4}` term. ## 5. Numerical evidence (exploration; not part of the proofs) -Gaussian kernel `e^{−|z|²/2}`, `d = 2`. The scripts are in the project archive +Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. The scripts are in the project archive `V2_2/frontiers_soft_rejected_pairs_20261002/exploration/` (numpy, scipy, mpmath, sympy); none is part of the repository. -1. *Theorem 1.* Quadrature of (0.2) under the contact law (`fastk.py`, `soft_d2.py`) gives the numbers of Remark 4 of §1. The - `b`-integrated tail is `κ∫𝒜^{rej}(b, κ) db → 0.0145472 = ∫F₀ db`. Also `∫∫𝒜^{rej}(b, s^{−4}) ds db = 0.014662`, and its `2π` - multiple is `I^{cand} − c₁ = 0.092124`. +1. *Theorem 1.* + - *`d = 2`.* Quadrature of (0.2) under the contact law (`fastk.py`, `soft_d2.py`) gives the numbers of Remark 4 of §1. + The `b`-integrated tail is `κ∫𝒜^{rej}(b, κ) db → 0.0145472 = ∫F₀ db`. Also `∫∫𝒜^{rej}(b, s^{−4}) ds db = 0.014662`, + and its `2π` multiple is `I^{cand} − c₁ = 0.092124`. + - *`d = 3`.* Two quadratures compute `G(κ) := ∫∫𝒜^{rej} db dσ` under the `b`-integrated law of Corollary 1′'s proof. + - `Gd.py` integrates `f₄` in closed form and `λ₁` on a logarithmic grid, for `κ ≤ 25`. It reproduces the `d = 2` + profile of `fastk.py` to `10^{−7}`. + - `Gphi.py` uses the substitution of Step 2, for `κ ≥ 5`. + - On the overlap the two agree to `2·10^{−5}`. + - *`d = 3` results.* `κG(κ) = 0.115005` at `κ = 10⁴`, against Corollary 1′'s `0.1150058`. + - `∫_0^∞G(s^{−4})ds = 0.072444` on a grid refined near the sharp maximum of `G` at `κ ≈ 0.41` (step `0.025` in `log s`; + `G3_merged_fine.log`). This matches `I^{cand} − c₁ = 0.0724443` (#207 §8, #216) to `10^{−6}`. + - A plain Monte Carlo at `κ = 1` agrees: `0.07707 ± 0.00020`, against `0.07702`. 2. *Lemma 2.* The one-dimensional decision (`dec1d.py`) was compared with a flood fill of `{G > L_S ± ε}` on an `(X, z)` grid (`model2d.py`, `validate_D2.py`). - 600 parameter points drawn as in the computation of `H`: no disagreement. @@ -423,14 +638,23 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2`. The scripts are in the project archiv | `e^{−12k²}` | 0.887 | 0.619 | 0.340 | 0.147 | 0.050 | 0.013 | 0.003 | 0.0005 | 0.000006 | 5. *`R_{2/3}`.* - - `Ĩ = −0.536` (`assemble2.py`; a dense trapezoid rule gives `−0.537`), so `R_{2/3} = 0.0914028·Ĩ = −0.049`. + - `Ĩ = −0.536` (`assemble2.py`). + - The 200-point grid `H_dense3.json` is written by `dense2.py --write` and used below. It has the same values of `H` as + v1's `H_dense2.json`, and its `R23` field comes from the quadrature of its interpolant, `Ĩ = −0.5361`. + - The check mode of `dense2.py` (`dense2.log`) shows trapezoid and Simpson rules on the same grid giving `−0.535` to + `−0.536`. + - v1 quoted `−0.537` for "a dense trapezoid rule". That was the `R23` field stored in `H_dense2.json`, produced by an + unrecorded rule (finding A-1, found by the author while archiving v1). + - So `R_{2/3} = 0.0914028·Ĩ = −0.049` in `d = 2` and `R_{2/3} = 0.1150058·Ĩ = −0.0617` in `d = 3`. - Sensitivity: a uniform shift of `H` by `±0.002` on `k ≥ 0.1` moves `Ĩ` by `±0.018`. Halving the small-`k` term `62k⁴` - moves it by `−0.021`, and multiplying it by `1.5` by `+0.021`. So `Ĩ = −0.54 ± 0.03` and `R_{2/3} = −0.049 ± 0.003`. + moves it by `−0.021`, and multiplying it by `1.5` by `+0.021`. So `Ĩ = −0.54 ± 0.03`, `R_{2/3} = −0.049 ± 0.003` in `d = 2` + and `R_{2/3} = −0.062 ± 0.004` in `d = 3`. - With `C₃` ignored (`H = H₀`), `Ĩ = −1.133` and `R_{2/3} = −0.104`. With the pin density alone (`H = e^{−12k²}`), `Ĩ = 12^{5/6}Γ(−5/6)/6 = −8.829` and `R_{2/3} = −0.807`. So the soft model's extra jets matter: `B` nearly cancels the pin density at small `k`, and `C₃` raises `H` at `k ≈ 0.2–0.7`. -6. *#216's Monte Carlo.* The raw `d = 2` records (`batchA2.npz`: 4,000 samples, 271,272 rejected adjacent pairs with `ℓ < 0.3`) - were compared with the composite (`composite.py`, `comp2d.py`). #216's row counts *adjacent* rejected pairs, which carry no +6. *#216's Monte Carlo.* The raw records were compared with the composite: `composite.py` and `comp2d.py` in `d = 2`, and + `compd.py` in `d = 3`. The records are `batchA2.npz` (`d = 2`: 4,000 samples, 271,272 rejected adjacent pairs with + `ℓ < 0.3`) and `batch3B.npz` (`d = 3`: 996 samples, 45,952 such pairs). #216's row counts *adjacent* rejected pairs, which carry no `B_{2,L}` (#218 Remark 3). By §4 (inputs (ii)–(iii), and the `ℓ^{3/4}` order), these totals do not test the global `ℓ^{2/3}` coefficient. What they do test, bin by bin in `s = r/ℓ^{1/4}`, is the fold-side suppression described by `H`. - *Small `s` (the fold side).* For `s < 0.8` and `ℓ ∈ [10^{−4}, 0.1)`, 37 pairs are observed, 35.3 predicted by the @@ -440,14 +664,31 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2`. The scripts are in the project archiv which is not modelled here (finite-`r` cusp corrections, the `ℓ^{3/4}` order). - *Totals.* On `[10^{−4}, 10^{−2}]` the ratio of the counts to `(I^{cand} − c₁)ℓ^{1/4}` alone is `0.959 ± 0.017`; to the composite, `1.040`; to the two-term law with `R_{2/3} = −0.049`, `1.018`. + - *`d = 3`, small `s`.* For `s < 0.8` and `ℓ ∈ [10^{−4}, 0.1)`, 11 pairs are observed, 10.7 predicted by the composite, + and 1,329 by the cusp kernel alone. For `s ∈ [0.8, 1)` the three numbers are 651, 540 and 2,240 (`comp3d_final4.log`). + - *The test discriminates `H`.* With the pin density alone in place of `H` (`H = e^{−12k²}`), the predictions would be 1.2 + (`s < 0.8`) and 82.7 (`s ∈ [0.8, 1)`) (`comp3d_pin.log`). + - *The transition bin `s ∈ [0.8, 1)`.* The data exceed the composite, by `+4.8σ` in `d = 3` and by 2,248 against 1,904 in + `d = 2`. In both dimensions the excess is concentrated at `ℓ ≥ 3·10^{−2}`, where the data exceed the composite anyway. + - *`d = 3` totals.* The script keeps bins with at least five counts, so the effective ranges are `[3.1·10^{−4}, 9.7·10^{−3}]` + (8 bins) and `[3.1·10^{−4}, 2.3·10^{−2}]` (10 bins). In `d = 2` the first range is `[1.3·10^{−4}, 9.7·10^{−3}]` (10 bins). + + | `d = 3` range | `(I^{cand} − c₁)ℓ^{1/4}` alone | composite | two-term law, `R_{2/3} = −0.0617` | χ²: `ℓ^{1/4}` alone / composite | + |---|---|---|---|---| + | `[3.1·10^{−4}, 9.7·10^{−3}]` | `0.884 ± 0.040` | `1.008` | `0.975` | 11.8 / 2.1 (8 bins) | + | `[3.1·10^{−4}, 2.3·10^{−2}]` | `0.858 ± 0.024` | `1.032` | `0.989` | 44.9 / 5.1 (10 bins) | + - *Over #216's whole range* `[10^{−4}, 0.3]` the composite does not describe the counts: data/composite rises to `1.63` + near `ℓ ≈ 0.25`. This is the region of finite-`r` cusp corrections and the `ℓ^{3/4}` order, as in `d = 2`. In particular, + #216's three-parameter fit of the `ℓ^{1/4}` coefficient on that range, `0.069 ± 0.006`, lies only `0.55σ` below + `0.0724`, so it is not evidence of a deficit either way. ## 6. Scope and relation to other packets - **#229 (R⁺.1).** It proves `ρ_rej = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} − ν_eld^{far,r_0^*} + O(ℓ^{3/7})`. Conjecture 7 would resolve - its remainder in `d = 2`, given inputs (i)–(ii) of §4 (and (iii) to drop the far term). Theorem 1, Lemma 5 and Proposition 4 do not change any statement - of #229. + its remainder in every `d ≥ 2`, given inputs (i)–(ii) of §4 (and (iii) to drop the far term). Theorem 1, Lemma 5 and + Proposition 4 do not change any statement of #229. - **#240 Remark 1.** It leaves the ends `κ → ∞` and `κ → 0` of Theorem N open. Theorem 1 describes the `κ → ∞` end of the - *rejected* cusp kernel in `d = 2`: it decays like `1/κ`, while `𝒜^{cand}` and `𝒜^{eld}` grow like `κ²`. Lemma Q′'s margins do + *rejected* cusp kernel in every `d ≥ 2`: it decays like `1/κ`, while `𝒜^{cand}` and `𝒜^{eld}` grow like `κ²`. Lemma Q′'s margins do not reach the soft window `λ ≍ γ²/κ` where the rejected weight sits. - **#237 Remark 4 and #240 Remark 4.** They found residuals of #216's Monte Carlo about the three-term laws that two-power fits could not attribute (`ℓ^{1/2}` against `ℓ^{2/3} + ℓ^{3/4}`). Conjecture 7 predicts a contribution `−R_{2/3}ℓ^{2/3}` to @@ -456,17 +697,18 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2`. The scripts are in the project archiv leading term of `r^{−2}A_r^{rej}` is `(r/k)F(k; b, u)`, and `F(k) → F₀ ≠ 0` as `k → 0` (Theorem 1). - **Not claimed.** - Conjectures 6 and 7 are not proved. The values of `H` and `R_{2/3}` are numerical and not certified. - - Nothing for `d ≥ 3`. There the soft direction is an eigenvector of a `(d − 1) × (d − 1)` matrix, and `F₀` needs the density - of its smallest eigenvalue at `0`; the `1/κ` form of Theorem 1 is expected to persist. + - In `d ≥ 3` the fold-scale statements rest on Proposition 2′. It is exact for the model, and its transfer to the field + is part of Conjecture 6. - Nothing about the candidate density's own `ℓ^{2/3}` term. - - Lemma 2, Lemma 3 and Proposition 4 are statements about the limit model (2.1). Their transfer to the field is - Conjecture 6. - - The Gaussian-kernel statements are about the model case on `R²`. + - Lemma 2, Lemma 3 and Proposition 4 are statements about the limit models (2.1) and (2.6). Their transfer to the field + is Conjecture 6. + - The Gaussian-kernel statements are about the model case on `R^d`. ## 7. Controls `soft_check.py` uses the standard library and exact rationals, except S8 (fixed floating-point cases). Its output is -`RESULTS.json`. +`RESULTS.json`, which also prints the closed forms of S12 in floating point. In mutant mode the checker prints the names +of the failing controls to stderr. | Control | Checks | |---|---| @@ -480,9 +722,12 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2`. The scripts are in the project archiv | S8 | Lemma 2 on 16 fixed parameter points with known answers, including case (D′), in floating point | | S9 | the error exponents of Lemma 5 | | S10 | the limit (2.1), including `G₁`, on three exactly pinned degree-6 fields with `A = −λ̃r/k`, at `r = 10^{−3}` and `10^{−4}` | +| S11 | Theorem 1 for `m = 2, 3` in rational eigenframes (Cayley rotations): `z = f₄ − 3γᵀA^{−1}γ = 3γ₁²/λ₁ + f̃₄`, `Y = (Δ/12)z`, `w_κ = 36κ²Δ²(1 − φ²)`, the substitution (1.1) with the factor `Π′`, `69984/72⁴ = 1/384`, and the double-soft implication of Step 1 | +| S12 | the Gaussian kernel in `d = 3` from Hermite numbers: given the even pins, `A = −bI + G` with `G` in the Gaussian orthogonal ensemble and `f₄ ~ N(−3b, 24)` independent; the `b`-integrated law (`t ~ N(0, 5/3)`, `f₄ \| t ~ N(6t/5, 138/5)`; `d = 2`: `[[8/3, 2], [2, 30]]`); the odd jets along `e = (3/5, 4/5)` (variances `(6, 2, 2, 6)`); Corollary 1′'s constants `25√3/(48π²)` and `125√30/(192π³)` | +| S13 | Proposition 2′: (2.6), including `G_{1/2}`, on two exactly pinned degree-6 fields in `d = 3` at `r = 10^{−6}` and `10^{−8}` | -Mutants `M1`–`M10` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, `M4` S5, `M5` S7, `M6` S8, `M7` S9, `M8` S2, -`M9` S6, `M10` S10); an unknown label exits 2. +Mutants `M1`–`M13` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, `M4` S5, `M5` S7, `M6` S8, `M7` S9, `M8` S2, +`M9` S6, `M10` S10, `M11` S11, `M12` S12, `M13` S13); an unknown label exits 2. ## 8. Sources (exact identities in `SOURCES.json`) @@ -491,7 +736,7 @@ Mutants `M1`–`M10` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, | [R] | `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (blob `247b3ecf`) | (R5): Gaussian decay of the pin density — consumed | | [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d`) | §2 finite-jet rank, through #220 — consumed | | [E1], [E2], [REC] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`, `reviews/d1_section9_borel_repair_20260925/REPAIR.md`, `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` | reading rules and the Borel elder mark, through #220 — consumed | -| #207 | `frontiers/cusp_second_order_20261001/PROOF.md` (blob `f6df5a73`) | §§0, 4, 6 (the cusp objects); Theorem CU.2(c) — consumed | +| #207 | `frontiers/cusp_second_order_20261001/PROOF.md` (blob `f6df5a73`) | §§0, 4, 6 (the cusp objects, and the parity factorization after (CU.2), used in Corollary 1′); Theorem CU.2(c) — consumed | | #218 | `frontiers/candidate_third_order_20261001/PROOF.md` (blob `70ca57ef`) | §0: `𝒜^{cand}`, `I^{cand}`, `B_{d,L}`; Remark 3 — consumed | | #220 | `frontiers/elder_third_order_20261001/PROOF.md` (blob `c8767dde`) | §0 ((0.1), (0.2)); §1 (window coordinates, (1.1), Lemma Q, (M1)–(M6)) — consumed | | #229 | `frontiers/third_order_rate_20261001/PROOF.md` (blob `110ed33a`) | (R⁺.1); §0 (`Y_r`); §5 — consumed | @@ -499,16 +744,17 @@ Mutants `M1`–`M10` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, | #187 | `frontiers/far_elder_rate_20260930/PROOF.md` (blob `07260114`) | (0.2): `ν_eld^{far} ≤ Cℓ^{2/3}` — cited | | #240 | `frontiers/elder_cusp_parity_20261002/PROOF.md` (open PR; blob `16a1db06`) | Lemma Q′, Step N6, Corollary N′, Remarks 1 and 4 — cited | | #237 | `frontiers/candidate_parity_rate_20261001/PROOF.md` (open PR) | Theorem P's remainder, Remark 4 — cited | -| #216 | `frontiers/third_order_coefficient_20261001/NOTE.md` (open PR) | the full-field Monte Carlo (§5) — cited | +| #216 | `frontiers/third_order_coefficient_20261001/NOTE.md` (open PR) | the full-field Monte Carlo in `d = 2, 3` (§5) — cited | | #188 | far elder density `O(ℓ^N)` (open PR) | §4 (iii) — cited | ## 9. Review slices -- **A** (§1): Theorem 1, its proof and Remarks 1–3. +- **A** (§§0–1): Theorem 1 in every `d`, its proof (spectral coordinates, the double-soft bound, the substitution), + Remarks 1–3, and Corollary 1′. - **B** (§2): the limit field (2.1) and its monomial bookkeeping, the normalization, typing and weight, Lemma 2 (including case - (D′)) and Lemma 3. -- **C** (§3): Proposition 4, namely the edge (3.3), the identification of the rejected set near `(0, 0)`, the series (3.4), the - averaging and the bound off `𝒢`. + (D′)), Lemma 3, and Proposition 2′. +- **C** (§3): Proposition 4, namely the definition (3.1) and the factorization in every `d`, the edge (3.3), the + identification of the rejected set near `(0, 0)`, the series (3.4), the averaging and the bound off `𝒢`. - **D** (§4): Lemma 5, and whether Conjectures 6 and 7 are stated precisely, with inputs (i)–(iii) complete and consistent with #187, #229 and #240. - **E** (§§5–7): the evidence is described accurately and kept separate from the proofs, and the controls match their diff --git a/frontiers/soft_rejected_pairs_20261002/README.md b/frontiers/soft_rejected_pairs_20261002/README.md index f0c726471..be2017358 100644 --- a/frontiers/soft_rejected_pairs_20261002/README.md +++ b/frontiers/soft_rejected_pairs_20261002/README.md @@ -1,26 +1,31 @@ -# Soft rejected pairs in `d = 2` +# Soft rejected pairs: the `1/κ` tail of the rejected cusp kernel in every dimension **Author-side proof candidate and conjectures (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** -Object `CL-SOFT-REJECTED-20261002-v1`. Full text: [`PROOF.md`](PROOF.md). +Object `CL-SOFT-REJECTED-20261002-v1.1`. Full text: [`PROOF.md`](PROOF.md). Version history: +- v1.1 was made before any nonauthor review. +- It extends Theorem 1 to every `d ≥ 2` and adds the `d = 3` constants (Corollary 1′) and the stiff-direction reduction + (Proposition 2′). It also adds the `d = 3` Monte Carlo test and fixes finding A-1. ## Result Math- #229 proves `ρ_rej(ℓ) = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} − ν_eld^{far,r_0^*}(ℓ) + O(ℓ^{3/7})`. Math- #240 Remark 1 leaves -the cusp end `κ = ℓ/r⁴ → ∞` open. In `d = 2` this note describes that end. +the cusp end `κ = ℓ/r⁴ → ∞` open. This note describes that end for the rejected pairs, in every dimension. | | Statement | Status | |---|---|---| -| **Theorem 1** | `κ𝒜^{rej}(b, κ, u) = F₀(b, u) + O(κ^{−1})`, with `F₀ = (5/24)π₀p_A(0 \| b)E[γ⁶]` (Gaussian kernel: `25π₀p_A(0 \| b)`). The rejected weight sits on `λ = −A ∈ [γ²/(24κ), γ²/(8κ)]`. | proof | -| **Lemma 2** | The elder decision in the fold-scale soft model `G_k` (new jets `B = ∂_uA`, `C₃ = ∂_Θ³f`), by a one-dimensional scan, including a case where the saddle is a slice minimum | proof (model) | +| **Theorem 1** | In every `d ≥ 2`, `κ𝒜^{rej}(b, κ, u) = F₀(b, u) + O(κ^{−1})`, where `F₀ = (5/24)π₀𝔇` and `𝔇` is the density at `0` of the smallest eigenvalue `λ₁` of `−A`, weighted by `γ₁⁶(λ₂⋯λ_m)²`. In `d = 2`, `𝔇 = p_A(0 \| b)E[γ⁶]`. The rejected weight sits on `λ₁ ∈ [γ₁²/(24κ), γ₁²/(8κ)]`. | proof | +| **Corollary 1′** | Gaussian kernel: `∫∫F₀ db dσ = 25√3/(48π²) ≈ 0.0914` (`d = 2`), `125√30/(192π³) ≈ 0.1150` (`d = 3`) | proof | +| **Lemma 2** | The elder decision in the fold-scale soft model `G_k` (new jets `B = ∂_uA`, `C₃ = ∂_Θ³f` along the soft direction), by a one-dimensional scan, including a case where the saddle is a slice minimum | proof (model) | +| **Proposition 2′** | `d ≥ 3`: the stiff directions live at scale `r^{3/2}` and decouple; the limit `G_k − (1/2k)Σλ_iη_i²` has the elder decision of `G_k` | proof (model) | | **Lemma 3** | Elder whenever `φ ≤ (1/20)min(1, \|t\|^{−1}, \|χ₀\|^{−2/3})`; so `I(t, χ₀) ≤ (8000/3)max(1, \|t\|³, χ₀²)` | proof (model) | -| **Proposition 4** | Gaussian kernel: the fold-scale rejection rate is `F₀(b)H(k)` with `H(k) = 1 + (12/25)k² + O(k³)`. The elder edge `φ_e = 1/3 + t/3 + 10t²/27` contributes `+312/25`, and the pin density `e^{−12k²}` contributes `−12`. | proof (given the model) | -| **Lemma 5** | The composite `∫∫∫𝒜^{rej}(b, ℓ/r⁴, u)H(ℓ/r³)` equals `(I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`, where `R_{2/3} = ∫∫∫u⁴[F(u^{−3}) − F₀]` | proof | +| **Proposition 4** | Gaussian kernel, every `d`: the fold-scale rejection rate is `F₀(b)H(k)`, with `H` independent of `b` and `d`, and `H(k) = 1 + (12/25)k² + O(k³)`. The elder edge `φ_e = 1/3 + t/3 + 10t²/27` contributes `+312/25`, and the pin density `e^{−12k²}` contributes `−12`. | proof (given the model) | +| **Lemma 5** | The composite `∫∫∫𝒜^{rej}(b, ℓ/r⁴, u)H(ℓ/r³)` equals `(I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`, where `R_{2/3} = ∫∫∫v⁴[F(v^{−3}) − F₀]` | proof | | **Conjecture 6** | `(k/r)r^{−2}A_r^{rej}(b, k, u) → F(k; b, u)` as `r → 0` at fixed `k` | conjecture | -| **Conjecture 7** | `ρ_rej + ν_eld^{far,r_0^*} = B_{2,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`; Gaussian kernel: `R_{2/3} ≈ −0.049 ± 0.003` | conjecture, with evidence | +| **Conjecture 7** | `ρ_rej + ν_eld^{far,r_0^*} = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`; Gaussian kernel: `R_{2/3} ≈ −0.049 ± 0.003` (`d = 2`), `−0.062 ± 0.004` (`d = 3`) | conjecture, with evidence | Conjecture 7 needs two inputs that are not yet available (PROOF §4): -- (i) the soft layer `λ ≍ r`; +- (i) the soft layer `λ₁ ≍ r`, together with Proposition 2′ for the field in `d ≥ 3`; - (ii) the intermediate separations, where the best bounds are `ℓ^{4/9}log(1/ℓ)` (#229) and `ℓ^{3/5}` (#237). A third input, (iii) the far elder density, is needed only to remove `ν_eld^{far,r_0^*}` from the left side. #187 bounds it by @@ -30,10 +35,12 @@ A third input, (iii) the far elder density, is needed only to remove `ν_eld^{fa | | Result | |---|---| -| Theorem 1 | `κ𝒜^{rej}(0, κ)/F₀(0) = 0.751, 0.977, 0.9985, 0.9999` at `κ = 1, 4, 16, 64`; the `1/κ` coefficient `c(b) = 0.3b` is confirmed at `b = ±1` | +| Theorem 1, `d = 2` | `κ𝒜^{rej}(0, κ)/F₀(0) = 0.751, 0.977, 0.9985, 0.9999` at `κ = 1, 4, 16, 64`; the `1/κ` coefficient `c(b) = 0.3b` is confirmed at `b = ±1` | +| Theorem 1, `d = 3` | `κ∫∫𝒜^{rej}/∫∫F₀ = 0.670, 0.9899, 0.99936, 0.99994` at `κ = 1, 10, 100, 1000`; `∫∫F₀` agrees with Corollary 1′ to `10^{−5}`; per `b` the ratio tends to `1` at rate `1/κ`; `∫G(s^{−4})ds = 0.072444` reproduces #207's `I^{cand} − c₁ = 0.0724443` | | Lemma 2 | agrees with a two-dimensional flood fill on 900 random typed parameter points (one near-degenerate point needs a smaller level offset) | -| `H` | `H(0.2) = 1.004`, `H(0.4) = 0.727`, `H(0.6) = 0.201`, `H(0.8) = 0.018`. With `C₃` ignored, `R_{2/3}` would be `−0.104`; with the pin density alone, `−0.807`. | -| #216's Monte Carlo | rejected adjacent pairs with `s = r/ℓ^{1/4} < 0.8`: 37 observed, 35.3 from the composite, 4,322 from the cusp kernel alone | +| `H` | `H(0.2) = 1.004`, `H(0.4) = 0.727`, `H(0.6) = 0.201`, `H(0.8) = 0.018`. With `C₃` ignored, `R_{2/3}` (`d = 2`) would be `−0.104`; with the pin density alone, `−0.807`. | +| #216's Monte Carlo, `d = 2` | rejected adjacent pairs with `s = r/ℓ^{1/4} < 0.8`: 37 observed, 35.3 from the composite, 4,322 from the cusp kernel alone | +| #216's Monte Carlo, `d = 3` | `s < 0.8`: 11 observed, 10.7 composite (1.2 with the pin density alone), 1,329 cusp kernel alone. On `[3·10^{−4}, 10^{−2}]` the counts are `0.884 ± 0.040` of the `ℓ^{1/4}` law and `1.008` of the composite (χ² 11.8 against 2.1 on 8 bins). Over #216's whole range `[10^{−4}, 0.3]` the composite does not describe the counts (finite-`r` corrections). | ## Controls @@ -52,38 +59,38 @@ Lemma 2. Its output is `RESULTS.json`, byte-identical under `-O`. | S8 | Lemma 2 on 16 fixed parameter points with known answers, including case (D′) | | S9 | Lemma 5's error exponents (all `3/4`) | | S10 | the limit field (2.1), including its `r`-correction, on exactly pinned degree-6 fields | +| S11 | Theorem 1 in `m = 2, 3` rational eigenframes: the soft-window algebra with the factor `(λ₂⋯λ_m)²` and the double-soft implication | +| S12 | the Gaussian kernel in `d = 3`: the GOE law of `A`, the laws used in Corollary 1′, the odd jets along a rotated direction, the prefactor `(2π)^{−d}/(6√π)` from exact covariances, and the two constants | +| S13 | the expansion (2.6) of Proposition 2′ on exactly pinned degree-6 fields in `d = 3`, including the `r^{1/2}`-correction | -Mutants `M1`–`M10` each fail only their own control, and an unknown label exits 2: +Mutants `M1`–`M13` each fail only their own control (the checker names it on stderr), and an unknown label exits 2: python3 -B -S soft_check.py # exit 0, output = RESULTS.json python3 -B -S soft_check.py --mutant M1 # exit 1 ## Review record -- **Same-family referee.** A clean-context referee (Anthropic Claude) read the note against its sources at the declared blobs. - Its verdict was **MAJOR REVISION**, with no BLOCKING finding: - - one MAJOR finding: Conjecture 7 must keep the far elder density and state the missing inputs (ii)–(iii); - - eight MINOR findings and nineteen NITs. - All of them are applied. Among them: - - the definition of `F` and `R_{2/3}` for the torus field; - - Lemma 2's genericity hypothesis; - - a false closed form for `I(t, 0)` on `[t*, 1)`; - - one number in Theorem 1's remarks; - - a new control S10 for the limit field (2.1). -- **The referee's independent checks.** A sympy derivation of (2.1); the contact law of the Gaussian kernel; an independent - quadrature of Theorem 1; a 500-point two-dimensional test of Lemma 2; the `I(t, 0)` table; `H₀`. -- **Delta check of the revision** (the same referee): **ACCEPT WITH MINOR FIXES**, with no BLOCKING or MAJOR finding. Its three - new MINOR findings and seven NITs are applied: - - Lemma 2's genericity hypothesis, in its final form; - - input (ii) now accounts for the composite's own part at small `κ`; - - `H₀` had truncated the small-`γ` end. Correcting it moves `R_{2/3}` from `−0.050` to `−0.049`, inside the quoted - uncertainty. -- **Independence.** The referee is the same provider and the same GitHub account as the author, so it carries zero +- **Same-family referee on v1.** A clean-context referee (Anthropic Claude) read v1 against its sources at the declared + blobs. + - Its verdict was **MAJOR REVISION**, with no BLOCKING finding: one MAJOR finding (Conjecture 7 must keep the far elder + density and state the missing inputs (ii)–(iii)), eight MINOR findings and nineteen NITs. All of them are applied. + - Its delta check was **ACCEPT WITH MINOR FIXES**; its three MINOR findings and seven NITs are applied. + - The referee also made independent checks: a sympy derivation of (2.1), the contact law of the Gaussian kernel, a + quadrature of Theorem 1, a 500-point two-dimensional test of Lemma 2, the `I(t, 0)` table, and `H₀`. +- **v1.1.** A second clean-context same-family referee checked the delta (`REFEREE_B.md` in the project archive). + - Its verdict was **ACCEPT WITH MINOR FIXES**: no BLOCKING or MAJOR finding, four MINOR findings and fourteen NITs. All + are applied. + - The MINOR findings were: Step 3 of Theorem 1 must bound the main term on the excluded sets; the `d = 3` `s`-integral + needed a refined grid; the `d = 3` Monte Carlo statement overread #216's fit; and the packaging. + - Its independent checks: the Gaussian-kernel laws by symbolic differentiation; both constants of Corollary 1′ by the + per-`b` route; a Monte Carlo of `𝔡(b)`; an independent quadrature of `G₃` (agreeing to `3·10^{−7}`); (2.6) symbolically + in `d = 3, 4`; the Hessian factor `(λ₂⋯λ_m)²`; and the mutant isolation matrix. +- **Independence.