From f215cb32f2ec908daeed6a332e6d4c9ca3b21145 Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 10:39:22 +0000 Subject: [PATCH 1/6] Fold-scale rejection limit in every dimension (Conjecture 6 of #242) Author-side proof candidate CL-FOLD-LIMIT-20261002-v1 (scientific effect NONE; nonauthor review required): - Theorem FL: (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u) for every d >= 2, uniformly for (b, k) in compacts; the limit defining F in #242 (3.1) exists, and F is continuous and positive. - Corollary FL.5: r^-3 (1 - p_r) -> F/(k A_*) in (0, inf), the constant in the Theta(r^3) selection loss of [P] Theorem A and the OA bound (A3). - Corollary FL.6: on compact mark windows, nu_cand - nu_eld = d l^(2/3) + o(l^(2/3)) with d = (1/3) int k^(-8/3) F > 0. - Proposition FL.4: #242 Lemma 2's decision is stable under C^2-small perturbations with the pins critical, for the global filtration of any field realizing the window (stiff directions included). - Lemmas FL.1-FL.3: the quantitative soft window, the full-measure generic set (with OpenAI Codex's axial path: a rejected maximum dies strictly above the saddle; case (D') needs chi > 0), and saddle crossing. #242 is consumed at 87912bd (PROOF blob 8b2f5fae) and bound through the API with a drift gate. Stdlib exact checker fold_check.py (controls F1-F6, mutants M1-M7), RESULTS.json, SOURCES.json, workflow soft-fold-limit.yml. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .github/workflows/soft-fold-limit.yml | 158 ++++ frontiers/soft_fold_limit_20261002/PROOF.md | 714 ++++++++++++++++ frontiers/soft_fold_limit_20261002/README.md | 93 +++ .../soft_fold_limit_20261002/RESULTS.json | 155 ++++ .../soft_fold_limit_20261002/SOURCES.json | 145 ++++ .../soft_fold_limit_20261002/fold_check.py | 770 ++++++++++++++++++ 6 files changed, 2035 insertions(+) create mode 100644 .github/workflows/soft-fold-limit.yml create mode 100644 frontiers/soft_fold_limit_20261002/PROOF.md create mode 100644 frontiers/soft_fold_limit_20261002/README.md create mode 100644 frontiers/soft_fold_limit_20261002/RESULTS.json create mode 100644 frontiers/soft_fold_limit_20261002/SOURCES.json create mode 100644 frontiers/soft_fold_limit_20261002/fold_check.py diff --git a/.github/workflows/soft-fold-limit.yml b/.github/workflows/soft-fold-limit.yml new file mode 100644 index 000000000..b06c0de43 --- /dev/null +++ b/.github/workflows/soft-fold-limit.yml @@ -0,0 +1,158 @@ +name: Soft fold limit exact controls + +'on': + pull_request: + paths: + - 'frontiers/soft_fold_limit_20261002/**' + - '.github/workflows/soft-fold-limit.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/MARKED_CYLINDER_CAP_PROOF.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/c7_total_bounded_20260929/PROOF.md' + - 'frontiers/elder_dimension_lift_20260928/PROOF.md' + - 'frontiers/third_order_rate_20261001/PROOF.md' + # the consumed unmerged #242 PROOF is bound through the API (recorded head, blob and the current PR head) + - 'frontiers/soft_rejected_pairs_20261002/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_fold_limit_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 / 'fold_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_fold_limit_20261002/PROOF.md b/frontiers/soft_fold_limit_20261002/PROOF.md new file mode 100644 index 000000000..c491b6e0f --- /dev/null +++ b/frontiers/soft_fold_limit_20261002/PROOF.md @@ -0,0 +1,714 @@ +# The fold-scale rejection limit in every dimension: Conjecture 6 of Math- #242, the constant in `1 − p_r ~ r³`, and the compact-window `ℓ^{2/3}` law + +Object: CL-FOLD-LIMIT-20261002-v1. +Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 2 October 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem FL, Corollaries FL.5 and FL.6, Proposition FL.4, Lemmas FL.1–FL.3). +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- #242 (open) studies the rejected pairs at the end `κ = ℓ/r⁴ → ∞` of the cusp scale. It defines the fold-scale +rejection rate `F(k; b, u)` through an explicit soft model `G_k` and the elder decision of its Lemma 2 (#242 (3.1)), and states +as **Conjecture 6** that the actual rejected kernel at a fixed gap `k = ℓ/r³` converges to it: + + lim_{r↓0} (k/r)·r^{−2}A_r^{rej}(b, k, u) = F(k; b, u). (0.0) + +This note proves (0.0) in every `d ≥ 2`. It is the fold-scale counterpart of #207's Theorem CU.4 (the cusp kernels at a fixed +`κ`). The new points are that the decision is #242's Lemma 2 (slices are cubic, and in one case the saddle is a slice +minimum), that the transverse curvature must be zoomed at the scale `r` (the soft layer), and that the domination comes from +the parent's deterministic cap region ([P] §7), which confines rejection to that layer. + +**What is new.** +- **Theorem FL** (§3). (0.0) holds for every `d ≥ 2`, `L > 0`, `b ∈ R`, `k > 0` and `u ∈ S^{d−1}`, uniformly for `(b, k)` in + compact subsets of `R × (0, ∞)`. The limit defining `F` in #242 (3.1) exists, and `F` is continuous and positive. +- **Corollary FL.5** (§4, the selection loss). `r^{−3}(1 − p_r(b, k, u)) → F(k; b, u)/(k·A_∗(b, k, u)) ∈ (0, ∞)`. [P]'s + Theorem A gives `1 − p_r ≤ Cr³` and the merged dimension lift gives `1 − p_r ≥ cr³`; this identifies the constant. +- **Corollary FL.6** (§4, the compact-window density difference). For the compact mark windows of [P]'s Theorem B, + `ν_cand(ℓ) − ν_eld(ℓ) = d_{𝐁,𝐊}ℓ^{2/3} + o(ℓ^{2/3})` with `d_{𝐁,𝐊} = (1/3)∫∫∫_{S^{d−1}×𝐁×𝐊} k^{−8/3}F(k; b, u) > 0`. This + sharpens [P] (1.2) from `O(ℓ^{2/3})` to an asymptotic law; it is the compact-window analogue of #242's Conjecture 7. +- **Proposition FL.4** (§2, stability of #242's Lemma 2). For every parameter `ϖ` in a full-measure set `𝒦`, every + `C²`-small perturbation of the model with the two pins as critical points has the elder decision of the model, for the + global superlevel filtration of any field that realizes it in a window. The certificates are explicit paths and traps + crossing the saddle along model curves. In `d ≥ 3` the stiff directions enter as a negative definite form. +- **Lemma FL.2** (§2). `𝒦` has full measure; the proof gives explicit Jacobians, `z⁴(X ± ½)/(96φ²)` for the level + conditions. In the model a rejected maximum always dies strictly above the saddle level (by an axial path, an observation + of OpenAI Codex on #242), and #242's case (D′) requires `χ > 0`: for `β > 2`, `χ < 0` the pair is always rejected. +- **Lemma FL.1** (§1) makes #242's expansions (2.1) and (2.6) quantitative: the soft window of a pinned `C⁵` field is + `C²`-close to the model, at rate `r` (`d = 2`) or `r^{1/2}` (`d ≥ 3`), with a constant linear in `‖f‖_{C⁵}` and independent + of the transverse eigenvalues. + +**What is not claimed.** Conjecture 7 of #242 (the `ℓ^{2/3}` term of the unrestricted `ρ_rej`) is not proved: it needs +(0.0) uniformly from the cusp scale to the fold scale with integrable errors, and the intermediate separations (#242 §4, +inputs (i)–(iii)). Theorem FL is a limit without rate, at fixed `k` (locally uniformly). No numerical value is certified. + +**Dependencies.** +- Consumed (merged): + - [P] (`imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, blob `dfed3b8d`): §1 (`Q`, `W_r`, `Z_r`, + `p_r`), §2 (finite-jet rank), §3 ((3.3)–(3.5)), §§5–6 ((5.3), (6.2)), §7 (the cap region and (7.7)), §8 (pinned + genericity and the maximin), §§10–12 ((10.2), (10.3), (11.1)–(11.2), (12.1), (12.3)). + - [CAP] (`imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md`, blob `0633aca3`): §§1 and 5, the deterministic cap + theorem, as used in [P] §7: on its cap region every `C⁴` field with the pins has maximin `d_f(M) = f(S)`. + - [R] (`frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, blob `247b3ecf`): (R2)–(R3) (the coupling; the pathwise + bound (3.2) is derived from them), (R5), (R11). + - #207 (`frontiers/cusp_second_order_20261001/PROOF.md`, blob `f6df5a73`): §0 (the elder mark and the maximin), Theorem + CU.1 (the pin identities), and the structure of Proposition CU.3 and Theorem CU.4. + - [C7-K] (`frontiers/c7_total_bounded_20260929/PROOF.md`, blob `28748b08`): (K2). +- Consumed (unmerged): #242 (`frontiers/soft_rejected_pairs_20261002/PROOF.md` at `87912bd`, blob `8b2f5fae`): §0, + (2.1)–(2.5), Lemmas 2 and 3, Proposition 2′ ((2.6)), (3.1). This note must be rebound if that blob changes. +- Consumed through #207 §0: [E2] (`reviews/d1_section9_borel_repair_20260925/REPAIR.md`, blob `fe9b9ce4`), for the Borel + measurability of the elder mark. Reading rules for [P]: [E1] (`imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`, + blob `213594d6`) and [REC] (`reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md`, blob `75da2597`). +- Cited only: OpenAI Codex's #242 Slice B review (comment 5948352439, the axial path in Lemma FL.2(c)); OA dimension lift + (`frontiers/elder_dimension_lift_20260928/PROOF.md`, blob `7303bd79`): (A3); #229 + (`frontiers/third_order_rate_20261001/PROOF.md`, blob `110ed33a`): (R⁺.1); #242 Theorem 1 and Proposition 4 (same blob as + above). + +## 0. Setting and notation + +Setting and notation are those of #242 §0 and §2, through #207 §0 and [P]. +- **The field and the pair.