** The referees are the same provider and the same GitHub account as the author, so they carry zero organizational independence. Nonauthor review is required; the review slices are in PROOF §9. ## Not claimed - Conjectures 6 and 7 are not proved, and `R_{2/3}` is not certified. -- Nothing for `d ≥ 3`. +- Proposition 2′ is a statement about the limit model; its transfer to the field is part of Conjecture 6. - Nothing about the candidate density's own `ℓ^{2/3}` term. - No change to #229, #240 or any other packet. diff --git a/frontiers/soft_rejected_pairs_20261002/RESULTS.json b/frontiers/soft_rejected_pairs_20261002/RESULTS.json index 7d696ca24..c7adbd16c 100644 --- a/frontiers/soft_rejected_pairs_20261002/RESULTS.json +++ b/frontiers/soft_rejected_pairs_20261002/RESULTS.json @@ -8,6 +8,97 @@ "fields": 3, "max_error_after_r1_term_at_r_1e-4": "0.0004916683775860822" }, + "S11": { + "instances": 8 + }, + "S12": { + "A_given_b_d3": [ + [ + "2", + "0", + "0", + "0" + ], + [ + "0", + "2", + "0", + "0" + ], + [ + "0", + "0", + "1", + "0" + ], + [ + "0", + "0", + "0", + "24" + ] + ], + "b_integrated_t_h_A12_f4": [ + [ + "5/3", + "0", + "0", + "2" + ], + [ + "0", + "1", + "0", + "0" + ], + [ + "0", + "0", + "1", + "0" + ], + [ + "2", + "0", + "0", + "30" + ] + ], + "f4_regression_on_t": "6/5", + "f4_residual_variance": "138/5", + "int_F0_d2": "25 sqrt(3)/(48 pi^2) = 0.0914028323", + "int_F0_d3": "125 sqrt(30)/(192 pi^3) = 0.1150058133", + "odd_jets_along_e": [ + [ + "6", + "0", + "0", + "0" + ], + [ + "0", + "2", + "0", + "0" + ], + [ + "0", + "0", + "2", + "0" + ], + [ + "0", + "0", + "0", + "6" + ] + ] + }, + "S13": { + "fields": 2, + "max_scaled_remainder_at_r_1e-8": "7.123741313250203" + }, "S2_instances": 60, "S3": { "conditional_covariance_third_jets": [ @@ -224,16 +315,19 @@ }, "controls": { "S10_limit_field": true, - "S1_theorem_S": true, + "S11_theorem_1_every_d": true, + "S12_gaussian_kernel_d3": true, + "S13_stiff_directions": true, + "S1_theorem_1": true, "S2_model_algebra": true, "S3_gaussian_kernel_jets": true, "S4_elder_edge": true, "S5_I_series": true, - "S6_proposition_H": true, - "S7_lemma_B": true, - "S8_lemma_D_cases": true, - "S9_lemma_M": true + "S6_proposition_4": true, + "S7_lemma_3": true, + "S8_lemma_2_cases": true, + "S9_lemma_5": true }, - "object": "CL-SOFT-REJECTED-20261002-v1", + "object": "CL-SOFT-REJECTED-20261002-v1.1", "scientific_effect": "NONE" } diff --git a/frontiers/soft_rejected_pairs_20261002/SOURCES.json b/frontiers/soft_rejected_pairs_20261002/SOURCES.json index 79af497cc..d119ac0d7 100644 --- a/frontiers/soft_rejected_pairs_20261002/SOURCES.json +++ b/frontiers/soft_rejected_pairs_20261002/SOURCES.json @@ -1,8 +1,8 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-SOFT-REJECTED-20261002-v1", - "kind": "author-side proof candidate and conjectures, d = 2: Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 pA(0|b) E gamma^6, with the weight on soft transverse curvatures lambda in [gamma^2/(24 kappa), gamma^2/(8 kappa)]); the fold-scale soft model G_k with the jets B = d_u A and C3 = d_Theta^3 f; Lemma 2 (the elder decision in the model by a one-dimensional scan, including a case where the saddle is a slice minimum); Lemma 3 (an a priori elder region); Proposition 4 (Gaussian kernel: the fold-scale rejection rate is F0(b) H(k), H = 1 + (12/25) k^2 + O(k^3)); Lemma 5 (the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4))); Conjecture 6 (the fold-scale limit) and Conjecture 7 (rho_rej + nu_eld^far = B_{2,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 for the Gaussian kernel, not certified)", + "object": "CL-SOFT-REJECTED-20261002-v1.1", + "kind": "author-side proof candidate and conjectures, every d >= 2: Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 D with D the density at 0 of the smallest eigenvalue lambda_1 of -A weighted by gamma_1^6 (lambda_2...lambda_m)^2; d = 2: D = pA(0|b) E gamma^6); Corollary 1' (Gaussian kernel: int int F0 = 25 sqrt3/(48 pi^2) in d = 2, 125 sqrt30/(192 pi^3) in d = 3); the fold-scale soft model G_k with the jets B = d_u A and C3 = d_Theta^3 f along the soft direction; Lemma 2 (the elder decision in the model by a one-dimensional scan, including a case where the saddle is a slice minimum); Proposition 2' (d >= 3: the stiff directions at scale r^(3/2) decouple; same elder decision); Lemma 3 (an a priori elder region); Proposition 4 (Gaussian kernel, every d: the fold-scale rejection rate is F0(b) H(k) with H independent of b and d, H = 1 + (12/25) k^2 + O(k^3)); Lemma 5 (the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4))); Conjecture 6 (the fold-scale limit) and Conjecture 7 (rho_rej + nu_eld^far = B_{d,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 (d = 2) and -0.062 (d = 3) for the Gaussian kernel, not certified)", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { @@ -55,7 +55,7 @@ "sha256": "cffc1b83ac364da58bd650c3c301ea1babd38e19af54f74010bdc120edce38a8", "git_blob": "f6df5a7356a54771b0052e42a864a140709903ad", "tag": "207", - "use": "Math- #207 (merged 2026-10-01 at 566b1a1): sections 0, 4, 6 (the cusp objects); Theorem CU.2(c) (the exact elder window of the cusp polynomial)" + "use": "Math- #207 (merged 2026-10-01 at 566b1a1): sections 0, 4, 6 (the cusp objects); the parity factorization of [P] section 15 as stated after (CU.2) (Corollary 1'); Theorem CU.1 (the transverse pins, Proposition 2'); Theorem CU.2(c) (the exact elder window)" }, { "path": "frontiers/candidate_third_order_20261001/PROOF.md", @@ -154,7 +154,10 @@ "M7": "S9", "M8": "S2", "M9": "S6", - "M10": "S10" + "M10": "S10", + "M11": "S11", + "M12": "S12", + "M13": "S13" }, "replayed": [ "soft_check.py" @@ -171,30 +174,37 @@ "report": "project archive V2_2/frontiers_soft_rejected_pairs_20261002/REFEREE_A_DELTA.md", "sha256": "5ccfb644e1a58abf4a022a1fe83a9c3969fdca448724de315a9eace5dc60ca01", "bytes": 13940 + }, + "v1.1_delta": { + "verdict": "ACCEPT WITH MINOR FIXES, no BLOCKING or MAJOR finding: four MINOR (Theorem 1 Step 3 main term on the excluded sets; the d = 3 s-integral needed a refined grid; the d = 3 Monte Carlo statement overread #216's fit; packaging) and fourteen NIT findings, all applied", + "report": "project archive V2_2/frontiers_soft_rejected_pairs_20261002/REFEREE_B.md", + "sha256": "095505d20ac264d6d8a4d3a39cb9d137999e7a3c7aac97105336f4561c7ed205", + "bytes": 27059, + "limits": "same provider and same GitHub account as the author: organizational-independence credit 0" } }, - "exploration": "project archive V2_2/frontiers_soft_rejected_pairs_20261002/exploration/ (numpy, scipy, mpmath, sympy; not part of the repository)", + "exploration": "project archive V2_2/frontiers_soft_rejected_pairs_20261002/exploration/ (numpy, scipy, mpmath, sympy; not part of the repository); the raw Monte Carlo records batchA2.npz and batch3B.npz are delivered as SIDE24_MC_raw_2026-10-01.zip", "delivered_under": "explicit owner instructions of Dylan Roy in claude.ai session session_01NMeKEismAyeqgdB4sy2NJU, given after OWNER_STOP (2026-09-27): 2026-09-30 \"Continue to Work on closing as many mathematical items as possible while working with and supporting the other models in the GitHub. Also advance the mathematics in every place that's justified based on what we currently know\", followed by \"Continue\"; standing: \"Any request or permissions are preapproved in advance by me Dylan Roy\", pull requests only, never self-merge, no register/GRAPH/STATUS/lemma_closed/prize/premise edits, OWNER_STOP.md untouched. Nothing is merged by its author.", "files": [ { "path": "PROOF.md", - "bytes": 43001, - "sha256": "19e911e99298daf9e7879a88cc2d727278472c68ed80f4fd6701f38596c98fe7" + "bytes": 63690, + "sha256": "c312e423990442cfaad44459a2a20b231029828cae97018ba68724c20613331b" }, { "path": "README.md", - "bytes": 6239, - "sha256": "8a98e9039bdde282ab6c4e23717d7ec5b096d9e1368e4d922bb1e0376df9f589" + "bytes": 8806, + "sha256": "64ac4b380a45ae77847e77ab5e792d5f9d84898d5ec00fb670f01f8cf2cd1e7e" }, { "path": "soft_check.py", - "bytes": 25199, - "sha256": "feb656ce94401a63f994dca47d09416cf130f520dc53d8ece68e18a044b4a9ab" + "bytes": 40790, + "sha256": "9c92b76c7a77ed6630370b343466977a18d503543436f00524e864aff4323c07" }, { "path": "RESULTS.json", - "bytes": 2395, - "sha256": "e1e0e8885d6e5a6d49865026d6c5d0a27ecc14f7fac2d61cb55f082a9d791d38" + "bytes": 3445, + "sha256": "86e733071854b0c7189e91e125cf7505b2c6eefa27b14a8ceadf7f1d7bf78562" } ] } diff --git a/frontiers/soft_rejected_pairs_20261002/soft_check.py b/frontiers/soft_rejected_pairs_20261002/soft_check.py index 9e6d38d79..d712228bb 100644 --- a/frontiers/soft_rejected_pairs_20261002/soft_check.py +++ b/frontiers/soft_rejected_pairs_20261002/soft_check.py @@ -1,10 +1,11 @@ #!/usr/bin/env python3 -"""Exact controls for CL-SOFT-REJECTED-20261002-v1 (Theorem 1, Lemmas 2, 3 and 5, Proposition 4; PROOF.md). +"""Exact controls for CL-SOFT-REJECTED-20261002-v1.1 (Theorem 1, Corollary 1', Lemmas 2, 3 and 5, Propositions 2' and 4; +PROOF.md). Standard library only; exact rationals except control S8 (a deterministic floating-point implementation of Lemma 2 on fixed cases). Output: RESULTS.json (sorted keys), byte-identical under -O. Usage: python3 -B -S soft_check.py # exit 0, prints RESULTS.json - python3 -B -S soft_check.py --mutant M3 # exit 1 (only its control fails) + python3 -B -S soft_check.py --mutant M3 # exit 1 (only its control fails; M1..M13) unknown mutant label -> exit 2 Controls @@ -32,13 +33,24 @@ S9 Lemma 5: the exponent bookkeeping of its three error terms (all 3/4) and the convergence exponents of I~ S10 the limit (2.1): on three exactly pinned degree-6 fields with A = -lt r/k, (f(rX, rk zeta) - b)/(k r^3) equals G_k + r G1 + O(r^2) at four points (G1 explicit), checked at r = 1/1000 and 1/10000 in exact arithmetic + S11 Theorem 1 in every d (m = d - 1 = 2, 3; rational eigenframes from Cayley rotations): z = f4 - 3 g^T A^{-1} g = + 3 g1^2/l1 + f4~, Y = (Delta/12) z, w = 36 kappa^2 Delta^2 (1 - phi^2), the substitution (1.1) with the factor + Pi' = (l2...lm)^2, 69984/72^4 = 1/384, and the double-soft implication 3 sum_{i>=2} g_i^2/l_i >= kappa/2 => + l2 <= 6|g|^2/kappa + S12 the Gaussian kernel in d = 3 (Hermite numbers in three coordinates): A = -b I + GOE and f4 ~ N(-3b, 24) given the + even pins; the b-integrated law (t ~ N(0, 5/3), f4 | t ~ N(6t/5, 138/5); d = 2: [[8/3, 2], [2, 30]]); the odd + jets along e = (3/5, 4/5) with variances (6, 2, 2, 6); the prefactor p_G(0) p_{V_u}(0) (2 pi)^{-1/2} tau_u^{-1} = + (2 pi)^{-d}/(6 sqrt pi) from the exact covariances of grad f, V_u and tau_u; Corollary 1': 25 sqrt3/(48 pi^2) and + 125 sqrt30/(192 pi^3), in exact arithmetic of the form q pi^(e/2) sqrt(r) (floating-point values printed only) + S13 Proposition 2': on two exactly pinned degree-6 fields in d = 3 with A = diag(-lt r/k, mu) at 0, + (f(rX, rk zeta, r^(3/2) eta) - b)/(k r^3) = G_k + (mu/2k) eta^2 + r^(1/2) G_half + O(r), at r = 1e-6 and 1e-8 """ import json import math import sys from fractions import Fraction as Fr -MUTANTS = ('M1', 'M2', 'M3', 'M4', 'M5', 'M6', 'M7', 'M8', 'M9', 'M10') +MUTANTS = ('M1', 'M2', 'M3', 'M4', 'M5', 'M6', 'M7', 'M8', 'M9', 'M10', 'M11', 'M12', 'M13') MUT = None @@ -481,6 +493,281 @@ def val(cc, x, y, dx=0, dy=0): return ok, {'fields': 3, 'max_error_after_r1_term_at_r_1e-4': str(float(worst))} +# ------------------------------------------------------------------------------------- S11: Theorem 1 in every d (algebra) +def mmul(A, B): + return [[sum(A[i][k] * B[k][j] for k in range(len(B))) for j in range(len(B[0]))] for i in range(len(A))] + + +def mT(A): + return [list(r) for r in zip(*A)] + + +def minv(A): + n = len(A) + M = [list(A[i]) + [Fr(int(i == j)) for j in range(n)] for i in range(n)] + for c in range(n): + piv = next(i for i in range(c, n) if M[i][c] != 0) + M[c], M[piv] = M[piv], M[c] + M[c] = [x / M[c][c] for x in M[c]] + for i in range(n): + if i != c and M[i][c] != 0: + f = M[i][c] + M[i] = [a - f * b for a, b in zip(M[i], M[c])] + return [row[n:] for row in M] + + +def mdet(A): + n = len(A) + M = [list(r) for r in A] + d = Fr(1) + for c in range(n): + piv = next((i for i in range(c, n) if M[i][c] != 0), None) + if piv is None: + return Fr(0) + if piv != c: + M[c], M[piv] = M[piv], M[c] + d = -d + d *= M[c][c] + for i in range(c + 1, n): + f = M[i][c] / M[c][c] + M[i] = [a - f * b for a, b in zip(M[i], M[c])] + return d + + +def cayley(K): + """Rational orthogonal matrix (I - K)^{-1}(I + K) for skew K.""" + n = len(K) + I = [[Fr(int(i == j)) for j in range(n)] for i in range(n)] + return mmul(minv([[I[i][j] - K[i][j] for j in range(n)] for i in range(n)]), [[I[i][j] + K[i][j] for j in range(n)] for i in range(n)]) + + +def control_S11(): + """Theorem 1 for m = d - 1 = 2, 3: in the eigenframe of Lambda = -A, z = f4 - 3 g^T A^{-1} g = 3 g1^2/l1 + f4~, + Y = (Delta/12) z, w = 36 kappa^2 Delta^2 (1 - phi^2); the soft-window substitution with the factor l2^2...lm^2; + and the double-soft implication 3 sum_{i>=2} g_i^2/l_i >= kappa/2 => l2 <= 6 |g|^2/kappa.""" + ok = True + rng = LCG(31011) + n = 0 + for m in (2, 3, 2, 3, 2, 3, 2, 3): + K = [[Fr(0)] * m for _ in range(m)] + for i in range(m): + for j in range(i + 1, m): + K[i][j] = rng.rat(5, 4); K[j][i] = -K[i][j] + O = cayley(K) + ok &= mmul(mT(O), O) == [[Fr(int(i == j)) for j in range(m)] for i in range(m)] + lam = sorted({abs(rng.rat(9, 7, True)) for _ in range(m + 3)})[:m] + if len(lam) < m: + continue + Lam = mmul(mmul(O, [[lam[i] if i == j else Fr(0) for j in range(m)] for i in range(m)]), mT(O)) + A = [[-x for x in row] for row in Lam] + g = [rng.rat(9, 4) for _ in range(m)] + f4 = rng.rat(19, 3); kap = abs(rng.rat(40, 3, True)) + gi = [sum(O[k][i] * g[k] for k in range(m)) for i in range(m)] # g_i = g . e_i, e_i = O[:, i] + Ainv = minv(A) + q = -sum(g[i] * Ainv[i][j] * g[j] for i in range(m) for j in range(m)) + ok &= q == sum(gi[i] ** 2 / lam[i] for i in range(m)) + Delta = mdet(A) + prod = Fr(1) + for x in lam: + prod *= x + ok &= Delta == (-1) ** m * prod + adj = [[Delta * x for x in row] for row in Ainv] if MUT != 'M11' else Ainv + Y = f4 / 12 * Delta - sum(g[i] * adj[i][j] * g[j] for i in range(m) for j in range(m)) / 4 + z = f4 + 3 * q + ok &= Y == Delta / 12 * z + phi = z / (72 * kap) + ok &= 36 * kap ** 2 * Delta ** 2 - Y * Y == 36 * kap ** 2 * Delta ** 2 * (1 - phi * phi) + # the substitution: f4~ = f4 + 3 sum_{i>=2} g_i^2/l_i, l1(phi) = 3 g1^2/(72 kappa phi - f4~) + f4t = f4 + 3 * sum(gi[i] ** 2 / lam[i] for i in range(1, m)) + Pp = Fr(1) + for x in lam[1:]: + Pp *= x * x + for ph in (Fr(1, 3), Fr(1, 2), Fr(5, 7), Fr(99, 100)): + den = 72 * kap * ph - f4t + if den <= 0 or gi[0] == 0: + continue + l1 = 3 * gi[0] ** 2 / den + ok &= f4t + 3 * gi[0] ** 2 / l1 == 72 * kap * ph + jac = 216 * kap * gi[0] ** 2 / den ** 2 + ok &= 36 * kap ** 2 * l1 ** 2 * Pp * (1 - ph * ph) * jac == 69984 * kap ** 3 * gi[0] ** 6 * Pp * (1 - ph * ph) / den ** 4 + # double-soft implication + if 3 * sum(gi[i] ** 2 / lam[i] for i in range(1, m)) >= kap / 2: + ok &= lam[1] <= 6 * sum(x * x for x in g) / kap + n += 1 + ok &= Fr(69984, 72 ** 4) == Fr(1, 384) + return ok, {'instances': n} + + +# ------------------------------------------------------------------------------------- S12: the Gaussian kernel in d = 3 +def cov3(a, b): + out = Fr((-1) ** sum(b)) + for x, y in zip(a, b): + out *= d0(x + y) + return out + + +def covL(J1, J2): + return sum(c1 * c2 * cov3(a, b) for a, c1 in J1.items() for b, c2 in J2.items()) + + +def directional(order_u, e, n_e): + """d_u^order_u d_e^n_e f for e = (e1, e2) in the transverse plane, as {multi-index (u, 1, 2): coefficient}.""" + out = {} + for j in range(n_e + 1): + c = Fr(math.comb(n_e, j)) * e[0] ** (n_e - j) * e[1] ** j + key = (order_u, n_e - j, j) + out[key] = out.get(key, Fr(0)) + c + return out + + +def condition(Xs, Ps): + """Conditional covariance of the jets Xs given Ps (lists of linear combinations), and regression coefficients.""" + SPP = [[covL(p, q) for q in Ps] for p in Ps] + Pinv = minv(SPP) + SXP = [[covL(x, p) for p in Ps] for x in Xs] + coef = mmul(SXP, Pinv) + C = [[covL(x, y) - sum(coef[i][k] * SXP[j][k] for k in range(len(Ps))) for j, y in enumerate(Xs)] for i, x in enumerate(Xs)] + return C, coef + + +class Sym: + """q * pi^(e/2) * sqrt(r), q and r rational, r squarefree-normalized lazily (comparison by square).""" + def __init__(self, q, e, r): + self.q, self.e, self.r = Fr(q), e, Fr(r) + + def __mul__(self, o): + return Sym(self.q * o.q, self.e + o.e, self.r * o.r) + + def same(self, o): + return self.e == o.e and self.q * self.q * self.r == o.q * o.q * o.r and (self.q >= 0) == (o.q >= 0) + + def value(self): + return float(self.q) * math.pi ** (self.e / 2) * math.sqrt(float(self.r)) + + +def control_S12(): + """The Gaussian kernel exp(-|x|^2/2) in d = 3, coordinates (u, theta_1, theta_2): + (i) given (f, f_uu, f_u1, f_u2) = (b, 0, 0, 0): A = -b I + G with Var G_11 = Var G_22 = 2, Cov = 0, Var G_12 = 1 + (GOE), and f4 = -3b + N(0, 24) independent of A; + (ii) b-integrated (f free): t = tr A/2 ~ N(0, 5/3), the traceless part has variance 1 per coordinate, and + f4 | A ~ N((6/5) t, 138/5); d = 2: (f_TT, f4) ~ N(0, [[8/3, 2], [2, 30]]); + (iii) given grad f = 0, along e = (3/5, 4/5): (f_uuu, gamma_e, B_ee, C_eee) are independent with variances + (6, 2, 2, 6), as in d = 2 (rotation invariance), so the pin factor exp(-12 k^2) and H are unchanged; + (iv) the constants int int F0 db dsigma = 25 K_d d_int: d = 2: 25 sqrt3/(48 pi^2), d = 3: 125 sqrt30/(192 pi^3).""" + ok = True + f, fuu = {(0, 0, 0): Fr(1)}, {(2, 0, 0): Fr(1)} + fu1, fu2 = {(1, 1, 0): Fr(1)}, {(1, 0, 1): Fr(1)} + A11, A22, A12, f4 = {(0, 2, 0): Fr(1)}, {(0, 0, 2): Fr(1)}, {(0, 1, 1): Fr(1)}, {(4, 0, 0): Fr(1)} + C, coef = condition([A11, A22, A12, f4], [f, fuu, fu1, fu2]) + target = [[2, 0, 0, 0], [0, 2, 0, 0], [0, 0, 1, 0], [0, 0, 0, 24]] + if MUT == 'M12': + target[2][2] = 2 + ok &= C == [[Fr(x) for x in row] for row in target] + ok &= [row[0] for row in coef] == [-1, -1, 0, -3] # mean coefficients on b + t = {(0, 2, 0): Fr(1, 2), (0, 0, 2): Fr(1, 2)} + h = {(0, 2, 0): Fr(1, 2), (0, 0, 2): Fr(-1, 2)} + C2, _ = condition([t, h, A12, f4], [fuu, fu1, fu2]) + ok &= C2 == [[Fr(5, 3), 0, 0, 2], [0, 1, 0, 0], [0, 0, 1, 0], [2, 0, 0, 30]] + reg = C2[0][3] / C2[0][0]; var = C2[3][3] - C2[0][3] ** 2 / C2[0][0] + ok &= reg == Fr(6, 5) and var == Fr(138, 5) + C1, _ = condition([A11, f4], [fuu, fu1]) # d = 2 (theta_2 absent: its jets are uncorrelated) + ok &= C1 == [[Fr(8, 3), 2], [2, 30]] + e = (Fr(3, 5), Fr(4, 5)) + odd = [directional(3, e, 0), directional(2, e, 1), directional(1, e, 2), directional(0, e, 3)] + grad = [{(1, 0, 0): Fr(1)}, {(0, 1, 0): Fr(1)}, {(0, 0, 1): Fr(1)}] + C3, _ = condition(odd, grad) + ok &= C3 == [[Fr(x) for x in row] for row in ([6, 0, 0, 0], [0, 2, 0, 0], [0, 0, 2, 0], [0, 0, 0, 6])] + ok &= all(covL(o, ev) == 0 for o in odd for ev in (f, fuu, fu1, fu2, A11, A22, A12, f4)) + # (iv) the prefactor of #207's parity factorization, p_G(0) p_{V_u}(0) (2 pi)^{-1/2} tau_u^{-1}, from the exact + # covariances: grad f ~ N(0, I_d); V_u = (f_uu, f_u1, f_u2) with variances (3, 1, 1), uncorrelated; tau_u^2 = 6. + Gd3 = [{(1, 0, 0): Fr(1)}, {(0, 1, 0): Fr(1)}, {(0, 0, 1): Fr(1)}] + ok &= [[covL(x, y) for y in Gd3] for x in Gd3] == [[Fr(int(i == j)) for j in range(3)] for i in range(3)] + Vu = [fuu, fu1, fu2] + CV = [[covL(x, y) for y in Vu] for x in Vu] + ok &= CV == [[3, 0, 0], [0, 1, 0], [0, 0, 1]] + tau2 = C3[0][0] + ok &= tau2 == 6 + + def gauss0(variances): # density at 0 of independent centred normals: (2 pi)^{-n/2} prod v^{-1/2} + prod = Fr(1) + for v in variances: + prod *= v + n = len(variances) + return Sym(Fr(1), -n, 1 / (Fr(2) ** n * prod)) # pi^{-n/2} sqrt(2^{-n} / prod v) + + pref3 = gauss0([Fr(1)] * 3) * gauss0([CV[i][i] for i in range(3)]) * gauss0([tau2]) + pref2 = gauss0([Fr(1)] * 2) * gauss0([CV[0][0], CV[1][1]]) * gauss0([tau2]) + ok &= pref3.same(Sym(Fr(1, 48), -7, 1)) and pref2.same(Sym(Fr(1, 24), -5, 1)) # (2 pi)^{-d} / (6 sqrt pi) + K3 = Sym(4, 2, 1) * pref3 # |S^2| = 4 pi + K2 = Sym(2, 2, 1) * pref2 # |S^1| = 2 pi + ok &= K3.same(Sym(Fr(1, 12), -5, 1)) and K2.same(Sym(Fr(1, 12), -3, 1)) + # the weighted densities: d = 2, the density of A ~ N(0, 8/3) at 0; d = 3, E[(2 rho)^2 p_t(-rho)] with t ~ N(0, 5/3), + # rho ~ Rayleigh(1): 4 (2 pi 5/3)^{-1/2} int rho^3 exp(-(1/2 + 3/10) rho^2) d rho = (25/8) (10 pi/3)^{-1/2} + vt = C2[0][0] + a = Fr(1, 2) + 1 / (2 * vt) + ok &= vt == Fr(5, 3) and a == Fr(4, 5) and 4 * (1 / (2 * a * a)) == Fr(25, 8) + d2 = gauss0([C1[0][0]]) # (2 pi 8/3)^{-1/2} + d3 = Sym(4 * (1 / (2 * a * a)), 0, 1) * gauss0([vt]) # (25/8) (2 pi 5/3)^{-1/2} + F2 = Sym(25, 0, 1) * K2 * d2 + F3 = Sym(25, 0, 1) * K3 * d3 + ok &= F2.same(Sym(Fr(25, 48), -4, 3)) and F3.same(Sym(Fr(125, 192), -6, 30)) + return ok, {'A_given_b_d3': [[str(x) for x in row] for row in C], 'b_integrated_t_h_A12_f4': [[str(x) for x in row] for row in C2], + 'f4_regression_on_t': str(reg), 'f4_residual_variance': str(var), 'odd_jets_along_e': [[str(x) for x in row] for row in C3], + 'int_F0_d2': '25 sqrt(3)/(48 pi^2) = %.10f' % F2.value(), 'int_F0_d3': '125 sqrt(30)/(192 pi^3) = %.10f' % F3.value()} + + +# ------------------------------------------------------------------------------------- S13: the stiff directions (d = 3) +def control_S13(): + """Exactly pinned degree-6 fields in (x, y, w) with A = diag(-lt r/k, mu) at 0 (the eigenframe): at the fold scale + y = rk zeta, w = r^(3/2) eta, + (f - b)/(k r^3) = G_k(X, zeta) + (mu/(2k)) eta^2 + r^(1/2) G_half + O(r), + G_half = (c201/(2k))(X^2 - 1/4) eta + c111 X zeta eta + (k c021/2) zeta^2 eta, + checked at r = 10^-6 and 10^-8 (so r^(1/2) is rational) in exact arithmetic.""" + ok = True + rng = LCG(313) + pinned = [(0, 0, 0), (1, 0, 0), (2, 0, 0), (3, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 1), (1, 0, 1)] + mono = [(i, j, l) for n in range(7) for i in range(n + 1) for j in range(n + 1 - i) for l in [n - i - j]] + worst = Fr(0) + + def val(cc, x, y, w, dx=0, dy=0, dw=0): + out = Fr(0) + for (i, j, l), v in cc.items(): + if i >= dx and j >= dy and l >= dw: + out += v * x ** (i - dx) * y ** (j - dy) * w ** (l - dw) / (FACT[i - dx] * FACT[j - dy] * FACT[l - dw]) + return out + for trial in range(2): + k = Fr(1 + rng.nxt() % 4, 1 + rng.nxt() % 3); lt = Fr(1 + rng.nxt() % 5, 1 + rng.nxt() % 4); b = rng.rat(3, 2) + mu = -Fr(1 + rng.nxt() % 5, 1 + rng.nxt() % 3) + free = {mm: rng.rat(5, 3) for mm in mono if mm not in pinned} + free[(0, 1, 1)] = Fr(0); free[(0, 0, 2)] = mu + errs = [] + for sr in (Fr(1, 10 ** 3), Fr(1, 10 ** 4)): + r = sr * sr + c = dict(free); c[(0, 2, 0)] = -lt * r / k + rows, rhs = [], [] + for (x, target, kind) in ((-r / 2, b, (0, 0, 0)), (r / 2, b - k * r ** 3, (0, 0, 0)), (-r / 2, 0, (1, 0, 0)), + (r / 2, 0, (1, 0, 0)), (-r / 2, 0, (0, 1, 0)), (r / 2, 0, (0, 1, 0)), + (-r / 2, 0, (0, 0, 1)), (r / 2, 0, (0, 0, 1))): + rows.append([val({mm: Fr(1)}, x, Fr(0), Fr(0), *kind) for mm in pinned]) + rhs.append(target - val(c, x, Fr(0), Fr(0), *kind)) + for mm, v in zip(pinned, solve(rows, rhs)): + c[mm] = v + g, B, C3 = c[(2, 1, 0)], c[(1, 2, 0)], c[(0, 3, 0)] + e = [] + for X, ze, et in ((Fr(-3, 2), Fr(1, 3), Fr(1, 2)), (Fr(-1, 2), Fr(2), Fr(-1)), (Fr(1, 3), Fr(-1), Fr(2)), (Fr(2), Fr(1, 2), Fr(1, 3))): + F = (val(c, r * X, r * k * ze, r * sr * et) - b) / (k * r ** 3) + G = Gk(X, ze, g, lt, k, B, C3) + mu / (2 * k) * et * et + Gh = (c[(2, 0, 1)] / (2 * k) * (X * X - Fr(1, 4)) * et + c[(1, 1, 1)] * X * ze * et + k * c[(0, 2, 1)] / 2 * ze * ze * et) + if MUT == 'M13': + Gh = 0 + e.append(abs((F - G - sr * Gh) / r)) + errs.append(max(e)) + # remainder O(r): the scaled error stays bounded as r drops by 100 + ok &= errs[1] <= 2 * errs[0] + 1 and errs[1] <= 100 + worst = max(worst, errs[1]) + return ok, {'fields': 2, 'max_scaled_remainder_at_r_1e-8': str(float(worst))} + + def main(argv): global MUT if len(argv) == 2 and argv[0] == '--mutant': @@ -489,7 +776,7 @@ def main(argv): return 2 MUT = argv[1] elif argv: - print('usage: soft_check.py [--mutant M1..M10]', file=sys.stderr) + print('usage: soft_check.py [--mutant M1..M13]', file=sys.stderr) return 2 s1, s1d = control_S1() s2, n2 = control_S2() @@ -501,14 +788,22 @@ def main(argv): s8, s8d = control_S8() s9, s9d = control_S9() s10, s10d = control_S10() - res = {'object': 'CL-SOFT-REJECTED-20261002-v1', 'scientific_effect': 'NONE', - 'controls': {'S1_theorem_S': s1, 'S2_model_algebra': s2, 'S3_gaussian_kernel_jets': s3, 'S4_elder_edge': s4, - 'S5_I_series': s5, 'S6_proposition_H': s6, 'S7_lemma_B': s7, 'S8_lemma_D_cases': s8, - 'S9_lemma_M': s9, 'S10_limit_field': s10}, - 'S1': s1d, 'S2_instances': n2, 'S3': s3d, 'S4': s4d, 'S5': s5d, 'S6': s6d, 'S7': s7d, 'S8': s8d, 'S9': s9d, 'S10': s10d} + s11, s11d = control_S11() + s12, s12d = control_S12() + s13, s13d = control_S13() + res = {'object': 'CL-SOFT-REJECTED-20261002-v1.1', 'scientific_effect': 'NONE', + 'controls': {'S1_theorem_1': s1, 'S2_model_algebra': s2, 'S3_gaussian_kernel_jets': s3, 'S4_elder_edge': s4, + 'S5_I_series': s5, 'S6_proposition_4': s6, 'S7_lemma_3': s7, 'S8_lemma_2_cases': s8, + 'S9_lemma_5': s9, 'S10_limit_field': s10, 'S11_theorem_1_every_d': s11, + 'S12_gaussian_kernel_d3': s12, 'S13_stiff_directions': s13}, + 'S1': s1d, 'S2_instances': n2, 'S3': s3d, 'S4': s4d, 'S5': s5d, 'S6': s6d, 'S7': s7d, 'S8': s8d, 'S9': s9d, 'S10': s10d, + 'S11': s11d, 'S12': s12d, 'S13': s13d} ok = all(res['controls'].values()) if MUT is None: print(json.dumps(res, indent=1, sort_keys=True)) + else: + print('mutant ' + MUT + ': failing controls ' + ', '.join(sorted(k for k, v in res['controls'].items() if not v)), + file=sys.stderr) return 0 if ok else 1 From 9fa8f063d677e122e9a9758681ae2fb646104ed1 Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 12:12:39 +0000 Subject: [PATCH 3/7] Soft rejected pairs v1.2: apply the v1.1 nonauthor reviews; exact (D') witness; prior work #170/#175 Applies every finding of OpenAI Codex's nonauthor reviews of v1.1 (87912bd): - Slice A (5390308599): A-01. The exact window endpoints; the localization is weighted, not pointwise. - Slice B, model (5948352439): B-01. Lemma 2's proof is rewritten at the exact level (the component U of {G > L_S}; the axial path / #170's critical chord). The statement now also says that beta > 2, chi <= 0 is always rejected. - Slice C (5390667993): C-01 (the k^(7/2) parenthetical becomes a citation of Codex's upper remainder) and C-02 (the -96 Jacobian belongs to the rescaled system). - Slice D (5390555860): D-01 (section 4's opening matching) and D-02 (B_{d,L}). - Slices B (formulas) and E (5391485205): B-02 (weighted, not eventwise, localization), B-03 (the absolute Jacobian, t = 1), Codex's remainder (B.5) cited, E-01 (section 5 labelled author-reported, files pinned in SOURCES.json exploration_manifest) and E-02 (descriptive uncertainties; #216's L = 64 and L = 16). Adds Codex's exact (D') witness G_(3/2,8/3,20/3)(X, z) = (1/9) G_(1/6,0,0)(X + 2z, 3z), checked by the new control S14 (mutant M14). Records prior work (AUTH-242-01): the merged #170 and #175 already prove the fold-scale limit's existence and the compact-window coefficient. Conjecture 6's status is stated: d = 2 from #170 and #243 Proposition FL.7; #243 (open) claims every d. Rebased onto main f534754. #240 is merged (blob unchanged) and moves to cited_only. A same-family delta referee returned ACCEPT WITH FIXES (9 MINOR, 8 NIT), all applied. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .github/workflows/soft-rejected-pairs.yml | 3 + .../soft_rejected_pairs_20261002/PROOF.md | 328 +++++++++--- .../soft_rejected_pairs_20261002/README.md | 53 +- .../soft_rejected_pairs_20261002/RESULTS.json | 16 +- .../soft_rejected_pairs_20261002/SOURCES.json | 476 +++++++++++++++++- .../soft_check.py | 79 ++- 6 files changed, 846 insertions(+), 109 deletions(-) diff --git a/.github/workflows/soft-rejected-pairs.yml b/.github/workflows/soft-rejected-pairs.yml index 223457911..9359bc11a 100644 --- a/.github/workflows/soft-rejected-pairs.yml +++ b/.github/workflows/soft-rejected-pairs.yml @@ -17,6 +17,9 @@ name: Soft rejected pairs exact controls - 'frontiers/third_order_rate_20261001/PROOF.md' - 'frontiers/c7_total_bounded_20260929/PROOF.md' - 'frontiers/far_elder_rate_20260930/PROOF.md' + - 'frontiers/elder_cusp_parity_20261002/PROOF.md' + - 'frontiers/local_elder_geometry_20260930/PROOF.md' + - 'frontiers/concave_fibre_elder_20260930/PROOF.md' workflow_dispatch: permissions: diff --git a/frontiers/soft_rejected_pairs_20261002/PROOF.md b/frontiers/soft_rejected_pairs_20261002/PROOF.md index 8b2f5fae2..ad4beb846 100644 --- a/frontiers/soft_rejected_pairs_20261002/PROOF.md +++ b/frontiers/soft_rejected_pairs_20261002/PROOF.md @@ -1,6 +1,33 @@ # Soft rejected pairs: the `1/κ` tail of the rejected cusp kernel in every dimension, the fold-scale rejection function, and a conjectured `ℓ^{2/3}` term -Object: CL-SOFT-REJECTED-20261002-v1.1. Version history: +Object: CL-SOFT-REJECTED-20261002-v1.2. Version history: +- v1.1 → v1.2, after the nonauthor reviews of v1.1 by OpenAI Codex: + - Slice A, 5390308599; + - Slice B (model), 5948352439; + - Slice C, 5390667993; + - Slice D, 5390555860; + - Slices B (formulas) and E, 5391485205. + + The changes: + - OA-242-A-01: Theorem 1's leading window is stated with its exact endpoints, as a weighted asymptotic localization. + - OA-242-B-01: Lemma 2's proof is rewritten at the exact level, following the review. The lemma now also states that for + `β > 2`, `χ < 0` the pair is always rejected. + - The exact (D′) witness found by Codex (5946792240) is inserted, with control S14. + - OA-242-C-01: the unsupported "true next correction" parenthetical in Proposition 4 is replaced by a citation of + Codex's upper remainder. + - OA-242-C-02: the Jacobian `−96` is attributed to its rescaled system. + - OA-242-D-01: §4's opening qualifies the matching of the two limits. + - OA-242-D-02: `B_{d,L}` in §4(ii). + - OA-242-B-02: §2's opening states the soft-layer localization as weighted, not eventwise. + - OA-242-B-03: the absolute Jacobian `|dλ̃/dφ|` and the convention at `t = 1`. + - (2.1)'s remainder is now quantified: Codex's estimate (B.5). + - OA-242-E-01 and OA-242-E-02: §5's numbers are labelled author-reported, their custody is pinned in `SOURCES.json`, + their uncertainties are labelled, and #216's domains are stated. + - A same-family referee checked the delta: ACCEPT WITH FIXES, with 9 MINOR findings and 8 NITs, all applied. + - AUTH-242-01 (author): v1.1 did not cite #170 and #175. These are merged author-side candidates, at their stated + conditional scope, and they prove that the fold-scale limit exists, as `kA∗(α₁ + α₂)`. §§3–4 and 6 now cite them, and + Conjecture 6 records its status (§4). + - #240 is merged; its blob is unchanged. - v1 → v1.1, made before any nonauthor review: - Theorem 1 holds in every `d ≥ 2`; the soft direction is the eigenvector of `−A` with the smallest eigenvalue. - Corollary 1′ gives the Gaussian-kernel constants in `d = 2, 3`. @@ -11,16 +38,18 @@ Object: CL-SOFT-REJECTED-20261002-v1.1. Version history: value of `Ĩ` that was not reproducible from the recorded scripts. - Controls S11–S13 are new. - A same-family referee checked the delta (`REFEREE_B.md`: ACCEPT WITH MINOR FIXES; 4 MINOR, 14 NIT, all applied). -- §2's Lemmas 2 and 3 and the `d = 2` proof of Proposition 4 are unchanged. +- In v1.1, §2's Lemmas 2 and 3 and the `d = 2` proof of Proposition 4 were unchanged from v1. Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 2 October 2026. Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem 1, Corollary 1′, Lemmas 2, 3 and 5, Propositions 2′ and 4); CONJECTURES 6 -and 7, with numerical evidence. Nonauthor review required. Scientific effect: NONE — no register, graph, STATUS, +and 7, with numerical evidence. For Conjecture 6, the existence of the limit is in the merged #170/#175, at their +conditional scope. Its identification with `F` follows in `d = 2` from #170 and #243 Proposition FL.7, and #243 (open) +claims a proof in every `d`. Nonauthor review required. Scientific effect: NONE — no register, graph, STATUS, PROOF_INDEX, prize or Boolean change; no numerical constant is certified. Same GitHub account as every lane; zero organizational independence. **Why.** Math- #229 (merged) proves -`ρ_rej(ℓ) = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} − ν_eld^{far,r_0^*}(ℓ) + O(ℓ^{3/7})` ((R⁺.1)). Math- #240 (open) proves that each +`ρ_rej(ℓ) = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} − ν_eld^{far,r_0^*}(ℓ) + O(ℓ^{3/7})` ((R⁺.1)). Math- #240 (merged) proves that each fixed cusp window `κ₀ ≤ ℓ/r⁴ ≤ κ₁` contributes a multiple of `ℓ^{1/4}` with error `O(ℓ^{3/4})`, and its Remark 1 leaves the ends `κ → ∞` and `κ → 0` open. This note studies the end `κ → ∞` for the rejected pairs. Formally it is the only source of the next term of `ρ_rej` (§4). @@ -31,14 +60,17 @@ the next term of `ρ_rej` (§4). - The constant is `F₀ = (5/24)π₀(u; v₀(b, 0))𝔇(b, u) > 0`. Here `𝔇` is the density at `0` of the smallest eigenvalue `λ₁` of `−A`, weighted by `γ₁⁶(λ₂⋯λ_m)²`, where `γ₁` is the component of `γ` along the soft eigenvector. - In `d = 2`, `𝔇 = p_A(0 | b)E[γ⁶]`. - - The rejected weight sits on *soft* transverse curvatures `λ₁ ∈ [γ₁²/(24κ), γ₁²/(8κ)]`, up to relative errors - `O(1/κ)`. + - The rejected weight sits on *soft* transverse curvatures. On Step 2's event it sits exactly on + `λ₁ ∈ (3γ₁²/(72κ − f̃₄), 3γ₁²/(24κ − f̃₄)]`. For fixed jets this window is `[γ₁²/(24κ), γ₁²/(8κ)]` up to relative errors + `O(1/κ)`. This is a weighted, asymptotic localization, not a pointwise one: the double-soft and large-jet events are + handled separately (Step 1). - **Corollary 1′** (Gaussian kernel). `∫∫F₀ db dσ = 25√3/(48π²) ≈ 0.09140` in `d = 2`, and `125√30/(192π³) ≈ 0.11501` in `d = 3`. -- **The fold-scale soft model** (§2). At fixed `k = ℓ/r³` and `λ₁ = λ̃r/k`, the window field divided by `κ = k/r` tends to an +- **The fold-scale soft model** (§2). It is the same function as #170's typed cubic, in other coordinates (#243 + Proposition FL.7). At fixed `k = ℓ/r³` and `λ₁ = λ̃r/k`, the window field divided by `κ = k/r` tends to an explicit polynomial `G_k`, in which two more jets enter: `B = ∂_uA` and `C₃ = ∂_Θ³f`, taken along the soft direction. - **Lemma 2** decides the elder mark in `G_k` exactly by a one-dimensional scan, including a case in which the saddle is - not on the ridge. + not on the ridge. For `β > 2` that case needs `χ > 0`. An exact witness is given. - **Lemma 3** gives an a priori elder region. - The typed weight is `(γ⁴/16)(φ^{−2} − (1 − 12kB/γ²)²)₊`. - **Proposition 2′** (`d ≥ 3`). The stiff transverse directions live at the scale `r^{3/2}` and decouple. The limit is @@ -76,7 +108,12 @@ the next term of `ρ_rej` (§4). - [P] §2, through #220. - Cited only: - [C7-K] (`frontiers/c7_total_bounded_20260929/PROOF.md`, blob `28748b08`): (K2). - - #240 (open): Lemma Q′, Step N6, Corollary N′, Remarks 1 and 4. + - #240 (merged at `7636cae`, blob unchanged): Lemma Q′, Step N6, Corollary N′, Remarks 1 and 4. + - #170 (`frontiers/local_elder_geometry_20260930/PROOF.md`, blob `ef2aa579`) and #175 + (`frontiers/concave_fibre_elder_20260930/PROOF.md`, blob `923d3236`), both merged: the failure law + `r^{−3}(1 − p_r) → α₁ + α₂` (Theorems S and F), the compact-window coefficient, #170 §3 (the critical chord), and #170 + Theorem E. These are prior work for §§2–4; both packets are author-side candidates at their stated conditional scope. + - #243 (open, head `0740b2c`): Theorem FL and Proposition FL.7 (Conjecture 6). - #237 (open): Theorem P, Remark 4. - #216 (open): the Monte Carlo in `d = 2, 3`. - #187 (merged) and #188 (open): the far elder density. @@ -218,8 +255,15 @@ Multiplying by `12π₀` and by `κ` gives (1.0), since `12·(20/3)/384 = 5/24`. *Remarks on Theorem 1.* 1. Theorem 1 is about the cusp kernel `𝒜^{rej}` itself, not about the non-uniformity of #240's Theorem N as `κ → ∞`. - - The rejected weight sits on `λ₁ ∈ [γ₁²/(24κ), γ₁²/(8κ)]` (Step 2), with the other eigenvalues of order one. So the - rejected pairs at large `κ` are soft in exactly one transverse direction. + - The rejected weight sits on `λ₁ ∈ (3γ₁²/(72κ − f̃₄), 3γ₁²/(24κ − f̃₄)]` (Step 2), with the other eigenvalues of order + one. For fixed jets this is `[γ₁²/(24κ), γ₁²/(8κ)]` with relative error `O(1/κ)`. The shorthand is a weighted + asymptotic localization, not a uniform one over all jets. + - *Example.* With `m = 1`, `f₄ = κ/2` and `γ = 1`, the upper endpoint is `48/47` times `γ²/(8κ)` for every `κ`. + - Such growing-jet configurations are exponentially suppressed (OA-242-A-01). Those with `|f̃₄| ≥ κ` are Step 1's + large-jet event, as are the double-soft ones. Those with `|f̃₄| < κ` (for example `f₄ = κ/4`, ratio `96/95`) are covered + by the error term (1.2) of Step 2. + + So the rejected pairs at large `κ` are soft in exactly one transverse direction. - In #220's elder decision they are the pairs with `|φ| ∈ (1/3, 1)` coming from `3γ₁²/λ₁ ≈ z`, not from `f₄`. 2. In `d = 2`, expanding (1.2) to first order gives `κ𝒜^{rej}/F₀ = 1 + c(b)/κ + O(κ^{−2})`, with `c(b) = (12/5)[E(f₄ | A = 0, b)/18 − ∂_a log p_A(0 | b)·E[γ⁸]/(24E[γ⁶])]`. Here `12/5` is the mean of `1/φ` under the @@ -275,9 +319,16 @@ rejected coefficient itself is smaller: `I^{cand} − c₁ = 0.07244` against `0 ## 2. The fold-scale soft model Theorem 1 shows where the rejected weight sits as `κ → ∞` along the cusp scale. The *fold scale* is the regime of fixed -`k = ℓ/r³` and `r → 0`, so again `κ = k/r → ∞`. There the pair is rejected only if the transverse curvature is of order `r` -([C7-K] (K2) bounds the rejected weight by `O(r³/k)`), and two more jets enter. This section is formal for the field (the -limit is Conjecture 6) and exact for the model. It is written for `d = 2`; Proposition 2′ at its end reduces `d ≥ 3` to it. +`k = ℓ/r³` and `r → 0`, so again `κ = k/r → ∞`. +- *Where the rejected weight sits.* [C7-K] (K2) bounds the rejected weight by `O(r³/k)`, and [P]'s cap region shows that in + weighted expectation the rejection comes from transverse curvatures of order `r`. +- *Weighted, not eventwise* (OA-242-B-02). This is a statement in weighted expectation, not an eventwise restriction over + all derivative jets. An event of probability `r³` can have curvature of order one. For example, the exact cubic with + `k = 1`, `λ̃ = 1/r`, `γ² = 12/r` and `B = C₃ = 0` has `φ = ½` and is rejected, with mixed jets growing like `r^{−1/2}`. +- *So the soft contribution is studied with `λ = (r/k)λ̃`,* and two more jets enter. + +This section is formal for the field (the limit is Conjecture 6) and exact for the model. It is written for `d = 2`; +Proposition 2′ at its end reduces `d ≥ 3` to it. *The limit window field.* In the window coordinates of #220 §1 (`Φ(X, Ξ) = rXu + r²ΞΘ`, `𝔉 = r^{−4}(f∘Φ − b)`), put `λ = λ̃/κ` and `Ξ = κζ`, so that `x = rX` and `y = rkζ`. Every monomial `x^iy^j` of the Taylor expansion at `0` contributes @@ -289,6 +340,11 @@ and `Ξ = κζ`, so that `x = rX` and `y = rkζ`. Every monomial `x^iy^j` of the Control S10 checks (2.1), including `G₁`, on exactly pinned polynomial fields. The pins are exact: `G_k(−½, 0) = 0`, `G_k(½, 0) = −1`, and `∇G_k = 0` at both. +- *The remainder is quantified* by OpenAI Codex's root-authored estimate (B.5) in its review 5391485205. For + `f ∈ C⁵([−1, 1]²)` with the pins, `M := max_{i+j=5}sup|∂_x^i∂_y^jf|`, `0 < r ≤ 1/3`, `|X| ≤ 3` and `rk|ζ| ≤ 1`, + the `O_k` term of (2.1) is at most `M r²[17203/(7680k) + (1303/384)|ζ| + (9k/4)ζ² + (3k²/4)|ζ|³ + (k³/8)ζ⁴ + (k⁴/120)|ζ|⁵]`. + This is a deterministic value estimate. It is uniform for `k` in compacts of `(0, ∞)` and bounded `M`. +- Such constants do not by themselves dominate a Gaussian integral over unbounded jets. *Normalization.* For `γ > 0` set `ζ = (γ/λ̃)z` and @@ -307,8 +363,9 @@ ridge `g/(24φκ)` of #220 §1 with `f₄ = 0`, and `φ` is #220's `φ` in the l 0 < φ < 1/|c′|, c′ := 1 − t. (2.4) -The scaled Kac–Rice weight is `36λ̃² − Y² = (γ⁴/16)(φ^{−2} − c′²)`. With `dλ̃ = (γ²/(24φ²))dφ`, the weight per unit `φ` is -`(γ⁶/384)φ^{−2}(φ^{−2} − c′²)`. By (2.4), `1 + β/2 − φ = 1 − φc′ > 0`, so the `z`-curvature `−(1 + β/2)` at `M` is negative. +The scaled Kac–Rice weight is `36λ̃² − Y² = (γ⁴/16)(φ^{−2} − c′²)`. Since `dλ̃/dφ = −γ²/(24φ²)`, with the absolute Jacobian +`|dλ̃/dφ| = γ²/(24φ²)` (OA-242-B-03), the weight per unit `φ` is `(γ⁶/384)φ^{−2}(φ^{−2} − c′²)`. At `t = 1` (`c′ = 0`), (2.4) +reads `φ > 0`, with upper endpoint `∞`. By (2.4), `1 + β/2 − φ = 1 − φc′ > 0`, so the `z`-curvature `−(1 + β/2)` at `M` is negative. *Slices.* For fixed `X`, `z ↦ G(X, z)` is a polynomial of degree at most three. Write `p := X² − ¼`, `a := 1 − βX` and `D := a² − χp`. Call `X` *open* if the slice has no local maximum (`χ = 0` and `a ≤ 0`, or `χ ≠ 0` and `D ≤ 0`). Otherwise @@ -323,40 +380,89 @@ Then `S` kills `M` in the superlevel filtration of `G` (the elder rule) iff one - (D) `1 − β/2 > 0`, the right end of `I_M` is `½`, its left end `X_L` is finite, and on `(X_L, ½)` we have `V < L_S` and `R < 0` for `X ≠ −½`. -- (D′) `1 − β/2 < 0` and `χ ≠ 0`, `½ ∈ I_M`, `I_M = (X_L, X_R)` is bounded, and on `I_M` we have `R < 0` for `X ≠ −½` and +- (D′) `1 − β/2 < 0` and `χ > 0`, `½ ∈ I_M`, `I_M = (X_L, X_R)` is bounded, and on `I_M` we have `R < 0` for `X ≠ −½` and `V < L_S` for `X ≠ ½`. -If `1 − β/2 < 0` and `χ = 0`, then `e = 0`. +If `1 − β/2 < 0` and `χ ≤ 0`, then `S` does not kill `M`. Below, `e := 1{S kills M}`. *Proof.* Fix a level `c`. On a non-open slice, `{z : G(X, z) > c}` has a bounded *ridge piece* around `z_r` when `R(X) > c`. When `χ ≠ 0` it also has an unbounded *far piece* beyond `z_v`, and the two are one interval iff `V(X) > c`. On an open slice -the set is unbounded (one or two half-lines, or `R`), and `G → +∞` along it, as along every far piece. -1. *Above `L_S`.* Let `c ↓ L_S`. Suppose no `X ∈ I_M` has `V(X) > c`. Then the component of `{G > c}` containing `M` is the union of - the ridge pieces over `I_M`. These vary continuously and shrink to points at finite ends of `I_M`, and the supremum of the - union is `max_{I_M} R`. So `M` is still the oldest point of its component iff `I_M` is bounded and `R < 0` on - `I_M ∖ {−½}`. - - If `I_M` is unbounded, then as `|X| → ∞` along it, either some limit point of `I_M` is open or `R → ±∞` (a finite limit is - excluded by hypothesis). The value `−∞` contradicts `R > L_S` on `I_M`, so the component reaches above `0`. - - If some `X ∈ I_M` has `V(X) > c`, the component meets a far piece. - In all of these cases `M` has died at a level above `L_S`, and `e = 0`. -2. *`1 − β/2 > 0`.* At `X = ½`, where `p = 0`, the slice has its local maximum at `z = 0`, with value `L_S`. So `S` lies on the - ridge, and the right end of `I_M` is at most `½`. - - If it is smaller, `R` drops below `L_S` between `M` and `S`, so `S` is not in the closure of `M`'s component above `L_S`, - and `e = 0`. - - Otherwise, for `c < L_S` close to `L_S`, the component of `M` contains `S` and the segment `{(X, 0) : X > ½}`. Along it - `G = A₀(X)` increases from `L_S` to `+∞`. So `S` merges `M`'s component with one containing points above `0`, and `M` dies - at `S`. -3. *`1 − β/2 < 0`.* Now `a(½) < 0`. - - If `χ = 0`, `a` vanishes at `X = 1/β ∈ (0, ½)`, and `R = A₀ + p²/(8a) → +∞` as `X ↑ 1/β`. Meanwhile - `R ≥ G(X, 0) = A₀ > L_S` on `(−½, 1/β)`. So `R > 0` somewhere on `I_M`, and `e = 0`. - - If `χ ≠ 0`, the slice at `X = ½` has its local minimum at `z = 0` (value `L_S`) and its local maximum at `z = 2a(½)/χ` - (value `L_S + (2/3)|a(½)|³/χ² > L_S`). So `S` is the valley point of this slice. `V` has a strict local maximum `L_S` at - `X = ½`, because the Schur complement of `∂_z²G(S) > 0` in the indefinite Hessian is negative. - - If `½ ∉ I_M`, `S` is not in the closure of `M`'s component above `L_S`, and `e = 0`. - - If `½ ∈ I_M` and Step 1 leaves `M` alive, then for `c < L_S` close to `L_S` the slice at `X = ½` is one interval. So `M`'s - component contains the far piece at `X = ½`, along which `G → +∞`, and `S` kills `M`. ∎ - -Case (D′) is not empty. Numerically (control S8), `(φ, β, χ) = (0.2401, 2.1425, 0.5536)` is elder, with `I_M ≈ (−1.032, 0.515)`. +the set is unbounded (one or two half-lines, or `R`), and `G → +∞` along it, as along every far piece. The argument works at +the exact level `L_S` (OA-242-B-01), following OpenAI Codex's review of v1.1 (5948352439). +1. *The exact level.* Let `U` be the component of `{G > L_S}` containing `M`. + - *The axial segment.* `{(X, 0) : −½ ≤ X ≤ 2}` carries `G = A₀`, with `A₀ − L_S = (X − ½)²(X + 1)/(12φ) ≥ 0` (equality only + at `S`). It ends at `A₀(2) = 25/(48φ) > 0`. This is the critical-chord bound of #170 §3. So the death level of `M` is + `≥ L_S`, and `(−½, ½) × {0} ⊂ U`. + - *If `U` contains a point above `0`,* a path in `U` reaches it, because `U` is open and connected. By compactness the + path has minimum `> L_S`, so `M` dies above `L_S`, and `e = 0`. + - *Otherwise* every path from `M` to a point above `0` leaves `U`, so the death level is exactly `L_S`. Since `S` is the + only critical point at that level, `S` kills `M`: `e = 1`. +2. *When `U` reaches above `0`.* `U` contains the ridge tube over `I_M`. It contains a point above `0` in each of three cases: + - `R > 0` somewhere on `I_M`; + - `V > L_S` somewhere on `I_M` (a far piece joins, along which `G → +∞`); + - `I_M` is unbounded. `R` is an algebraic function of `X`, so it has a limit in `[−∞, ∞]` along `I_M`. A finite limit is + excluded by hypothesis, and `−∞` contradicts `R > L_S`. + + *An open end of `I_M` reduces to these cases.* + - If `χ ≠ 0`, then as `X` tends to an open end inside `I_M`, `R` and `V` tend to the common inflection value, which is + `≥ L_S` because `R > L_S` on `I_M`. + - If that value exceeds `L_S`, then `V > L_S` nearby (the second case). + - If it equals `L_S`, then either `V > L_S` nearby, or the inflection point is a critical point at level `L_S` other + than `S`. It is not `S`, because `D(½) = a(½)² > 0` when `β ≠ 2`. + - If `χ = 0`, an open end is a zero of `a`, where `R → +∞` (the first case). + + If none of the three cases holds, then `I_M` is bounded with non-open ends, `U` is the ridge tube over `I_M`, and + `sup_U G = 0`, attained at `M`. Here the equality cases are excluded by the hypotheses: + - a ridge value `0` at some `X ≠ −½`, with `R ≤ 0` nearby, would be a critical point at level `0`; + - a valley value `L_S`, with `V ≤ L_S` nearby, would be a critical point at level `L_S`, hence `S`. + + So `e = 1` iff `I_M` is bounded, `R < 0` on `I_M ∖ {−½}`, and `V < L_S` on `I_M` except at `S`. In particular its ends are + then non-open. +3. *A slice fact.* Let `|X| < ½` and `a(X) > 0`, with a non-open slice. + - If `χ > 0`, then `D = a² + χ|p| > a²`, so `z_r < 0 < z_v`. + - If `χ < 0`, then `z_v < z_r < 0`. + - If `χ = 0`, then `z_r = p/(2a) < 0`. + + In every case the slice decreases strictly from `z_r` to `0` (between `z_r` and `z_v`, or beyond `z_r` away from `z_v`), so + `R(X) > G(X, 0) = A₀(X) > L_S`. +4. *`1 − β/2 > 0`.* On (2.4), `β > −2`, so `a > 0` on `[−½, ½]`. By Step 3, the first open slice to the right of `−½`, if it + lies in `(−½, ½)`, is an open end of `I_M`, and then `e = 0` by Step 2. Otherwise `(−½, ½) ⊂ I_M`. + - At `X = ½`, where `p = 0`, the slice has its local maximum at `z = 0`, with value `L_S`. So `S` is the ridge point there, + and the right end of `I_M` is `½`. + - Inside `(X_L, ½)` a point with `V = L_S` would be a critical point at level `L_S` other than `S`. + + With Step 2 this is (D). +5. *`1 − β/2 < 0`.* Now `a(½) < 0`, and `a` vanishes at `X = 1/β ∈ (0, ½)`. + - *`χ = 0`.* `R = A₀ + p²/(8a) → +∞` as `X ↑ 1/β`, while `R ≥ G(X, 0) = A₀ > L_S` on `(−½, 1/β)`. So `R > 0` somewhere + on `I_M`, and `e = 0`. + - *`χ < 0`.* At `X = 1/β`, `D = −χp < 0`, so that slice is open. By Step 3, the first open slice in `(−½, 1/β]` is an open + end of `I_M`, and `e = 0` by Step 2. + - *`χ > 0`.* For `|X| < ½`, `D > a² ≥ 0` and `z_r < 0 < z_v` whatever the sign of `a`, so `R > A₀ > L_S` on `(−½, ½)`. + - At `X = ½` the slice has its local minimum at `z = 0` (value `L_S`) and its local maximum at `z = 2a(½)/χ` (value + `L_S + (2/3)|a(½)|³/χ² > L_S`). So `[−½, ½] ⊂ I_M`, and `S` is the valley point at `X = ½`. + - `V` has a strict local maximum `L_S` there, because the Schur complement of `∂_z²G(S) > 0` in the indefinite Hessian + is negative. + + With Step 2 this is (D′). ∎ + +#243's Lemma FL.2(c) gives the same case analysis. Off #170's null sets `Σ_k ∪ Δ_k`, #170 Theorem E(1) and #243 +Proposition FL.7(iii) let the decision be read off the critical values: `e = 0` iff `G` has a critical point with value in +`(L_S, 0)`. + +Case (D′) is not empty. +- *An exact witness,* found by OpenAI Codex (5946792240) (control S14): + + G_{(3/2, 8/3, 20/3)}(X, z) = (1/9)·G_{(1/6, 0, 0)}(X + 2z, 3z). + + - The right side has concave slices and `φ = 1/6 < 1/3`, so `R = (X + ½)²(X² + 3X − 15/4)/8` and + `R − L_S = (X − ½)²(X² + 5X + 17/4)/8`. It satisfies (D), with `I_M = ((−5 + 2√2)/2, ½)`, because the convex quadratic + `X² + 3X − 15/4` is negative on `[−2, ½]`. + - The linear map fixes `M` and `S`, and a positive factor preserves elder decisions. So the witness, which has `β = 8/3 > 2` + and `χ = 20/3 > 0`, is elder: case (D′). + - It has exactly three critical points, with values `0`, `L_S = −1/36` and `−125/384`. In its own coordinates + `I_M ≈ (−1.50494, 0.51622)`. +- *A numerical example* (control S8): `(φ, β, χ) = (0.2401, 2.1425, 0.5536)` is elder, with `I_M ≈ (−1.032, 0.515)`. + The margins vanish quadratically at the special points `X = −½` (for `R`) and `X = ½` (for `V`), as the lemma requires. A flood fill of `{G > L_S ± ε}` on a grid agrees with Lemma 2 at 900 random typed parameter points, once one near-degenerate point is rechecked with a smaller `ε` (§5). @@ -423,7 +529,10 @@ The other monomials, and the Taylor remainder, are `O(r)` on the window. This pr - For (b), the fiber of `{G_k^{(d)} > c}` over `(X, ζ)` is the open ellipsoid `{Σλ_iη_i² < 2k(G_k(X, ζ) − c)}`, which is nonempty iff `G_k(X, ζ) > c`. The projection is continuous and open, its fibers are connected, and its image is `{G_k > c}`, so it induces a bijection of components. The supremum over a fiber is attained at `η = 0`. -- (c) follows from (a) and (b), and from the block form of the Hessians. ∎ +- (c) follows from (a) and (b), and from the block form of the Hessians. In physical coordinates the pin Hessian is + `[[rS + O(r²), rC + O(r²)], [rCᵀ + O(r²), D + O(r)]]` with `D = −diag(λ₂, …, λ_m)` fixed and invertible. Its Schur + complement is `rS + O(r²)`, so `det = r²det(D)det(S) + O(r³)`, as OpenAI Codex's Slice B review (§3) spells out. This is not + uniform as a second transverse eigenvalue tends to `0`. ∎ Control S13 checks (2.6), including `G_{1/2}`, on exactly pinned degree-6 fields in `d = 3`, at `r = 10^{−6}` and `10^{−8}` (so that `r^{1/2}` is rational). Like (2.1) and #207 Theorem CU.1, (2.6) is a deterministic Taylor statement for every `C⁵` @@ -447,8 +556,23 @@ This motivates with `(γ₁, B₁, C₁) := (γ_{e₁}, B_{e₁}, C_{e₁})` and the expectation under the contact law at `v₀(b, k)`. In `d = 2` this is `12π₀(u; v₀(b, k))p_A(0 | b, k)E_{v₀(b,k)}[(γ⁶/384)I(12kB/γ², 576k²C₃/γ³) | A = 0]`. Formally, -`r^{−2}A_r^{rej}(b, k, u) = (r/k)F(k; b, u) + o(r)` (Conjecture 6 below). As `k → 0`, `I → I(0, 0) = 20/3`, and (3.1) tends -to the `F₀` of Theorem 1. +`r^{−2}A_r^{rej}(b, k, u) = (r/k)F(k; b, u) + o(r)` (Conjecture 6 below). Formally, as `k → 0`, `I → I(0, 0) = 20/3`, and +(3.1) tends to the `F₀` of Theorem 1. For the Gaussian kernel this is proved by Proposition 4 (`H → 1`). + +*Prior work (AUTH-242-01; not cited in v1.1).* Here `A_r^{rej} := A_r(1 − p_r)`, with `A_r` the candidate kernel of [P] +§§10–11 and `A∗ := lim A_r` ([P] (10.3), where it is written `A_0`). The merged packets #170 (`d = 2`, Theorem S) and #175 +(fixed `d ≥ 3`, Theorem F) are author-side candidates at their stated conditional scope. They prove that `r^{−3}(1 − p_r)` +converges at fixed marks. The limit is `a_fail = α₁ + α₂`, a Gaussian integral over the window-saddle classifier of #170's +typed cubic. +- So the fold-scale limit of `(k/r)r^{−2}A_r^{rej} = kA_r·r^{−3}(1 − p_r)` exists, and equals `kA∗a_fail`. Their §§9 and 7 + also give the compact-window `ℓ^{2/3}` coefficient `C_fail`. +- #242's `G_k` is #170's cubic `P_θ(X, kζ)/k` at `θ = (−λ̃/k, γ, B, C₃)`, with the same typed domain, weight and maximin + decision (#243 Proposition FL.7). + - *In `d = 2`,* (3.1) equals `kA∗a_fail` by a model-level computation (#243 Proposition FL.7(iv)). With #170 Theorem S + this gives Conjecture 6 at each fixed `(b, k, u)`, conditional on #170's interfaces. + - *In `d ≥ 3`* the identity uses #243's Theorem FL (open) and uniqueness of limits. +- The remaining content of Conjecture 6 is local uniformity in `k`, and the `d ≥ 3` case. #243 claims both, by a partly + different route. **Proposition 4 (Gaussian kernel).** Let `C(x) = e^{−|x|²/2}` on `R^d`, `d ≥ 2`. Then: 1. `F(k; b) = F₀(b)H(k)`, where `H(k) := e^{−12k²}E[γ⁶I(t, χ₀)]/(E[γ⁶]·20/3)` depends on neither `b` nor `d`. Here @@ -473,9 +597,11 @@ Hence the expectation in (3.1) factorizes as `𝔡(b)` times the soft-jet averag R − L_S = (X − ½)²[2(X + 1) + 3φ(X + ½)²]/(24φ) (t = 0), (3.2) whose bracket has discriminant `4 − 12φ` in `X + ½`. At `φ = 1/3`, (3.2) is `((X − ½)(X + 3/2))²/(24φ)`, with a nondegenerate -double zero at `X = −3/2`. The elder edge solves `R(X) = L_S`, `R′(X) = 0` in `(X, φ)` at `β = 2tφ`. The Jacobian at -`(−3/2, 1/3, t = 0)` is `−96 ≠ 0`, so the implicit function theorem gives analytic `X_e(t)` and `φ_e(t)`. Their series -(control S4) are +double zero at `X = −3/2`. The elder edge solves `R(X) = L_S`, `R′(X) = 0` in `(X, φ)` at `β = 2tφ`. Equivalently it solves +`N = N_X = 0` for the rescaled `N := 24φa(R − L_S)`. +- The Jacobian of `(N, N_X)` in `(X, φ)` at `(−3/2, 1/3, t = 0)` is `−96 ≠ 0`. For the literal system `(R − L_S, R′)` it is + `−3/2`, and for `Q := N/(X − ½)²` it is `−6` (OA-242-C-02). +- So the implicit function theorem gives analytic `X_e(t)` and `φ_e(t)`. Their series (control S4) are X_e(t) = −3/2 − t/3 − t²/3 + O(t³), φ_e(t) = 1/3 + t/3 + (10/27)t² + O(t³). (3.3) @@ -507,7 +633,10 @@ Hence E[γ⁶I(t, χ₀)] = 120·(20/3) + (52/3)·576k² + O(k³), H(k) = e^{−12k²}(1 + (312/25)k²) + O(k³) = 1 + (12/25)k² + O(k³). -(The true next correction is of order `k^{7/2}`, since `E[γ⁶t⁴] = ∞`.) +The divergence of `E[γ⁶t⁴]` does not by itself show a term of order `k^{7/2}` (OA-242-C-01). +- OpenAI Codex's Slice C review (5390667993, §5) proves the sharper upper remainder `H(k) = 1 + (12/25)k² + O(k^{7/2})`, by + sign averaging and a truncated inverse moment. That support is Codex's own and is not reviewed here. +- No nonzero `k^{7/2}` coefficient is claimed. (3) Lemma 3 gives `γ⁶I ≤ C(γ⁶ + 1728k³|B|³ + 331776k⁴C₃²)`. ∎ @@ -523,9 +652,12 @@ coefficients are `−144/(2σ₃²)` and `(1872/75)σ_B²/σ_γ⁴`, with `σ₃ ## 4. The composite density and the conjectured `ℓ^{2/3}` term -The cusp kernel is the `k → 0` limit, and `(r/k)F(k; b, u)` the `κ → ∞` limit, of the same rejected kernel. Theorem 1 says -that they agree in the overlap: `F(k)/F₀ → 1`. Put `H(k; b, u) := F(k; b, u)/F₀(b, u)`; for the Gaussian kernel this is the `H(k)` -of Proposition 4. The simplest density built from both limits is the *composite* +Theorem 1 describes the cusp kernel's tail as `κ → ∞`: `κ𝒜^{rej} → F₀`. At the fold scale the leading term of the same +rejected kernel is, by Conjecture 6, `(r/k)F(k; b, u)`. Its existence is #170/#175's failure law, and its identification +with (3.1) is discussed after (3.1). That the two limits agree in the overlap, `F(k)/F₀ → 1` as `k → 0`, is a further statement about (3.1). It needs (3.1) +plus domination, and for the Gaussian kernel Proposition 4 proves it (`H → 1`). Theorem 1 alone does not give it +(OA-242-D-01). Put `H(k; b, u) := F(k; b, u)/F₀(b, u)`; for the Gaussian kernel this is the `H(k)` of Proposition 4. The +simplest density built from both limits is the *composite* ν^c(ℓ) := ∫_0^∞∫_R∫_{S^{d−1}} 𝒜^{rej}(b, ℓ/r⁴, u) H(ℓ/r³; b, u) dσ(u) db dr. (4.1) @@ -551,6 +683,13 @@ Integrating over `b` and `u` gives (4.2). ∎ **Conjecture 6 (the fold-scale limit).