** Fix `d ≥ 2`, `m := d − 1`, `L > 0`, and the [P] field on `X = R^d/(LZ^d)`. The near pins are + `M = −ru/2` and `S = ru/2`, at heights `b` and `b − kr³`, with zero gradients; the gap `k > 0` is **fixed**. `Q = Q_{r,b,k,u}` + is the pinned law, `π_r(v_r)` the pin density, `W_r = |det H_M det H_S|1{H_M < 0, index H_S = m}` the typed weight, and + `e = 1{d_f(M) = f(S)}` the elder mark for the maximin `d_f(M) = sup{min_t f(γ(t)) : γ(0) = M, f(γ(1)) > f(M)}` (#207 §0; + Borel by [E2]). The kernels are + + A_r := 12π_r(v_r)E_Q[W_r/r²], A_r^{rej} := 12π_r(v_r)E_Q[(W_r/r²)(1 − e)], 1 − p_r = A_r^{rej}/A_r, (0.1) + + with `p_r = p_r(b, k, u)` the selection probability of [P] §1 (there written `p_r(b, k, R)`, with frame `R` and `u = Re_1`), + and `A_r → A_∗(b, k, u) := 12π_0(u; v_0(b, k))z_0(b, k, u) > 0` ([P] (10.3), where it is written `A_0`; [R] (R11)). +- **The contact law** at `v_0(b, k)` is the law of the field given `f(0) = b`, `∂_uf = ∂_u²f = 0`, `∂_u³f = 12k`, + `∇_Θf = ∇_Θ∂_uf = 0` at `0` ([P] (3.3)–(3.4)); `E_{v_0(b,k)}` is its expectation. Its transverse Hessian `𝔸 := D_Θ²f(0)` is a + nondegenerate Gaussian matrix with density `p_𝔸 ≤ C_0e^{−c_0|·|²}` on `Sym(m)` ([P] §§3, 5). +- **Jets and the soft eigenvalue.** For `f ∈ C⁵` let `A := D_Θ²f(0)`, and let `λ_1 ≤ λ_2 ≤ ⋯ ≤ λ_m` be the eigenvalues of + `−A` (of either sign) with an orthonormal eigenbasis `e_1, …, e_m`. Put + + λ̃ := kλ_1/r, γ := ∂_u²∂_{e_1}f(0), B := ∂_u∂_{e_1}²f(0), C_3 := ∂_{e_1}³f(0), Y := 3kB − γ²/4. (0.2) + + When `λ̃ > 0` and `γ ≠ 0`, the normalized parameter of #242 (2.2) is `ϖ = ϖ(f) := (φ, β, χ)` with + + φ := γ²/(24λ̃), β := kB/λ̃ = 2tφ, χ := k²C_3γ/λ̃² = χ_0φ², t := 12kB/γ², χ_0 := 576k²C_3/γ³. (0.3) + + These do not depend on the sign of `e_1`. +- **The model** (#242 (2.3)). `G_ϖ(X, z) = A₀(X) + ½(X² − ¼)z − ½(1 − βX)z² + (χ/6)z³`, `A₀(X) = (X + ½)²(X − 1)/(12φ)`, + `L_S := −1/(24φ)`, with critical points `M̂ = (−½, 0)` (value `0`) and `Ŝ = (½, 0)` (value `L_S`). It is *typed* iff + `ϖ ∈ 𝒯 := {φ > 0, |φ − β/2| < 1}`; this is #242 (2.4). The slice notation `p = X² − ¼`, `a = 1 − βX`, `D = a² − χp`, the + ridge and valley values `R(X)`, `V(X)` at the slice critical points `z_r(X)`, `z_v(X)`, open slices, and the interval `I_M` + are those of #242 §2. For typed `ϖ`, `e(G_ϖ) := 1{d(M̂) = L_S}` is the actual elder decision of `G_ϖ` on `R²`, with + `d(M̂)` the maximin of #207 §0 for `G_ϖ`. On the generic set `𝒦` of §2 it is the decision of #242 Lemma 2, and #242's + rejected set `𝓡(t, χ_0)` and integral `I(t, χ_0)` ((2.5) there) are read with this decision. +- **Windows.** The soft window and its normalized form are + + Φ(X, ζ, η) := rXu + rkζe_1 + r^{3/2}Σ_{i≥2}η_ie_i, 𝔉(X, ζ, η) := (f(Φ(X, ζ, η)) − b)/(kr³), + 𝔊(X, z, η) := (λ̃/γ²)𝔉(X, (γ/λ̃)z, η). (0.4) + + The model of #242 (2.1), (2.6) is `G_k^{(d)}(X, ζ, η) := G_k(X, ζ) − (1/(2k))Σ_{i≥2}λ_iη_i²`, with `G_k` built from + `(λ̃, γ, B, C_3)`. By #242 (2.3), `(λ̃/γ²)G_k^{(d)}(X, (γ/λ̃)z, η) = G_ϖ(X, z) − 𝔮(η)`, where + `𝔮(η) := ½ηᵀ𝒬η` and `𝒬 := (λ̃/(kγ²))diag(λ_2, …, λ_m)`. + +## 1. The soft window + +**Lemma FL.1 (the soft window).** Let `w ≥ 1` and `0 < k_− ≤ k_+`. There are `C = C(d, w, k_±)` and `r_0 = r_0(d, w, k_+, L)` +with the following property. Let `0 < r ≤ r_0`, `k ∈ [k_−, k_+]`, and let `f ∈ C⁵(X)` have critical points `M = −ru/2` and +`S = ru/2` with `f(M) = b` and `f(S) = b − kr³`. Then on `𝒲_w := {|X| ≤ w, |ζ| ≤ w, |η| ≤ w}`, + + ‖𝔉 − G_k^{(d)}‖_{C²(𝒲_w)} ≤ C‖f‖_{C⁵(X)}ϑ(r), ϑ(r) := r (d = 2), r^{1/2} (d ≥ 3). (1.1) + +The bound holds for every sign and size of `λ̃` and of the `λ_i`, which enter `G_k^{(d)}` exactly. + +*Proof.* Put `N := ‖f‖_{C⁵}`, `x = rX`, `y_1 = rkζ` and `y_i = r^{3/2}η_i` (`i ≥ 2`), in the eigenframe of `A`. Expand `f` at +`0` to total order `4`. The remainder `ρ_5` satisfies `|∂^αρ_5| ≤ C_dN(|x| + |y|)^{5−|α|}` for `|α| ≤ 2`. On `𝒲_w`, +`|x| + |y| ≤ c_wr`, and `∂_X = r∂_x`, `∂_ζ = rk∂_{y_1}`, `∂_{η_i} = r^{3/2}∂_{y_i}`. So `(kr³)^{−1}ρ_5∘Φ` is `O(Nr²)` in +`C²(𝒲_w)`. The Taylor monomial `x^iy_1^jy′^l` (`l` a multi-index over `i′ ≥ 2`) contributes +`r^{ex(i,j,l)}k^{j−1}X^iζ^jη^l` times its coefficient, with + + ex(i, j, l) := i + j + 3|l|/2 − 3. (1.2) + +Apart from the monomials listed below, every monomial of degree `≤ 4` has `ex ≥ 1` when `|l| ≠ 1` and `ex ≥ ½` when +`|l| = 1`, and its coefficient is a derivative of order `3` or `4`, bounded by `N`. The listed monomials are pinned or +exact: +- `|l| = 0`, `i + j ≤ 3`. The axial pins give, exactly as in #207 Theorem CU.1 with `κr⁴ = kr³`, + `f(0) − b = −kr³/2 + f_4r⁴/384 + O(Nr⁵)`, `∂_uf(0) = −(3/2)kr² + O(Nr⁴)`, `∂_u²f(0) = −(r²/24)f_4 + O(Nr³)` and + `∂_u³f(0) = 12k + O(Nr²)`. The transverse pins give `∂_{e_1}f(0) = −(r²/8)γ + O(Nr⁴)` and `∂_u∂_{e_1}f(0) = O(Nr²)`. And + `∂_{e_1}²f(0) = −λ_1 = −λ̃r/k` exactly. Inserting these, the monomials with `i + j ≤ 3` give #242 (2.1): + `G_k(X, ζ) + O(Nr)`. The `O(Nr)` collects `f_4r(1/16 − X²/2)/(24k)` and `∂_u∂_{e_1}f(0)Xζ/r`; with the `x⁴` monomial the + `f_4`-terms form #242's `rf_4(X² − ¼)²/(24k)`. +- `|l| = 2`, `i = j = 0` (`ex = 0`): `½Σ∂_{e_i}∂_{e_{i′}}f(0)y_iy_{i′} = −½Σλ_iy_i²`, which gives `−(1/(2k))Σλ_iη_i²` + exactly. +- `|l| = 1`, `i + j ≤ 1` (`ex ≤ −½`). The transverse pins along `e_i` give `∂_{e_i}f(0) = −(r²/8)γ_{e_i} + O(Nr⁴)` and + `∂_u∂_{e_i}f(0) = O(Nr²)`; the eigenframe gives `∂_{e_1}∂_{e_i}f(0) = 0`. They contribute + `−(r^{1/2}/(8k))γ_{e_i}η_i + O(Nr^{3/2})`, which is `O(Nr^{1/2})`. + +In `d = 2` only `|l| = 0` occurs, so every unlisted monomial has `ex ≥ 1`. This proves (1.1). ∎ + +These are #242's expansions (2.1) and (2.6), read as deterministic Taylor statements with an explicit remainder, as #207 +Theorem CU.1 does at the cusp scale. Control F3 checks the exponent bookkeeping (1.2) and the rates. If `λ̃ > 0` and `γ ≠ 0`, then (0.4) +gives, on `𝒲′ := {|X| ≤ w, |z| ≤ wλ̃/|γ|, |η| ≤ w}`, + + ‖𝔊 − (G_ϖ − 𝔮)‖_{C²(𝒲′)} ≤ (λ̃/γ²)max(1, |γ|/λ̃)²·C‖f‖_{C⁵}ϑ(r). (1.3) + +## 2. The model decision is stable + +### 2.1 Generic parameters and the shape of a rejection + +Let `𝒦` be the set of `ϖ ∈ 𝒯` such that: +- (G0) `β ≠ 2`, `χ ≠ 0` and `β² ≠ χ`; +- (G1) `M̂` is the only critical point of `G_ϖ` at level `0`, and `Ŝ` the only one at level `L_S`; +- (G2) `R` has no finite limit along an unbounded `I_M` (the third hypothesis of #242 Lemma 2); +- (G3) no open slice has its inflection value equal to `L_S`: there is no `X` with `D(X) = 0` and + `A₀(X) + a(X)³/(6χ²) = L_S`. + +**Lemma FL.2 (generic parameters).** +- (a) `𝒯 ∖ 𝒦` is a Lebesgue null set, and on `𝒦` the hypotheses of #242 Lemma 2 hold. +- (b) For `ϖ ∈ 𝒦`, every finite end `X_e` of `I_M` is either a non-open slice with `R(X_e) = L_S`, or an open slice whose + inflection value `A₀(X_e) + a(X_e)³/(6χ²)` (the common limit of `R` and `V`) exceeds `L_S`. +- (c) For every typed `ϖ`, `d(M̂) ≥ L_S`; so `e(G_ϖ) = 0` if and only if `M` dies strictly above `L_S`. For `ϖ ∈ 𝒦` with + `e(G_ϖ) = 1`, either `β < 2` and case (D) of #242 Lemma 2 holds, or `β > 2`, `χ > 0` and case (D′) holds. + +So in the model a rejected maximum never survives below the saddle level, and (D′) requires `χ > 0`. + +*Proof of (a).* (G0) excludes three null sets. + +(G1). Write `Γ_0 := (∂_XG_ϖ, ∂_zG_ϖ, G_ϖ)` and `Γ_S := (∂_XG_ϖ, ∂_zG_ϖ, G_ϖ − L_S)` as functions of `(X, z, ϖ)`. Since +`∂_φA₀ = −A₀/φ` and `∂_φL_S = 1/(24φ²)`, their Jacobians in `ϖ = (φ, β, χ)` have determinants (control F2) + + det ∂_ϖΓ_0 = z⁴(X + ½)/(96φ²), det ∂_ϖΓ_S = z⁴(X − ½)/(96φ²). (2.1) + +So on `{z ≠ 0, X ≠ −½}` the map `Γ_0` is a submersion. Its zero set there is a two-dimensional submanifold of `R² × 𝒯`, and +the projection of that set to `𝒯 ⊂ R³` is null (Sard). On the line `X = −½` a critical point with `z ≠ 0` has +`z = (2 + β)/χ = 1/β` and value `G_ϖ(−½, z) = −z²(2 + β)/12 ≠ 0`, since `2 + β > 2φ > 0` on `𝒯`. On `{z = 0}`, +`∂_zG_ϖ = ½(X² − ¼)` vanishes only at `M̂` and `Ŝ`. The same argument for `Γ_S` uses the line `X = ½`, where a critical +point with `z ≠ 0` has value `G_ϖ − L_S = −z²(2 − β)/12 ≠ 0`. + +(G2). If `β² < χ`, then `D → −∞` as `|X| → ∞` and `I_M` is bounded. If `β² > χ`, put `ω := (β² − χ)^{1/2}`. Then +`D(X) ~ ω²X²`, the slices are non-open for large `|X|`, and along them `R(X) = A₀ + (p/3)z_r − (a/6)z_r²` with +`z_r ~ X/(ω − β)` as `X → +∞` and `z_r ~ −X/(ω + β)` as `X → −∞` (`ω ≠ ±β` by (G0)). Hence `R(X) = κ_±X³ + O(X²)` as +`X → ±∞`, with + + κ_+ := 1/(12φ) + (2ω − β)/(6(ω − β)²), κ_− := 1/(12φ) − (2ω + β)/(6(ω + β)²). (2.2) + +For fixed `(β, χ)` each `κ_±` is strictly monotone in `φ`, so it vanishes for at most one `φ`. Off the null set +`{κ_+κ_− = 0}`, `R → ±∞` along every unbounded direction, so (G2) holds. (The witness of control F4 has `κ_+ = 0` exactly; its +`I_M` is bounded, so (G2) holds for it all the same.) + +(G3). On `{D = 0}` the slice has a double critical point `z = a/χ`, with value `A₀ + a³/(6χ²)`. The map +`(X, ϖ) ↦ (D, A₀ + a³/(6χ²) − L_S)` has Jacobian determinant in `(φ, χ)` equal to `−p(A₀ − L_S)/φ` on `{D = 0}`, and +`A₀ − L_S = (X − ½)²(X + 1)/(12φ)` (control F2). Off `X ∈ {±½, −1}` the map is a submersion, and its zero set projects to a +null set. At `X = −1` the conditions force `a(−1) = 0` and then `χ = 0`; at `X = ½` they force `β = 2`; at `X = −½`, +`D = (1 + β/2)² > 0`. + +*Proof of (b).* At a non-open end `R(X_e) = L_S` by continuity. At an open end the ridge and valley values tend to the +inflection value, which is therefore `≥ L_S`; equality is excluded by (G3). + +*Proof of (c).