** In every `d ≥ 2`, with `F` as in (3.1), for every `k > 0`, `b` and `u`, `lim_{r↓0} (k/r)·r^{−2}A_r^{rej}(b, k, u) = F(k; b, u)`, locally uniformly in `k`. +*Status (v1.2).* +- *Existence of the limit, as `kA∗(α₁ + α₂)`.* This is in #170 (`d = 2`) and #175 (`d ≥ 3`): merged author-side candidates, + at their conditional scope. +- *`d = 2`, at fixed `(b, k, u)`.* This follows from #170 Theorem S and #243 Proposition FL.7(iv), a model-level identity. +- *Local uniformity in `k`, and `d ≥ 3`.* #243 (open; author-side, nonauthor review pending) claims a proof (Theorem FL). +- The label is kept here until #243 is reviewed. + **Conjecture 7 (the next term of the rejected density).** In every `d ≥ 2`, ρ_rej(ℓ) + ν_eld^{far,r_0^*}(ℓ) = B_{d,L} + (I^{cand} − c₁) ℓ^{1/4} + R_{2/3} ℓ^{2/3} + O(ℓ^{3/4}), (4.3) @@ -566,7 +705,7 @@ with `R_{2/3}` as in (4.2). errors integrating to `O(ℓ^{3/4})`. In fixed cusp windows this is #240's Theorem N (`O(r²)`). The new part is the window field's elder decision in the soft scaling `λ ≍ r`: Lemma 2 with margins for the field, as #240 Lemma Q′ gives them with `ε̃` for `λ ≍ 1`. In `d ≥ 3` it also needs Proposition 2′ for the field, uniformly in the stiff eigenvalues. -- (ii) *The end `κ → 0`.* On the intermediate separations `ℓ^{1/4} ≪ r ≤ r_0^*`, the rejected kernel must match `B_{2,L}`'s part +- (ii) *The end `κ → 0`.* On the intermediate separations `ℓ^{1/4} ≪ r ≤ r_0^*`, the rejected kernel must match `B_{d,L}`'s part plus the composite's own part, up to errors integrating to `O(ℓ^{3/4})`. (The composite is not negligible there: `𝒜^{rej} ≍ κ³` as `κ → 0`, so `r ≥ Sℓ^{1/4}` contributes about `ℓ^{1/4}S^{−11}`.) The best available bounds on these separations are #229's `O(ℓ^{4/9}log(1/ℓ))` (elder) and #237's `O(ℓ^{3/5})` (candidate), both larger than `ℓ^{2/3}`. @@ -576,7 +715,7 @@ with `R_{2/3}` as in (4.2). Formally, a matched-asymptotics count gives cusp-side powers `ℓ^{(j+1)/4}` and fold-side powers `ℓ^{(j+1)/3}` (`j = 0, 1, 2, …`). In `ρ_rej` the -fold-scale kernel is `O(r³/k)` ([C7-K] (K2)), with leading coefficient `F` (formally, Theorem 1 and Proposition 4). This +fold-scale kernel is `O(r³/k)` ([C7-K] (K2)), with leading coefficient `F` (formally, Conjecture 6; existence by #170/#175). This makes `ℓ^{2/3}` the first fold-side power and `ℓ^{3/4}` the next cusp-side one (#240 Corollary N′). *Remarks.* @@ -591,8 +730,13 @@ makes `ℓ^{2/3}` the first fold-side power and `ℓ^{3/4}` the next cusp-side o ## 5. Numerical evidence (exploration; not part of the proofs) -Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. The scripts are in the project archive -`V2_2/frontiers_soft_rejected_pairs_20261002/exploration/` (numpy, scipy, mpmath, sympy); none is part of the repository. +Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. +- *Status.* The numbers below are author-reported exploration, not independently reproduced (OA-242-E-01). +- *Where the scripts are.* They are in the project archive `V2_2/frontiers_soft_rejected_pairs_20261002/exploration/` + (numpy, scipy, mpmath, sympy); none is part of the repository. +- *Pinned identities.* `SOURCES.json` (`exploration_manifest`) pins every script, table and log there by its SHA-256, + including those of the Monte Carlo comparison. #216's raw records are pinned by name in #216's archive + `SIDE24_MC_raw_2026-10-01.zip`. 1. *Theorem 1.* - *`d = 2`.* Quadrature of (0.2) under the contact law (`fastk.py`, `soft_d2.py`) gives the numbers of Remark 4 of §1. @@ -654,8 +798,15 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. The scripts a density at small `k`, and `C₃` raises `H` at `k ≈ 0.2–0.7`. 6. *#216's Monte Carlo.* The raw records were compared with the composite: `composite.py` and `comp2d.py` in `d = 2`, and `compd.py` in `d = 3`. The records are `batchA2.npz` (`d = 2`: 4,000 samples, 271,272 rejected adjacent pairs with - `ℓ < 0.3`) and `batch3B.npz` (`d = 3`: 996 samples, 45,952 such pairs). #216's row counts *adjacent* rejected pairs, which carry no - `B_{2,L}` (#218 Remark 3). By §4 (inputs (ii)–(iii), and the `ℓ^{3/4}` order), these totals do not test the global + `ℓ < 0.3`) and `batch3B.npz` (`d = 3`: 996 samples, 45,952 such pairs). + - *#216's domains.* The fields are periodized on the torus of side `L = 64` (`d = 2`) and `L = 16` (`d = 3`). The + comparison with the composite of the `R^d` Gaussian kernel is exploratory, not a certified transfer. + - *What the row counts.* #216's row counts *adjacent* rejected pairs, which carry no `B_{d,L}` (#218 Remark 3). + - *Uncertainties (OA-242-E-02).* They are descriptive. + - Each `±` on a ratio is the Poisson relative error `1/√n` of the observed count. + - "`+4.8σ`" is the Pearson residual `(651 − 540)/√540`. + - The `χ²` values are Pearson sums over the bins with at least five counts, a selection that depends on the data. + - Pairs from one field realization can be correlated, so none of these is a calibrated significance. By §4 (inputs (ii)–(iii), and the `ℓ^{3/4}` order), these totals do not test the global `ℓ^{2/3}` coefficient. What they do test, bin by bin in `s = r/ℓ^{1/4}`, is the fold-side suppression described by `H`. - *Small `s` (the fold side).* For `s < 0.8` and `ℓ ∈ [10^{−4}, 0.1)`, 37 pairs are observed, 35.3 predicted by the composite, and 4,322 by the cusp kernel alone. In the bin `ℓ ∈ [3·10^{−3}, 10^{−2})`, `s ∈ [0.6, 0.8)` the three numbers @@ -668,8 +819,8 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. The scripts a and 1,329 by the cusp kernel alone. For `s ∈ [0.8, 1)` the three numbers are 651, 540 and 2,240 (`comp3d_final4.log`). - *The test discriminates `H`.* With the pin density alone in place of `H` (`H = e^{−12k²}`), the predictions would be 1.2 (`s < 0.8`) and 82.7 (`s ∈ [0.8, 1)`) (`comp3d_pin.log`). - - *The transition bin `s ∈ [0.8, 1)`.* The data exceed the composite, by `+4.8σ` in `d = 3` and by 2,248 against 1,904 in - `d = 2`. In both dimensions the excess is concentrated at `ℓ ≥ 3·10^{−2}`, where the data exceed the composite anyway. + - *The transition bin `s ∈ [0.8, 1)`.* The data exceed the composite. In `d = 3` the Pearson residual is `+4.8` + (descriptive; see above), and in `d = 2` the counts are 2,248 against 1,904. In both dimensions the excess is concentrated at `ℓ ≥ 3·10^{−2}`, where the data exceed the composite anyway. - *`d = 3` totals.* The script keeps bins with at least five counts, so the effective ranges are `[3.1·10^{−4}, 9.7·10^{−3}]` (8 bins) and `[3.1·10^{−4}, 2.3·10^{−2}]` (10 bins). In `d = 2` the first range is `[1.3·10^{−4}, 9.7·10^{−3}]` (10 bins). @@ -694,14 +845,22 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. The scripts a fits could not attribute (`ℓ^{1/2}` against `ℓ^{2/3} + ℓ^{3/4}`). Conjecture 7 predicts a contribution `−R_{2/3}ℓ^{2/3}` to `ν_eld`, in addition to any `ℓ^{2/3}` term of `ν_cand`, with a coefficient computed independently of the Monte Carlo. - **[C7-K] (K2)** bounds the fold-scale rejected weight by `O(r³/k)`. Formally (Conjecture 6) the bound is attained: the - leading term of `r^{−2}A_r^{rej}` is `(r/k)F(k; b, u)`, and `F(k) → F₀ ≠ 0` as `k → 0` (Theorem 1). + leading term of `r^{−2}A_r^{rej}` is `(r/k)F(k; b, u)`. That leading term exists by #170/#175. For the Gaussian kernel, + `F(k) → F₀ ≠ 0` as `k → 0` (Proposition 4). +- **#170 and #175 (merged author-side candidates, at their stated conditional scope).** They prove the fold-scale failure + law `r^{−3}(1 − p_r) → α₁ + α₂` and the compact-window `ℓ^{2/3}` coefficient (AUTH-242-01). #170 Theorem E decides the + window cubic. That is the same decision as Lemma 2, computed by another method: critical chords and [CUB]'s classifier. +- **#243 (open).** Theorem FL and Proposition FL.7 claim a proof of Conjecture 6. - **Not claimed.** - - Conjectures 6 and 7 are not proved. The values of `H` and `R_{2/3}` are numerical and not certified. - - In `d ≥ 3` the fold-scale statements rest on Proposition 2′. It is exact for the model, and its transfer to the field - is part of Conjecture 6. + - Conjectures 6 and 7 are not proved here. For Conjecture 6, see its status in §4. The values of `H` and `R_{2/3}` are + numerical and not certified. + - In `d ≥ 3` the fold-scale statements rest on Proposition 2′. It is exact for the model. Its transfer to the field is + #175 Theorem H for #170's cubic, and #243 Proposition FL.4 here. - Nothing about the candidate density's own `ℓ^{2/3}` term. - Lemma 2, Lemma 3 and Proposition 4 are statements about the limit models (2.1) and (2.6). Their transfer to the field is Conjecture 6. + - No priority is claimed for the existence of the fold-scale limit, or for the compact-window `ℓ^{2/3}` coefficient + (#170, #175). - The Gaussian-kernel statements are about the model case on `R^d`. ## 7. Controls @@ -715,19 +874,20 @@ of the failing controls to stderr. | S1 | Theorem 1: `69984/72⁴ = 1/384`; `∫_{1/3}^1(φ^{−4} − φ^{−2})dφ = 20/3`; `12·(20/3)/384 = 5/24`; `F₀ = 25π₀p_A(0 \| b)`; the soft window at `f₄ = 0` | | S2 | the algebra of `G_k` at 60 random rational points: its derivatives (exact four-point stencils), pins, `det H_M = 6λ̃ + Y`, `det H_S = −6λ̃ + Y`, the normalization (both signs of `γ`), weight, measure and typed window | | S3 | the Gaussian kernel's jet covariances from Hermite numbers: (0.3), and `a′ = 144/(2·6) = 12` | -| S4 | (3.3) as exact truncated series; the double zero at `t = 0`; the Jacobian `−96` | +| S4 | (3.3) as exact truncated series; the double zero at `t = 0`; the Jacobian `−96` (computed for `24φ(R − L_S)`, which equals `N` at `t = 0`) | | S5 | (3.4), and the antiderivative of the weight | | S6 | `E[γ⁶] = 120`, `E[γ⁶t²] = 576k²`, `312/25`, `12/25`, and the moment identities of Proposition 4's proof | | S7 | every numerical inequality of Lemma 3's proof, including the critical-value formulas as identities | -| S8 | Lemma 2 on 16 fixed parameter points with known answers, including case (D′), in floating point | +| S8 | Lemma 2 on 16 fixed parameter points with known answers, including case (D′), in floating point; the scan is not reliable within about `4·10^{−3}` of `β = 2` (none of the 16 points is there) | | S9 | the error exponents of Lemma 5 | | S10 | the limit (2.1), including `G₁`, on three exactly pinned degree-6 fields with `A = −λ̃r/k`, at `r = 10^{−3}` and `10^{−4}` | | S11 | Theorem 1 for `m = 2, 3` in rational eigenframes (Cayley rotations): `z = f₄ − 3γᵀA^{−1}γ = 3γ₁²/λ₁ + f̃₄`, `Y = (Δ/12)z`, `w_κ = 36κ²Δ²(1 − φ²)`, the substitution (1.1) with the factor `Π′`, `69984/72⁴ = 1/384`, and the double-soft implication of Step 1 | | S12 | the Gaussian kernel in `d = 3` from Hermite numbers: given the even pins, `A = −bI + G` with `G` in the Gaussian orthogonal ensemble and `f₄ ~ N(−3b, 24)` independent; the `b`-integrated law (`t ~ N(0, 5/3)`, `f₄ \| t ~ N(6t/5, 138/5)`; `d = 2`: `[[8/3, 2], [2, 30]]`); the odd jets along `e = (3/5, 4/5)` (variances `(6, 2, 2, 6)`); Corollary 1′'s constants `25√3/(48π²)` and `125√30/(192π³)` | | S13 | Proposition 2′: (2.6), including `G_{1/2}`, on two exactly pinned degree-6 fields in `d = 3` at `r = 10^{−6}` and `10^{−8}` | +| S14 | Lemma 2's exact (D′) witness: `G_{(3/2, 8/3, 20/3)}(X, z) = (1/9)G_{(1/6, 0, 0)}(X + 2z, 3z)` on a unisolvent `4 × 4` rational grid (hence identically); typing; the three critical points `(±½, 0)`, `(−3, 35/8)` of `G_{(1/6,0,0)}` (gradient zero, and the factorization `∂_XG(X, p/2) = (X² − ¼)(3/2 + X/2)`), with witness values `0`, `−1/36 = L_S`, `−125/384`; the (D) certificate at `(1/6, 0, 0)`: the two factorizations `R = (X + ½)²(X² + 3X − 15/4)/8` and `R − L_S = (X − ½)²(X² + 5X + 17/4)/8`, with the signs of their quadratic factors; and, for `β > 2`, `χ < 0`, the open slice `D(1/β) = −χp(1/β) < 0` on rational points | Mutants `M1`–`M13` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, `M4` S5, `M5` S7, `M6` S8, `M7` S9, `M8` S2, -`M9` S6, `M10` S10, `M11` S11, `M12` S12, `M13` S13); an unknown label exits 2. +`M9` S6, `M10` S10, `M11` S11, `M12` S12, `M13` S13, `M14` S14); an unknown label exits 2. ## 8. Sources (exact identities in `SOURCES.json`) @@ -742,7 +902,10 @@ Mutants `M1`–`M13` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, | #229 | `frontiers/third_order_rate_20261001/PROOF.md` (blob `110ed33a`) | (R⁺.1); §0 (`Y_r`); §5 — consumed | | [C7-K] | `frontiers/c7_total_bounded_20260929/PROOF.md` (blob `28748b08`) | (K2) — cited | | #187 | `frontiers/far_elder_rate_20260930/PROOF.md` (blob `07260114`) | (0.2): `ν_eld^{far} ≤ Cℓ^{2/3}` — cited | -| #240 | `frontiers/elder_cusp_parity_20261002/PROOF.md` (open PR; blob `16a1db06`) | Lemma Q′, Step N6, Corollary N′, Remarks 1 and 4 — cited | +| #240 | `frontiers/elder_cusp_parity_20261002/PROOF.md` (merged at `7636cae`; blob `16a1db06`, unchanged) | Lemma Q′, Step N6, Corollary N′, Remarks 1 and 4 — cited | +| #170 | `frontiers/local_elder_geometry_20260930/PROOF.md` (merged; blob `ef2aa579`) | §3 (the critical chord), Theorem E, Theorem S, §9 — cited (prior work, AUTH-242-01) | +| #175 | `frontiers/concave_fibre_elder_20260930/PROOF.md` (merged; blob `923d3236`) | Theorems H and F, §7 — cited (prior work, AUTH-242-01) | +| #243 | `frontiers/soft_fold_limit_20261002/PROOF.md` (open PR, head `0740b2c`; blob `caedaef6`) | Theorem FL, Proposition FL.7, Lemma FL.2(c) — cited (Conjecture 6) | | #237 | `frontiers/candidate_parity_rate_20261001/PROOF.md` (open PR) | Theorem P's remainder, Remark 4 — cited | | #216 | `frontiers/third_order_coefficient_20261001/NOTE.md` (open PR) | the full-field Monte Carlo in `d = 2, 3` (§5) — cited | | #188 | far elder density `O(ℓ^N)` (open PR) | §4 (iii) — cited | @@ -759,3 +922,26 @@ Mutants `M1`–`M13` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, #187, #229 and #240. - **E** (§§5–7): the evidence is described accurately and kept separate from the proofs, and the controls match their claims. + +**Changed bytes in v1.2** (for a delta check against the v1.1 reviews): +- *Header:* the version history, the Disposition, "Why" (#240 merged), "What is new" (the window, A-01; the soft model and + Lemma 2 notes), and the Dependencies (#240, #170, #175, #243). +- *§1:* Remark 1 (A-01). +- *§2:* + - the opening (B-02); + - the remainder estimate after (2.1); + - the absolute Jacobian and `t = 1` (B-03); + - Lemma 2's statement (`χ > 0` in (D′), `χ ≤ 0` rejected, `e` defined) and its proof (B-01); + - the exact witness and the remark after the proof; + - the Schur complement in Proposition 2′'s proof. +- *§3:* the sentence on `k → 0` after (3.1), the prior-work paragraph, the Jacobian normalization (C-02), and the `k^{7/2}` + sentence (C-01). +- *§4:* the opening (D-01), Conjecture 6's status, input (ii) (D-02), and the matched-asymptotics sentence. +- *§5:* the status, custody and uncertainty labels (E-01, E-02), and #216's domains. +- *§§6–8:* + - the [C7-K] item, the #170/#175/#243 items and "Not claimed"; + - the S4, S8 and S14 rows; + - the sources. + +§§0–1's proofs, Lemma 3, Proposition 2′, Proposition 4's statement and proof (apart from C-01 and C-02), and Lemma 5 are +unchanged. diff --git a/frontiers/soft_rejected_pairs_20261002/README.md b/frontiers/soft_rejected_pairs_20261002/README.md index be2017358..90c5576ad 100644 --- a/frontiers/soft_rejected_pairs_20261002/README.md +++ b/frontiers/soft_rejected_pairs_20261002/README.md @@ -2,7 +2,16 @@ **Author-side proof candidate and conjectures (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** -Object `CL-SOFT-REJECTED-20261002-v1.1`. Full text: [`PROOF.md`](PROOF.md). Version history: +Object `CL-SOFT-REJECTED-20261002-v1.2`. Full text: [`PROOF.md`](PROOF.md). Version history: +- v1.2 applies the findings of the nonauthor reviews of v1.1 (OpenAI Codex, Slices A, B (model), C, D, and B (formulas) with + E): OA-242-A-01, B-01, B-02, B-03, C-01, C-02, D-01, D-02, E-01 and E-02. + - It adds Codex's exact (D′) witness with control S14, and states that for `β > 2`, `χ < 0` the pair is always rejected. + - It records prior work it had not cited (AUTH-242-01): the merged #170/#175 already prove that the fold-scale limit + exists, as `kA∗(α₁ + α₂)`. Conjecture 6's identification with `F` is #243 (open). + - It labels §5's numbers as author-reported exploration, pins their files in `SOURCES.json` (`exploration_manifest`), and + labels their uncertainties as descriptive. + - A same-family referee checked the delta: ACCEPT WITH FIXES (9 MINOR, 8 NIT), all applied. + - PROOF §9 lists the changed bytes. - v1.1 was made before any nonauthor review. - It extends Theorem 1 to every `d ≥ 2` and adds the `d = 3` constants (Corollary 1′) and the stiff-direction reduction (Proposition 2′). It also adds the `d = 3` Monte Carlo test and fixes finding A-1. @@ -14,14 +23,14 @@ the cusp end `κ = ℓ/r⁴ → ∞` open. This note describes that end for the | | Statement | Status | |---|---|---| -| **Theorem 1** | In every `d ≥ 2`, `κ𝒜^{rej}(b, κ, u) = F₀(b, u) + O(κ^{−1})`, where `F₀ = (5/24)π₀𝔇` and `𝔇` is the density at `0` of the smallest eigenvalue `λ₁` of `−A`, weighted by `γ₁⁶(λ₂⋯λ_m)²`. In `d = 2`, `𝔇 = p_A(0 \| b)E[γ⁶]`. The rejected weight sits on `λ₁ ∈ [γ₁²/(24κ), γ₁²/(8κ)]`. | proof | +| **Theorem 1** | In every `d ≥ 2`, `κ𝒜^{rej}(b, κ, u) = F₀(b, u) + O(κ^{−1})`, where `F₀ = (5/24)π₀𝔇` and `𝔇` is the density at `0` of the smallest eigenvalue `λ₁` of `−A`, weighted by `γ₁⁶(λ₂⋯λ_m)²`. In `d = 2`, `𝔇 = p_A(0 \| b)E[γ⁶]`. The rejected weight sits exactly on `λ₁ ∈ (3γ₁²/(72κ − f̃₄), 3γ₁²/(24κ − f̃₄)]`; for fixed jets this is `[γ₁²/(24κ), γ₁²/(8κ)]` up to `O(1/κ)`. | proof | | **Corollary 1′** | Gaussian kernel: `∫∫F₀ db dσ = 25√3/(48π²) ≈ 0.0914` (`d = 2`), `125√30/(192π³) ≈ 0.1150` (`d = 3`) | proof | -| **Lemma 2** | The elder decision in the fold-scale soft model `G_k` (new jets `B = ∂_uA`, `C₃ = ∂_Θ³f` along the soft direction), by a one-dimensional scan, including a case where the saddle is a slice minimum | proof (model) | +| **Lemma 2** | The elder decision in the fold-scale soft model `G_k` (new jets `B = ∂_uA`, `C₃ = ∂_Θ³f` along the soft direction), by a one-dimensional scan, proved at the exact level. It includes a case (D′, `β > 2`, `χ > 0`) where the saddle is a slice minimum, with an exact witness. For `β > 2`, `χ ≤ 0` the pair is always rejected. `G_k` is #170's typed cubic in other coordinates. | proof (model) | | **Proposition 2′** | `d ≥ 3`: the stiff directions live at scale `r^{3/2}` and decouple; the limit `G_k − (1/2k)Σλ_iη_i²` has the elder decision of `G_k` | proof (model) | | **Lemma 3** | Elder whenever `φ ≤ (1/20)min(1, \|t\|^{−1}, \|χ₀\|^{−2/3})`; so `I(t, χ₀) ≤ (8000/3)max(1, \|t\|³, χ₀²)` | proof (model) | -| **Proposition 4** | Gaussian kernel, every `d`: the fold-scale rejection rate is `F₀(b)H(k)`, with `H` independent of `b` and `d`, and `H(k) = 1 + (12/25)k² + O(k³)`. The elder edge `φ_e = 1/3 + t/3 + 10t²/27` contributes `+312/25`, and the pin density `e^{−12k²}` contributes `−12`. | proof (given the model) | +| **Proposition 4** | Gaussian kernel, every `d`: the fold-scale rejection rate is `F₀(b)H(k)`, with `H` independent of `b` and `d`, and `H(k) = 1 + (12/25)k² + O(k³)`. Codex's Slice C review proves the upper remainder `O(k^{7/2})`. The elder edge `φ_e = 1/3 + t/3 + 10t²/27` contributes `+312/25`, and the pin density `e^{−12k²}` contributes `−12`. | proof (given the model) | | **Lemma 5** | The composite `∫∫∫𝒜^{rej}(b, ℓ/r⁴, u)H(ℓ/r³)` equals `(I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`, where `R_{2/3} = ∫∫∫v⁴[F(v^{−3}) − F₀]` | proof | -| **Conjecture 6** | `(k/r)r^{−2}A_r^{rej}(b, k, u) → F(k; b, u)` as `r → 0` at fixed `k` | conjecture | +| **Conjecture 6** | `(k/r)r^{−2}A_r^{rej}(b, k, u) → F(k; b, u)` as `r → 0` at fixed `k`. The limit exists by #170/#175 (merged); its identification with `F` is #243 (open). | proved author-side in #243 | | **Conjecture 7** | `ρ_rej + ν_eld^{far,r_0^*} = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`; Gaussian kernel: `R_{2/3} ≈ −0.049 ± 0.003` (`d = 2`), `−0.062 ± 0.004` (`d = 3`) | conjecture, with evidence | Conjecture 7 needs two inputs that are not yet available (PROOF §4): @@ -31,7 +40,7 @@ Conjecture 7 needs two inputs that are not yet available (PROOF §4): A third input, (iii) the far elder density, is needed only to remove `ν_eld^{far,r_0^*}` from the left side. #187 bounds it by `Cℓ^{2/3}`, and #188 (open) claims `O(ℓ^N)`. -## Evidence (exploration; outside the repository) +## Evidence (exploration; outside the repository; author-reported, not independently reproduced) | | Result | |---|---| @@ -40,7 +49,9 @@ A third input, (iii) the far elder density, is needed only to remove `ν_eld^{fa | Lemma 2 | agrees with a two-dimensional flood fill on 900 random typed parameter points (one near-degenerate point needs a smaller level offset) | | `H` | `H(0.2) = 1.004`, `H(0.4) = 0.727`, `H(0.6) = 0.201`, `H(0.8) = 0.018`. With `C₃` ignored, `R_{2/3}` (`d = 2`) would be `−0.104`; with the pin density alone, `−0.807`. | | #216's Monte Carlo, `d = 2` | rejected adjacent pairs with `s = r/ℓ^{1/4} < 0.8`: 37 observed, 35.3 from the composite, 4,322 from the cusp kernel alone | -| #216's Monte Carlo, `d = 3` | `s < 0.8`: 11 observed, 10.7 composite (1.2 with the pin density alone), 1,329 cusp kernel alone. On `[3·10^{−4}, 10^{−2}]` the counts are `0.884 ± 0.040` of the `ℓ^{1/4}` law and `1.008` of the composite (χ² 11.8 against 2.1 on 8 bins). Over #216's whole range `[10^{−4}, 0.3]` the composite does not describe the counts (finite-`r` corrections). | +| #216's Monte Carlo, `d = 3` | `s < 0.8`: 11 observed, 10.7 composite (1.2 with the pin density alone), 1,329 cusp kernel alone. On `[3·10^{−4}, 10^{−2}]` the counts are `0.884 ± 0.040` of the `ℓ^{1/4}` law and `1.008` of the composite (Pearson χ² 11.8 against 2.1 on 8 bins). Over #216's whole range `[10^{−4}, 0.3]` the composite does not describe the counts (finite-`r` corrections). | + +#216's fields are on the torus with `L = 64` (`d = 2`) and `L = 16` (`d = 3`). The `±` values are Poisson relative errors `1/√n`, and the `χ²` values are Pearson sums over bins with at least five counts. All of these are descriptive, not calibrated significances (PROOF §5). ## Controls @@ -62,8 +73,9 @@ Lemma 2. Its output is `RESULTS.json`, byte-identical under `-O`. | S11 | Theorem 1 in `m = 2, 3` rational eigenframes: the soft-window algebra with the factor `(λ₂⋯λ_m)²` and the double-soft implication | | S12 | the Gaussian kernel in `d = 3`: the GOE law of `A`, the laws used in Corollary 1′, the odd jets along a rotated direction, the prefactor `(2π)^{−d}/(6√π)` from exact covariances, and the two constants | | S13 | the expansion (2.6) of Proposition 2′ on exactly pinned degree-6 fields in `d = 3`, including the `r^{1/2}`-correction | +| S14 | Lemma 2's exact (D′) witness `G_{(3/2, 8/3, 20/3)}(X, z) = (1/9)G_{(1/6,0,0)}(X + 2z, 3z)`: the identity, its three critical points and values, and the (D) certificate at `(1/6, 0, 0)`; and the open slice at `1/β` for `β > 2`, `χ < 0` | -Mutants `M1`–`M13` each fail only their own control (the checker names it on stderr), and an unknown label exits 2: +Mutants `M1`–`M14` each fail only their own control (the checker names it on stderr), and an unknown label exits 2: python3 -B -S soft_check.py # exit 0, output = RESULTS.json python3 -B -S soft_check.py --mutant M1 # exit 1 @@ -85,12 +97,29 @@ Mutants `M1`–`M13` each fail only their own control (the checker names it on s - Its independent checks: the Gaussian-kernel laws by symbolic differentiation; both constants of Corollary 1′ by the per-`b` route; a Monte Carlo of `𝔡(b)`; an independent quadrature of `G₃` (agreeing to `3·10^{−7}`); (2.6) symbolically in `d = 3, 4`; the Hessian factor `(λ₂⋯λ_m)²`; and the mutant isolation matrix. -- **Independence.** The referees are the same provider and the same GitHub account as the author, so they carry zero - organizational independence. Nonauthor review is required; the review slices are in PROOF §9. +- **Nonauthor reviews of v1.1 (OpenAI Codex, same GitHub account, organizational independence 0).** All are at + `87912bd`, PROOF blob `8b2f5fae`: + - Slice A (5390308599): PASS_TECHNICAL_SCOPED. One nonblocking finding, A-01. + - Slice B, model geometry and Proposition 2′ (5948352439): PASS_TECHNICAL. One finding, B-01. + - Slice C (5390667993): PASS_TECHNICAL_SCOPED. Findings C-01 and C-02. + - Slice D (5390555860): PASS_TECHNICAL_SCOPED for Lemma 5. Findings D-01 and D-02. + - Slices B (the remaining formulas) and E (5391485205): PASS_TECHNICAL_SCOPED for the limit-field algebra, normalization, + typing and weight; PASS for the checker and mutant correspondence. Findings B-02, B-03, E-01 and E-02. It also gives + the root-authored remainder estimate (B.5), now cited after (2.1). + - Every finding is applied in v1.2. A delta check of the changed bytes is requested. +- **v1.2 delta (same family).** A clean-context referee checked the changed bytes. + - Verdict: **ACCEPT WITH FIXES**, with no BLOCKING or MAJOR finding: 9 MINOR findings and 8 NITs, all applied. + - Its own checks: the witness by sympy, the Jacobians `−96`, `−3/2` and `−6`, and the `48/47` ratio. + - It also ran 20,000 typed samples with `β > 2`, `χ < 0`, and found no elder case. +- **Independence.** The same-family referees are the same provider and the same GitHub account as the author, so they + carry zero organizational independence. The review slices are in PROOF §9. ## Not claimed -- Conjectures 6 and 7 are not proved, and `R_{2/3}` is not certified. -- Proposition 2′ is a statement about the limit model; its transfer to the field is part of Conjecture 6. +- Conjectures 6 and 7 are not proved here, and `R_{2/3}` is not certified. Conjecture 6 is proved author-side in #243, and + its existence part is merged (#170/#175). +- Proposition 2′ is a statement about the limit model. Its transfer to the field is #175 Theorem H (for #170's cubic), and + #243 Proposition FL.4. +- No priority for the existence of the fold-scale limit or for the compact-window `ℓ^{2/3}` coefficient (#170, #175). - Nothing about the candidate density's own `ℓ^{2/3}` term. - No change to #229, #240 or any other packet. diff --git a/frontiers/soft_rejected_pairs_20261002/RESULTS.json b/frontiers/soft_rejected_pairs_20261002/RESULTS.json index c7adbd16c..140d622b2 100644 --- a/frontiers/soft_rejected_pairs_20261002/RESULTS.json +++ b/frontiers/soft_rejected_pairs_20261002/RESULTS.json @@ -99,6 +99,19 @@ "fields": 2, "max_scaled_remainder_at_r_1e-8": "7.123741313250203" }, + "S14": { + "critical_values": [ + "-125/384", + "-1/36", + "0" + ], + "open_slice_cases_beta_gt_2_chi_lt_0": 3, + "witness": [ + "3/2", + "8/3", + "20/3" + ] + }, "S2_instances": 60, "S3": { "conditional_covariance_third_jets": [ @@ -318,6 +331,7 @@ "S11_theorem_1_every_d": true, "S12_gaussian_kernel_d3": true, "S13_stiff_directions": true, + "S14_exact_dprime_witness": true, "S1_theorem_1": true, "S2_model_algebra": true, "S3_gaussian_kernel_jets": true, @@ -328,6 +342,6 @@ "S8_lemma_2_cases": true, "S9_lemma_5": true }, - "object": "CL-SOFT-REJECTED-20261002-v1.1", + "object": "CL-SOFT-REJECTED-20261002-v1.2", "scientific_effect": "NONE" } diff --git a/frontiers/soft_rejected_pairs_20261002/SOURCES.json b/frontiers/soft_rejected_pairs_20261002/SOURCES.json index d119ac0d7..3aa24e5b9 100644 --- a/frontiers/soft_rejected_pairs_20261002/SOURCES.json +++ b/frontiers/soft_rejected_pairs_20261002/SOURCES.json @@ -1,12 +1,12 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-SOFT-REJECTED-20261002-v1.1", - "kind": "author-side proof candidate and conjectures, every d >= 2: Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 D with D the density at 0 of the smallest eigenvalue lambda_1 of -A weighted by gamma_1^6 (lambda_2...lambda_m)^2; d = 2: D = pA(0|b) E gamma^6); Corollary 1' (Gaussian kernel: int int F0 = 25 sqrt3/(48 pi^2) in d = 2, 125 sqrt30/(192 pi^3) in d = 3); the fold-scale soft model G_k with the jets B = d_u A and C3 = d_Theta^3 f along the soft direction; Lemma 2 (the elder decision in the model by a one-dimensional scan, including a case where the saddle is a slice minimum); Proposition 2' (d >= 3: the stiff directions at scale r^(3/2) decouple; same elder decision); Lemma 3 (an a priori elder region); Proposition 4 (Gaussian kernel, every d: the fold-scale rejection rate is F0(b) H(k) with H independent of b and d, H = 1 + (12/25) k^2 + O(k^3)); Lemma 5 (the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4))); Conjecture 6 (the fold-scale limit) and Conjecture 7 (rho_rej + nu_eld^far = B_{d,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 (d = 2) and -0.062 (d = 3) for the Gaussian kernel, not certified)", + "object": "CL-SOFT-REJECTED-20261002-v1.2", + "kind": "author-side proof candidate and conjectures, every d >= 2 (v1.2: applies OA-242-A-01, B-01, B-02, B-03, C-01, C-02, D-01, D-02, E-01, E-02; exact (D-prime) witness; Lemma 2 now states that beta > 2, chi <= 0 is always rejected; prior work #170/#175 recorded, AUTH-242-01; Conjecture 6 status: existence merged (#170/#175), identification #243): Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 D with D the density at 0 of the smallest eigenvalue lambda_1 of -A weighted by gamma_1^6 (lambda_2...lambda_m)^2; d = 2: D = pA(0|b) E gamma^6); Corollary 1' (Gaussian kernel: int int F0 = 25 sqrt3/(48 pi^2) in d = 2, 125 sqrt30/(192 pi^3) in d = 3); the fold-scale soft model G_k with the jets B = d_u A and C3 = d_Theta^3 f along the soft direction; Lemma 2 (the elder decision in the model by a one-dimensional scan, including a case where the saddle is a slice minimum); Proposition 2' (d >= 3: the stiff directions at scale r^(3/2) decouple; same elder decision); Lemma 3 (an a priori elder region); Proposition 4 (Gaussian kernel, every d: the fold-scale rejection rate is F0(b) H(k) with H independent of b and d, H = 1 + (12/25) k^2 + O(k^3)); Lemma 5 (the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4))); Conjecture 6 (the fold-scale limit) and Conjecture 7 (rho_rej + nu_eld^far = B_{d,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 (d = 2) and -0.062 (d = 3) for the Gaussian kernel, not certified)", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { - "branch_base": "6f74f7a77bae15d6aa07506bab38977e34dcc449" + "branch_base": "f5347541a6ef7b84de96120dcce6137b816b8033" }, "consumed": [ { @@ -99,19 +99,33 @@ "git_blob": "072601145601a1021037a2e91bf8826c1c971ecc", "tag": "187", "use": "Math- #187 (merged): (0.2), the far elder density is O(l^(2/3)) (why Conjecture 7 keeps the far term)" - } - ], - "cited_unmerged": [ + }, { "path": 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section 3 (the critical chord, used in Lemma 2 Step 1), Theorem E (the same window decision as Lemma 2), Theorem S (r^-3 (1 - p_r) -> alpha_1 + alpha_2), section 9 (C_fail)" + }, + { + "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", + "bytes": 23416, + "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e", + "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", + "tag": "175", + "use": "Math- #175 (merged author-side candidate, OpenAI, at its stated conditional scope; prior work, AUTH-242-01): Theorem H (the hard-fibre transfer, the field version of Proposition 2-prime for the cubic), Theorem F, section 7" + } + ], + "cited_unmerged": [ { "path": "frontiers/candidate_parity_rate_20261001/PROOF.md", "bytes": 53499, @@ -141,6 +155,16 @@ "use": "Math- #188 (open): the far elder density O(l^N) (Conjecture 7: only to remove the far term)", "head": "85f0586a468c2b2f73af0e340d251e33a52caade", "pr": "Math- #188" + }, + { + "path": 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AI review; same GitHub account; organizational independence 0)", + "review": "pullrequestreview-5390308599", + "at": "87912bd", + "verdict": "PASS_TECHNICAL_SCOPED", + "findings": [ + "OA-242-A-01" + ], + "applied_in": "v1.2" + }, + { + "slice": "B (model geometry, Proposition 2-prime)", + "reviewer": "OpenAI Codex", + "comment": "5948352439", + "at": "87912bd", + "verdict": "PASS_TECHNICAL", + "findings": [ + "OA-242-B-01" + ], + "applied_in": "v1.2" + }, + { + "slice": "C", + "reviewer": "OpenAI Codex", + "review": "pullrequestreview-5390667993", + "at": "87912bd", + "verdict": "PASS_TECHNICAL_SCOPED", + "findings": [ + "OA-242-C-01", + "OA-242-C-02" + ], + "applied_in": "v1.2" + }, + { + "slice": "D", + "reviewer": "OpenAI Codex", + "review": "pullrequestreview-5390555860", + "at": "87912bd", + "verdict": "PASS_TECHNICAL_SCOPED (Lemma 5)", + "findings": [ + "OA-242-D-01", + "OA-242-D-02" + ], + "applied_in": "v1.2" + }, + { + "slice": "B (formulas) and E", + "reviewer": "OpenAI Codex", + "review": "pullrequestreview-5391485205", + "at": "87912bd", + "verdict": "PASS_TECHNICAL_SCOPED; PASS for the checker correspondence", + "findings": [ + "OA-242-B-02", + "OA-242-B-03", + "OA-242-E-01", + "OA-242-E-02" + ], + "applied_in": "v1.2" } ] } diff --git a/frontiers/soft_rejected_pairs_20261002/soft_check.py b/frontiers/soft_rejected_pairs_20261002/soft_check.py index d712228bb..5aec5c0c9 100644 --- a/frontiers/soft_rejected_pairs_20261002/soft_check.py +++ b/frontiers/soft_rejected_pairs_20261002/soft_check.py @@ -1,11 +1,11 @@ #!/usr/bin/env python3 -"""Exact controls for CL-SOFT-REJECTED-20261002-v1.1 (Theorem 1, Corollary 1', Lemmas 2, 3 and 5, Propositions 2' and 4; +"""Exact controls for CL-SOFT-REJECTED-20261002-v1.2 (Theorem 1, Corollary 1', Lemmas 2, 3 and 5, Propositions 2' and 4; PROOF.md). Standard library only; exact rationals except control S8 (a deterministic floating-point implementation of Lemma 2 on fixed cases). Output: RESULTS.json (sorted keys), byte-identical under -O. Usage: python3 -B -S soft_check.py # exit 0, prints RESULTS.json - python3 -B -S soft_check.py --mutant M3 # exit 1 (only its control fails; M1..M13) + python3 -B -S soft_check.py --mutant M3 # exit 1 (only its control fails; M1..M14) unknown mutant label -> exit 2 Controls @@ -44,13 +44,21 @@ 125 sqrt30/(192 pi^3), in exact arithmetic of the form q pi^(e/2) sqrt(r) (floating-point values printed only) S13 Proposition 2': on two exactly pinned degree-6 fields in d = 3 with A = diag(-lt r/k, mu) at 0, (f(rX, rk zeta, r^(3/2) eta) - b)/(k r^3) = G_k + (mu/2k) eta^2 + r^(1/2) G_half + O(r), at r = 1e-6 and 1e-8 + S14 Lemma 2's exact (D') witness (OpenAI Codex, #242 comment 5946792240): G_(3/2,8/3,20/3)(X, z) = + (1/9) G_(1/6,0,0)(X + 2z, 3z) on a 4 x 4 rational grid (unisolvent for bidegree <= (3, 3), hence an identity); the + witness is typed with beta > 2, chi > 0; G_(1/6,0,0) has gradient zero at (+-1/2, 0) and (-3, 35/8), and + d_X G(X, p/2) = (X^2 - 1/4)(3/2 + X/2) (so these are all its critical points); the witness's critical values + 0, -1/36 = L_S, -125/384; and the (D) certificate at (1/6, 0, 0): R = (X + 1/2)^2 (X^2 + 3X - 15/4)/8 < 0 off -1/2 on + [-2, 1/2] (convexity, endpoint signs), R - L_S = (X - 1/2)^2 (X^2 + 5X + 17/4)/8 at six points (quartic identity), + and the edge factor changes sign once on (-2, -1); and, for beta > 2, chi < 0 (rejected by Lemma 2), the slice at + X = 1/beta is open: a = 0 and D = -chi p < 0, on three typed rational points """ import json import math import sys from fractions import Fraction as Fr -MUTANTS = ('M1', 'M2', 'M3', 'M4', 'M5', 'M6', 'M7', 'M8', 'M9', 'M10', 'M11', 'M12', 'M13') +MUTANTS = ('M1', 'M2', 'M3', 'M4', 'M5', 'M6', 'M7', 'M8', 'M9', 'M10', 'M11', 'M12', 'M13', 'M14') MUT = None @@ -768,6 +776,62 @@ def val(cc, x, y, w, dx=0, dy=0, dw=0): return ok, {'fields': 2, 'max_scaled_remainder_at_r_1e-8': str(float(worst))} +# ------------------------------------------------------------------------------------- S14: the exact (D') witness +def control_S14(): + ok = True + H = Fr(1, 2) + wit = (Fr(3, 2), Fr(8, 3), Fr(20, 3)) + base = (Fr(1, 6), Fr(0), Fr(0)) + scale = Fr(1, 9) if MUT != 'M14' else Fr(1, 8) + grid = [Fr(j, 3) - 1 for j in range(4)] + for X in grid: + for z in grid: + ok &= Gn(X, z, *wit) == scale * Gn(X + 2 * z, 3 * z, *base) + phi, beta, chi = wit + ok &= phi > 0 and abs(phi - beta / 2) < 1 and beta > 2 and chi > 0 + # critical points of G_(1/6,0,0): grad zero, and d_X G(X, p/2) = (X^2 - 1/4)(1/(4 phi) + X/2) with 1/(4 phi) = 3/2 + g = lambda X, z: Gn(X, z, *base) + def grad(X, z): + e = Fr(1, 10 ** 6) + # exact derivatives of a cubic by the 4-point stencil (exact for degree <= 4) + gx = (-g(X + 2 * e, z) + 8 * g(X + e, z) - 8 * g(X - e, z) + g(X - 2 * e, z)) / (12 * e) + gz = (-g(X, z + 2 * e) + 8 * g(X, z + e) - 8 * g(X, z - e) + g(X, z - 2 * e)) / (12 * e) + return gx, gz + pts = [(-H, Fr(0)), (H, Fr(0)), (Fr(-3), Fr(35, 8))] + for X, z in pts: + ok &= grad(X, z) == (0, 0) + for X in (Fr(-7, 2), Fr(-1), Fr(0), Fr(2, 3), Fr(5, 4)): + p = X * X - Fr(1, 4) + ok &= grad(X, p / 2)[0] == p * (Fr(3, 2) + X / 2) + ok &= grad(X, p / 2)[1] == 0 + # the witness's critical points are the images under (X, z) -> (X - 2 z/3, z/3) of these, with values / 9 + vals = sorted(scale * g(X, z) for X, z in pts) + ok &= vals == sorted([Fr(0), Fr(-1, 36), Fr(-125, 384)]) and Fr(-1, 24) / phi == Fr(-1, 36) + for X, z in pts: + Xw, zw = X - 2 * z / 3, z / 3 + ok &= Gn(Xw, zw, *wit) == scale * g(X, z) + # the (D) certificate at (1/6, 0, 0): concave slices, R = (X + 1/2)^2 (X^2 + 3X - 15/4)/8 + for X in (Fr(-2), Fr(-3, 2), Fr(-1), Fr(0), H): + R = g(X, (X * X - Fr(1, 4)) / 2) + ok &= R == (X + H) ** 2 * (X * X + 3 * X - Fr(15, 4)) / 8 + q = lambda X: X * X + 3 * X - Fr(15, 4) + ok &= q(Fr(-2)) < 0 and q(H) < 0 # convex, so negative on [-2, 1/2] + e2 = lambda X: X * X + 5 * X + Fr(17, 4) # R - L_S = (X - 1/2)^2 e2(X)/8 at phi = 1/6 (L_S = -1/4) + for X in (Fr(-3), Fr(-2), Fr(-1), Fr(0), Fr(1, 3), Fr(2)): + R = g(X, (X * X - Fr(1, 4)) / 2) + ok &= R + Fr(1, 4) == (X - H) ** 2 * e2(X) / 8 + ok &= e2(Fr(-2)) < 0 and e2(Fr(-1)) > 0 and e2(H) > 0 + # beta > 2, chi < 0: the slice at X = 1/beta is open, D = a^2 - chi p = -chi p(1/beta) < 0 (Lemma 2, Step 5) + nopen = 0 + for (ph, be, ch) in ((Fr(1, 2), Fr(11, 5), Fr(-1, 2)), (Fr(3, 2), Fr(8, 3), Fr(-20, 3)), (Fr(2), Fr(5, 2), Fr(-1, 10))): + ok &= ph > 0 and abs(ph - be / 2) < 1 and be > 2 and ch < 0 + X = 1 / be + a = 1 - be * X; p = X * X - Fr(1, 4) + ok &= a == 0 and p < 0 and a * a - ch * p < 0 + nopen += 1 + return ok, {'witness': [str(v) for v in wit], 'critical_values': [str(v) for v in vals], 'open_slice_cases_beta_gt_2_chi_lt_0': nopen} + + def main(argv): global MUT if len(argv) == 2 and argv[0] == '--mutant': @@ -776,7 +840,7 @@ def main(argv): return 2 MUT = argv[1] elif argv: - print('usage: soft_check.py [--mutant M1..M13]', file=sys.stderr) + print('usage: soft_check.py [--mutant M1..M14]', file=sys.stderr) return 2 s1, s1d = control_S1() s2, n2 = control_S2() @@ -791,13 +855,14 @@ def main(argv): s11, s11d = control_S11() s12, s12d = control_S12() s13, s13d = control_S13() - res = {'object': 'CL-SOFT-REJECTED-20261002-v1.1', 'scientific_effect': 'NONE', + s14, s14d = control_S14() + res = {'object': 'CL-SOFT-REJECTED-20261002-v1.2', 'scientific_effect': 'NONE', 'controls': {'S1_theorem_1': s1, 'S2_model_algebra': s2, 'S3_gaussian_kernel_jets': s3, 'S4_elder_edge': s4, 'S5_I_series': s5, 'S6_proposition_4': s6, 'S7_lemma_3': s7, 'S8_lemma_2_cases': s8, 'S9_lemma_5': s9, 'S10_limit_field': s10, 'S11_theorem_1_every_d': s11, - 'S12_gaussian_kernel_d3': s12, 'S13_stiff_directions': s13}, + 'S12_gaussian_kernel_d3': s12, 'S13_stiff_directions': s13, 'S14_exact_dprime_witness': s14}, 'S1': s1d, 'S2_instances': n2, 'S3': s3d, 'S4': s4d, 'S5': s5d, 'S6': s6d, 'S7': s7d, 'S8': s8d, 'S9': s9d, 'S10': s10d, - 'S11': s11d, 'S12': s12d, 'S13': s13d} + 'S11': s11d, 'S12': s12d, 'S13': s13d, 'S14': s14d} ok = all(res['controls'].values()) if MUT is None: print(json.dumps(res, indent=1, sort_keys=True)) From 1e5517a31f32b0ad7ccd4e8548894a5e9138ca7d Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 13:21:40 +0000 Subject: [PATCH 4/7] Soft rejected pairs v1.2 README amendment: OA-242-V12-01 and V12-02 README only (with its SOURCES.json entry). PROOF.md, soft_check.py and RESULTS.json are the v1.2 bytes, so #243's binding of the PROOF blob is unaffected. - V12-01: the Theorem 1 row states the exact window on Step 2's event, as a weighted asymptotic localization, as the PROOF overview does. - V12-02: the R_{2/3} ranges are labelled as sensitivity ranges; the Monte Carlo +- is identified as the relative error 1/sqrt(n) that compd.py prints (0.884 +- 0.040 means 4.0%, about +-0.035 additive). - Review record: the v1.2 readback (5952577050) and the complementary delta review (5391986303), with its root-authored D.1 noted. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .../soft_rejected_pairs_20261002/README.md | 12 +++-- .../soft_rejected_pairs_20261002/SOURCES.json | 44 ++++++++++++++++++- 2 files changed, 50 insertions(+), 6 deletions(-) diff --git a/frontiers/soft_rejected_pairs_20261002/README.md b/frontiers/soft_rejected_pairs_20261002/README.md index 90c5576ad..2fe406c62 100644 --- a/frontiers/soft_rejected_pairs_20261002/README.md +++ b/frontiers/soft_rejected_pairs_20261002/README.md @@ -12,6 +12,7 @@ Object `CL-SOFT-REJECTED-20261002-v1.2`. Full text: [`PROOF.md`](PROOF.md). Vers labels their uncertainties as descriptive. - A same-family referee checked the delta: ACCEPT WITH FIXES (9 MINOR, 8 NIT), all applied. - PROOF §9 lists the changed bytes. +- v1.2 README amendment: OA-242-V12-01 (the Theorem 1 row now states the window on Step 2's event, as the PROOF overview does) and OA-242-V12-02 (the two kinds of `±` are distinguished, and the Monte Carlo `±` is identified as the relative error `1/√n` printed by `compd.py`). Only this README and `SOURCES.json` change; `PROOF.md`, `soft_check.py` and `RESULTS.json` are the v1.2 bytes. - v1.1 was made before any nonauthor review. - It extends Theorem 1 to every `d ≥ 2` and adds the `d = 3` constants (Corollary 1′) and the stiff-direction reduction (Proposition 2′). It also adds the `d = 3` Monte Carlo test and fixes finding A-1. @@ -23,7 +24,7 @@ the cusp end `κ = ℓ/r⁴ → ∞` open. This note describes that end for the | | Statement | Status | |---|---|---| -| **Theorem 1** | In every `d ≥ 2`, `κ𝒜^{rej}(b, κ, u) = F₀(b, u) + O(κ^{−1})`, where `F₀ = (5/24)π₀𝔇` and `𝔇` is the density at `0` of the smallest eigenvalue `λ₁` of `−A`, weighted by `γ₁⁶(λ₂⋯λ_m)²`. In `d = 2`, `𝔇 = p_A(0 \| b)E[γ⁶]`. The rejected weight sits exactly on `λ₁ ∈ (3γ₁²/(72κ − f̃₄), 3γ₁²/(24κ − f̃₄)]`; for fixed jets this is `[γ₁²/(24κ), γ₁²/(8κ)]` up to `O(1/κ)`. | proof | +| **Theorem 1** | In every `d ≥ 2`, `κ𝒜^{rej}(b, κ, u) = F₀(b, u) + O(κ^{−1})`, where `F₀ = (5/24)π₀𝔇` and `𝔇` is the density at `0` of the smallest eigenvalue `λ₁` of `−A`, weighted by `γ₁⁶(λ₂⋯λ_m)²`. In `d = 2`, `𝔇 = p_A(0 \| b)E[γ⁶]`. On Step 2's event the rejected weight sits exactly on `λ₁ ∈ (3γ₁²/(72κ − f̃₄), 3γ₁²/(24κ − f̃₄)]`; for fixed jets this is `[γ₁²/(24κ), γ₁²/(8κ)]` up to `O(1/κ)`. This is a weighted, asymptotic localization, not a pointwise one: rejected weight outside this window exists, and the double-soft and large-jet events are handled separately (PROOF §1, Step 1). | proof | | **Corollary 1′** | Gaussian kernel: `∫∫F₀ db dσ = 25√3/(48π²) ≈ 0.0914` (`d = 2`), `125√30/(192π³) ≈ 0.1150` (`d = 3`) | proof | | **Lemma 2** | The elder decision in the fold-scale soft model `G_k` (new jets `B = ∂_uA`, `C₃ = ∂_Θ³f` along the soft direction), by a one-dimensional scan, proved at the exact level. It includes a case (D′, `β > 2`, `χ > 0`) where the saddle is a slice minimum, with an exact witness. For `β > 2`, `χ ≤ 0` the pair is always rejected. `G_k` is #170's typed cubic in other coordinates. | proof (model) | | **Proposition 2′** | `d ≥ 3`: the stiff directions live at scale `r^{3/2}` and decouple; the limit `G_k − (1/2k)Σλ_iη_i²` has the elder decision of `G_k` | proof (model) | @@ -31,7 +32,7 @@ the cusp end `κ = ℓ/r⁴ → ∞` open. This note describes that end for the | **Proposition 4** | Gaussian kernel, every `d`: the fold-scale rejection rate is `F₀(b)H(k)`, with `H` independent of `b` and `d`, and `H(k) = 1 + (12/25)k² + O(k³)`. Codex's Slice C review proves the upper remainder `O(k^{7/2})`. The elder edge `φ_e = 1/3 + t/3 + 10t²/27` contributes `+312/25`, and the pin density `e^{−12k²}` contributes `−12`. | proof (given the model) | | **Lemma 5** | The composite `∫∫∫𝒜^{rej}(b, ℓ/r⁴, u)H(ℓ/r³)` equals `(I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`, where `R_{2/3} = ∫∫∫v⁴[F(v^{−3}) − F₀]` | proof | | **Conjecture 6** | `(k/r)r^{−2}A_r^{rej}(b, k, u) → F(k; b, u)` as `r → 0` at fixed `k`. The limit exists by #170/#175 (merged); its identification with `F` is #243 (open). | proved author-side in #243 | -| **Conjecture 7** | `ρ_rej + ν_eld^{far,r_0^*} = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`; Gaussian kernel: `R_{2/3} ≈ −0.049 ± 0.003` (`d = 2`), `−0.062 ± 0.004` (`d = 3`) | conjecture, with evidence | +| **Conjecture 7** | `ρ_rej + ν_eld^{far,r_0^*} = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`; Gaussian kernel: `R_{2/3} ≈ −0.049 ± 0.003` (`d = 2`), `−0.062 ± 0.004` (`d = 3`); these `±` are quadrature and interpolation sensitivity ranges (PROOF §5(5)), not sampling errors | conjecture, with evidence | Conjecture 7 needs two inputs that are not yet available (PROOF §4): - (i) the soft layer `λ₁ ≍ r`, together with Proposition 2′ for the field in `d ≥ 3`; @@ -51,7 +52,7 @@ A third input, (iii) the far elder density, is needed only to remove `ν_eld^{fa | #216's Monte Carlo, `d = 2` | rejected adjacent pairs with `s = r/ℓ^{1/4} < 0.8`: 37 observed, 35.3 from the composite, 4,322 from the cusp kernel alone | | #216's Monte Carlo, `d = 3` | `s < 0.8`: 11 observed, 10.7 composite (1.2 with the pin density alone), 1,329 cusp kernel alone. On `[3·10^{−4}, 10^{−2}]` the counts are `0.884 ± 0.040` of the `ℓ^{1/4}` law and `1.008` of the composite (Pearson χ² 11.8 against 2.1 on 8 bins). Over #216's whole range `[10^{−4}, 0.3]` the composite does not describe the counts (finite-`r` corrections). | -#216's fields are on the torus with `L = 64` (`d = 2`) and `L = 16` (`d = 3`). The `±` values are Poisson relative errors `1/√n`, and the `χ²` values are Pearson sums over bins with at least five counts. All of these are descriptive, not calibrated significances (PROOF §5). +#216's fields are on the torus with `L = 64` (`d = 2`) and `L = 16` (`d = 3`). In the Monte Carlo rows, the number after a count ratio's `±` is the Poisson *relative* error `1/√n` of the observed count `n`, as `compd.py` prints it. So `0.884 ± 0.040` means a relative error of 4.0%, that is about `±0.035` on the ratio itself. The `χ²` values are Pearson sums over bins with at least five counts. All of these are descriptive, not calibrated significances (PROOF §5). The `±` ranges for `R_{2/3}` in the table above are of a different kind: sensitivity ranges of the quadrature and interpolation (PROOF §5(5)). ## Controls @@ -106,7 +107,10 @@ Mutants `M1`–`M14` each fail only their own control (the checker names it on s - Slices B (the remaining formulas) and E (5391485205): PASS_TECHNICAL_SCOPED for the limit-field algebra, normalization, typing and weight; PASS for the checker and mutant correspondence. Findings B-02, B-03, E-01 and E-02. It also gives the root-authored remainder estimate (B.5), now cited after (2.1). - - Every finding is applied in v1.2. A delta check of the changed bytes is requested. + - Every finding is applied in v1.2. +- **Nonauthor reviews of v1.2 (OpenAI Codex), at `1f86fea`, PROOF blob `ad4beb84`:** + - Bounded readback of B-01, D-01, the exact (D′) witness and the strengthened `β > 2`, `χ ≤ 0` rejection (5952577050): PASS_TECHNICAL_SCOPED, no new finding. It adds a direct finite-path proof of that rejection. + - Complementary delta review (5391986303): PASS_TECHNICAL_SCOPED for the changed proof corrections and the conditional source alignment. Two minor README findings, V12-01 and V12-02, are applied in the v1.2 README amendment. Its §3 is root-authored support (D.1): a direct model-integral proof of `F = kA∗a_fail` in every fixed dimension, not reviewed here. - **v1.2 delta (same family).** A clean-context referee checked the changed bytes. - Verdict: **ACCEPT WITH FIXES**, with no BLOCKING or MAJOR finding: 9 MINOR findings and 8 NITs, all applied. - Its own checks: the witness by sympy, the Jacobians `−96`, `−3/2` and `−6`, and the `48/47` ratio. diff --git a/frontiers/soft_rejected_pairs_20261002/SOURCES.json b/frontiers/soft_rejected_pairs_20261002/SOURCES.json index 3aa24e5b9..735281b9e 100644 --- a/frontiers/soft_rejected_pairs_20261002/SOURCES.json +++ b/frontiers/soft_rejected_pairs_20261002/SOURCES.json @@ -224,8 +224,8 @@ }, { "path": "README.md", - "bytes": 12214, - "sha256": "5d84f48f65e6f77438397ed10eae3d90012eb6ff453ef341d55f7e8febba09b6" + "bytes": 13878, + "sha256": "bd43ee1b5833a6069381aadbd12159316e08569b105be348bd640c3db0e01272" }, { "path": "soft_check.py", @@ -645,6 +645,46 @@ "OA-242-E-02" ], "applied_in": "v1.2" + }, + { + "slice": "v1.2 bounded readback (B-01, D-01, exact D-prime witness, strengthened beta > 2, chi <= 0 rejection)", + "reviewer": "OpenAI Codex (agent review_soft_pair_scope; delegated AI review; same GitHub account; organizational independence 0)", + "comment": "5952577050", + "at": "1f86fea", + "verdict": "PASS_TECHNICAL_SCOPED", + "findings": [], + "applied_in": null + }, + { + "slice": "v1.2 complementary delta (A-01, B-02, B-03, C-01, C-02, D-02, E-01, E-02, Schur paragraph, prior-work interfaces)", + "reviewer": "OpenAI Codex /root (delegated AI review; same GitHub account; organizational independence 0)", + "review": "pullrequestreview-5391986303", + "at": "1f86fea", + "verdict": "PASS_TECHNICAL_SCOPED", + "findings": [ + "OA-242-V12-01", + "OA-242-V12-02" + ], + "applied_in": "v1.2 README amendment", + "note": "its section 3 (D.1) is root-authored support: a direct model-integral proof of F = k A_* a_fail in every fixed d; not reviewed here" + } + ], + "amendments": [ + { + "label": "v1.2 README amendment", + "files_changed": [ + "README.md", + "SOURCES.json" + ], + "unchanged": [ + "PROOF.md", + "soft_check.py", + "RESULTS.json" + ], + "findings": [ + "OA-242-V12-01", + "OA-242-V12-02" + ] } ] } From d504cdb15851b4a9140755fee6a295282c503ed4 Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 15:42:47 +0000 Subject: [PATCH 5/7] Soft rejected pairs v1.3: correct section 5's numbers (AUTH-242-02) Author correction, disclosed on the PR in 5953336662 and 5955380848; xAI (5955184422) asked that it be in the landed note. - PROOF section 5(2) and the README record two artifacts of the scanner dec1d.py in thin layers on either side of beta = 2 (a missed fold just above; a loose 1e-3 tolerance just below with chi < 0). They are errors of the implementation, not of Lemma 2, and move H by at most 2.0e-4. - Section 5(3)-(5) recompute H and H0, Delta_chi/k^4 -> 66.56 (was the fitted 62), I~ = -0.5337 and R_{2/3} = -0.0488 (d = 2), -0.0614 (d = 3) from the closed form of Math- #244 (open; conditional on #170 Theorem E(1) and [CUB] Theorem C). The v1.2 sensitivity ranges stand. - Section 4: Remark 2's v >= 2 contribution, the numerical sentence after Conjecture 7, and Conjecture 6's status (#243's delivered reviews). - SOURCES.json: v1.3 object, supersedes v1.2 (b82b0ca), #243 bound at its reviewed head 11a4cd4, #244 added (head 1781c0f), the v1.3 exploration files pinned, the D.1 review and the xAI receipt recorded. PROOF sections 0-3, soft_check.py and RESULTS.json are byte-identical to v1.2. No statement, proof, conjecture or control changes. Scientific effect: NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .../soft_rejected_pairs_20261002/PROOF.md | 140 +++++++++++---- .../soft_rejected_pairs_20261002/README.md | 25 ++- .../soft_rejected_pairs_20261002/SOURCES.json | 162 ++++++++++++++++-- 3 files changed, 277 insertions(+), 50 deletions(-) diff --git a/frontiers/soft_rejected_pairs_20261002/PROOF.md b/frontiers/soft_rejected_pairs_20261002/PROOF.md index ad4beb846..5fd29e63e 100644 --- a/frontiers/soft_rejected_pairs_20261002/PROOF.md +++ b/frontiers/soft_rejected_pairs_20261002/PROOF.md @@ -1,6 +1,17 @@ # Soft rejected pairs: the `1/κ` tail of the rejected cusp kernel in every dimension, the fold-scale rejection function, and a conjectured `ℓ^{2/3}` term -Object: CL-SOFT-REJECTED-20261002-v1.2. Version history: +Object: CL-SOFT-REJECTED-20261002-v1.3. Version history: +- v1.2 → v1.3, an author correction of §5's numbers (AUTH-242-02). It was disclosed in PR comments 5953336662 and + 5955380848; xAI asked in 5955184422 that it be in the landed note. No statement, proof or control changes. + - §5(2) records two artifacts of the one-dimensional scanner `dec1d.py`, in thin layers on either side of `β = 2`. They + are in the implementation, not in Lemma 2, and they move `H` by at most `2.0·10^{−4}`. + - §5(4) gives the `H` and `H₀` rows recomputed, the accuracy sentence, and the small-`k` coefficient of `Δ_χ` + (`66.56`, exact, in place of the fitted `62`). + - §5(5), §4 Remark 2, and the numerical remarks in "What is new" and Conjecture 7: `Ĩ = −0.5337`, so + `R_{2/3} = −0.0488` (`d = 2`) and `−0.0614` (`d = 3`), inside the v1.2 sensitivity ranges, which stand. + - The corrected values use the closed form of `I` in Math- #244 (open; conditional on #170 Theorem E(1) and [CUB] + Theorem C). They are exploration, not certified. + - §§0–3 are byte-identical to v1.2, as are `soft_check.py` and `RESULTS.json`. §9 lists the changed bytes. - v1.1 → v1.2, after the nonauthor reviews of v1.1 by OpenAI Codex: - Slice A, 5390308599; - Slice B (model), 5948352439; @@ -83,7 +94,7 @@ the next term of `ρ_rej` (§4). - The two-scale composite density has the expansion `(I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`, with `R_{2/3}` given by `F` and `F₀`. - Conjecture 7 says that `ρ_rej + ν_eld^{far,r_0^*} − B_{d,L}` has the same expansion. - - For the Gaussian kernel, numerically `R_{2/3} ≈ −0.049` in `d = 2` and `≈ −0.062` in `d = 3` (§5). + - For the Gaussian kernel, numerically `R_{2/3} ≈ −0.049` in `d = 2` and `≈ −0.061` in `d = 3` (§5). - **Evidence** (§5, exploration): - the tail of Theorem 1, by quadrature in `d = 2` and `d = 3`; - Lemma 2 against a two-dimensional flood fill; @@ -113,7 +124,9 @@ the next term of `ρ_rej` (§4). (`frontiers/concave_fibre_elder_20260930/PROOF.md`, blob `923d3236`), both merged: the failure law `r^{−3}(1 − p_r) → α₁ + α₂` (Theorems S and F), the compact-window coefficient, #170 §3 (the critical chord), and #170 Theorem E. These are prior work for §§2–4; both packets are author-side candidates at their stated conditional scope. - - #243 (open, head `0740b2c`): Theorem FL and Proposition FL.7 (Conjecture 6). + - #243 (open; its five review slices are at head `11a4cd4`): Theorem FL and Proposition FL.7 (Conjecture 6). + - #244 (open, head `1781c0f`): the closed form of `I` (Theorem A), `t* = 2/3` (Corollary A.1) and the expansion of `H` to + order `k⁴` (Theorem B), conditional on #170 Theorem E(1) and [CUB] Theorem C. §5's v1.3 numbers use it. - #237 (open): Theorem P, Remark 4. - #216 (open): the Monte Carlo in `d = 2, 3`. - #187 (merged) and #188 (open): the far elder density. @@ -687,8 +700,10 @@ Integrating over `b` and `u` gives (4.2). ∎ - *Existence of the limit, as `kA∗(α₁ + α₂)`.