* The axial segment `{(X, 0) : −½ ≤ X ≤ 2}` carries `G_ϖ = A₀`, with `A₀ − L_S = (X − ½)²(X + 1)/(12φ) ≥ 0` +(equality only at `Ŝ`), and it ends at `A₀(2) = 25/(48φ) > 0`. So `d(M̂) ≥ L_S` for every typed `ϖ`. This observation is +OpenAI Codex's, in its review of #242 Lemma 2 (comment 5948352439). Hence `e(G_ϖ) = 1` iff `d(M̂) ≤ L_S`, i.e. iff for every +`c > L_S` the point `M` is the highest point of its component of `{G_ϖ > c}`. Assume this, with `ϖ ∈ 𝒦`. As in Step 1 of the +proof of Lemma 2: +- (i) `R < 0` on `I_M ∖ {−½}`. A point `X*` with `R(X*) > 0` would put a point above `0` into `M`'s component at the level + `min_{[−½, X*]}R > L_S`. A point with `R(X*) = 0` and `R ≤ 0` nearby would be a critical point at level `0`, excluded by (G1). +- (ii) `V ≤ L_S` on `I_M`; otherwise a ridge piece and a far piece merge above `L_S`. +- (iii) `I_M` is bounded. `R` is an algebraic function of `X`, so it has a limit in `[−∞, ∞]` along an unbounded `I_M`; by + (G2) the limit is `±∞`. `R → +∞` would give (i)'s first alternative, and `R → −∞` contradicts `R > L_S`. +- (iv) No end of `I_M` is open. By (b), at an open end `R` and `V` tend to a value `> L_S`, so `V > L_S` near it inside + `I_M`, against (ii). + +Now look at the slices with `|X| < ½` and `a(X) > 0`. If `χ > 0`, then `D = a² + χ|p| > a²`, so `z_r < 0 < z_v`. If `χ < 0` +and `D > 0`, then `z_v < z_r < 0`. In both cases `z = 0` lies on the ridge side of the slice, so + + R(X) > G_ϖ(X, 0) = A₀(X) > L_S, (2.3) + +because `A₀ − L_S = (X − ½)²(X + 1)/(12φ) > 0` on `(−1, ½)`. +- *`β < 2`.* On `𝒯`, `β > −2`, so `a > 0` on `[−½, ½]`. By (iv) no slice in `(−½, ½)` is open: the first open slice to the right + of `−½` would be an open end of `I_M`. By (2.3), `(−½, ½) ⊂ I_M`. Since `R(½) = L_S`, the right end of `I_M` is `½`. + Moreover `V < L_S` on `(X_L, ½)`: a point with `V = L_S` would be a local maximum of `V`, hence a critical point at level + `L_S` other than `Ŝ`. With (i)–(iii), this is case (D) of Lemma 2. +- *`β > 2`, `χ < 0`.* At `X = 1/β ∈ (0, ½)`, `a = 0` and `p < 0`, so `D = −χp < 0`: the slice is open. The first open slice to + the right of `−½` lies in `(−½, 1/β]` and is an open end of `I_M`, against (iv). So this case does not occur: for `β > 2`, + `χ < 0` the pair is always rejected. +- *`β > 2`, `χ > 0`.* For `|X| < ½`, `D = a² + χ|p| > a² ≥ 0`, so `z_r < 0 < z_v` whatever the sign of `a`, and (2.3) holds + on `(−½, ½)`. At `½`, `R(½) = L_S + (2/3)|a(½)|³/χ² > L_S`. So `[−½, ½] ⊂ I_M`. As before `V < L_S` on `I_M ∖ {½}`; at `½` the + valley point is `Ŝ`. With (i)–(iii), this is case (D′). ∎ + +### 2.2 Crossing a saddle + +**Lemma FL.3 (crossing a nondegenerate critical point).** Let `H_0` be a polynomial on `R^n` with a critical point `s`, let +`c ∈ C²(B̄_{ρ_0}; R^n)`, `B̄_{ρ_0} ⊂ R^j` a closed ball of some radius `ρ_0 > 0`, with `c(0) = s` and `Dc(0)` injective, and +suppose the form `Dc(0)ᵀHess H_0(s)Dc(0)` is positive definite (respectively negative definite). There are `ρ_3 ∈ (0, ρ_0]` +and `ε_0 > 0` such that: for every `H ∈ C²(B(s, 2ρ_c))` with `∇H(s) = 0` and `‖H − H_0‖_{C²(B(s, 2ρ_c))} ≤ ε_0`, where +`ρ_c := ρ_3 sup|Dc|`, the function `H∘c − H(s)` is positive (respectively negative) on `0 < |w| ≤ ρ_3`. + +*Proof.* Put `h := H∘c`. Then `∇h(0) = Dc(0)ᵀ∇H(s) = 0` and +`Hess h(w) = Dc(w)ᵀHess H(c(w))Dc(w) + Σ_i∂_iH(c(w))Hess c_i(w)`. For `|w| ≤ ρ_3`: +`Hess H(c(w)) = Hess H_0(s) + O(ρ_3 + ε_0)`, because `H_0` has a Lipschitz Hessian on compacts; +`|∇H(c(w))| ≤ |∇H_0(c(w))| + ε_0 = O(ρ_3 + ε_0)`; and `Dc(w) = Dc(0) + O(ρ_3)`. So `Hess h` stays definite, with a fixed margin, +on `|w| ≤ ρ_3` when `ρ_3` and `ε_0` are small. Taylor's formula with integral remainder at `0` gives the sign. ∎ + +### 2.3 The stability proposition + +**Proposition FL.4 (stability of the model decision).** Let `ϖ ∈ 𝒦` and let `𝒬` be a positive definite +`(m − 1) × (m − 1)` matrix (nothing if `m = 1`), `𝔮(η) := ½ηᵀ𝒬η`, `q_− := λ_min(𝒬)`. There are a compact rectangle +`W = W_ϖ ⊂ R²` containing `M̂` and `Ŝ` in its interior, a radius `R_η = R_η(ϖ, 𝒬)` and an `ε = ε(ϖ, 𝒬) > 0` with the +following property. Let `𝔊 ∈ C²(W × B̄_{R_η})` satisfy + +- (S1) `M̂_0 := (−½, 0, 0)` and `Ŝ_0 := (½, 0, 0)` are critical points of `𝔊`; +- (S2) `‖𝔊 − (G_ϖ − 𝔮)‖_{C²(W × B̄_{R_η})} ≤ ε`. + +Let `f` be continuous on `X`, and let `Ψ : W × B̄_{R_η} → X` be continuous and injective, with `Ψ(M̂_0) = M`, `Ψ(Ŝ_0) = S` and +`f∘Ψ = b + s𝔊` for constants `b ∈ R` and `s > 0`. Then + + 1{d_f(M) = f(S)} = e(G_ϖ). (2.4) + +If moreover `f` is `C²` near `M` and `S` and `Ψ` is a `C²` diffeomorphism near `M̂_0` and `Ŝ_0`, then `M` is a nondegenerate +local maximum of `f` and `S` a nondegenerate critical point of index `m`. + +*Proof.* Constants `δ, ρ, τ` below depend on `ϖ` and `𝒬` only. + +*Step 1 (types).* By (S2), `Hess 𝔊(M̂_0)` is within `ε` of `diag(Hess G_ϖ(M̂), −𝒬)`, which is negative definite (on `𝒯`, +`det Hess G_ϖ(M̂) = (1 + β/2 − φ)/(4φ) > 0` and `∂_z²G_ϖ(M̂) = −(1 + β/2) < 0`). Likewise `Hess 𝔊(Ŝ_0)` is within `ε` of +`diag(Hess G_ϖ(Ŝ), −𝒬)`, which has index `m` (`det Hess G_ϖ(Ŝ) = −(1 − β/2 + φ)/(4φ) < 0`). For small `ε` the inertia +persists (Weyl), and it transfers to `f` through `Ψ`. + +*Step 2 (paths and traps).* Two elementary facts, valid for any `c′ ∈ R`. +- (Path) If `Γ ⊂ W × B̄_{R_η}` is a path from `M̂_0` to a point `𝔭̂` with `𝔊(𝔭̂) > 𝔊(M̂_0)`, and `𝔊 ≥ c′` on `Γ`, then + `d_f(M) ≥ b + sc′`. Indeed `Ψ∘Γ` is a path from `M` to a point above `f(M)` along which `f ≥ b + sc′`. +- (Trap) Let `𝒟` be open in `R^d`, with `𝒟 ⊂ int(W × B̄_{R_η})` and `𝒟̄ ⊂ W × B̄_{R_η}`. Suppose `M̂_0 ∈ 𝒟`, `𝔊 ≤ c′` on + `∂𝒟`, and `𝔊 < 𝔊(M̂_0)` on `𝒟 ∖ {M̂_0}`. Then `d_f(M) ≤ b + sc′`. + + Indeed, by invariance of domain `Ψ(𝒟)` is open in `X`. Its closure lies in the compact set `Ψ(𝒟̄)`, so its boundary lies in + `Ψ(∂𝒟)`, where `f ≤ b + sc′`. The path component `C` of `{f > b + sc′}` containing `M` therefore lies in `Ψ(𝒟)`, so + `f < f(M)` on `C ∖ {M}`. A path from `M` to a point above `f(M)` must leave `C`, and so it meets `{f ≤ b + sc′}`. + +*Step 3 (model certificates).* By Lemma FL.2(c), `ϖ ∈ 𝒦` falls in exactly one of three cases. Each comes with a margin +`δ > 0` and a radius `ρ > 0`. Let `ρ_3` be the smallest of the radii that Lemma FL.3 gives for the three maps of Step 4; we +take `2ρ ≤ ρ_3` (`ρ` may be decreased at the cost of `δ`). + +- **(R)** `e(G_ϖ) = 0`, so `M` dies above `L_S`. Then there is a compact path `Γ` from `M̂` to a point `𝔭` with + `G_ϖ(𝔭) > 2δ`, along which `G_ϖ ≥ L_S + 2δ`. (By the definition of the maximin, and because components of open subsets of + `R²` are path connected.) +- **(D), (D′)** `e(G_ϖ) = 1`. Then there are: + - `C²` curves `σ_±` through `Ŝ` (`σ_±(0) = Ŝ`), with `σ_+′(0)` ascending and `σ_−′(0)` descending for `Hess G_ϖ(Ŝ)`; + - a compact path `Γ` from `M̂` to a point `𝔭` with `G_ϖ(𝔭) > 2δ`, containing the arc `σ_+([−ρ, ρ])`, with + `G_ϖ ≥ L_S + 2δ` on `Γ ∖ σ_+((−ρ, ρ))`; + - a bounded open `𝒟_2 ∋ M̂` whose boundary contains `σ_−([−ρ, ρ])`, with `G_ϖ ≤ L_S − 2δ` on + `∂𝒟_2 ∖ σ_−((−ρ, ρ))` and `G_ϖ ≤ −2δ` on `𝒟̄_2 ∖ B(M̂, ρ)`. + +*The certificates exist in cases (D) and (D′).* Write `z_∂(X) := z_r(X) − Z·sgn χ` for a large constant `Z`; this is on +the side of the ridge point away from the valley. On that side the slice decreases to `−∞`, so `G_ϖ(X, z_∂(X)) < L_S − 1` +on any compact set of non-open slices once `Z` is large. For a compact interval `J` of non-open slices put + + 𝒟_2(J) := {(X, z) : X ∈ int J, z strictly between z_∂(X) and z_v(X)}. (2.5) + +On each closed slice segment of `𝒟̄_2(J)` the slice rises from `z_∂` to its maximum `R(X)` at `z_r` and falls to `V(X)` at +`z_v`, so `G_ϖ ≤ R(X)` there. + +By Lemma FL.2(b)–(c), the finite ends of `I_M` are non-open with `R = L_S`. They differ from `½` except for the right end in +case (D), so there `R′ ≠ 0` by (G1): a zero would make `(X_e, z_r(X_e))` a critical point at level `L_S`. Hence `R < L_S` +just outside `I_M`. (Here `R′` is the derivative of `R`.) +- *(D)* (`β < 2`). `I_M = (X_L, ½)`, with `V < L_S` and `R < 0` (for `X ≠ −½`) on `(X_L, ½)`. At `½`, `z_r(½) = 0`, + `R(½) = L_S` and `V(½) = L_S − (2/3)a(½)³/χ² < L_S`. + - *Curves.* `σ_−(t) := (½, t)` is descending: `∂_z²G_ϖ(Ŝ) = −a(½) < 0`. `σ_+(t) := (½ + t, z_r(½ + t))` is the ridge + curve; it is ascending, because `σ_+′(0)ᵀHess G_ϖ(Ŝ)σ_+′(0) = R″(½) = det Hess G_ϖ(Ŝ)/∂_z²G_ϖ(Ŝ) > 0`. + - *Trap.* `𝒟_2 := 𝒟_2([X_L − τ, ½])` for small `τ > 0`. Its right side is the segment + `{½} × [z_∂(½), z_v(½)] ⊃ σ_−([−ρ, ρ])`, on which `G_ϖ < L_S` except at `Ŝ`. Its left side carries + `G_ϖ ≤ R(X_L − τ) < L_S`, and its curved sides carry `G_ϖ < L_S`. Inside, `G_ϖ ≤ R(X) < 0` except at `M̂`. + - *Path.* Follow the ridge from `M̂` to `σ_+(ρ_1)`. There `R ≥ L_S + 2δ` on `[−½, ½ − ρ]`, and `R > L_S` on `(½, ½ + ρ_1]`. + Then go down the segment `{X_1} × [0, z_r(X_1)]`, `X_1 := ½ + ρ_1`. Since `p(X_1), a(X_1) > 0`, the slice is increasing + on `[0, z_r(X_1)]`: for `χ > 0`, `0 < z_r < z_v`, and for `χ < 0`, `z_v < 0 < z_r`. So `G_ϖ ≥ A₀(X_1) > L_S` there. + Finish along `{z = 0}` to + `𝔭 := (2, 0)`. There `G_ϖ = A₀` and `A₀ − L_S = (X − ½)²(X + 1)/(12φ)` increases on `[X_1, 2]`, and + `A₀(2) = 25/(48φ) > 0`. +- *(D′)* (`β > 2`, `χ > 0`). `½ ∈ I_M = (X_L, X_R)`, with `R < 0` on `I_M ∖ {−½}` and `V < L_S` on `I_M ∖ {½}`. At `½`, + `z_v(½) = 0`, `V(½) = L_S`, `V″(½) = det Hess G_ϖ(Ŝ)/∂_z²G_ϖ(Ŝ) < 0`, and `z_r(½) = 2a(½)/χ` with `R(½) > L_S`. + - *Curves.