* This is in #170 (`d = 2`) and #175 (`d ≥ 3`): merged author-side candidates, at their conditional scope. - *`d = 2`, at fixed `(b, k, u)`.* This follows from #170 Theorem S and #243 Proposition FL.7(iv), a model-level identity. -- *Local uniformity in `k`, and `d ≥ 3`.* #243 (open; author-side, nonauthor review pending) claims a proof (Theorem FL). -- The label is kept here until #243 is reviewed. +- *Local uniformity in `k`, and `d ≥ 3`.* #243 (open; author-side) claims a proof (Theorem FL). Its five nonauthor slice + reviews (same account, organizational independence 0) are delivered at head `11a4cd4`, with four minor clarifications + to apply. +- The label is kept here until #243 lands. **Conjecture 7 (the next term of the rejected density).** In every `d ≥ 2`, @@ -698,7 +713,7 @@ with `R_{2/3}` as in (4.2). - Equivalently, `ρ_rej + ν_eld^{far,r_0^*} − B_{d,L} − ν^c = O(ℓ^{3/4})`. This is (R⁺.1) of #229 with its remainder resolved. - If `ν_eld^{far,r_0^*}(ℓ) = O(ℓ^{3/4})` (#188, open, claims `O(ℓ^N)`), then `ρ_rej = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`. -- For the Gaussian kernel, numerically `R_{2/3} ≈ −0.049` in `d = 2` and `R_{2/3} ≈ −0.062` in `d = 3` (§5). +- For the Gaussian kernel, numerically `R_{2/3} ≈ −0.049` in `d = 2` and `R_{2/3} ≈ −0.061` in `d = 3` (§5). *What a proof needs.* Inputs (i) and (ii) are missing; (iii) is needed only to drop the far term. - (i) *The soft layer.* A two-scale version of Conjecture 6, uniform from the cusp scale to the fold scale (`κ ≥ κ₁`), with @@ -724,7 +739,8 @@ makes `ℓ^{2/3}` the first fold-side power and `ℓ^{3/4}` the next cusp-side o #237's remainder `ℓ^{3/5}` allows both. Only `ρ_rej`, or the rejected Monte Carlo row binned in `s`, tests `R_{2/3}` alone, and only subject to (ii)–(iii). 2. For the Gaussian kernel, `Ĩ` is dominated by `v ∈ [1, 1.5]`, that is `k ∈ [0.3, 1]`, where `H` falls from `≈ 1` to `≈ 0`. The - contributions are `−0.200` (`v < 1`), `−0.563` (`1 ≤ v ≤ 1.5`), `+0.003` (`1.5 ≤ v ≤ 2`) and `+0.223` (`v ≥ 2`). + contributions are `−0.200` (`v < 1`), `−0.563` (`1 ≤ v ≤ 1.5`), `+0.003` (`1.5 ≤ v ≤ 2`) and `+0.227` (`v ≥ 2`) (v1.3, + by `itilde_split.py` of §5's v1.3 custody item; v1.2 had `+0.223` for `v ≥ 2`). 3. `R_{2/3}/(I^{cand} − c₁) ≈ −0.53` in `d = 2` and `≈ −0.85` in `d = 3`. Since `ℓ^{5/12} = 0.147` at `ℓ = 10^{−2}`, the `ℓ^{2/3}` term there is about 8% (`d = 2`) and 12% (`d = 3`) of the `ℓ^{1/4}` term. @@ -737,6 +753,9 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. - *Pinned identities.* `SOURCES.json` (`exploration_manifest`) pins every script, table and log there by its SHA-256, including those of the Monte Carlo comparison. #216's raw records are pinned by name in #216's archive `SIDE24_MC_raw_2026-10-01.zip`. +- *v1.3.* The scripts and logs behind the corrected values are in `exploration/v13/` of the same archive, pinned in + `SOURCES.json` (`exploration_manifest`, `v13_files`). They evaluate #244's closed form; #244's own exploration is + pinned in its `SOURCES.json`. 1. *Theorem 1.* - *`d = 2`.* Quadrature of (0.2) under the contact law (`fastk.py`, `soft_d2.py`) gives the numbers of Remark 4 of §1. @@ -758,11 +777,34 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. default `ε`; with `ε_rel = 5·10^{−4}` the flood fill agrees. - The referee's independent comparison (500 points, 40 of them (D′)-elder) agrees after resolving two numerical artifacts. + - *Two artifacts of `dec1d.py` (v1.3, AUTH-242-02).* Both are in thin layers on either side of `β = 2`. They are errors + of the implementation, not of Lemma 2, and the comparison with the closed form of #244 found them. + - *Just above `β = 2`, false rejections.* Here `a(½) = 1 − β/2 < 0`. The slices just right of `X = ½` lose their + maximum at a fold, at distance `≈ a(½)²/χ` from `½`, which can be far below the grid step. `dec1d.py` treats a first + grid point past `½` whose slice has no maximum as an open end. But if the fold value `A₀ + a³/(6χ²)` is below `L_S`, + the band closes before the fold, and `S` is the death saddle. Example: at `(t, χ₀) = (9.005434, 193.926947)`, + `dec1d.py` rejects `φ ∈ (0.11104, 0.12491)`, and the rejected set is `(0.11245, 0.12491)`. + - *Just below `β = 2`, with `χ < 0`, false acceptances.* The tolerance `δ = 10^{−3}` of `dec1d.py` treats a band that + ends within `10^{−3}` of `X = ½` as ending at `S`. But the slices just left of `X = ½` open (at distance + `≈ a(½)²/|χ|`), and the axial segment joins `M` to the far region above `L_S`, so the pair is rejected. + - The flood fill errs the other way here: its level offset `ε` exceeds the relevant level gaps. So the comparisons + above did not expose the artifacts. + - The artifacts move `H` by at most `2.0·10^{−4}` (at `k = 0.4`). Items 4–5 give the corrected values. Control S8 + stays clear of this region (§7). 3. *The function `I`.* - A `101 × 121` table of `I(t, χ₀)` on `|t| ≤ 30`, `|χ₀| ≤ 2000` (`Itab2d_v2.py`), and an exact evaluator for other points (`Ifast.py`). - `I(t, 0)/|t|³ = 1.95, 1.51, 1.39` at `t = −10², −10³, −10⁴`, tending to `4/3`. - `I(0, χ₀)/χ₀² = 0.0228, 0.0212, 0.0209` at `χ₀ = 10³, 10⁴, 10⁵`, tending to `1/48`. + - *The closed form (v1.3).* #244 (open) derives `I` exactly, conditional on #170 Theorem E(1) and [CUB] Theorem C, + through #243 Proposition FL.7's identification. With `ψ = 1/φ` and `c′ = 1 − t`, off a null set the pair is elder iff + `ψ ≥ 2c′` and `(χ₀ + 8 − 12t)² ≤ 16(ψ − 2c′)²(ψ + c′)`. So `I = |c′|δ² + δ³/3`, where `δ = ψ_e − |c′|` and `ψ_e` is the + largest root of that cubic. + - The table above agrees with it to a median relative difference of `1.8·10^{−7}`. The 97 of its 12,221 entries that + differ by more than `10^{−3}` are explained: 7 tiny intervals the grid misses, 54 table resolution, and 36 the + artifacts of item 2. + - In particular `t* = 2/3` exactly (§2 has `t* ∈ (0.65, 0.68)`), and the two limits above have the exact next terms + `I(t, 0) = (4/3)|t|³ + 3√3|t|^{5/2} + O(t²)` and `I(0, χ₀) = χ₀²/48 + 16^{−2/3}|χ₀|^{4/3} + O(|χ₀|)`. 4. *The function `H`.* `H = H₀ + Δ_χ`. - `H₀`, the value with `C₃` ignored, comes from a 639-point table of `I(t, 0)` on `|t| ≤ 400`, its asymptotic forms beyond (`|t|³(4/3 + 5.74|t|^{−1/2})` and `t − 2/3`), and adaptive quadrature in `t` and `γ²` down to `γ → 0` (`Hk0b.py`). Its @@ -770,32 +812,54 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. - `Δ_χ` is the difference of two direct quadratures over `(|γ|, B, C₃)`, with and without `C₃` (`Hdirect4.py`). These use 32 Gauss–Legendre nodes in `|γ|` on `[0, 7.5]` and 24 Gauss–Hermite nodes in `B` and in `C₃`, with the table inside its range and the exact evaluator outside it. - - Accuracy checks: without the table, `Δ_χ(0.2)` changes by `2·10^{−5}`; with 48 nodes in `|γ|`, `H` changes by - `5·10^{−5}`. For small `k`, `Δ_χ ≈ 62k⁴`. The quadrature `H` agrees with a table-free direct quadrature at `k = 0.2` to - `1.3·10^{−4}`. - - Values: + - Accuracy checks (v1.2): without the table, `Δ_χ(0.2)` changes by `2·10^{−5}`; with 48 nodes in `|γ|`, `H` changes by + `5·10^{−5}`; and the quadrature `H` agrees with a table-free direct quadrature at `k = 0.2` to `1.3·10^{−4}`. All three + checks use `dec1d.py`, so none of them measures its bias (item 2). That bias reaches `2.0·10^{−4}` at `k = 0.4`. + - *Recomputed values (v1.3).* The rows below come from #244's closed form of `I`. `H` and `H₀` are evaluated by nested + Gauss–Legendre quadrature over `(|γ|, B, C₃)`, split at the kinks of the integrand. Raising the node counts from `60³` + to `100³` changes them by less than `10^{−10}`, and #244's referee reproduces the `H` row with an independent + quadrature. + - The v1.2 rows differed by at most `2.0·10^{−4}` (`H`) and `1.3·10^{−5}` (`H₀`). + - At four decimals the v1.2 rows had `H = 1.0039, 0.9345, 0.7271, 0.4397, 0.2007` at `k = 0.1, 0.3, 0.4, 0.5, 0.6` + and `H₀ = 0.4663` at `k = 0.4`. + - *Small `k`.* `Δ_χ/k⁴ → 53248/800 = 66.56` (#244 Theorem B, Step 3); v1.2 used the fitted value `62`. #244 Theorem B + also gives `H = 1 + (12/25)k² + h_{7/2}k^{7/2} + (728/25)k⁴ + o(k⁴)`, with + `h_{7/2} = 16Γ(9/4)(1728√6 − 4332√2 − 2721)/(2625π) ≈ −10.144`. So the upper remainder `O(k^{7/2})` from Codex's + Slice C review, cited in the proof of Proposition 4(2), is attained (OA-242-C-01). This packet itself still claims no + `k^{7/2}` coefficient. + - Values (v1.3): | `k` | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 1.0 | |---|---|---|---|---|---|---|---|---|---| - | `H₀` | 0.9976 | 0.9334 | 0.7403 | 0.4663 | 0.2268 | 0.0845 | 0.0241 | 0.0053 | 0.0001 | - | `H` | 1.0039 | 1.0041 | 0.9345 | 0.7271 | 0.4397 | 0.2007 | 0.0687 | 0.0177 | 0.0005 | + | `H₀` | 0.9976 | 0.9334 | 0.7403 | 0.4662 | 0.2268 | 0.0845 | 0.0241 | 0.0053 | 0.0001 | + | `H` | 1.0040 | 1.0041 | 0.9344 | 0.7269 | 0.4395 | 0.2006 | 0.0687 | 0.0177 | 0.0005 | | `e^{−12k²}` | 0.887 | 0.619 | 0.340 | 0.147 | 0.050 | 0.013 | 0.003 | 0.0005 | 0.000006 | 5. *`R_{2/3}`.* - - `Ĩ = −0.536` (`assemble2.py`). - - The 200-point grid `H_dense3.json` is written by `dense2.py --write` and used below. It has the same values of `H` as - v1's `H_dense2.json`, and its `R23` field comes from the quadrature of its interpolant, `Ĩ = −0.5361`. - - The check mode of `dense2.py` (`dense2.log`) shows trapezoid and Simpson rules on the same grid giving `−0.535` to - `−0.536`. - - v1 quoted `−0.537` for "a dense trapezoid rule". That was the `R23` field stored in `H_dense2.json`, produced by an - unrecorded rule (finding A-1, found by the author while archiving v1). - - So `R_{2/3} = 0.0914028·Ĩ = −0.049` in `d = 2` and `R_{2/3} = 0.1150058·Ĩ = −0.0617` in `d = 3`. - - Sensitivity: a uniform shift of `H` by `±0.002` on `k ≥ 0.1` moves `Ĩ` by `±0.018`. Halving the small-`k` term `62k⁴` - moves it by `−0.021`, and multiplying it by `1.5` by `+0.021`. So `Ĩ = −0.54 ± 0.03`, `R_{2/3} = −0.049 ± 0.003` in `d = 2` - and `R_{2/3} = −0.062 ± 0.004` in `d = 3`. - - With `C₃` ignored (`H = H₀`), `Ĩ = −1.133` and `R_{2/3} = −0.104`. With the pin density alone (`H = e^{−12k²}`), - `Ĩ = 12^{5/6}Γ(−5/6)/6 = −8.829` and `R_{2/3} = −0.807`. So the soft model's extra jets matter: `B` nearly cancels the pin - density at small `k`, and `C₃` raises `H` at `k ≈ 0.2–0.7`. + - *v1.3.* With `H` from #244's closed form, `Ĩ = −0.53367`. The quadrature is Gauss–Legendre in `q = k^{1/3}` on + `[0.1, 1.2]`, with #244 Theorem B's expansion below `q = 0.1` and `H = 0` above `q = 1.2`, where `H < 2·10^{−13}`. With + 16 and 24 nodes per segment the results agree to `2.5·10^{−8}`. + - So `R_{2/3} = 0.0914028·Ĩ = −0.04878` in `d = 2` and `R_{2/3} = 0.1150058·Ĩ = −0.06137` in `d = 3`. Both lie inside + the sensitivity ranges below, which stand. + - *Why v1.2 differed.* Its `Ĩ` (`−0.5361` on the dense grid, `−0.5364` by `assemble2.py`) was off by about + `−0.0025`. By the decomposition in #244 §5, about `−0.0019` comes from v1.2's interpolation of `Δ_χ` between the + nodes `k = 0.1, 0.15, 0.2`, and about `−0.0008` from its small-`k` model `62k⁴`. Only about `+0.0002` comes from the + artifacts of item 2 (`k = 0.25–0.7`). + - *The v1.2 computation, kept as a record.* + - `Ĩ = −0.536` (`assemble2.py`). + - The 200-point grid `H_dense3.json` is written by `dense2.py --write`. It has the same values of `H` as v1's + `H_dense2.json`, and its `R23` field comes from the quadrature of its interpolant, `Ĩ = −0.5361`. + - The check mode of `dense2.py` (`dense2.log`) shows trapezoid and Simpson rules on the same grid giving `−0.535` to + `−0.536`. + - v1 quoted `−0.537` for "a dense trapezoid rule". That was the `R23` field stored in `H_dense2.json`, produced by an + unrecorded rule (finding A-1, found by the author while archiving v1). + - So v1.2 had `R_{2/3} = 0.0914028·Ĩ = −0.049` in `d = 2` and `R_{2/3} = 0.1150058·Ĩ = −0.0617` in `d = 3`. + - *Sensitivity (v1.2; the ranges stand).* A uniform shift of `H` by `±0.002` on `k ≥ 0.1` moves `Ĩ` by `±0.018`. Halving + v1.2's small-`k` term `62k⁴` moves it by `−0.021`, and multiplying it by `1.5` by `+0.021`. So `Ĩ = −0.54 ± 0.03`, + `R_{2/3} = −0.049 ± 0.003` in `d = 2` and `R_{2/3} = −0.062 ± 0.004` in `d = 3`. + - *Comparisons.* With `C₃` ignored (`H = H₀`), `Ĩ = −1.134` and `R_{2/3} = −0.104` (v1.3; v1.2 had `Ĩ = −1.133`). With + the pin density alone (`H = e^{−12k²}`), `Ĩ = 12^{5/6}Γ(−5/6)/6 = −8.829` and `R_{2/3} = −0.807`. So the soft model's + extra jets matter: `B` nearly cancels the pin density at small `k`, and `C₃` raises `H` at `k ≈ 0.2–0.7`. 6. *#216's Monte Carlo.* The raw records were compared with the composite: `composite.py` and `comp2d.py` in `d = 2`, and `compd.py` in `d = 3`. The records are `batchA2.npz` (`d = 2`: 4,000 samples, 271,272 rejected adjacent pairs with `ℓ < 0.3`) and `batch3B.npz` (`d = 3`: 996 samples, 45,952 such pairs). @@ -851,6 +915,9 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. law `r^{−3}(1 − p_r) → α₁ + α₂` and the compact-window `ℓ^{2/3}` coefficient (AUTH-242-01). #170 Theorem E decides the window cubic. That is the same decision as Lemma 2, computed by another method: critical chords and [CUB]'s classifier. - **#243 (open).** Theorem FL and Proposition FL.7 claim a proof of Conjecture 6. +- **#244 (open).** Conditional on #170 Theorem E(1) and [CUB] Theorem C, and through #243 Proposition FL.7's + identification, it gives `𝓡(t, χ₀)` and `I(t, χ₀)` of (2.5) in closed form, `t* = 2/3`, and + `H = 1 + (12/25)k² + h_{7/2}k^{7/2} + (728/25)k⁴ + o(k⁴)`. It changes no statement here. §5's v1.3 numbers use it. - **Not claimed.** - Conjectures 6 and 7 are not proved here. For Conjecture 6, see its status in §4. The values of `H` and `R_{2/3}` are numerical and not certified. @@ -878,7 +945,7 @@ of the failing controls to stderr. | S5 | (3.4), and the antiderivative of the weight | | S6 | `E[γ⁶] = 120`, `E[γ⁶t²] = 576k²`, `312/25`, `12/25`, and the moment identities of Proposition 4's proof | | S7 | every numerical inequality of Lemma 3's proof, including the critical-value formulas as identities | -| S8 | Lemma 2 on 16 fixed parameter points with known answers, including case (D′), in floating point; the scan is not reliable within about `4·10^{−3}` of `β = 2` (none of the 16 points is there) | +| S8 | Lemma 2 on 16 fixed parameter points with known answers, including case (D′), in floating point; the scan is not reliable within about `4·10^{−3}` of `β = 2` (none of the 16 points is there; §5(2) describes the two artifacts there, and #244's control C9 confirms the 16 decisions from the closed form) | | S9 | the error exponents of Lemma 5 | | S10 | the limit (2.1), including `G₁`, on three exactly pinned degree-6 fields with `A = −λ̃r/k`, at `r = 10^{−3}` and `10^{−4}` | | S11 | Theorem 1 for `m = 2, 3` in rational eigenframes (Cayley rotations): `z = f₄ − 3γᵀA^{−1}γ = 3γ₁²/λ₁ + f̃₄`, `Y = (Δ/12)z`, `w_κ = 36κ²Δ²(1 − φ²)`, the substitution (1.1) with the factor `Π′`, `69984/72⁴ = 1/384`, and the double-soft implication of Step 1 | @@ -905,7 +972,8 @@ Mutants `M1`–`M13` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, | #240 | `frontiers/elder_cusp_parity_20261002/PROOF.md` (merged at `7636cae`; blob `16a1db06`, unchanged) | Lemma Q′, Step N6, Corollary N′, Remarks 1 and 4 — cited | | #170 | `frontiers/local_elder_geometry_20260930/PROOF.md` (merged; blob `ef2aa579`) | §3 (the critical chord), Theorem E, Theorem S, §9 — cited (prior work, AUTH-242-01) | | #175 | `frontiers/concave_fibre_elder_20260930/PROOF.md` (merged; blob `923d3236`) | Theorems H and F, §7 — cited (prior work, AUTH-242-01) | -| #243 | `frontiers/soft_fold_limit_20261002/PROOF.md` (open PR, head `0740b2c`; blob `caedaef6`) | Theorem FL, Proposition FL.7, Lemma FL.2(c) — cited (Conjecture 6) | +| #243 | `frontiers/soft_fold_limit_20261002/PROOF.md` (open PR, head `11a4cd4`; blob `3d9faa87`) | Theorem FL, Proposition FL.7, Lemma FL.2(c) — cited (Conjecture 6) | +| #244 | `frontiers/soft_closed_form_20261002/PROOF.md` (open PR, head `1781c0f`; blob `310ef828`) | Theorems A and B, Corollary A.1, §5 — cited (§5's v1.3 numbers) | | #237 | `frontiers/candidate_parity_rate_20261001/PROOF.md` (open PR) | Theorem P's remainder, Remark 4 — cited | | #216 | `frontiers/third_order_coefficient_20261001/NOTE.md` (open PR) | the full-field Monte Carlo in `d = 2, 3` (§5) — cited | | #188 | far elder density `O(ℓ^N)` (open PR) | §4 (iii) — cited | @@ -923,6 +991,20 @@ Mutants `M1`–`M13` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, - **E** (§§5–7): the evidence is described accurately and kept separate from the proofs, and the controls match their claims. +**Changed bytes in v1.3** (for a delta check against the v1.2 reviews; AUTH-242-02): +- *Header:* the object label and version history, the closing numerical item of "What is new" (`≈ −0.061` in `d = 3`), + and the Dependencies (#243's reviewed head, #244). +- *§4:* Conjecture 6's status (#243's reviews), the numerical sentence after Conjecture 7 (`≈ −0.061`), and the + `v ≥ 2` contribution in Remark 2 (`+0.227`). +- *§5:* the v1.3 custody item; item 2's account of the two `dec1d.py` artifacts; item 3's closed-form paragraph; item 4's + accuracy sentence, recomputation paragraph, small-`k` paragraph and both value rows; item 5's v1.3 paragraph, and the + regrouping of its v1.2 record, sensitivity and comparisons (`Ĩ = −1.134` with `C₃` ignored). +- *§§6–8:* the #244 item, the S8 row, and the #243 and #244 source rows. +- *§9:* this list. + +§§0–3 (byte-identical), Lemma 5, Conjectures 6 and 7 themselves, §5(1) and §5(6), `soft_check.py` and `RESULTS.json` +are unchanged. + **Changed bytes in v1.2** (for a delta check against the v1.1 reviews): - *Header:* the version history, the Disposition, "Why" (#240 merged), "What is new" (the window, A-01; the soft model and Lemma 2 notes), and the Dependencies (#240, #170, #175, #243). diff --git a/frontiers/soft_rejected_pairs_20261002/README.md b/frontiers/soft_rejected_pairs_20261002/README.md index 2fe406c62..79d80d0ec 100644 --- a/frontiers/soft_rejected_pairs_20261002/README.md +++ b/frontiers/soft_rejected_pairs_20261002/README.md @@ -2,7 +2,18 @@ **Author-side proof candidate and conjectures (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** -Object `CL-SOFT-REJECTED-20261002-v1.2`. Full text: [`PROOF.md`](PROOF.md). Version history: +Object `CL-SOFT-REJECTED-20261002-v1.3`. Full text: [`PROOF.md`](PROOF.md). Version history: +- v1.3 is an author correction of PROOF §5's numbers (AUTH-242-02), first disclosed in PR comments 5953336662 and + 5955380848. xAI asked (5955184422) that it be in the landed note; it is in PROOF §5 and below. + - The one-dimensional scanner `dec1d.py` behind §5's `I` table and `H` has two artifacts, in thin layers on either side + of `β = 2`. Just above, it misses a fold and falsely rejects; just below, with `χ < 0`, its tolerance `10^{−3}` falsely + accepts. These are errors of the implementation, not of Lemma 2. They move `H` by at most `2.0·10^{−4}`. + - The corrected values come from the closed form of `I` in Math- #244 (open; conditional on #170 Theorem E(1) and [CUB] + Theorem C): `H` and `H₀` rows, `Δ_χ/k⁴ → 66.56` (v1.2 fitted `62`), `Ĩ = −0.5337`, and `R_{2/3} = −0.0488` + (`d = 2`) and `−0.0614` (`d = 3`). These are inside v1.2's sensitivity ranges, which stand. Most of the change in + `Ĩ` comes from v1.2's small-`k` interpolation and model, not from the scanner. + - No statement, proof or control changes: PROOF §§0–3 (byte-identical), `soft_check.py` and `RESULTS.json` are + unchanged. PROOF §9 lists the changed bytes. - v1.2 applies the findings of the nonauthor reviews of v1.1 (OpenAI Codex, Slices A, B (model), C, D, and B (formulas) with E): OA-242-A-01, B-01, B-02, B-03, C-01, C-02, D-01, D-02, E-01 and E-02. - It adds Codex's exact (D′) witness with control S14, and states that for `β > 2`, `χ < 0` the pair is always rejected. @@ -29,10 +40,10 @@ the cusp end `κ = ℓ/r⁴ → ∞` open. This note describes that end for the | **Lemma 2** | The elder decision in the fold-scale soft model `G_k` (new jets `B = ∂_uA`, `C₃ = ∂_Θ³f` along the soft direction), by a one-dimensional scan, proved at the exact level. It includes a case (D′, `β > 2`, `χ > 0`) where the saddle is a slice minimum, with an exact witness. For `β > 2`, `χ ≤ 0` the pair is always rejected. `G_k` is #170's typed cubic in other coordinates. | proof (model) | | **Proposition 2′** | `d ≥ 3`: the stiff directions live at scale `r^{3/2}` and decouple; the limit `G_k − (1/2k)Σλ_iη_i²` has the elder decision of `G_k` | proof (model) | | **Lemma 3** | Elder whenever `φ ≤ (1/20)min(1, \|t\|^{−1}, \|χ₀\|^{−2/3})`; so `I(t, χ₀) ≤ (8000/3)max(1, \|t\|³, χ₀²)` | proof (model) | -| **Proposition 4** | Gaussian kernel, every `d`: the fold-scale rejection rate is `F₀(b)H(k)`, with `H` independent of `b` and `d`, and `H(k) = 1 + (12/25)k² + O(k³)`. Codex's Slice C review proves the upper remainder `O(k^{7/2})`. The elder edge `φ_e = 1/3 + t/3 + 10t²/27` contributes `+312/25`, and the pin density `e^{−12k²}` contributes `−12`. | proof (given the model) | +| **Proposition 4** | Gaussian kernel, every `d`: the fold-scale rejection rate is `F₀(b)H(k)`, with `H` independent of `b` and `d`, and `H(k) = 1 + (12/25)k² + O(k³)`. Codex's Slice C review proves the upper remainder `O(k^{7/2})`. The elder edge `φ_e = 1/3 + t/3 + 10t²/27` contributes `+312/25`, and the pin density `e^{−12k²}` contributes `−12`. #244 (open, conditional) gives the exact `k^{7/2}` and `k⁴` terms. | proof (given the model) | | **Lemma 5** | The composite `∫∫∫𝒜^{rej}(b, ℓ/r⁴, u)H(ℓ/r³)` equals `(I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`, where `R_{2/3} = ∫∫∫v⁴[F(v^{−3}) − F₀]` | proof | | **Conjecture 6** | `(k/r)r^{−2}A_r^{rej}(b, k, u) → F(k; b, u)` as `r → 0` at fixed `k`. The limit exists by #170/#175 (merged); its identification with `F` is #243 (open). | proved author-side in #243 | -| **Conjecture 7** | `ρ_rej + ν_eld^{far,r_0^*} = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`; Gaussian kernel: `R_{2/3} ≈ −0.049 ± 0.003` (`d = 2`), `−0.062 ± 0.004` (`d = 3`); these `±` are quadrature and interpolation sensitivity ranges (PROOF §5(5)), not sampling errors | conjecture, with evidence | +| **Conjecture 7** | `ρ_rej + ν_eld^{far,r_0^*} = B_{d,L} + (I^{cand} − c₁)ℓ^{1/4} + R_{2/3}ℓ^{2/3} + O(ℓ^{3/4})`; Gaussian kernel: `R_{2/3} ≈ −0.049 ± 0.003` (`d = 2`), `−0.062 ± 0.004` (`d = 3`); these `±` are quadrature and interpolation sensitivity ranges (PROOF §5(5)), not sampling errors. The v1.3 point values are `−0.0488` and `−0.0614`. | conjecture, with evidence | Conjecture 7 needs two inputs that are not yet available (PROOF §4): - (i) the soft layer `λ₁ ≍ r`, together with Proposition 2′ for the field in `d ≥ 3`; @@ -47,8 +58,8 @@ A third input, (iii) the far elder density, is needed only to remove `ν_eld^{fa |---|---| | Theorem 1, `d = 2` | `κ𝒜^{rej}(0, κ)/F₀(0) = 0.751, 0.977, 0.9985, 0.9999` at `κ = 1, 4, 16, 64`; the `1/κ` coefficient `c(b) = 0.3b` is confirmed at `b = ±1` | | Theorem 1, `d = 3` | `κ∫∫𝒜^{rej}/∫∫F₀ = 0.670, 0.9899, 0.99936, 0.99994` at `κ = 1, 10, 100, 1000`; `∫∫F₀` agrees with Corollary 1′ to `10^{−5}`; per `b` the ratio tends to `1` at rate `1/κ`; `∫G(s^{−4})ds = 0.072444` reproduces #207's `I^{cand} − c₁ = 0.0724443` | -| Lemma 2 | agrees with a two-dimensional flood fill on 900 random typed parameter points (one near-degenerate point needs a smaller level offset) | -| `H` | `H(0.2) = 1.004`, `H(0.4) = 0.727`, `H(0.6) = 0.201`, `H(0.8) = 0.018`. With `C₃` ignored, `R_{2/3}` (`d = 2`) would be `−0.104`; with the pin density alone, `−0.807`. | +| Lemma 2 | agrees with a two-dimensional flood fill on 900 random typed parameter points (one near-degenerate point needs a smaller level offset). **Disclosure (v1.3):** the scanner `dec1d.py` that implements Lemma 2 for the numbers errs in thin layers on either side of `β = 2`: just above, it misses a fold and falsely rejects; just below, with `χ < 0`, its tolerance `10^{−3}` falsely accepts. The flood fill's level offset was too large to expose this. The closed form of #244 found it. It moves `H` by at most `2.0·10^{−4}`; the values below are corrected (PROOF §5(2), (4)). | +| `H` | `H(0.2) = 1.004`, `H(0.4) = 0.727`, `H(0.6) = 0.201`, `H(0.8) = 0.018`; v1.3 recomputes them from #244's closed form (v1.2's fourth decimals at `k = 0.1, 0.3–0.6` differed by up to `2·10^{−4}`). For small `k`, `H − H₀ ≈ 66.56k⁴` (v1.2 fitted `62k⁴`). `Ĩ = −0.5337`, so `R_{2/3} = −0.0488` (`d = 2`) and `−0.0614` (`d = 3`); v1.2 had `Ĩ = −0.536`. With `C₃` ignored, `R_{2/3}` (`d = 2`) would be `−0.104`; with the pin density alone, `−0.807`. | | #216's Monte Carlo, `d = 2` | rejected adjacent pairs with `s = r/ℓ^{1/4} < 0.8`: 37 observed, 35.3 from the composite, 4,322 from the cusp kernel alone | | #216's Monte Carlo, `d = 3` | `s < 0.8`: 11 observed, 10.7 composite (1.2 with the pin density alone), 1,329 cusp kernel alone. On `[3·10^{−4}, 10^{−2}]` the counts are `0.884 ± 0.040` of the `ℓ^{1/4}` law and `1.008` of the composite (Pearson χ² 11.8 against 2.1 on 8 bins). Over #216's whole range `[10^{−4}, 0.3]` the composite does not describe the counts (finite-`r` corrections). | @@ -111,6 +122,8 @@ Mutants `M1`–`M14` each fail only their own control (the checker names it on s - **Nonauthor reviews of v1.2 (OpenAI Codex), at `1f86fea`, PROOF blob `ad4beb84`:** - Bounded readback of B-01, D-01, the exact (D′) witness and the strengthened `β > 2`, `χ ≤ 0` rejection (5952577050): PASS_TECHNICAL_SCOPED, no new finding. It adds a direct finite-path proof of that rejection. - Complementary delta review (5391986303): PASS_TECHNICAL_SCOPED for the changed proof corrections and the conditional source alignment. Two minor README findings, V12-01 and V12-02, are applied in the v1.2 README amendment. Its §3 is root-authored support (D.1): a direct model-integral proof of `F = kA∗a_fail` in every fixed dimension, not reviewed here. + - D.1 was then reviewed by a different Codex agent, on #243 (5954510413): PASS_TECHNICAL_SCOPED, no finding, against this PROOF at blob `ad4beb84` (§§2–3, unchanged in v1.3). +- **xAI / Grok on the v1.2 README amendment (`b82b0ca`).** Head receipt 5393329837 and comment 5955184422: NOT READY. The branch was behind `main`; Conjecture 7 is open; `R_{2/3}` is a sensitivity range, not a certified enclosure; and the scanner disclosure must be in the landed note. v1.3 is rebased on `main` and puts the disclosure in PROOF §5 and in this README. It does not change Conjecture 7's status or the ranges. - **v1.2 delta (same family).** A clean-context referee checked the changed bytes. - Verdict: **ACCEPT WITH FIXES**, with no BLOCKING or MAJOR finding: 9 MINOR findings and 8 NITs, all applied. - Its own checks: the witness by sympy, the Jacobians `−96`, `−3/2` and `−6`, and the `48/47` ratio. @@ -122,6 +135,8 @@ Mutants `M1`–`M14` each fail only their own control (the checker names it on s - Conjectures 6 and 7 are not proved here, and `R_{2/3}` is not certified. Conjecture 6 is proved author-side in #243, and its existence part is merged (#170/#175). +- The v1.3 values rest on #244's closed form, which is conditional on #170 Theorem E(1) and [CUB] Theorem C, and on + floating-point quadrature. They are exploration. - Proposition 2′ is a statement about the limit model. Its transfer to the field is #175 Theorem H (for #170's cubic), and #243 Proposition FL.4. - No priority for the existence of the fold-scale limit or for the compact-window `ℓ^{2/3}` coefficient (#170, #175). diff --git a/frontiers/soft_rejected_pairs_20261002/SOURCES.json b/frontiers/soft_rejected_pairs_20261002/SOURCES.json index 735281b9e..a0a1bf684 100644 --- a/frontiers/soft_rejected_pairs_20261002/SOURCES.json +++ b/frontiers/soft_rejected_pairs_20261002/SOURCES.json @@ -1,12 +1,12 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-SOFT-REJECTED-20261002-v1.2", - "kind": "author-side proof candidate and conjectures, every d >= 2 (v1.2: applies OA-242-A-01, B-01, B-02, B-03, C-01, C-02, D-01, D-02, E-01, E-02; exact (D-prime) witness; Lemma 2 now states that beta > 2, chi <= 0 is always rejected; prior work #170/#175 recorded, AUTH-242-01; Conjecture 6 status: existence merged (#170/#175), identification #243): Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 D with D the density at 0 of the smallest eigenvalue lambda_1 of -A weighted by gamma_1^6 (lambda_2...lambda_m)^2; d = 2: D = pA(0|b) E gamma^6); Corollary 1' (Gaussian kernel: int int F0 = 25 sqrt3/(48 pi^2) in d = 2, 125 sqrt30/(192 pi^3) in d = 3); the fold-scale soft model G_k with the jets B = d_u A and C3 = d_Theta^3 f along the soft direction; Lemma 2 (the elder decision in the model by a one-dimensional scan, including a case where the saddle is a slice minimum); Proposition 2' (d >= 3: the stiff directions at scale r^(3/2) decouple; same elder decision); Lemma 3 (an a priori elder region); Proposition 4 (Gaussian kernel, every d: the fold-scale rejection rate is F0(b) H(k) with H independent of b and d, H = 1 + (12/25) k^2 + O(k^3)); Lemma 5 (the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4))); Conjecture 6 (the fold-scale limit) and Conjecture 7 (rho_rej + nu_eld^far = B_{d,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 (d = 2) and -0.062 (d = 3) for the Gaussian kernel, not certified)", + "object": "CL-SOFT-REJECTED-20261002-v1.3", + "kind": "author-side proof candidate and conjectures, every d >= 2 (v1.3: author correction AUTH-242-02 of section 5 only: two artifacts of the scanner dec1d.py in thin layers on either side of beta = 2 are disclosed, and H, H0, Delta_chi/k^4 -> 66.56, I~ = -0.5337 and R_{2/3} = -0.0488 (d = 2), -0.0614 (d = 3) are recomputed from the closed form of Math- #244, inside the v1.2 ranges; no statement, proof or control changes; v1.2: applies OA-242-A-01, B-01, B-02, B-03, C-01, C-02, D-01, D-02, E-01, E-02; exact (D-prime) witness; Lemma 2 now states that beta > 2, chi <= 0 is always rejected; prior work #170/#175 recorded, AUTH-242-01; Conjecture 6 status: existence merged (#170/#175), identification #243): Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 D with D the density at 0 of the smallest eigenvalue lambda_1 of -A weighted by gamma_1^6 (lambda_2...lambda_m)^2; d = 2: D = pA(0|b) E gamma^6); Corollary 1' (Gaussian kernel: int int F0 = 25 sqrt3/(48 pi^2) in d = 2, 125 sqrt30/(192 pi^3) in d = 3); the fold-scale soft model G_k with the jets B = d_u A and C3 = d_Theta^3 f along the soft direction; Lemma 2 (the elder decision in the model by a one-dimensional scan, including a case where the saddle is a slice minimum); Proposition 2' (d >= 3: the stiff directions at scale r^(3/2) decouple; same elder decision); Lemma 3 (an a priori elder region); Proposition 4 (Gaussian kernel, every d: the fold-scale rejection rate is F0(b) H(k) with H independent of b and d, H = 1 + (12/25) k^2 + O(k^3)); Lemma 5 (the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4))); Conjecture 6 (the fold-scale limit) and Conjecture 7 (rho_rej + nu_eld^far = B_{d,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 (d = 2) and -0.061 (d = 3) for the Gaussian kernel, not certified)", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { - "branch_base": "f5347541a6ef7b84de96120dcce6137b816b8033" + "branch_base": "a808ba25de9db22110db8b7555a64852f38e031d" }, "consumed": [ { @@ -158,13 +158,23 @@ }, { "path": "frontiers/soft_fold_limit_20261002/PROOF.md", - "bytes": 72241, - "sha256": "9047cc30a37af0aed43ed289d6f94ce31a2a3607bd67c71195158390a48d67d2", - "git_blob": "caedaef6cbb0ebaad647c2f091a6835bf3228697", + "bytes": 72742, + "sha256": "7716a113bcf8c1725f9ec771d58ef033767f0a7bb50d592b841ca0ffe86f31af", + "git_blob": "3d9faa87d717fbd68b784b2ae408505fb2cc56dc", "tag": "243", - "use": "Math- #243 (open): Theorem FL and Proposition FL.7 (Conjecture 6: the identification of the fold-scale limit with F, continuity, local uniformity), Lemma FL.2(c)", - "head": "0740b2cae546866c5f5f27e770ba023bde64bbe8", + "use": "Math- #243 (open; its five nonauthor slice reviews are at this head): Theorem FL and Proposition FL.7 (Conjecture 6: the identification of the fold-scale limit with F, continuity, local uniformity), Lemma FL.2(c)", + "head": "11a4cd4e3b3f927eb9d4a743d8a43e387f1a66da", "pr": "Math- #243" + }, + { + "path": "frontiers/soft_closed_form_20261002/PROOF.md", + "bytes": 46949, + "sha256": "a6c29a529af87da69ea7929bccd2ad81cad313cecb235ad8bf2dea274d2acb96", + "git_blob": "310ef82832a51bc876494f72fc3f002ea2bfaea0", + "tag": "244", + "use": "Math- #244 (open; conditional on #170 Theorem E(1) and [CUB] Theorem C): Theorem A (the closed form of the rejected set and of I), Corollary A.1 (t* = 2/3), Theorem B (H to order k^4, the exact k^(7/2) and k^4 coefficients), section 5 (its numerics); section 5 here uses it for the v1.3 values", + "head": "1781c0fd37469a0840f3c768d2b505ef6319133e", + "pr": "Math- #244" } ], "cited_external": [], @@ -219,13 +229,13 @@ "files": [ { "path": "PROOF.md", - "bytes": 79394, - "sha256": "ab952b2a2c7648beb5f7473b3f028cdc495c8317c52522bee144d26eecc8e2bb" + "bytes": 87672, + "sha256": "2d6981593fc32533fd3f3a25c8c424bff829c7b2baa47da148b9502520e62467" }, { "path": "README.md", - "bytes": 13878, - "sha256": "bd43ee1b5833a6069381aadbd12159316e08569b105be348bd640c3db0e01272" + "bytes": 16712, + "sha256": "ff9cccc50084953833c3b43760388989e9be32c721f915ac3c170c4b26740ae8" }, { "path": "soft_check.py", @@ -239,9 +249,10 @@ } ], "supersedes": { - "object": "CL-SOFT-REJECTED-20261002-v1.1", - "head": "87912bd8f28fe0d31fcb525670d1fd71a00e0db1", - "proof_blob": "8b2f5fae2a0014a401003463d7c2b2221e7124f5" + "object": "CL-SOFT-REJECTED-20261002-v1.2", + "head": "b82b0ca16926666adb8895c21801b6e85d7e18df", + "proof_blob": "ad4beb8467a775c7c76f85d5584c13338d0e66be", + "note": "b82b0ca is the v1.2 README amendment; the v1.2 PROOF commit was 1f86fea (same PROOF blob)" }, "exploration_manifest": { "location": "claude.ai project archive V2_2/frontiers_soft_rejected_pairs_20261002/ (and the local archive zip SIDE24_V2_2_2026-09-29.zip); not part of the repository", @@ -583,7 +594,85 @@ } ], "raw_monte_carlo": "batchA2.npz (d = 2, L = 64) and batch3B.npz (d = 3, L = 16), records of Math- #216, delivered as SIDE24_MC_raw_2026-10-01.zip; not pinned here by hash", - "status": "author-reported exploration, not independently reproduced (OA-242-E-01)" + "status": "author-reported exploration, not independently reproduced (OA-242-E-01)", + "v13_files": [ + { + "path": "exploration/v13/Hcf.py", + "bytes": 2651, + "sha256": "36787ccaf7dbd42ffcf21b4c138e2266aeb6e47a0e3f5b81553782124841ad96" + }, + { + "path": "exploration/v13/Hcf_grid.log", + "bytes": 465, + "sha256": "b42481ae7c05551d0b7ee815ad7ab0f08ac337f44bf564a32f1a5c1b0f7852a1" + }, + { + "path": "exploration/v13/Hcf_grid_nochi_v2.log", + "bytes": 465, + "sha256": "58dac2db3aa20d2f93eb5519f79448f27a852c071b690e4fdd339deffe1efe83" + }, + { + "path": "exploration/v13/Itilde.py", + "bytes": 1306, + "sha256": "86f7bb01ceaa534550ab01f53ee2c7a42f11cc1202175300460e13e758f925be" + }, + { + "path": "exploration/v13/Itilde_16_60.log", + "bytes": 258, + "sha256": "9703c947dbf2e027715f58a937ee4ed102bcb50bbe542cd58b53b3f26b3a222d" + }, + { + "path": "exploration/v13/Itilde_24_80.log", + "bytes": 258, + "sha256": "545bcf4a273236cce107daef85f6f81fcf2c698673f51c510a30676be5db678f" + }, + { + "path": "exploration/v13/Itilde_headfix.log", + "bytes": 480, + "sha256": "84fa0caa234243d9a780613c94de06a592bfa8f5d2c91f0ce696641f524dbbe4" + }, + { + "path": "exploration/v13/Itilde_headfix.py", + "bytes": 778, + "sha256": "ee003377068c55ac8db454801b9d62ef99401f9b089e78367fbaa534373f4f46" + }, + { + "path": "exploration/v13/README.md", + "bytes": 2365, + "sha256": "2ef3f9b975b93d635a0f4f9b25f230bdf0a042e0b88d0731f717d2ccab1104f6" + }, + { + "path": "exploration/v13/cf.py", + "bytes": 1784, + "sha256": "5861010443118e0407fa78ba30e8ae9eaeb044a6fe1676ab4db02bb6be8f36af" + }, + { + "path": "exploration/v13/g6i.py", + "bytes": 2261, + "sha256": "ec81a1631304f2dc17b254a61452a46f012e22797aedccc4d00fba20537ce1c3" + }, + { + "path": "exploration/v13/itilde_split.py", + "bytes": 2042, + "sha256": "f2603cc204cc0f6ce07130d8d07100b62ba2df6be8f4a69a19eec471e1a8cefd" + }, + { + "path": "exploration/v13/split_chi_16_60.log", + "bytes": 383, + "sha256": "55d0eaba57e0067bb6fc7807320352b1e46b3cd05967914cd6c368b782e8a73f" + }, + { + "path": "exploration/v13/split_nochi_16_60.log", + "bytes": 381, + "sha256": "43273dc5535e41010c9338b89588abcf24e6b490ed5f4fd2a2e5b08e76b3205a" + }, + { + "path": "exploration/v13/split_nochi_24_80.log", + "bytes": 381, + "sha256": "739406306a03d914c02c87185d013d215cbe79c4fc6b66e270380f1e37d94521" + } + ], + "v13_note": "v1.3 (AUTH-242-02): the scripts and logs behind section 5's corrected values (H and H0 rows, the I~ split by v-range, I~ with and without C3), evaluating the closed form of Math- #244; the comparison with the I table, the classification of its 97 outliers and the two dec1d.py artifacts are #244's exploration, pinned in #244's SOURCES.json" }, "nonauthor_reviews": [ { @@ -667,6 +756,25 @@ ], "applied_in": "v1.2 README amendment", "note": "its section 3 (D.1) is root-authored support: a direct model-integral proof of F = k A_* a_fail in every fixed d; not reviewed here" + }, + { + "slice": "D.1 supplement (root-authored section 3 of review 5391986303): the direct model identity F = k A_* a_fail in every fixed d", + "reviewer": "OpenAI Codex agent prioritize_program_closure_work (delegated AI review; nonauthor of D.1; same GitHub account; organizational independence 0)", + "comment": "Math- #243 5954510413", + "at": "b82b0ca (PROOF blob ad4beb84)", + "verdict": "PASS_TECHNICAL_SCOPED", + "findings": [], + "note": "reviews D.1 against this PROOF's sections 2-3 (unchanged in v1.3); not a review of this packet's proofs" + }, + { + "slice": "head receipt on the v1.2 README amendment", + "reviewer": "xAI / Grok team (same GitHub account; organizational independence 0)", + "review": "pullrequestreview-5393329837", + "comment": "5955184422", + "at": "b82b0ca", + "verdict": "NOT READY (not a proof acceptance or a merge vote): behind main; Conjecture 7 open; R_{2/3} a sensitivity range, not a certified enclosure; the scanner disclosure must be in the landed note", + "findings": [], + "applied_in": "v1.3 (rebased on main; the disclosure is in PROOF section 5 and the README)" } ], "amendments": [ @@ -685,6 +793,28 @@ "OA-242-V12-01", "OA-242-V12-02" ] + }, + { + "label": "v1.3", + "files_changed": [ + "PROOF.md", + "README.md", + "SOURCES.json" + ], + "unchanged": [ + "soft_check.py", + "RESULTS.json" + ], + "proof_sections_unchanged": [ + "0", + "1", + "2", + "3" + ], + "findings": [ + "AUTH-242-02 (author: two dec1d.py artifacts near beta = 2; section 5 numbers corrected; disclosed in 5953336662, corrected in 5955380848)" + ], + "requested_by": "xAI 5955184422 (disclosure in the landed note); Codex 5954054126 (PR242 held for the promised section 5 correction)" } ] } From ffc7005509554a378edd81136aa6e1f4fbd0eff8 Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 16:45:57 +0000 Subject: [PATCH 6/7] Soft rejected pairs v1.4: apply the C78 delta review of v1.3 (OA-242-V13-01/02/03) Reporting corrections in PROOF section 5 and the header, as requested in Codex review 5394316763 (findings first sent in 5956579220): - V13-01: the 2.0e-4 comparison of H is the largest difference observed at the 15 tabulated k of v1.2, not a certified uniform bound; it also contains v1.2's quadrature and table-interpolation errors. - V13-02: section 5(3)'s expansions state their limits (t -> -infinity and |chi0| -> infinity); for t >= 1, I(t, 0) = t - 2/3. - V13-03: the inherited sensitivity is +-0.002*10^(5/3)/5 = +-0.0186 (v1.2 printed +-0.018); the ranges stand. - C78's rounding note: section 5(5) prints I~ = -0.5336676. SOURCES.json pins the raw Monte Carlo archive by hash. Sections 0-4 and 6-8, soft_check.py and RESULTS.json are byte-identical to v1.3. Scientific effect NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .../soft_rejected_pairs_20261002/PROOF.md | 52 +++++++++-- .../soft_rejected_pairs_20261002/README.md | 16 +++- .../soft_rejected_pairs_20261002/SOURCES.json | 88 ++++++++++++++++--- 3 files changed, 133 insertions(+), 23 deletions(-) diff --git a/frontiers/soft_rejected_pairs_20261002/PROOF.md b/frontiers/soft_rejected_pairs_20261002/PROOF.md index 5fd29e63e..c15929151 100644 --- a/frontiers/soft_rejected_pairs_20261002/PROOF.md +++ b/frontiers/soft_rejected_pairs_20261002/PROOF.md @@ -1,10 +1,24 @@ # Soft rejected pairs: the `1/κ` tail of the rejected cusp kernel in every dimension, the fold-scale rejection function, and a conjectured `ℓ^{2/3}` term -Object: CL-SOFT-REJECTED-20261002-v1.3. Version history: +Object: CL-SOFT-REJECTED-20261002-v1.4. Version history: +- v1.3 → v1.4 applies the three findings of OpenAI Codex's C78 delta review of v1.3 (review 5394316763; the findings + were first sent in 5956579220). It changes the header and §5 only. No statement, proof or control changes, and no + tabulated value changes. + - OA-242-V13-01: the `2.0·10^{−4}` comparison of `H` is now stated as what it is. It is the largest difference observed + at the 15 values of `k` in v1.2's table, not a certified uniform bound, and it also contains v1.2's quadrature and + table-interpolation errors. + - OA-242-V13-02: §5(3)'s two expansions now state their limits, `t → −∞` and `|χ₀| → ∞`. + - OA-242-V13-03: the inherited sensitivity is computed exactly. A shift of `H` by `±0.002` on `k ≥ 0.1` moves `Ĩ` by + `±0.002·10^{5/3}/5 = ±0.0186`, not `±0.018`. The ranges stand. + - C78 also noted (not a finding) that `0.1150058 × 0.53367` rounds to `0.06138`, not to the printed `0.06137`. §5(5) now + prints `Ĩ = −0.5336676`, the value from which `R_{2/3}` was computed. + - `SOURCES.json` also pins the raw Monte Carlo archive by hash. §§0–4, `soft_check.py` and `RESULTS.json` are + byte-identical to v1.3, and §9 lists the changed bytes. - v1.2 → v1.3, an author correction of §5's numbers (AUTH-242-02). It was disclosed in PR comments 5953336662 and 5955380848; xAI asked in 5955184422 that it be in the landed note. No statement, proof or control changes. - §5(2) records two artifacts of the one-dimensional scanner `dec1d.py`, in thin layers on either side of `β = 2`. They - are in the implementation, not in Lemma 2, and they move `H` by at most `2.0·10^{−4}`. + are in the implementation, not in Lemma 2. At the 15 values of `k` in v1.2's table, v1.2's `H` differs from the + recomputed `H` by at most `2.0·10^{−4}`. That is an observed difference, not a certified bound (v1.4). - §5(4) gives the `H` and `H₀` rows recomputed, the accuracy sentence, and the small-`k` coefficient of `Δ_χ` (`66.56`, exact, in place of the fitted `62`). - §5(5), §4 Remark 2, and the numerical remarks in "What is new" and Conjecture 7: `Ĩ = −0.5337`, so @@ -789,8 +803,11 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. `≈ a(½)²/|χ|`), and the axial segment joins `M` to the far region above `L_S`, so the pair is rejected. - The flood fill errs the other way here: its level offset `ε` exceeds the relevant level gaps. So the comparisons above did not expose the artifacts. - - The artifacts move `H` by at most `2.0·10^{−4}` (at `k = 0.4`). Items 4–5 give the corrected values. Control S8 - stays clear of this region (§7). + - At the 15 values of `k` in v1.2's table (`k = 0.1, 0.15, …, 0.6` and `0.7, 0.8, 0.9, 1.0`), v1.2's `H` differs from + the recomputed `H` of item 4 by at most `2.0·10^{−4}`, at `k = 0.4`. This is the largest observed difference at + those points, not a certified uniform bound on the artifacts' effect, and it also contains v1.2's quadrature and + table-interpolation errors (OA-242-V13-01). + - Items 4–5 give the corrected values. Control S8 stays clear of this region (§7). 3. *The function `I`.* - A `101 × 121` table of `I(t, χ₀)` on `|t| ≤ 30`, `|χ₀| ≤ 2000` (`Itab2d_v2.py`), and an exact evaluator for other points (`Ifast.py`). @@ -804,7 +821,10 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. differ by more than `10^{−3}` are explained: 7 tiny intervals the grid misses, 54 table resolution, and 36 the artifacts of item 2. - In particular `t* = 2/3` exactly (§2 has `t* ∈ (0.65, 0.68)`), and the two limits above have the exact next terms - `I(t, 0) = (4/3)|t|³ + 3√3|t|^{5/2} + O(t²)` and `I(0, χ₀) = χ₀²/48 + 16^{−2/3}|χ₀|^{4/3} + O(|χ₀|)`. + `I(t, 0) = (4/3)|t|³ + 3√3|t|^{5/2} + O(t²)` as `t → −∞` and + `I(0, χ₀) = χ₀²/48 + 16^{−2/3}|χ₀|^{4/3} + O(|χ₀|)` as `|χ₀| → ∞` (OA-242-V13-02). For `t ≥ 1`, instead, + `I(t, 0) = t − 2/3` exactly (#244 Corollary A.1). At `t = 0`, Theorem A's formula depends on `χ₀` only through + `(χ₀ + 8)²`, so the second expansion holds in both directions. 4. *The function `H`.* `H = H₀ + Δ_χ`. - `H₀`, the value with `C₃` ignored, comes from a 639-point table of `I(t, 0)` on `|t| ≤ 400`, its asymptotic forms beyond (`|t|³(4/3 + 5.74|t|^{−1/2})` and `t − 2/3`), and adaptive quadrature in `t` and `γ²` down to `γ → 0` (`Hk0b.py`). Its @@ -814,12 +834,14 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. range and the exact evaluator outside it. - Accuracy checks (v1.2): without the table, `Δ_χ(0.2)` changes by `2·10^{−5}`; with 48 nodes in `|γ|`, `H` changes by `5·10^{−5}`; and the quadrature `H` agrees with a table-free direct quadrature at `k = 0.2` to `1.3·10^{−4}`. All three - checks use `dec1d.py`, so none of them measures its bias (item 2). That bias reaches `2.0·10^{−4}` at `k = 0.4`. + checks use `dec1d.py`, so none of them measures its bias (item 2). Against the recomputed values, v1.2's `H` is off + by up to `2.0·10^{−4}` at the tabulated `k` (at `k = 0.4`; an observed difference, item 2). - *Recomputed values (v1.3).* The rows below come from #244's closed form of `I`. `H` and `H₀` are evaluated by nested Gauss–Legendre quadrature over `(|γ|, B, C₃)`, split at the kinks of the integrand. Raising the node counts from `60³` to `100³` changes them by less than `10^{−10}`, and #244's referee reproduces the `H` row with an independent quadrature. - - The v1.2 rows differed by at most `2.0·10^{−4}` (`H`) and `1.3·10^{−5}` (`H₀`). + - At the 15 values of `k` in v1.2's table, the v1.2 values differ from these by at most `2.0·10^{−4}` (`H`, at + `k = 0.4`) and `1.3·10^{−5}` (`H₀`, at `k = 0.35`). These are observed differences, not certified bounds. - At four decimals the v1.2 rows had `H = 1.0039, 0.9345, 0.7271, 0.4397, 0.2007` at `k = 0.1, 0.3, 0.4, 0.5, 0.6` and `H₀ = 0.4663` at `k = 0.4`. - *Small `k`.* `Δ_χ/k⁴ → 53248/800 = 66.56` (#244 Theorem B, Step 3); v1.2 used the fitted value `62`. #244 Theorem B @@ -836,7 +858,7 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. | `e^{−12k²}` | 0.887 | 0.619 | 0.340 | 0.147 | 0.050 | 0.013 | 0.003 | 0.0005 | 0.000006 | 5. *`R_{2/3}`.* - - *v1.3.* With `H` from #244's closed form, `Ĩ = −0.53367`. The quadrature is Gauss–Legendre in `q = k^{1/3}` on + - *v1.3.* With `H` from #244's closed form, `Ĩ = −0.5336676`. The quadrature is Gauss–Legendre in `q = k^{1/3}` on `[0.1, 1.2]`, with #244 Theorem B's expansion below `q = 0.1` and `H = 0` above `q = 1.2`, where `H < 2·10^{−13}`. With 16 and 24 nodes per segment the results agree to `2.5·10^{−8}`. - So `R_{2/3} = 0.0914028·Ĩ = −0.04878` in `d = 2` and `R_{2/3} = 0.1150058·Ĩ = −0.06137` in `d = 3`. Both lie inside @@ -854,7 +876,9 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. - v1 quoted `−0.537` for "a dense trapezoid rule". That was the `R23` field stored in `H_dense2.json`, produced by an unrecorded rule (finding A-1, found by the author while archiving v1). - So v1.2 had `R_{2/3} = 0.0914028·Ĩ = −0.049` in `d = 2` and `R_{2/3} = 0.1150058·Ĩ = −0.0617` in `d = 3`. - - *Sensitivity (v1.2; the ranges stand).* A uniform shift of `H` by `±0.002` on `k ≥ 0.1` moves `Ĩ` by `±0.018`. Halving + - *Sensitivity (v1.2; the ranges stand).* A uniform shift of `H` by `±0.002` on `k ≥ 0.1` (that is, `v ≤ 10^{1/3}`) + moves `Ĩ` by exactly `±0.002·∫_0^{10^{1/3}}v⁴dv = ±0.002·10^{5/3}/5 = ±0.0186` (v1.2 printed `±0.018`; + OA-242-V13-03). Halving v1.2's small-`k` term `62k⁴` moves it by `−0.021`, and multiplying it by `1.5` by `+0.021`. So `Ĩ = −0.54 ± 0.03`, `R_{2/3} = −0.049 ± 0.003` in `d = 2` and `R_{2/3} = −0.062 ± 0.004` in `d = 3`. - *Comparisons.* With `C₃` ignored (`H = H₀`), `Ĩ = −1.134` and `R_{2/3} = −0.104` (v1.3; v1.2 had `Ĩ = −1.133`). With @@ -991,6 +1015,16 @@ Mutants `M1`–`M13` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, - **E** (§§5–7): the evidence is described accurately and kept separate from the proofs, and the controls match their claims. +**Changed bytes in v1.4** (for a delta check against the C78 review of v1.3, 5394316763): +- *Header:* the object label, the new v1.4 entry, and the `2.0·10^{−4}` sentence of the v1.3 entry (V13-01). +- *§5:* the last item of item 2 (V13-01); the expansions in item 3 (V13-02); item 4's accuracy sentence and the + comparison sentence after the recomputation (V13-01); item 5's first sentence (`Ĩ` printed to seven digits, C78's + rounding note) and its sensitivity sentence (V13-03). +- *§9:* this list. + +§§0–4, §§6–8, every value in §5's tables, every other value in §5, `soft_check.py` and `RESULTS.json` are unchanged. +The only changed numbers are the sensitivity `±0.0186` (V13-03) and the extra digits of `Ĩ` in item 5. + **Changed bytes in v1.3** (for a delta check against the v1.2 reviews; AUTH-242-02): - *Header:* the object label and version history, the closing numerical item of "What is new" (`≈ −0.061` in `d = 3`), and the Dependencies (#243's reviewed head, #244). diff --git a/frontiers/soft_rejected_pairs_20261002/README.md b/frontiers/soft_rejected_pairs_20261002/README.md index 79d80d0ec..ac2668f97 100644 --- a/frontiers/soft_rejected_pairs_20261002/README.md +++ b/frontiers/soft_rejected_pairs_20261002/README.md @@ -2,12 +2,22 @@ **Author-side proof candidate and conjectures (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** -Object `CL-SOFT-REJECTED-20261002-v1.3`. Full text: [`PROOF.md`](PROOF.md). Version history: +Object `CL-SOFT-REJECTED-20261002-v1.4`. Full text: [`PROOF.md`](PROOF.md). Version history: +- v1.4 applies the three findings of OpenAI Codex's C78 delta review of v1.3 (review 5394316763, AMEND; findings first + sent in 5956579220). The changes are to the reporting in PROOF §5 and the header. + - OA-242-V13-01: the `2.0·10^{−4}` comparison is the largest difference observed at v1.2's tabulated `k`, not a + certified bound. + - OA-242-V13-02: the limits of §5(3)'s expansions are stated. + - OA-242-V13-03: the sensitivity `±0.018` is computed exactly, as `±0.0186`. + - C78's rounding note: PROOF §5(5) prints `Ĩ = −0.5336676`, from which `R_{2/3}` is computed. + - `SOURCES.json` pins the raw Monte Carlo archive by hash. PROOF §§0–4, `soft_check.py` and `RESULTS.json` are + unchanged. - v1.3 is an author correction of PROOF §5's numbers (AUTH-242-02), first disclosed in PR comments 5953336662 and 5955380848. xAI asked (5955184422) that it be in the landed note; it is in PROOF §5 and below. - The one-dimensional scanner `dec1d.py` behind §5's `I` table and `H` has two artifacts, in thin layers on either side of `β = 2`. Just above, it misses a fold and falsely rejects; just below, with `χ < 0`, its tolerance `10^{−3}` falsely - accepts. These are errors of the implementation, not of Lemma 2. They move `H` by at most `2.0·10^{−4}`. + accepts. These are errors of the implementation, not of Lemma 2. At v1.2's tabulated `k`, v1.2's `H` differs from the + recomputed `H` by at most `2.0·10^{−4}`. That is an observed difference, not a certified bound (v1.4). - The corrected values come from the closed form of `I` in Math- #244 (open; conditional on #170 Theorem E(1) and [CUB] Theorem C): `H` and `H₀` rows, `Δ_χ/k⁴ → 66.56` (v1.2 fitted `62`), `Ĩ = −0.5337`, and `R_{2/3} = −0.0488` (`d = 2`) and `−0.0614` (`d = 3`). These are inside v1.2's sensitivity ranges, which stand. Most of the change in @@ -58,7 +68,7 @@ A third input, (iii) the far elder density, is needed only to remove `ν_eld^{fa |---|---| | Theorem 1, `d = 2` | `κ𝒜^{rej}(0, κ)/F₀(0) = 0.751, 0.977, 0.9985, 0.9999` at `κ = 1, 4, 16, 64`; the `1/κ` coefficient `c(b) = 0.3b` is confirmed at `b = ±1` | | Theorem 1, `d = 3` | `κ∫∫𝒜^{rej}/∫∫F₀ = 0.670, 0.9899, 0.99936, 0.99994` at `κ = 1, 10, 100, 1000`; `∫∫F₀` agrees with Corollary 1′ to `10^{−5}`; per `b` the ratio tends to `1` at rate `1/κ`; `∫G(s^{−4})ds = 0.072444` reproduces #207's `I^{cand} − c₁ = 0.0724443` | -| Lemma 2 | agrees with a two-dimensional flood fill on 900 random typed parameter points (one near-degenerate point needs a smaller level offset). **Disclosure (v1.3):** the scanner `dec1d.py` that implements Lemma 2 for the numbers errs in thin layers on either side of `β = 2`: just above, it misses a fold and falsely