* `σ_+(t) := (½, t)` is ascending: `∂_z²G_ϖ(Ŝ) = −a(½) > 0`. `σ_−(t) := (½ + t, z_v(½ + t))` is the valley curve, + which is descending: `σ_−′(0)ᵀHess G_ϖ(Ŝ)σ_−′(0) = V″(½) < 0`. + - *Trap.* `𝒟_2 := 𝒟_2([X_L − τ, X_R + τ])` for small `τ > 0`. Its upper (valley) boundary carries `G_ϖ = V ≤ L_S`, with + equality only at `Ŝ`, and `V ≤ L_S − 2δ` off `σ_−((−ρ, ρ))`. The other sides carry `G_ϖ < L_S`, and inside + `G_ϖ ≤ R(X) < 0` except at `M̂`. + - *Path.* Follow the ridge from `M̂` to `(½, z_r(½))`; there `R > L_S` on `[−½, ½]`. Then go along the slice `{½} × ·` through + `Ŝ` to a point `𝔭 = (½, z_𝔭)` on the far side of `0`, with `G_ϖ(𝔭) > 2δ`. The slice at `½` has its only critical points at + `z_r(½)` (maximum) and `0` (minimum, value `L_S`), and it tends to `+∞` on the far side. + +In both cases compactness gives `δ` and `ρ`. + +*Step 4 (persistence).* The three maps are `t ↦ (σ_+(t), 0)` and `(t, η) ↦ (σ_−(t), η)` at `Ŝ_0`, and the identity at +`M̂_0`. The forms of `H_0 := G_ϖ − 𝔮` on their tangent spaces are `σ_+′ᵀ𝐇_Sσ_+′ > 0`, `diag(σ_−′ᵀ𝐇_Sσ_−′, −𝒬) < 0` and +`diag(Hess G_ϖ(M̂), −𝒬) < 0`, with `𝐇_S := Hess G_ϖ(Ŝ)`. Let `ρ_3` and `ε_0` be the smallest radius and tolerance that +Lemma FL.3 gives for the three maps (with `2ρ ≤ ρ_3` as in Step 3), and `ρ_c` the largest of their radii `ρ_3 sup|Dc|`. +Choose `ε ≤ min(δ/2, ε_0, q_−ρ²/8)`, with `ε < 1`, so small that Step 1 applies. Then fix a compact rectangle `W` whose +interior contains `Γ`, `𝒟̄_2` and the closed `2ρ_c`-discs about `M̂` and `Ŝ`, and a radius `R_η ≥ 2ρ_c` with +`q_−R_η²/2 ≥ max_W G_ϖ − L_S + 3`. So `W × B̄_{R_η}` contains the balls of Lemma FL.3, and `δ, ρ, ρ_3, ε_0, ε` do not +depend on `W` or `R_η`. By (S2), `|𝔊(M̂_0)| ≤ ε` and `|𝔊(Ŝ_0) − L_S| ≤ ε`. + +- **(R).** On `Γ × {0}`, `𝔊 ≥ L_S + 2δ − ε > L_S + ε ≥ 𝔊(Ŝ_0)`, and `𝔊(𝔭, 0) > 2δ − ε > 𝔊(M̂_0)`. By (Path), + `d_f(M) ≥ b + s(L_S + 2δ − ε) > f(S)`. So `1{d_f(M) = f(S)} = 0`. +- **(D), (D′).** + - *Path.* Take `Γ × {0}`. Off the crossing arc, `𝔊 ≥ L_S + 2δ − ε > 𝔊(Ŝ_0)`. On `σ_+([−ρ, ρ]) × {0}`, `𝔊 ≥ 𝔊(Ŝ_0)` by Lemma + FL.3. At the end, `𝔊(𝔭, 0) > 𝔊(M̂_0)`. By (Path), `d_f(M) ≥ f(S)`. + - *Trap.* Take `𝒟 := 𝒟_2 × B_{R_η}`. First, `𝔊 < 𝔊(M̂_0)` on `𝒟 ∖ {M̂_0}`: + - off `B(M̂, ρ) × B_{R_η}`, `𝔊 ≤ −2δ + ε < −ε`; + - on `B(M̂, ρ) × {|η| ≥ ρ}`, `𝔊 ≤ −q_−ρ²/2 + ε < −ε`; + - on `B(M̂, ρ) × {|η| < ρ} ⊂ B(M̂_0, 2ρ)`, by Lemma FL.3 for the identity map (`2ρ ≤ ρ_3`). + + Second, `𝔊 ≤ 𝔊(Ŝ_0)` on `∂𝒟`: + - on `𝒟̄_2 × ∂B_{R_η}`, `𝔊 ≤ max_W G_ϖ − q_−R_η²/2 + ε < L_S − 2`; + - on `(∂𝒟_2 ∖ σ_−((−ρ, ρ))) × B̄_{R_η}`, `𝔊 ≤ L_S − 2δ + ε < 𝔊(Ŝ_0)`; + - on `σ_−([−ρ, ρ]) × {|η| ≤ ρ}`, where `|(t, η)| ≤ 2ρ ≤ ρ_3`, `𝔊 ≤ 𝔊(Ŝ_0)` by Lemma FL.3; + - on `σ_−([−ρ, ρ]) × {|η| ≥ ρ}`, `𝔊 ≤ L_S − q_−ρ²/2 + ε < 𝔊(Ŝ_0)`, because `G_ϖ∘σ_− ≤ L_S` near `0` (Lemma FL.3 with + `H = H_0`). + + By (Trap) with `c′ := 𝔊(Ŝ_0)`, `d_f(M) ≤ f(S)`. + + So `d_f(M) = f(S)`, and `1{d_f(M) = f(S)} = 1`. ∎ + +*Remarks.* +1. As in #207, only window information is used: every conclusion is an explicit path inside `Ψ(W × B̄_{R_η})` or a trap whose + closure lies inside it. So the landscape outside the window cannot change the decision. +2. The field ridge and valley are never needed. The certificates are the model's curves, and at `Ŝ` a crossing curve, which + Lemma FL.3 makes robust using only `C²` closeness and the criticality of `Ŝ_0`. +3. `W`, `δ`, `ρ` and `ε` are not uniform in `ϖ`. Theorem FL needs only the pointwise statement, because it passes to the limit + by dominated convergence at each fixed parameter. +4. The proof works verbatim for a continuous function on `R²` in place of `f`, with `Ψ = id` (Steps 2–4 only use + invariance of domain). Applied to `G_{ϖ′}` (with `𝔮 = 0`), it shows: for every `ϖ ∈ 𝒦` there is a neighbourhood `U` of + `ϖ` such that the actual decision `e(G_{ϖ′})` equals `e(G_ϖ)` for every typed `ϖ′ ∈ U`, generic or not. So + `ϖ ↦ e(G_ϖ)` is continuous at every point of the full-measure set `𝒦`, hence Lebesgue measurable. The `(R)` part needs + only `C⁰` closeness on `Γ`, and so holds without genericity. + +## 3. The fold-scale limit + +**Theorem FL.** Let `d ≥ 2`, `L > 0` and `u ∈ S^{d−1}`. For every `b ∈ R` and `k > 0`, + + lim_{r↓0} (k/r)·r^{−2}A_r^{rej}(b, k, u) = F(k; b, u), (3.1) + +where `F` is #242 (3.1), and the limit there exists. `F` is continuous and positive, and the convergence is uniform for +`(b, k)` in compact subsets of `R × (0, ∞)`. + +*Proof.* Fix `u` and a compact set `𝒞 ⊂ R × (0, ∞)` of `(b, k)`. Constants are uniform over `𝒞`; `P := 1 + |b| + k`. + +*Step 1 (coupling and conditioning).* Use the coupling (R3) of [R]: on one probability space `(Ω, ℙ)` an unconditioned copy +`𝖥` of the field gives `F_r := 𝖥 + C_rΣ_r^{−1}(v_r − U_r)` with law `Q`, and `F_0 := 𝖥 + C_0Σ_0^{−1}(v_0 − U_0)` with the +contact law. By (R2) and Taylor's formula for the centred difference quotients `U_r` (e.g. `U_3 = ∂_u³𝖥(0) + +(r²/40)∂_u⁵𝖥(0) + ⋯`), pathwise, + + ‖F_r − F_0‖_{C⁵} ≤ r²T, ‖F_0‖_{C⁵} ≤ T, T := C(P + ‖𝖥‖_{C⁵}), (3.2) + +and `T` has all moments ([P] §2). Let `𝔸 := D_Θ²F_0(0)`. The pair `(F_r, 𝔸)` is jointly Gaussian, so + + F_r = μ_r + 𝔅_r(𝔸 − E𝔸) + g_r (r ≥ 0), (3.3) + +with deterministic `μ_r := EF_r` and `𝔅_r := Cov(F_r, 𝔸)Cov(𝔸)^{−1}`, and `g_r` centred and independent of `𝔸`. From +(3.2), `‖μ_r − μ_0‖_{C⁵} + ‖𝔅_r − 𝔅_0‖_{C⁵} ≤ Cr²P`. For `a ∈ Sym(m)` put `F_r^a := μ_r + 𝔅_r(a − E𝔸) + g_r`. +- *Pins.* Every `F_r^a` has the pins. The pin functionals `ℓ_j` satisfy `ℓ_j(F_r) = v_{r,j}` on `Ω`, so `ℓ_j(F_r)` is + deterministic. Hence `ℓ_j(𝔅_r) = Cov(ℓ_j(F_r), 𝔸)Cov(𝔸)^{−1} = 0`, `ℓ_j(μ_r) = v_{r,j}`, and `ℓ_j(g_r) = 0` identically. +- *Contact Hessian.* Similarly, since `D_Θ²F_0(0) = 𝔸`, `D_Θ²𝔅_0(0)` is the identity of `Sym(m)`, `D_Θ²μ_0(0) = E𝔸` and + `D_Θ²g_0(0) = 0`. So `D_Θ²F_0^a(0) = a` exactly. +- *Representation.* By independence and Fubini, for Borel `Ξ ≥ 0`, + `E_Q[Ξ(f)] = ∫_{Sym(m)}p_𝔸(a)E[Ξ(F_r^a)]da`. +- *Pathwise bounds.* `‖F_r^a − F_0^a‖_{C⁵} ≤ Cr²(T + P(1 + |a| + |𝔸|)) =: r²T_a`. Since `F_0^a = F_0 + 𝔅_0(a − 𝔸)`, + also `‖F_r^a‖_{C⁵} ≤ C(T + P + |a| + |𝔸|)`. + +*Step 2 (the exceptional set, and the zoom).* Let `M_3`, `M_4` be the third- and fourth-derivative suprema of [P] on the +local cylinder. By [P] (7.7), `(k/r)E_Q[(W_r/r⁴)1{rM_4 > 3k/10}] ≤ (k/r³)·Cr⁴ → 0`. Write the rest in Weyl coordinates on +`Sym(m)`: `a = −O diag(μ_1, …, μ_m)Oᵀ` with `μ_1 < ⋯ < μ_m`, `da = c_m𝒱(μ)dμ dO`, `𝒱(μ) := Π_{i 0`). +- (iii) `(γ, B, C_3)(f_r) → (γ*, B*, C_3*)`, the jets of `f*` at `0` along `Oε_1`. Also `Y(f_r) → Y* := 3kB* − γ*²/4`. +- (iv) By Lemma FL.1, `‖𝔉_r − G_k^{(d)}[f_r]‖_{C²(𝒲_w)} ≤ C‖f_r‖_{C⁵}ϑ(r) → 0`. And `G_k^{(d)}[f_r] → G_k^{(d)}[*]`, the model + built from `(μ̃, γ*, B*, C_3*)` and `μ_2, …, μ_m`. Here `𝔉_r` is the window (0.4) of `f_r` in its own eigenframe. + +Consequences: +- *The `M_4` indicator* tends to `1`, since `rM_4(f_r) ≤ Cr‖f_r‖_{C⁴}`. +- *Types and weight.* Since `f_r∘Φ_r = b + kr³𝔉_r` with `Φ_r` affine, the Hessians of `f_r` at `M`, `S` are congruent to + those of `𝔉_r` at `M̂_0`, `Ŝ_0` up to a positive factor. Computing the determinant of `Φ_r`'s linear part, + + W_r(f_r)/r⁴ = k^{2(m−1)}|det Hess 𝔉_r(M̂_0)·det Hess 𝔉_r(Ŝ_0)|·1{typed} (exactly). (3.5) + + The limit Hessians are block diagonal. At `M̂_0` the planar block is `[[−6, −γ*/2], [−γ*/2, −μ̃ − kB*/2]]`, with + determinant `6μ̃ + Y*`. At `Ŝ_0` it is `[[6, γ*/2], [γ*/2, −μ̃ + kB*/2]]`, with determinant `−6μ̃ + Y*`. The stiff block is + `−diag(μ_2, …, μ_m)/k`. + + So off the null set `{6μ̃ = ±Y*}` the inertia converges. The limit is typed iff `μ_2 > 0` and `|Y*| < 6μ̃`, which forces + `μ̃ > 0`. And `W_r(f_r)/r⁴ → (μ_2⋯μ_m)²(36μ̃² − Y*²)1{μ_2 > 0, |Y*| < 6μ̃}`. +- *Decision.* Suppose `μ_2 > 0`, `μ̃ > 0`, `γ* ≠ 0` and `ϖ* := ϖ(μ̃, γ*, B*, C_3*) ∈ 𝒦`. Apply Proposition FL.4 with + `(ϖ, 𝒬) := (ϖ*, 𝒬*)`, `𝒬* := (μ̃/(kγ*²))diag(μ_2, …, μ_m)`, to the normalized window `𝔊_r` of `f_r`: + - (S1) holds exactly; + - (S2) holds for small `r`, by (1.3), (iv), `ϖ(f_r) → ϖ*` and `𝒬(f_r) → 𝒬*`; + - `Ψ_r(X, z, η) := Φ_r(X, (γ(f_r)/λ̃(f_r))z, η)` is affine and injective on the bounded window for small `r`, and + `f_r∘Ψ_r = b + s_r𝔊_r` with `s_r := kr³γ(f_r)²/λ̃(f_r) > 0`. + + So `e(f_r) = e(G_{ϖ*})` for small `r`. +- *Full measure.* For fixed `μ̃ > 0`, `μ_2, …, μ_m > 0` and `O`, the vector `(γ*, B*, C_3*)` is Gaussian, being affine in + `g_0`. Its covariance is nondegenerate: these are three distinct third-order functionals, and the conditional covariance + given the contact pins and `𝔸` is positive definite by [P] §2. The map `(γ, B, C_3) ↦ ϖ = (γ²/(24μ̃), kB/μ̃, k²C_3γ/μ̃²)` is + a local diffeomorphism on `{γ ≠ 0}`, with Jacobian determinant `k³γ²/(12μ̃⁴)`. So `ϖ*` has an absolutely continuous law, + and by Lemma FL.2, `ℙ(ϖ* ∈ 𝒯 ∖ 𝒦) = ℙ(γ* = 0) = ℙ(|Y*| = 6μ̃) = 0`. + +Hence, almost surely, `Ξ_r(f_r) → Ξ* := (μ_2⋯μ_m)²(36μ̃² − Y*²)1{μ_2 > 0, |Y*| < 6μ̃}(1 − e(G_{ϖ*}))`. The weight bound +(3.6) below gives `sup_r E[Ξ_r(f_r)²] < ∞`, so `h_r → h := E[Ξ*]` (Vitali). Also `p_𝔸(a_r) → p_𝔸(a*)`, +`𝒱(μ̃r/k, μ_2, …) → 𝒱(0, μ_2, …)`, and `1{μ̃r/k < μ_2} → 1{μ_2 > 0}`. + +*Step 4 (domination).