rejects; just below, with `χ < 0`, its tolerance `10^{−3}` falsely accepts. The flood fill's level offset was too large to expose this. The closed form of #244 found it. It moves `H` by at most `2.0·10^{−4}`; the values below are corrected (PROOF §5(2), (4)). | +| Lemma 2 | agrees with a two-dimensional flood fill on 900 random typed parameter points (one near-degenerate point needs a smaller level offset). **Disclosure (v1.3):** the scanner `dec1d.py` that implements Lemma 2 for the numbers errs in thin layers on either side of `β = 2`: just above, it misses a fold and falsely rejects; just below, with `χ < 0`, its tolerance `10^{−3}` falsely accepts. The flood fill's level offset was too large to expose this. The closed form of #244 found it. At v1.2's tabulated `k`, v1.2's `H` differs from the corrected values by at most `2.0·10^{−4}` (an observed difference, not a certified bound); the values below are corrected (PROOF §5(2), (4)). | | `H` | `H(0.2) = 1.004`, `H(0.4) = 0.727`, `H(0.6) = 0.201`, `H(0.8) = 0.018`; v1.3 recomputes them from #244's closed form (v1.2's fourth decimals at `k = 0.1, 0.3–0.6` differed by up to `2·10^{−4}`). For small `k`, `H − H₀ ≈ 66.56k⁴` (v1.2 fitted `62k⁴`). `Ĩ = −0.5337`, so `R_{2/3} = −0.0488` (`d = 2`) and `−0.0614` (`d = 3`); v1.2 had `Ĩ = −0.536`. With `C₃` ignored, `R_{2/3}` (`d = 2`) would be `−0.104`; with the pin density alone, `−0.807`. | | #216's Monte Carlo, `d = 2` | rejected adjacent pairs with `s = r/ℓ^{1/4} < 0.8`: 37 observed, 35.3 from the composite, 4,322 from the cusp kernel alone | | #216's Monte Carlo, `d = 3` | `s < 0.8`: 11 observed, 10.7 composite (1.2 with the pin density alone), 1,329 cusp kernel alone. On `[3·10^{−4}, 10^{−2}]` the counts are `0.884 ± 0.040` of the `ℓ^{1/4}` law and `1.008` of the composite (Pearson χ² 11.8 against 2.1 on 8 bins). Over #216's whole range `[10^{−4}, 0.3]` the composite does not describe the counts (finite-`r` corrections). | diff --git a/frontiers/soft_rejected_pairs_20261002/SOURCES.json b/frontiers/soft_rejected_pairs_20261002/SOURCES.json index a0a1bf684..2c6d43eae 100644 --- a/frontiers/soft_rejected_pairs_20261002/SOURCES.json +++ b/frontiers/soft_rejected_pairs_20261002/SOURCES.json @@ -1,8 +1,8 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-SOFT-REJECTED-20261002-v1.3", - "kind": "author-side proof candidate and conjectures, every d >= 2 (v1.3: author correction AUTH-242-02 of section 5 only: two artifacts of the scanner dec1d.py in thin layers on either side of beta = 2 are disclosed, and H, H0, Delta_chi/k^4 -> 66.56, I~ = -0.5337 and R_{2/3} = -0.0488 (d = 2), -0.0614 (d = 3) are recomputed from the closed form of Math- #244, inside the v1.2 ranges; no statement, proof or control changes; v1.2: applies OA-242-A-01, B-01, B-02, B-03, C-01, C-02, D-01, D-02, E-01, E-02; exact (D-prime) witness; Lemma 2 now states that beta > 2, chi <= 0 is always rejected; prior work #170/#175 recorded, AUTH-242-01; Conjecture 6 status: existence merged (#170/#175), identification #243): Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 D with D the density at 0 of the smallest eigenvalue lambda_1 of -A weighted by gamma_1^6 (lambda_2...lambda_m)^2; d = 2: D = pA(0|b) E gamma^6); Corollary 1' (Gaussian kernel: int int F0 = 25 sqrt3/(48 pi^2) in d = 2, 125 sqrt30/(192 pi^3) in d = 3); the fold-scale soft model G_k with the jets B = d_u A and C3 = d_Theta^3 f along the soft direction; Lemma 2 (the elder decision in the model by a one-dimensional scan, including a case where the saddle is a slice minimum); Proposition 2' (d >= 3: the stiff directions at scale r^(3/2) decouple; same elder decision); Lemma 3 (an a priori elder region); Proposition 4 (Gaussian kernel, every d: the fold-scale rejection rate is F0(b) H(k) with H independent of b and d, H = 1 + (12/25) k^2 + O(k^3)); Lemma 5 (the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4))); Conjecture 6 (the fold-scale limit) and Conjecture 7 (rho_rej + nu_eld^far = B_{d,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 (d = 2) and -0.061 (d = 3) for the Gaussian kernel, not certified)", + "object": "CL-SOFT-REJECTED-20261002-v1.4", + "kind": "author-side proof candidate and conjectures, every d >= 2 (v1.4: applies the C78 delta review of v1.3 (OA-242-V13-01, V13-02, V13-03): wording of section 5 and the header only; the 2.0e-4 comparison of H is the largest difference observed at the 15 tabulated k, not a certified bound; the limits of the section 5(3) expansions are stated; the inherited sensitivity is +-0.0186 (exact), the ranges stand; the raw Monte Carlo archive is pinned by hash; no statement, proof, control or tabulated value changes; v1.3: author correction AUTH-242-02 of section 5 only: two artifacts of the scanner dec1d.py in thin layers on either side of beta = 2 are disclosed, and H, H0, Delta_chi/k^4 -> 66.56, I~ = -0.5337 and R_{2/3} = -0.0488 (d = 2), -0.0614 (d = 3) are recomputed from the closed form of Math- #244, inside the v1.2 ranges; no statement, proof or control changes; v1.2: applies OA-242-A-01, B-01, B-02, B-03, C-01, C-02, D-01, D-02, E-01, E-02; exact (D-prime) witness; Lemma 2 now states that beta > 2, chi <= 0 is always rejected; prior work #170/#175 recorded, AUTH-242-01; Conjecture 6 status: existence merged (#170/#175), identification #243): Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 D with D the density at 0 of the smallest eigenvalue lambda_1 of -A weighted by gamma_1^6 (lambda_2...lambda_m)^2; d = 2: D = pA(0|b) E gamma^6); Corollary 1' (Gaussian kernel: int int F0 = 25 sqrt3/(48 pi^2) in d = 2, 125 sqrt30/(192 pi^3) in d = 3); the fold-scale soft model G_k with the jets B = d_u A and C3 = d_Theta^3 f along the soft direction; Lemma 2 (the elder decision in the model by a one-dimensional scan, including a case where the saddle is a slice minimum); Proposition 2' (d >= 3: the stiff directions at scale r^(3/2) decouple; same elder decision); Lemma 3 (an a priori elder region); Proposition 4 (Gaussian kernel, every d: the fold-scale rejection rate is F0(b) H(k) with H independent of b and d, H = 1 + (12/25) k^2 + O(k^3)); Lemma 5 (the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4))); Conjecture 6 (the fold-scale limit) and Conjecture 7 (rho_rej + nu_eld^far = B_{d,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 (d = 2) and -0.061 (d = 3) for the Gaussian kernel, not certified)", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { @@ -229,13 +229,13 @@ "files": [ { "path": "PROOF.md", - "bytes": 87672, - "sha256": "2d6981593fc32533fd3f3a25c8c424bff829c7b2baa47da148b9502520e62467" + "bytes": 90777, + "sha256": "f45ce06a0a78169fb5003cb91705f344c5e5867f6bd8271210253b423b53f9a6" }, { "path": "README.md", - "bytes": 16712, - "sha256": "ff9cccc50084953833c3b43760388989e9be32c721f915ac3c170c4b26740ae8" + "bytes": 17657, + "sha256": "84883a6e1ef2b064c4c9fcc3ab5d518ff9683a71c3f935489c164eb7cdf6bfac" }, { "path": "soft_check.py", @@ -249,10 +249,10 @@ } ], "supersedes": { - "object": "CL-SOFT-REJECTED-20261002-v1.2", - "head": "b82b0ca16926666adb8895c21801b6e85d7e18df", - "proof_blob": "ad4beb8467a775c7c76f85d5584c13338d0e66be", - "note": "b82b0ca is the v1.2 README amendment; the v1.2 PROOF commit was 1f86fea (same PROOF blob)" + "object": "CL-SOFT-REJECTED-20261002-v1.3", + "head": "d504cdb15851b4a9140755fee6a295282c503ed4", + "proof_blob": "5fd29e63e81980a968009c7e8c3ba8e0af9e46a8", + "note": "v1.3 PROOF sections 0-4, soft_check.py and RESULTS.json are byte-identical in v1.4" }, "exploration_manifest": { "location": "claude.ai project archive V2_2/frontiers_soft_rejected_pairs_20261002/ (and the local archive zip SIDE24_V2_2_2026-09-29.zip); not part of the repository", @@ -593,7 +593,29 @@ "sha256": "cd5084d643bfebc71b7b9bdcafae77ed2a9dddd351b9a79725ebec6ce333e78f" } ], - "raw_monte_carlo": "batchA2.npz (d = 2, L = 64) and batch3B.npz (d = 3, L = 16), records of Math- #216, delivered as SIDE24_MC_raw_2026-10-01.zip; not pinned here by hash", + "raw_monte_carlo": { + "archive": "SIDE24_MC_raw_2026-10-01.zip", + "bytes": 30456459, + "sha256": "85ec255ec653377765aaf940050bbd22c7f889843e2196c005258db11cb6b3e3", + "members": [ + { + "path": "batchA2.npz", + "bytes": 26880641, + "sha256": "d34030095cbeaf4b6ff2810d89f2828c13c28d617734ae8dfb9f425b7cd4340e" + }, + { + "path": "batch3B.npz", + "bytes": 3580675, + "sha256": "20f53f39aaaf5da6f073f3f6223e45e4177df149a688f1954079043e5bb3461b" + }, + { + "path": "README.md", + "bytes": 4770, + "sha256": "592552635479e8f5a7a50ecb67b65a4b98fd84138cf48735c28c6d3225ad120e" + } + ], + "note": "batchA2.npz (d = 2, L = 64) and batch3B.npz (d = 3, L = 16): the raw records of Math- #216 used in section 5(6); delivered to the owner as SIDE24_MC_raw_2026-10-01.zip (claude.ai outputs, 1 October 2026), not part of the repository; pinned by hash in v1.4 (v1.1-v1.3: named, not pinned)" + }, "status": "author-reported exploration, not independently reproduced (OA-242-E-01)", "v13_files": [ { @@ -775,6 +797,21 @@ "verdict": "NOT READY (not a proof acceptance or a merge vote): behind main; Conjecture 7 open; R_{2/3} a sensitivity range, not a certified enclosure; the scanner disclosure must be in the landed note", "findings": [], "applied_in": "v1.3 (rebased on main; the disclosure is in PROOF section 5 and the README)" + }, + { + "slice": "C78 bounded delta review of v1.3 (section 9 changed bytes, section 5 numerical disclosures and source bindings, README/SOURCES, unchanged checker)", + "reviewer": "OpenAI Codex root and /root/c78_numerical_delta_review (delegated AI review; same GitHub account; organizational independence 0)", + "comment": "5956579220", + "pickup": "5956488449", + "at": "d504cdb (PROOF blob 5fd29e63)", + "verdict": "AMEND (three minor reporting corrections); PASS_TECHNICAL for the unchanged-slice correspondence, the source identities checked and the executable-control correspondence", + "findings": [ + "OA-242-V13-01", + "OA-242-V13-02", + "OA-242-V13-03" + ], + "applied_in": "v1.4", + "review": "pullrequestreview-5394316763" } ], "amendments": [ @@ -815,6 +852,35 @@ "AUTH-242-02 (author: two dec1d.py artifacts near beta = 2; section 5 numbers corrected; disclosed in 5953336662, corrected in 5955380848)" ], "requested_by": "xAI 5955184422 (disclosure in the landed note); Codex 5954054126 (PR242 held for the promised section 5 correction)" + }, + { + "label": "v1.4", + "files_changed": [ + "PROOF.md", + "README.md", + "SOURCES.json" + ], + "unchanged": [ + "soft_check.py", + "RESULTS.json" + ], + "proof_sections_unchanged": [ + "0", + "1", + "2", + "3", + "4", + "6", + "7", + "8" + ], + "findings": [ + "OA-242-V13-01", + "OA-242-V13-02", + "OA-242-V13-03" + ], + "requested_by": "Codex C78 delta review 5394316763 (findings first sent in 5956579220)", + "also": "C78 rounding note (not a finding): section 5(5) prints I~ = -0.5336676" } ] } From ee56652e1b30f2bfbb476da269fab15eb24cecba Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 17:32:37 +0000 Subject: [PATCH 7/7] Soft rejected pairs v1.5: apply the C80 readback of v1.4 (OA-242-V14-01/02) Reporting and custody only, after Codex review 5394603905: - V14-01: the sensitivity is written +-0.002*10^(5/3)/5 ~ +-0.0186 (an exact expression and its rounding), and the H0 comparison reads about 1.33e-5 (the observed maximum is 1.326e-5). - V14-02: SOURCES.json records Codex root as C78's final reviewer and the helper c78_numerical_delta_review as interrupted preliminary support. Sections 0-4 and 6-8, soft_check.py and RESULTS.json are byte-identical to v1.4. Scientific effect NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .../soft_rejected_pairs_20261002/PROOF.md | 23 +++++-- .../soft_rejected_pairs_20261002/README.md | 9 ++- .../soft_rejected_pairs_20261002/SOURCES.json | 65 +++++++++++++++---- 3 files changed, 79 insertions(+), 18 deletions(-) diff --git a/frontiers/soft_rejected_pairs_20261002/PROOF.md b/frontiers/soft_rejected_pairs_20261002/PROOF.md index c15929151..271412dbc 100644 --- a/frontiers/soft_rejected_pairs_20261002/PROOF.md +++ b/frontiers/soft_rejected_pairs_20261002/PROOF.md @@ -1,6 +1,14 @@ # Soft rejected pairs: the `1/κ` tail of the rejected cusp kernel in every dimension, the fold-scale rejection function, and a conjectured `ℓ^{2/3}` term -Object: CL-SOFT-REJECTED-20261002-v1.4. Version history: +Object: CL-SOFT-REJECTED-20261002-v1.5. Version history: +- v1.4 → v1.5 applies the two minor amendments of OpenAI Codex's C80 readback of v1.4 (review 5394603905). It changes the + header, one sentence each in §5(4) and §5(5), §9, the README and `SOURCES.json`. No proof, control or tabulated value + changes. + - OA-242-V14-01: the sensitivity is written `±0.002·10^{5/3}/5 ≈ ±0.0186`, the exact expression and its rounding. The + `H₀` comparison reads about `1.33·10^{−5}`; the largest observed difference is `1.326·10^{−5}`, which exceeds the + `1.3·10^{−5}` printed before. + - OA-242-V14-02: `SOURCES.json` records Codex root as C78's final reviewer. The helper `c78_numerical_delta_review` + supplied preliminary notes and a script and then stopped before a report; it is recorded as interrupted support. - v1.3 → v1.4 applies the three findings of OpenAI Codex's C78 delta review of v1.3 (review 5394316763; the findings were first sent in 5956579220). It changes the header and §5 only. No statement, proof or control changes, and no tabulated value changes. @@ -9,7 +17,7 @@ Object: CL-SOFT-REJECTED-20261002-v1.4. Version history: table-interpolation errors. - OA-242-V13-02: §5(3)'s two expansions now state their limits, `t → −∞` and `|χ₀| → ∞`. - OA-242-V13-03: the inherited sensitivity is computed exactly. A shift of `H` by `±0.002` on `k ≥ 0.1` moves `Ĩ` by - `±0.002·10^{5/3}/5 = ±0.0186`, not `±0.018`. The ranges stand. + `±0.002·10^{5/3}/5 ≈ ±0.0186`, not `±0.018`. The ranges stand. - C78 also noted (not a finding) that `0.1150058 × 0.53367` rounds to `0.06138`, not to the printed `0.06137`. §5(5) now prints `Ĩ = −0.5336676`, the value from which `R_{2/3}` was computed. - `SOURCES.json` also pins the raw Monte Carlo archive by hash. §§0–4, `soft_check.py` and `RESULTS.json` are @@ -841,7 +849,7 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. to `100³` changes them by less than `10^{−10}`, and #244's referee reproduces the `H` row with an independent quadrature. - At the 15 values of `k` in v1.2's table, the v1.2 values differ from these by at most `2.0·10^{−4}` (`H`, at - `k = 0.4`) and `1.3·10^{−5}` (`H₀`, at `k = 0.35`). These are observed differences, not certified bounds. + `k = 0.4`) and about `1.33·10^{−5}` (`H₀`, at `k = 0.35`). These are observed differences, not certified bounds. - At four decimals the v1.2 rows had `H = 1.0039, 0.9345, 0.7271, 0.4397, 0.2007` at `k = 0.1, 0.3, 0.4, 0.5, 0.6` and `H₀ = 0.4663` at `k = 0.4`. - *Small `k`.* `Δ_χ/k⁴ → 53248/800 = 66.56` (#244 Theorem B, Step 3); v1.2 used the fitted value `62`. #244 Theorem B @@ -877,7 +885,7 @@ Gaussian kernel `e^{−|z|²/2}`, `d = 2` unless stated otherwise. unrecorded rule (finding A-1, found by the author while archiving v1). - So v1.2 had `R_{2/3} = 0.0914028·Ĩ = −0.049` in `d = 2` and `R_{2/3} = 0.1150058·Ĩ = −0.0617` in `d = 3`. - *Sensitivity (v1.2; the ranges stand).* A uniform shift of `H` by `±0.002` on `k ≥ 0.1` (that is, `v ≤ 10^{1/3}`) - moves `Ĩ` by exactly `±0.002·∫_0^{10^{1/3}}v⁴dv = ±0.002·10^{5/3}/5 = ±0.0186` (v1.2 printed `±0.018`; + moves `Ĩ` by exactly `±0.002·∫_0^{10^{1/3}}v⁴dv = ±0.002·10^{5/3}/5 ≈ ±0.0186` (v1.2 printed `±0.018`; OA-242-V13-03). Halving v1.2's small-`k` term `62k⁴` moves it by `−0.021`, and multiplying it by `1.5` by `+0.021`. So `Ĩ = −0.54 ± 0.03`, `R_{2/3} = −0.049 ± 0.003` in `d = 2` and `R_{2/3} = −0.062 ± 0.004` in `d = 3`. @@ -1015,6 +1023,13 @@ Mutants `M1`–`M13` each break exactly one control (`M1` S1, `M2` S3, `M3` S4, - **E** (§§5–7): the evidence is described accurately and kept separate from the proofs, and the controls match their claims. +**Changed bytes in v1.5** (for a delta check against C80's readback of v1.4, 5394603905): +- *Header:* the object label, the new v1.5 entry, and `≈` in the v1.4 entry (V14-01). +- *§5:* the `H₀` figure in item 4's comparison sentence, and `≈` in item 5's sensitivity sentence (V14-01). +- *§9:* this list. + +Everything else in `PROOF.md`, and `soft_check.py` and `RESULTS.json`, is unchanged. + **Changed bytes in v1.4** (for a delta check against the C78 review of v1.3, 5394316763): - *Header:* the object label, the new v1.4 entry, and the `2.0·10^{−4}` sentence of the v1.3 entry (V13-01). - *§5:* the last item of item 2 (V13-01); the expansions in item 3 (V13-02); item 4's accuracy sentence and the diff --git a/frontiers/soft_rejected_pairs_20261002/README.md b/frontiers/soft_rejected_pairs_20261002/README.md index ac2668f97..9f8502b3e 100644 --- a/frontiers/soft_rejected_pairs_20261002/README.md +++ b/frontiers/soft_rejected_pairs_20261002/README.md @@ -2,13 +2,18 @@ **Author-side proof candidate and conjectures (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** -Object `CL-SOFT-REJECTED-20261002-v1.4`. Full text: [`PROOF.md`](PROOF.md). Version history: +Object `CL-SOFT-REJECTED-20261002-v1.5`. Full text: [`PROOF.md`](PROOF.md). Version history: +- v1.5 applies the two minor amendments of OpenAI Codex's C80 readback of v1.4 (review 5394603905). The changes are to + reporting and custody only. + - OA-242-V14-01: the sensitivity is `±0.002·10^{5/3}/5 ≈ ±0.0186`, an exact expression and its rounding. The `H₀` + comparison is about `1.33·10^{−5}`. + - OA-242-V14-02: `SOURCES.json` names C78's final reviewer (Codex root) and records the interrupted helper as support. - v1.4 applies the three findings of OpenAI Codex's C78 delta review of v1.3 (review 5394316763, AMEND; findings first sent in 5956579220). The changes are to the reporting in PROOF §5 and the header. - OA-242-V13-01: the `2.0·10^{−4}` comparison is the largest difference observed at v1.2's tabulated `k`, not a certified bound. - OA-242-V13-02: the limits of §5(3)'s expansions are stated. - - OA-242-V13-03: the sensitivity `±0.018` is computed exactly, as `±0.0186`. + - OA-242-V13-03: the sensitivity is `±0.002·10^{5/3}/5 ≈ ±0.0186` (v1.2 printed `±0.018`). - C78's rounding note: PROOF §5(5) prints `Ĩ = −0.5336676`, from which `R_{2/3}` is computed. - `SOURCES.json` pins the raw Monte Carlo archive by hash. PROOF §§0–4, `soft_check.py` and `RESULTS.json` are unchanged. diff --git a/frontiers/soft_rejected_pairs_20261002/SOURCES.json b/frontiers/soft_rejected_pairs_20261002/SOURCES.json index 2c6d43eae..50f8371f9 100644 --- a/frontiers/soft_rejected_pairs_20261002/SOURCES.json +++ b/frontiers/soft_rejected_pairs_20261002/SOURCES.json @@ -1,8 +1,8 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-SOFT-REJECTED-20261002-v1.4", - "kind": "author-side proof candidate and conjectures, every d >= 2 (v1.4: applies the C78 delta review of v1.3 (OA-242-V13-01, V13-02, V13-03): wording of section 5 and the header only; the 2.0e-4 comparison of H is the largest difference observed at the 15 tabulated k, not a certified bound; the limits of the section 5(3) expansions are stated; the inherited sensitivity is +-0.0186 (exact), the ranges stand; the raw Monte Carlo archive is pinned by hash; no statement, proof, control or tabulated value changes; v1.3: author correction AUTH-242-02 of section 5 only: two artifacts of the scanner dec1d.py in thin layers on either side of beta = 2 are disclosed, and H, H0, Delta_chi/k^4 -> 66.56, I~ = -0.5337 and R_{2/3} = -0.0488 (d = 2), -0.0614 (d = 3) are recomputed from the closed form of Math- #244, inside the v1.2 ranges; no statement, proof or control changes; v1.2: applies OA-242-A-01, B-01, B-02, B-03, C-01, C-02, D-01, D-02, E-01, E-02; exact (D-prime) witness; Lemma 2 now states that beta > 2, chi <= 0 is always rejected; prior work #170/#175 recorded, AUTH-242-01; Conjecture 6 status: existence merged (#170/#175), identification #243): Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 D with D the density at 0 of the smallest eigenvalue lambda_1 of -A weighted by gamma_1^6 (lambda_2...lambda_m)^2; d = 2: D = pA(0|b) E gamma^6); Corollary 1' (Gaussian kernel: int int F0 = 25 sqrt3/(48 pi^2) in d = 2, 125 sqrt30/(192 pi^3) in d = 3); the fold-scale soft model G_k with the jets B = d_u A and C3 = d_Theta^3 f along the soft direction; Lemma 2 (the elder decision in the model by a one-dimensional scan, including a case where the saddle is a slice minimum); Proposition 2' (d >= 3: the stiff directions at scale r^(3/2) decouple; same elder decision); Lemma 3 (an a priori elder region); Proposition 4 (Gaussian kernel, every d: the fold-scale rejection rate is F0(b) H(k) with H independent of b and d, H = 1 + (12/25) k^2 + O(k^3)); Lemma 5 (the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4))); Conjecture 6 (the fold-scale limit) and Conjecture 7 (rho_rej + nu_eld^far = B_{d,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 (d = 2) and -0.061 (d = 3) for the Gaussian kernel, not certified)", + "object": "CL-SOFT-REJECTED-20261002-v1.5", + "kind": "author-side proof candidate and conjectures, every d >= 2 (v1.5: applies the C80 readback of v1.4 (OA-242-V14-01: the sensitivity written +-0.002*10^(5/3)/5 ~ +-0.0186 and the H0 comparison about 1.33e-5; OA-242-V14-02: C78 final-reviewer attribution); reporting and custody only; v1.4: applies the C78 delta review of v1.3 (OA-242-V13-01, V13-02, V13-03): wording of section 5 and the header only; the 2.0e-4 comparison of H is the largest difference observed at the 15 tabulated k, not a certified bound; the limits of the section 5(3) expansions are stated; the inherited sensitivity is +-0.002*10^(5/3)/5 ~ +-0.0186, the ranges stand; the raw Monte Carlo archive is pinned by hash; no statement, proof, control or tabulated value changes; v1.3: author correction AUTH-242-02 of section 5 only: two artifacts of the scanner dec1d.py in thin layers on either side of beta = 2 are disclosed, and H, H0, Delta_chi/k^4 -> 66.56, I~ = -0.5337 and R_{2/3} = -0.0488 (d = 2), -0.0614 (d = 3) are recomputed from the closed form of Math- #244, inside the v1.2 ranges; no statement, proof or control changes; v1.2: applies OA-242-A-01, B-01, B-02, B-03, C-01, C-02, D-01, D-02, E-01, E-02; exact (D-prime) witness; Lemma 2 now states that beta > 2, chi <= 0 is always rejected; prior work #170/#175 recorded, AUTH-242-01; Conjecture 6 status: existence merged (#170/#175), identification #243): Theorem 1 (the rejected cusp kernel decays like F0/kappa, F0 = (5/24) pi0 D with D the density at 0 of the smallest eigenvalue lambda_1 of -A weighted by gamma_1^6 (lambda_2...lambda_m)^2; d = 2: D = pA(0|b) E gamma^6); Corollary 1' (Gaussian kernel: int int F0 = 25 sqrt3/(48 pi^2) in d = 2, 125 sqrt30/(192 pi^3) in d = 3); the fold-scale soft model G_k with the jets B = d_u A and C3 = d_Theta^3 f along the soft direction; Lemma 2 (the elder decision in the model by a one-dimensional scan, including a case where the saddle is a slice minimum); Proposition 2' (d >= 3: the stiff directions at scale r^(3/2) decouple; same elder decision); Lemma 3 (an a priori elder region); Proposition 4 (Gaussian kernel, every d: the fold-scale rejection rate is F0(b) H(k) with H independent of b and d, H = 1 + (12/25) k^2 + O(k^3)); Lemma 5 (the composite density = (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4))); Conjecture 6 (the fold-scale limit) and Conjecture 7 (rho_rej + nu_eld^far = B_{d,L} + (I^cand - c1) l^(1/4) + R_{2/3} l^(2/3) + O(l^(3/4)); numerically R_{2/3} = -0.049 (d = 2) and -0.061 (d = 3) for the Gaussian kernel, not certified)", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { @@ -229,13 +229,13 @@ "files": [ { "path": "PROOF.md", - "bytes": 90777, - "sha256": "f45ce06a0a78169fb5003cb91705f344c5e5867f6bd8271210253b423b53f9a6" + "bytes": 91967, + "sha256": "ee2930c1434bb765d11da0690a2abf3ab330d544ce7ede1ae076a4c29e206c75" }, { "path": "README.md", - "bytes": 17657, - "sha256": "84883a6e1ef2b064c4c9fcc3ab5d518ff9683a71c3f935489c164eb7cdf6bfac" + "bytes": 18111, + "sha256": "17dbd735b22a01949148a29b8012fc6318ed7f8dce11451e35334877e7efe17d" }, { "path": "soft_check.py", @@ -249,10 +249,10 @@ } ], "supersedes": { - "object": "CL-SOFT-REJECTED-20261002-v1.3", - "head": "d504cdb15851b4a9140755fee6a295282c503ed4", - "proof_blob": "5fd29e63e81980a968009c7e8c3ba8e0af9e46a8", - "note": "v1.3 PROOF sections 0-4, soft_check.py and RESULTS.json are byte-identical in v1.4" + "object": "CL-SOFT-REJECTED-20261002-v1.4", + "head": "ffc7005509554a378edd81136aa6e1f4fbd0eff8", + "proof_blob": "c15929151e9f82813b7993a31b6735bcff8dc6fa", + "note": "v1.4 PROOF sections 0-4 and 6-8, soft_check.py and RESULTS.json are byte-identical in v1.5" }, "exploration_manifest": { "location": "claude.ai project archive V2_2/frontiers_soft_rejected_pairs_20261002/ (and the local archive zip SIDE24_V2_2_2026-09-29.zip); not part of the repository", @@ -800,7 +800,7 @@ }, { "slice": "C78 bounded delta review of v1.3 (section 9 changed bytes, section 5 numerical disclosures and source bindings, README/SOURCES, unchanged checker)", - "reviewer": "OpenAI Codex root and /root/c78_numerical_delta_review (delegated AI review; same GitHub account; organizational independence 0)", + "reviewer": "OpenAI Codex root (final reviewer of review 5394316763; delegated AI review; same GitHub account; organizational independence 0)", "comment": "5956579220", "pickup": "5956488449", "at": "d504cdb (PROOF blob 5fd29e63)", @@ -811,7 +811,21 @@ "OA-242-V13-03" ], "applied_in": "v1.4", - "review": "pullrequestreview-5394316763" + "review": "pullrequestreview-5394316763", + "support": "the helper /root/c78_numerical_delta_review supplied preliminary findings and the script reconstruct.py, then hit a usage limit before completing a report: interrupted preliminary support, not a completed separate review (OA-242-V14-02)" + }, + { + "slice": "C80 bounded readback of the v1.4 reporting fixes", + "reviewer": "OpenAI Codex root (delegated AI review; same GitHub account; organizational independence 0)", + "review": "pullrequestreview-5394603905", + "pickup": "5957362074", + "at": "ffc7005 (PROOF blob c1592915)", + "verdict": "PASS_TECHNICAL_SCOPED for the corrected scope statements, tail directions, sensitivity formula and unchanged-proof/control correspondence; two minor amendments", + "findings": [ + "OA-242-V14-01", + "OA-242-V14-02" + ], + "applied_in": "v1.5" } ], "amendments": [ @@ -881,6 +895,33 @@ ], "requested_by": "Codex C78 delta review 5394316763 (findings first sent in 5956579220)", "also": "C78 rounding note (not a finding): section 5(5) prints I~ = -0.5336676" + }, + { + "label": "v1.5", + "files_changed": [ + "PROOF.md", + "README.md", + "SOURCES.json" + ], + "unchanged": [ + "soft_check.py", + "RESULTS.json" + ], + "proof_sections_unchanged": [ + "0", + "1", + "2", + "3", + "4", + "6", + "7", + "8" + ], + "findings": [ + "OA-242-V14-01", + "OA-242-V14-02" + ], + "requested_by": "Codex C80 readback 5394603905" } ] }