* Let `U_a := 1 + ‖F_r^a‖_{C⁵} ≤ C(T + P + |a| + |𝔸|)`. Then `M_3, M_4 ≤ CU_a`, and +`‖A_M − a‖ ≤ (r/2)M_3 + ‖D_Θ²(F_r^a − F_0^a)(0)‖ ≤ CrU_a + r²T_a`, where `A_M := D_Θ²F_r^a(M)`. +- *Weight.* On the typed support, `A_M < 0`. By [P] (6.2) and Weyl's inequality (`λ_j(−A_M) ≤ |μ_j| + ‖A_M − a‖`), with + `h = M_3`, + + W_r/r⁴ ≤ (h²/4)(λ_1/r)(λ_1/r + 3h/2)Π_{j≥2}λ_j(λ_j + rh) ≤ CU′²(|μ̃| + U′)²Π_{j≥2}(|μ_j| + U′)², (3.6) + + where `U′ := U_a + rT_a`. +- *Soft layer.* The deterministic cap theorem ([CAP] §§1 and 5, as used in [P] §7) shows that every `C⁴` field with the + pins has `d_f(M) = f(S)`, i.e. `e = 1`, on the cap region `{λ_min(−A_M) > (4/(3k))rM_3², rM_4 ≤ 3k/10}`. It uses no Morse + or genericity hypothesis, because our mark is the maximin itself. So on `{typed, e = 0, rM_4 ≤ 3k/10}`, + `λ_min(−A_M) ≤ (4/(3k))rM_3²`. Hence `μ̃r/k = λ_1(−a) ≤ CrU′²`. On the typed support, `λ_min(−A_M) > 0`, so `μ̃ ≥ −CU′`. + Thus the support of `Ξ_r(f_r)` lies in `{−CU′ ≤ μ̃ ≤ CU′²}`. +- *Removing `μ̃` from `U′`.* We have `|a_r| ≤ |μ̃|r/k + Σ_{j≥2}|μ_j|`. + - `m ≥ 2`: on the domain of (3.4), `μ̃r/k < μ_2`. If `μ̃ ≥ 0`, then `|μ̃|r/k < |μ_2|`. If `μ̃ < 0`, the support bound + gives `|μ̃| ≤ C(P + Σ|μ_j| + T + |𝔸| + |μ̃|r/k)`, so `|μ̃| ≤ 2C(P + Σ|μ_j| + T + |𝔸|)` once `Cr/k ≤ ½`. Either way, on + the support, `U′ ≤ Ū := C(P + Σ_{j≥2}|μ_j| + T + |𝔸|)`. + With `1{|μ̃| ≤ CŪ²} ≤ (1 + CŪ²)^q(1 + |μ̃|)^{−q}`, `p_𝔸(a_r) ≤ C_0e^{−c_0Σ_{j≥2}μ_j²}` and + `𝒱 ≤ C(1 + |μ̃| + Σ|μ_j|)^{m(m−1)/2}`, the integrand of (3.4) is at most + `C_q(1 + |μ̃|)^{m(m−1)/2+2−q}(1 + Σ_{j≥2}|μ_j|)^{c_q}e^{−c_0Σ_{j≥2}μ_j²}`. With `q := m(m−1)/2 + 4` this is + integrable and independent of `r`. + - `m = 1`: `|a_r| = |μ̃|r/k`. For `μ̃ < 0` the support bound gives `|μ̃| ≤ 2CV`, `V := P + T + |𝔸|`, as for `m ≥ 2`. For + `μ̃ ≥ 0` it gives `μ̃ ≤ C(V + μ̃r/k)²`. So either `μ̃ ≤ C′V²` (*near branch*), which is dominated as for `m ≥ 2`, or + `μ̃ ≥ k²/(C′r²)` (*far branch*). On the far + branch, `p_𝔸(−μ̃r/k) ≤ C_0e^{−c_0μ̃²r²/k²}` and (3.6) is polynomial in `μ̃` and `r^{−1}`. So the far branch contributes + `O(r^{−c}e^{−c′/r²}) → 0`, uniformly. + +*Step 5 (conclusion).* By dominated convergence in (3.4), and since `12π_r(v_r) → 12π_0(u; v_0(b, k))` ([R] (R5)), + + (k/r)r^{−2}A_r^{rej} → 12π_0(u; v_0(b,k)) c_m∫_{0<μ_2<⋯<μ_m}∫_O p_𝔸(a*)𝒱(0, μ_2, …) ∫_0^∞ h dμ̃ dμ_2⋯dO. (3.7) + +By #242 §2, the change of variables `μ̃ = γ*²/(24φ)` turns `36μ̃² − Y*²` into `(γ*⁴/16)(φ^{−2} − (1 − t*)²)` and `dμ̃` into +`(γ*²/(24φ²))dφ`. So by Fubini, `∫_0^∞h dμ̃ = (μ_2⋯μ_m)²E[(γ*⁶/384)I(t*, χ_0*)]`, with `t*`, `χ_0*` as in (0.3) and `I` as +in #242 (2.5). + +It remains to identify (3.7) with #242 (3.1). By Weyl's formula, +`ε^{−1}E_{v_0}[Υ1{𝔸 < 0, λ_1 < ε}] = c_m∫∫ε^{−1}∫_0^ε p_𝔸(a)𝒱(λ)E[Υ(F_0^a)]dλ_1 dλ_2⋯dO`, where +`Υ := (γ_1⁶/384)(λ_2⋯λ_m)²I(t, χ_0)`. The inner average tends to its value at `λ_1 = 0`, for three reasons: +- by #242 Lemma 3, `γ⁶I ≤ C(γ⁶ + k³|B|³ + k⁴C_3²)`, which is integrable with Gaussian weights; +- `I` is continuous at almost every `(t, χ_0)`. By Remark 4 of §2, the actual decision is constant near each point of `𝒦`; + for almost every `(t, χ_0)` the set `{φ : (φ, 2tφ, χ_0φ²) ∉ 𝒦}` is null, so `1{φ ∈ 𝓡(t′, χ_0′)} → 1{φ ∈ 𝓡(t, χ_0)}` for + almost every `φ` as `(t′, χ_0′) → (t, χ_0)`. Lemma 3 bounds `φ` away from `0` on the rejected set, locally uniformly, + which gives domination in `φ`; +- the jets of `F_0^a` along `e_1(a)` depend continuously on `a` and have a nondegenerate Gaussian law. + +So the limit in #242 (3.1) exists, and (3.7) equals `F(k; b, u)`. + +*Uniformity and continuity.* The proof applies verbatim along any sequence `(r_n, b_n, k_n) → (0, b, k)` in `(0, 1] × 𝒞`. +The contact law depends on `(b, k)` only through an affine shift of the mean ([P] (3.3)), so `p_𝔸` and the jets move +continuously. All bounds above are uniform on `𝒞`. A diagonal argument then gives both the continuity of `F` on `𝒞` and the +uniform convergence. Positivity is proved in Corollary FL.5. (Uniformity in `u` would need `u_n → u` in this argument as +well; it is not claimed.) ∎ + +## 4. The selection loss and the compact-window density difference + +**Corollary FL.5 (the constant in `1 − p_r ~ r³`).** For fixed `u ∈ S^{d−1}`, uniformly for `(b, k)` in compact subsets of +`R × (0, ∞)`, + + r^{−3}(1 − p_r(b, k, u)) → F(k; b, u)/(k·A_∗(b, k, u)) ∈ (0, ∞) (r → 0). (4.1) + +*Proof.* By (0.1), `r^{−3}(1 − p_r) = r^{−3}A_r^{rej}/A_r = k^{−1}[(k/r)r^{−2}A_r^{rej}]/A_r`. Theorem FL and +`A_r → A_∗ > 0` uniformly ([P] (10.3)) give (4.1). + +*Positivity of `F`.* At `ϖ = (φ, 0, 0)` the slices are concave (`a = 1`, `χ = 0`), `R(X) = A₀ + p²/8`, and exactly +(control F6) + + R(X) − L_S = (X − ½)²[2(X + 1) + 3φ(X + ½)²]/(24φ). (4.2) + +The bracket is a quadratic in `X` with discriminant `4 − 12φ`, so for `φ > 1/3` it is positive: `R > L_S` for all `X ≠ ½`, +and `R ~ X⁴/8 → +∞` as `X → −∞`. So `I_M ⊃ (−∞, ½)` and `M` dies above `L_S`, along the ridge into `X → −∞`. By Remark 4 of §2 +this `(R)` certificate persists, with only `C⁰` closeness on a compact path, for every `ϖ′` near `(φ, 0, 0)`; by compactness, +uniformly for `φ ∈ [1/3 + δ_5, 1 − δ_5]`. So `𝓡(t, χ_0) ⊃ [1/3 + δ_5, 1 − δ_5]` for `|t| + |χ_0| ≤ θ`, and there +`I(t, χ_0) ≥ ∫_{1/3+δ_5}^{1−δ_5}φ^{−2}(φ^{−2} − (1 − t)²)dφ > 0` (for `θ` small). Given `𝔸 = a*`, the event +`{|t*| + |χ_0*| ≤ θ, γ* ≠ 0}` has positive probability, since the jets are nondegenerate Gaussian. Also `p_𝔸 > 0`, +`π_0 > 0` and `𝒱 > 0` almost everywhere. Hence `F > 0`. ∎ + +[P]'s Theorem A gives `1 − p_r ≤ Cr³`, and the OA dimension lift (A3) gives `1 − p_r ≥ cr³`, both uniformly. (4.1) +identifies the constant at each `(b, k, u)`. + +**Corollary FL.6 (the compact-window density difference).** Let `𝐁` (births) and `𝐊 = [k_−, k_+] ⊂ (0, ∞)` (gap marks) be +compact intervals of positive length, and let `ν_cand`, `ν_eld` be the compact-window densities of [P]'s Theorem B. Then + + ν_cand(ℓ) − ν_eld(ℓ) = d_{𝐁,𝐊}ℓ^{2/3} + o(ℓ^{2/3}), d_{𝐁,𝐊} := (1/3)∫_{S^{d−1}}∫_𝐁∫_𝐊 k^{−8/3}F(k; b, u)dk db dσ(u) ∈ (0, ∞). (4.3) + +Consequently `E[N_cand(0, t] − N_eld(0, t]] = (3/5)d_{𝐁,𝐊}t^{5/3} + o(t^{5/3})`. And for every real `q > −5/3`, +`E Σ_{candidate, not selected, ℓ ≤ t}ℓ^q = d_{𝐁,𝐊}t^{q+5/3}/(q + 5/3) + o(t^{q+5/3})`. These sharpen [P] (1.2), (12.1) +and (12.3) from upper bounds to asymptotic laws. + +*Proof.* By [P] (11.1)–(11.2), at `r = (ℓ/k)^{1/3}`, +`ν_cand(ℓ) − ν_eld(ℓ) = ℓ^{−1/3}∫A_r(1 − p_r)/(3k^{2/3}) db dk dσ = ℓ^{−1/3}∫A_r^{rej}/(3k^{2/3})`. Since `r³ = ℓ/k`, +`A_r^{rej} = (ℓ/k²)·[(k/r)r^{−2}A_r^{rej}]`, so + + ℓ^{−2/3}(ν_cand(ℓ) − ν_eld(ℓ)) = (1/3)∫_{S^{d−1}×𝐁×𝐊} k^{−8/3}[(k/r)r^{−2}A_r^{rej}(b, k, u)] dk db dσ. (4.4) + +By [C7-K] (K2) (`r ≤ k_−`), `(k/r)r^{−2}A_r^{rej} ≤ 12π_r(v_r)CP^N` is bounded on the compact window. Theorem FL and +dominated convergence give (4.3), and `d_{𝐁,𝐊} > 0` by the positivity of `F`. The cumulative and moment statements follow +by integrating (4.3) against `ℓ^q dℓ` on `(0, t]`, as in [P] §12. ∎ + +*Remarks.* +1. #242 Proposition 4 computes `F = F_0(b)H(k)` for the Gaussian kernel `e^{−|x|²/2}` on `R^d`, the model case, not for the + torus field of Theorem FL. For that model the formula reads `d_{𝐁,𝐊} = (1/3)|S^{d−1}|∫_𝐁F_0 db·∫_𝐊 k^{−8/3}H(k)dk`; for the + periodized torus field the jets' laws differ by `O(e^{−L²/8})` (#242 §0). +2. Without the lower cutoff `k_−`, the integral in (4.3) diverges at `k → 0`, where `F(k) → F_0 > 0`. Cutting it at the cusp + scale `k ≍ ℓ^{1/4}` produces `ℓ^{2/3}·ℓ^{−5/12} = ℓ^{1/4}`, the order of #229's term `(I^{cand} − c_1)ℓ^{1/4}`. This is why + the unrestricted `ℓ^{2/3}` coefficient (#242 Conjecture 7) needs the two-scale composite and the intermediate separations, + while the compact-window coefficient (4.3) does not. + +## 5. Numerical evidence (exploration; not part of the proofs) + +The scripts and records are in the project archive (`exploration/`; numpy and scipy, outside the repository). + +**Decision and weight convergence on exactly pinned fields** (`fold_d2.py`, `fold_d3.py`). +- *Setup.* Gaussian kernel `e^{−|x|²/2}`, `b = 0`, `k = 0.4`. Jets of order `≤ 6` at `0` are drawn from the contact law with + `𝔸 = 0` (`d = 2`) or `𝔸 = −diag(0, μ_2)`, `μ_2 ~ U(0.5, 3)` (`d = 3`), and `γ` from the `γ⁶`-tilted law. +- *Fields.* For each sample, and each `φ` on a logarithmic grid over the typed range (40 points in `d = 2`, 24 in `d = 3`), + take the exactly pinned degree-6 polynomial at separation `r` with `∂_{e_1}²f(0) = −μ̃r/k`, `μ̃ = γ²/(24φ)`. +- *Decisions.* Its normalized window (0.4) is flood-filled at the levels `L_S ± 2·10⁻³|L_S|`, on a 601² grid (`d = 2`) or a + 181² × 41 grid (`d = 3`). The model `G_ϖ` (`G_ϖ − 𝔮` in `d = 3`) is flood-filled on the same grid. +- *Weights and integrals.* `W_r/r⁴` comes from the exact Hessians. Per sample, the integral + `I_r := (384/γ⁶)∫(W_r/r⁴)(1 − e_r)dμ̃` is compared with the model integral (including the factor `μ_2²` in `d = 3`). + +| `d` | samples | `r` | mean `I_r` / mean model integral (same grid) | decision mismatches | +|---|---|---|---|---| +| 2 | 400 | 0.1 | 0.918 ± 0.054 | 378 / 15,171 (2.5%) | +| 2 | 400 | 0.03 | 0.998 ± 0.005 | 110 / 15,215 (0.72%) | +| 2 | 400 | 0.01 | 1.0015 ± 0.0026 | 35 / 15,223 (0.23%) | +| 2 | 400 | 0.003 | 0.9999 ± 0.0023 | 10 / 15,230 (0.066%) | +| 3 | 120 | 0.1 | 1.20 ± 0.14 | 80 / 2,696 (3.0%) | +| 3 | 120 | 0.01 | 0.9927 ± 0.0037 | 5 / 2,733 (0.18%) | +| 3 | 120 | 0.001 | 0.99938 ± 0.00031 | 0 / 2,737 | + +- *Mismatch rates.* In `d = 2` the mismatch rate falls in proportion to `r`, the rate of Lemma FL.1. +- *Weights.* The median relative weight error `|W_r/r⁴ ÷ lim − 1|` in `d = 3` is `1.8·10⁻²`, `1.1·10⁻³` and `1.1·10⁻⁴` + at `r = 0.1, 0.01, 0.001`. That is rate `r`, not `r^{1/2}`: the `r^{1/2}`-terms of #242 (2.6) are linear in `η`, so they + move the pin Hessians' determinants only at order `r`. +- *Why the same grid.* The comparison is made on the same grid because the grid decision itself differs from the + one-dimensional implementation of #242 Lemma 2 (`dec1d`, a fixed-tolerance numerical scan) at 37 of 15,232 points in + `d = 2` and 26 of 2,737 in `d = 3`. Against `dec1d` the ratios settle at `0.992` (`d = 2`) and `0.967` (`d = 3`), which are + the grid model's own offsets. + +**Lemma FL.2(c).** The first half (`d(M̂) ≥ L_S`) is the one-line axial-path argument. The case statement was also tested. +- *Author.* On 1,500 typed points from #242's validation law (500 elder, 1,000 rejected), a two-dimensional flood fill shows + `M` dead at `L_S + 2·10⁻³|L_S|` for 998 rejected points. Of the other two: + - one is near-degenerate: `V` exceeds `L_S` by `10⁻⁴` near `X = 0.39`, and the point dies above `L_S` at a finer offset; + - the other has `|χ| < 10⁻²`, and its far valley lies outside the box. +- *Referee (same family, independent scripts).* 14,040 typed points were tested in seven families: broad, Gaussian jets, + `β > 2`, `|β − 2| ∈ [10⁻⁶, 10⁻¹]`, `|χ| ∈ [10⁻⁷, 10⁻³]`, near the typed edge, and near the elder edge. The death level was + computed by bisection on the exact slice structure. No point had `d(M̂) < L_S`, and no elder point had `β > 2`, `χ < 0`. + The bisection agreed with a two-dimensional flood fill on 800 cross-check points, after one grid artifact was resolved. + +**Lemma FL.1 and Proposition FL.4 beyond polynomials (referee, independent).** +- *Lemma FL.1.* The fields were exactly pinned non-polynomial fields (trigonometric base plus polynomial correction), + evaluated in 60-digit arithmetic. The `C²` error of the window falls by `100.0` per factor `100` in `r` in `d = 2`, and by + `10.1` in `d = 3`. It is identical for `λ̃ ∈ {3/2, 100, −5, 1/100}` and for the stiff eigenvalues tested. +- *Proposition FL.4.* The perturbations were non-polynomial and `C²`-small (sums of sines, corrected to keep `M̂` and `Ŝ` + critical). They were applied to five model points in `d = 2`, and to three in `d = 3` with a stiff direction coupled to the + plane. The decision persisted up to the largest amplitude tested (`10⁻²` in `d = 2`, `3·10⁻³` in `d = 3`), with one + exception: #242's near-degenerate (D′) point flips at amplitudes `≥ 3·10⁻³`. This is consistent with `ε` depending on the + margins. + +## 6. Scope and relation to other packets + +- **#242.** Conjecture 6 is Theorem FL. The definition (3.1) of `F` is the actual fold-scale limit of the rejected kernel in + every dimension. For the Gaussian kernel on `R^d` (#242's model case), #242 Proposition 4 describes `F`: `F = F_0H`, + `H(k) = 1 + (12/25)k² + O(k³)`. + Proposition 2′ of #242 (the stiff directions) is transferred to the field here, as part of Proposition FL.4. +- **#242 Conjecture 7** is not proved. It needs inputs (i)–(ii) of #242 §4. Input (i) asks for (3.1) with an error that, + integrated against `dr` from the cusp scale to the fold scale, is `O(ℓ^{3/4})` after the composite is subtracted. That + needs quantitative margins in Proposition FL.4 near the decision boundary, which the present pointwise argument does not + give. Input (ii) (the intermediate separations) is untouched. +- **[P], the dimension lift, and [C7-K].** Corollaries FL.5 and FL.6 identify the constants in the `Θ(r³)` selection loss + and the `Θ(ℓ^{2/3})` compact-window density difference. Their orders were known from [P] Theorems A and B, the OA lower + bound (A3), and [C7-K] (K2). +- **Matching.** #242 Theorem 1 gives `κ𝒜^{rej} → F_0` as `κ → ∞` at the cusp scale. Theorem FL gives + `(k/r)r^{−2}A_r^{rej} → F(k)`, with `F(k) → F_0` as `k → 0` (#242 §3). The two limits agree in the overlap, which is what + the composite of #242 Lemma 5 assumes. +- **Not claimed:** any rate in (3.1); uniformity as `k → 0` or `k → ∞`; any statement about `ν_cand`'s own corrections. + +## 7. Controls + +`fold_check.py` (standard library, exact rationals; F4 also prints floating-point margins on fixed points) produces +`RESULTS.json`, byte-identical under `-O`. +- **F1.** The model at the pins: + - `det Hess G_ϖ(M̂) = (1 + β/2 − φ)/(4φ)`, `∂_z²G_ϖ(M̂) = −(1 + β/2)` and `det Hess G_ϖ(Ŝ) = −(1 − β/2 + φ)/(4φ)`, by exact + central differences (exact for cubics), on random rationals; + - the typed region `𝒯`; + - the unnormalized determinants `6λ̃ + Y` and `−6λ̃ + Y`, and the normalization (0.4); + - the window determinant identity behind (3.5), `|det Hess f| = k^{m−1}r²|det Hess 𝔉|`, for `m = 1, 2, 3`. +- **F2.** Lemma FL.2: + - the Jacobians (2.1), as polynomial identities in `(X, z, 1/φ, β, χ)`, with the derivatives taken by exact polynomial + differentiation; + - the critical values on `X = ±½`, the factorization of `A₀ − L_S` and the end value `A₀(2) = 25/(48φ)` of the axial path, + the double root `z = a/χ` and the inflection value on `D = 0`, and the `1/φ`-scaling of `A₀ − L_S`; + - the coefficients `κ_±` of (2.2), against the leading-order slice equation and against `R(X)/X³` at `X = ±10⁶` + (floating point); `κ_+ = 0` for the witness; + - the Jacobian `k³γ²/(12μ̃⁴)` of `(γ, B, C_3) ↦ ϖ`, by exact differentiation. +- **F3.** Lemma FL.1: + - the exponent table (1.2) for every monomial of degree `≤ 4` in `d = 2, 3, 4`, and the order-5 remainder; + - on exactly pinned degree-6 fields with rational coefficients, the measured `C²` error of the window against + `G_k^{(d)}` at four points. It drops by `99.87` (`d = 2`) and by `10.0001` (`d = 3`) when `r` drops by `100`: rates `r` + and `r^{1/2}`. (The referee's check on non-polynomial fields gave the same rates; §5.) +- **F4.** Proposition FL.4: + - the exact (D′) witness `G_{(3/2, 8/3, 20/3)}(X, z) = (1/9)G_{(1/6, 0, 0)}(X + 2z, 3z)` (found by OpenAI Codex on #242), as a + polynomial identity; + - a polynomial factorization showing that `G_{(1/6, 0, 0)}` has exactly the critical points `(±½, 0)` and `(−3, 35/8)`, so + the witness has exactly three, with values `0`, `−1/36` and `−125/384`; + - the crossing forms at `Ŝ` on rational points in cases (D) and (D′); + - the two elementary facts used in Lemma FL.2(c): for `β > 2`, `χ < 0` the slice at `1/β` is open; for `χ > 0`, `D > a²` on + `|X| < ½`; + - the floating-point margins of the certificates of Step 3 on three fixed points, one per case. +- **F5.** Theorem FL: the change of variables `μ̃ ↔ φ`, the constant `384`, and `χ = k²C_3γ/μ̃² = χ_0φ²`, on random rationals. +- **F6.** The corollaries: + - the factorization (4.2), as a polynomial identity in `(X, 1/φ)`, and its discriminant `4 − 12φ`; + - the concave slice at `ϖ = (φ, 0, 0)` (`G = R` at `z_r = p/2`); + - exact Sturm certificates of `(R)` along the ridge for `φ ∈ {2/5, 1/2, 3/4, 9/10}`: no root of `R − L_S − 1/200` on + `[−3, −½]`, and `R(−3) > 0`; + - the exponents of the pushforward (4.4) and of Remark 2. + +Mutants `M1`–`M7` each fail only their own control (`M1` F1, `M2` F2, `M3` F3, `M4` and `M7` F4, `M5` F5, `M6` F6); the checker +names the failing control on stderr, and an unknown label exits 2. + +**What the controls do not test.** Proposition FL.4 for non-polynomial perturbations, the transversality and Sard arguments +themselves (only their Jacobians), Lemma FL.2(c) beyond its two elementary facts (it is tested numerically in §5), and the +probabilistic steps of Theorem FL (coupling, conditioning, domination). These are argued in the text and are the subject of +review slices B–D. + +## 8. Sources (exact identities in `SOURCES.json`) + +| Tag | Path (blob) | Use | +|---|---|---| +| [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (`dfed3b8d`) | §§1–3, 5–8, 10–12 — consumed | +| [R] | `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (`247b3ecf`) | (R2)–(R3), (R5), (R11) — consumed | +| #207 | `frontiers/cusp_second_order_20261001/PROOF.md` (`f6df5a73`) | §0, Theorem CU.1, the structure of CU.3–CU.4 — consumed | +| [C7-K] | `frontiers/c7_total_bounded_20260929/PROOF.md` (`28748b08`) | (K2) — consumed | +| [CAP] | `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` (`0633aca3`) | §§1, 5: the deterministic cap theorem — consumed (as in [P] §7) | +| [E1], [REC] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` (`213594d6`), `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` (`75da2597`) | reading rules for [P] | +| [E2] | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` (`fe9b9ce4`) | the elder mark is Borel — consumed | +| #242 | `frontiers/soft_rejected_pairs_20261002/PROOF.md` at `87912bd` (`8b2f5fae`), unmerged | §0, (2.1)–(2.6), Lemmas 2–3, Proposition 2′, (3.1) — consumed | +| OA lift | `frontiers/elder_dimension_lift_20260928/PROOF.md` (`7303bd79`) | (A3) — cited | +| #229 | `frontiers/third_order_rate_20261001/PROOF.md` (`110ed33a`) | (R⁺.1) — cited | + +## 9. Review slices + +- **A** — §§0–1: the setting, Lemma FL.1 and (1.3); in particular the exponent table and the pinned coefficients in + `d ≥ 3`. +- **B** — §2: Lemma FL.2 (the transversality arguments, (G2), (G3), and the trichotomy (c)), Lemma FL.3, and Proposition + FL.4 (the facts (Path) and (Trap), the certificates in cases (R), (D), (D′), and the persistence in Step 4, including + the stiff directions). +- **C** — §3, Steps 1–3: the coupling and the conditioning on `𝔸`, the Weyl zoom (3.4), the window identity (3.5), and the + almost-everywhere convergence (full measure of `ϖ* ∈ 𝒦`). +- **D** — §3, Steps 4–5: the domination (cap region, (3.6), the `m = 1` far branch), and the identification with #242 (3.1). +- **E** — §4: Corollaries FL.5 and FL.6 (the positivity of `F`, and the use of [P] (11.2) and [C7-K] (K2)), with §§5–6. + +A reviewer should record, per slice: ACCEPT, ACCEPT WITH FIXES (list), or REJECT (with the failing step). diff --git a/frontiers/soft_fold_limit_20261002/README.md b/frontiers/soft_fold_limit_20261002/README.md new file mode 100644 index 000000000..5c178fa11 --- /dev/null +++ b/frontiers/soft_fold_limit_20261002/README.md @@ -0,0 +1,93 @@ +# The fold-scale rejection limit in every dimension + +**Author-side proof candidate (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** + +Object `CL-FOLD-LIMIT-20261002-v1`. Full text: [`PROOF.md`](PROOF.md). + +## Result + +Math- #242 (open) defines the fold-scale rejection rate `F(k; b, u)` through an explicit soft model and the elder decision +of its Lemma 2, and states as Conjecture 6 that the rejected kernel at a fixed gap `k = ℓ/r³` converges to it. This note +proves that conjecture in every dimension and draws two consequences for the parent's Theorems A and B. + +| | Statement | Status | +|---|---|---| +| **Theorem FL** | `(k/r)r^{−2}A_r^{rej}(b, k, u) → F(k; b, u)` as `r → 0`, every `d ≥ 2`, uniformly for `(b, k)` in compacts; the limit in #242 (3.1) exists, and `F` is continuous and positive | proof | +| **Corollary FL.5** | `r^{−3}(1 − p_r) → F/(k·A_∗) ∈ (0, ∞)` (fixed `u`, uniformly for `(b, k)` in compacts): the constant in the `Θ(r³)` selection loss of [P] Theorem A and the OA lower bound (A3) | proof | +| **Corollary FL.6** | compact mark windows: `ν_cand − ν_eld = d_{𝐁,𝐊}ℓ^{2/3} + o(ℓ^{2/3})`, `d_{𝐁,𝐊} = (1/3)∫∫∫k^{−8/3}F > 0`; so also `E[N_cand − N_eld](0, t] ~ (3/5)d_{𝐁,𝐊}t^{5/3}` | proof | +| **Proposition FL.4** | stability of #242 Lemma 2's decision: every `C²`-small perturbation with the pins as critical points has the model's decision, for the global superlevel filtration of any field realizing it in a window; stiff directions included | proof | +| **Lemma FL.2** | the generic set `𝒦` has full measure; a rejected model maximum always dies strictly above the saddle level (an axial path, observed by OpenAI Codex on #242), and case (D′) needs `χ > 0` | proof | +| **Lemma FL.3** | a nondegenerate saddle can be crossed along a model curve, robustly under `C²`-small perturbation | proof | +| **Lemma FL.1** | the soft window of a pinned `C⁵` field is `C²`-close to the model: rate `r` (`d = 2`), `r^{1/2}` (`d ≥ 3`), constant linear in `‖f‖_{C⁵}` | proof | + +How the proof goes: +- **Conditioning.** The proof conditions the coupled pinned field on the contact Hessian `𝔸`. +- **Zoom.** It writes `𝔸` in Weyl coordinates and zooms its smallest eigenvalue at the scale `r`. +- **Pointwise limit.** For almost every zoomed configuration, the field's decision is the model's (Proposition FL.4), and + the weight converges by an exact window identity. +- **Domination.** It uses dominated convergence. The domination comes from [P]'s deterministic cap region (pairing holds + when `λ_min(−A_M) > (4/(3k))rM_3²`) and [P]'s weight bound (6.2). + +## Evidence (exploration; outside the repository) + +The tests use exactly pinned degree-6 fields with contact-law jets (Gaussian kernel, `b = 0`, `k = 0.4`). For each sample +they compare the finite-`r` rejected integral `(384/γ⁶)∫(W_r/r⁴)(1 − e_r)dμ̃` with its model limit, using the same flood-fill +grid for both (PROOF §5): + +| `d` | `r` | ratio | decision mismatches | +|---|---|---|---| +| 2 | 0.1 / 0.03 / 0.01 / 0.003 | 0.918 / 0.998 / 1.0015 / 0.9999 (± 0.054 … 0.0023) | 2.5% / 0.72% / 0.23% / 0.066% | +| 3 | 0.1 / 0.01 / 0.001 | 1.20 / 0.9927 / 0.99938 (± 0.14 … 0.0003) | 3.0% / 0.18% / 0 of 2,737 | + +- **Lemma FL.2(c).** No counterexample on 1,500 (author) or 14,040 (referee) typed parameter points. +- **Beyond polynomials.** The referee also checked Lemma FL.1 on non-polynomial fields: rate `r` in `d = 2`, `r^{1/2}` in + `d = 3`, with the same error for every transverse eigenvalue tested. It checked Proposition FL.4 under non-polynomial + `C²`-small perturbations, in `d = 2` and in `d = 3`. + +## Controls + +`fold_check.py` uses the standard library and exact rationals; control F4 also prints floating-point certificate margins +on three fixed points. Its output is `RESULTS.json`, byte-identical under `-O`. + +| Control | Checks | +|---|---| +| F1 | the model's Hessians at the pins, the typed region, the unnormalized blocks `±6λ̃ + Y`, the normalization, and the window determinant identity `|det Hess f| = k^{m−1}r²|det Hess 𝔉|` for `m = 1, 2, 3` | +| F2 | Lemma FL.2: the transversality Jacobians `z⁴(X ± ½)/(96φ²)` as polynomial identities, the critical values on `X = ±½`, `A₀ − L_S = (X − ½)²(X + 1)/(12φ)` and the axial path, the inflection value, the asymptotic coefficients `κ_±`, and the Jacobian of `(γ, B, C_3) ↦ ϖ` (by differentiation) | +| F3 | Lemma FL.1: the exponent table for every monomial of degree `≤ 4` in `d = 2, 3, 4`, and the measured `C²` rates on exactly pinned fields (`r` in `d = 2`, `r^{1/2}` in `d = 3`) | +| F4 | Proposition FL.4: the exact (D′) witness `G_{(3/2, 8/3, 20/3)}(X, z) = (1/9)G_{(1/6, 0, 0)}(X + 2z, 3z)`, a factorization proving it has exactly three critical points, the crossing forms at `Ŝ`, the two facts used in Lemma FL.2(c), and floating-point certificate margins on three fixed points | +| F5 | Theorem FL: the change of variables `μ̃ ↔ φ`, the constant `384`, and `χ = χ_0φ²` | +| F6 | the corollaries: the factorization (4.2) of `R − L_S` at `ϖ = (φ, 0, 0)` and its discriminant, exact Sturm certificates of rejection for `φ ∈ {2/5, 1/2, 3/4, 9/10}`, and the exponents of the pushforward | + +Mutants `M1`–`M7` each fail only their own control (the checker names it on stderr), and an unknown label exits 2: + + python3 -B -S fold_check.py # exit 0, output = RESULTS.json + python3 -B -S fold_check.py --mutant M1 # exit 1 + +## Not claimed + +- No rate in Theorem FL, and no uniformity as `k → 0` or `k → ∞`. +- #242 Conjecture 7 (the unrestricted `ℓ^{2/3}` term) is not proved; it still needs #242 §4's inputs (i)–(ii). +- No numerical constant is certified. +- No change to #242, [P] or any other packet; #242 is consumed at a pinned blob. + +## Review record + +One clean-context same-family referee (an Anthropic Claude subagent) read the note, the checker and the consumed sources at +their declared blobs (#242 at `87912bd`). It re-derived the identities with its own symbolic scripts, including the powers of +`k` and `r` in (3.5), and tested Lemma FL.1 and Proposition FL.4 on non-polynomial fields and Lemma FL.2(c) on 14,040 typed +parameter points (`PROOF.md` §5). Verdict: **ACCEPT WITH MINOR FIXES**, with no BLOCKING or MAJOR finding. Seven MINOR +findings and fourteen NITs, all applied before submission: +- MINOR: the cap step of the domination (now the deterministic cap theorem, [CAP] §§1 and 5, which needs no Morse + hypothesis because the mark is the maximin); the dependence of Proposition FL.4's constants on the stiff form `𝒬`; the + radius bookkeeping of Lemma FL.3 and Step 4; Remark 4 (the actual decision is locally constant near `𝒦`, hence + measurable), as used in Step 5; the description of the controls, with Lemma FL.2(c) now tested in §5; `SOURCES.json` and the + wording of [R] (R2)–(R3); uniformity in `u` (no longer claimed). + +The same referee then checked the fixes: **ACCEPT WITH MINOR FIXES**. Every fix was correct. It found one new MINOR finding +(a stale definition of `q_−` in Step 3 of Theorem FL, with no mathematical effect) and six NITs: the name of the Hessian at +`Ŝ`, the domain of Lemma FL.3, two descriptions in §§5 and 7, the `kind` summary and the reading rules [E1], [REC], and the +description of [CAP]. All seven were applied. After that check, Step 4 of Proposition FL.4 was reordered so that `W` and +`R_η` are fixed after `ε` and visibly contain the balls of Lemma FL.3; the argument is unchanged. + +Same GitHub account and same provider as the author: zero organizational-independence credit. Nonauthor review is +required for every slice (`PROOF.md` §9). diff --git a/frontiers/soft_fold_limit_20261002/RESULTS.json b/frontiers/soft_fold_limit_20261002/RESULTS.json new file mode 100644 index 000000000..2df93a56e --- /dev/null +++ b/frontiers/soft_fold_limit_20261002/RESULTS.json @@ -0,0 +1,155 @@ +{ + "F1": { + "random_parameter_points": 40, + "window_identity_dims_checked": [ + 1, + 2, + 3 + ] + }, + "F2": { + "kappa_pm_float": [ + [ + "1/2", + "1/3", + "-2", + 0.508700864, + -0.002528024 + ], + [ + "3/2", + "8/3", + "20/3", + 0.0, + -0.004444444 + ], + [ + "1/5", + "-2", + "3", + 0.490740741, + 0.416666667 + ], + [ + "2/3", + "5/2", + "-1/4", + 176.841338236, + 0.075328431 + ] + ], + "witness_kappa_plus": "0" + }, + "F3": { + "exponent_table": { + "2": { + "checked_monomials": 5, + "listed_pinned_or_exact": 10 + }, + "3": { + "checked_monomials": 21, + "listed_pinned_or_exact": 14 + }, + "4": { + "checked_monomials": 21, + "listed_pinned_or_exact": 14 + } + }, + "pinned_fields": { + "2": { + "C2_error_at_two_r": [ + "0.05243003801440329", + "0.0005249930840657922" + ], + "expected_ratio": "100", + "ratio": "99.86805465771194" + }, + "3": { + "C2_error_at_two_r": [ + "0.07666748457359383", + "0.007666667498445729" + ], + "expected_ratio": "10", + "ratio": "10.00010559857157" + } + } + }, + "F4": { + "certificate_margins_float": { + "D": { + "A0_minus_LS_at_X1": 0.004213, + "I_M": [ + -1.086, + 0.5 + ], + "boundary_max_minus_LS": -0.015882, + "interior_max_off_M": -0.003175, + "ridge_path_min_minus_LS_off_S": 0.004525, + "right_side_max_minus_LS_off_S": -0.003165, + "tau": 0.02 + }, + "D'": { + "I_M": [ + -1.505, + 0.517 + ], + "boundary_max_minus_LS": -0.000167, + "interior_max_off_M": -0.000191, + "ridge_path_min_minus_LS": 0.000556, + "slice_path_end_value": 71377.75, + "slice_path_min_minus_LS_off_S": 0.00054, + "tau": 0.005 + }, + "R": { + "endpoint_value": 0.050253, + "path_min_minus_LS": 0.070321 + } + }, + "crossing_forms_points": { + "D": 8, + "D'": 3 + }, + "trichotomy_points": { + "beta>2,chi<0 open at 1/beta": 40, + "chi>0: D > a^2 on |X| < 1/2": 24 + }, + "witness_critical_values": [ + "0", + "-1/36", + "-125/384" + ] + }, + "F5": { + "random_points": 30 + }, + "F6": { + "R_certificates_phi_0_0": { + "1/2": { + "R(-3)": "2075/384", + "roots_of_R_minus_LS_minus_delta_on_[-3,-1/2]": 0 + }, + "2/5": { + "R(-3)": "1675/384", + "roots_of_R_minus_LS_minus_delta_on_[-3,-1/2]": 0 + }, + "3/4": { + "R(-3)": "7825/1152", + "roots_of_R_minus_LS_minus_delta_on_[-3,-1/2]": 0 + }, + "9/10": { + "R(-3)": "25075/3456", + "roots_of_R_minus_LS_minus_delta_on_[-3,-1/2]": 0 + } + } + }, + "controls": { + "F1_model_pins_and_weight": true, + "F2_lemma_FL2_generic": true, + "F3_lemma_FL1_window": true, + "F4_proposition_FL4_certificates": true, + "F5_theorem_FL_bookkeeping": true, + "F6_corollaries": true + }, + "object": "CL-FOLD-LIMIT-20261002-v1", + "scientific_effect": "NONE" +} diff --git a/frontiers/soft_fold_limit_20261002/SOURCES.json b/frontiers/soft_fold_limit_20261002/SOURCES.json new file mode 100644 index 000000000..80b88dd82 --- /dev/null +++ b/frontiers/soft_fold_limit_20261002/SOURCES.json @@ -0,0 +1,145 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "object": "CL-FOLD-LIMIT-20261002-v1", + "kind": "author-side proof candidate, every d >= 2: Theorem FL (the fold-scale limit (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u), Conjecture 6 of Math- #242, uniformly on compacts of (b, k); the limit defining F exists, F is continuous and positive); Corollary FL.5 (r^-3 (1 - p_r) -> F/(k A_*), A_* the limit of A_r); Corollary FL.6 (compact mark windows BB x KK: nu_cand - nu_eld = d l^(2/3) + o(l^(2/3)), d = (1/3) int k^(-8/3) F > 0); Proposition FL.4 (stability of the model decision of #242 Lemma 2 under C^2-small perturbations, stiff directions included); Lemma FL.2 (the generic set has full measure; a rejected model maximum dies strictly above the saddle (an axial path, observed by OpenAI Codex on #242); case (D-prime) needs chi > 0); Lemma FL.3 (crossing a nondegenerate saddle); Lemma FL.1 (the soft window, rate r in d = 2 and r^(1/2) in d >= 3)", + "author": "Anthropic Claude", + "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", + "read_at": { + "branch_base": "7636cae5c386f9725f9eec127dfd4ea5ec630435" + }, + "consumed": [ + { + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "tag": "P", + "use": "section 1 (Q, W_r, Z_r, p_r), section 2 (finite-jet rank), section 3 ((3.3)-(3.5)), sections 5-6 ((5.3), (6.2)), section 7 (the cap region where pairing holds, and (7.7)), section 8 (pinned genericity and the maximin), sections 10-12 ((10.2), (10.3), (11.1)-(11.2), (12.1), (12.3))" + }, + { + "path": "frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md", + "bytes": 17734, + "sha256": "380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a", + "git_blob": "247b3ecf80bfbe896948d5d489b2d5842a81c481", + "tag": "R", + "use": "(R2)-(R3) (the covariance bounds and the coupling, from which the pathwise bound (3.2) is derived by Taylor's formula), (R5) (the pin density), (R11) (A_r -> A_*, written A_0 there)" + }, + { + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "bytes": 15160, + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc", + "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "tag": "CAP", + "use": "sections 1 and 5: the deterministic cap theorem, as used in [P] section 7 (on its cap region every C^4 field with the pins has maximin d_f(M) = f(S)); consumed in Step 4 of Theorem FL" + }, + { + "path": "frontiers/cusp_second_order_20261001/PROOF.md", + "bytes": 52605, + "sha256": "cffc1b83ac364da58bd650c3c301ea1babd38e19af54f74010bdc120edce38a8", + "git_blob": "f6df5a7356a54771b0052e42a864a140709903ad", + "tag": "207", + "use": "Math- #207 (merged): section 0 (the elder mark and the maximin), Theorem CU.1 (the pin identities used in Lemma FL.1), and the structure of Proposition CU.3 and Theorem CU.4" + }, + { + "path": "frontiers/c7_total_bounded_20260929/PROOF.md", + "bytes": 14982, + "sha256": "16c0efe372c1b1f26460aaa109287fed78e6fbd2dd00a0530998db296278dd01", + "git_blob": "28748b086ec6761ef467dc67cba475fbdf8b7455", + "tag": "C7-K", + "use": "(K2): the bound E_Q[(W_r/r^2) 1_{G_r^c}] <= C (r^3/k) P^N used for dominated convergence in Corollary FL.6" + }, + { + "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "bytes": 9062, + "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f", + "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", + "tag": "E2", + "use": "section 9 replacement (the elder mark is Borel), through #207 section 0" + }, + { + "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "bytes": 1782, + "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028", + "git_blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0", + "tag": "E1", + "use": "reading rule for [P]" + }, + { + "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "bytes": 23312, + "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da", + "git_blob": "75da2597971510f843f8d90c743950cb8c177342", + "tag": "REC", + "use": "reading rule for [P]" + } + ], + "consumed_unmerged": [ + { + "path": "frontiers/soft_rejected_pairs_20261002/PROOF.md", + "bytes": 63690, + "sha256": "c312e423990442cfaad44459a2a20b231029828cae97018ba68724c20613331b", + "git_blob": "8b2f5fae2a0014a401003463d7c2b2221e7124f5", + "tag": "242", + "use": "Math- #242 (open): section 0, (2.1)-(2.6) (the soft model and its normalization), Lemma 2 (the model decision), Lemma 3 (the a priori elder region, used for domination in the identification of the limit), Proposition 2-prime (the stiff directions), (3.1) (the definition of F); Theorem 1 and Proposition 4 are cited", + "head": "87912bd8f28fe0d31fcb525670d1fd71a00e0db1", + "pr": "Math- #242" + } + ], + "cited_only": [ + { + "path": "frontiers/elder_dimension_lift_20260928/PROOF.md", + "bytes": 23971, + "sha256": "b529fe3780e1909014b3cb144b9a74156d976bbdc19b6711d0c5b1a1da9e5669", + "git_blob": "7303bd791a68a1139251f0f6e403a9f7cc89b006", + "tag": "OA-lift", + "use": "OA elder dimension lift (merged): (A3), 1 - p_r >= c r^3, compared with Corollary FL.5" + }, + { + "path": "frontiers/third_order_rate_20261001/PROOF.md", + "bytes": 37626, + "sha256": "5de5a6c7b942a71675c812000ade30c16766f02ee388e8dc3e1c51dde5297f7d", + "git_blob": "110ed33a57902bef11873807e722fde3888b59fb", + "tag": "229", + "use": "Math- #229 (merged): (R+.1), the l^(1/4) term of the rejected density (Remark 2 after Corollary FL.6)" + } + ], + "cited_unmerged": [], + "cited_external": [], + "mutants": { + "M1": "F1", + "M2": "F2", + "M3": "F3", + "M4": "F4", + "M5": "F5", + "M6": "F6", + "M7": "F4" + }, + "replayed": [ + "fold_check.py" + ], + "exploration": "project archive V2_2/frontiers_soft_fold_limit_20261002/exploration/ (numpy, scipy; 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": 59974, + "sha256": "5f5a411d1333c8f1eb83ec02794e154d6793a41f7d822cf8faca82cba020ea9f" + }, + { + "path": "README.md", + "bytes": 7764, + "sha256": "2c0a3dc154b973f42aa7f9326ea6d354362d227d6fc31e15e56af2db86a28dac" + }, + { + "path": "fold_check.py", + "bytes": 37203, + "sha256": "680442d62ba7b146e9a55ad99797e7520ffe97a96943fc0a4982cba604e8d7cf" + }, + { + "path": "RESULTS.json", + "bytes": 2796, + "sha256": "61ca77e1eb57e22d60eb37bd8c97cd388c1814245406856c379c4ca51966c762" + } + ] +} diff --git a/frontiers/soft_fold_limit_20261002/fold_check.py b/frontiers/soft_fold_limit_20261002/fold_check.py new file mode 100644 index 000000000..af3384bd7 --- /dev/null +++ b/frontiers/soft_fold_limit_20261002/fold_check.py @@ -0,0 +1,770 @@ +#!/usr/bin/env python3 +"""Exact controls for CL-FOLD-LIMIT-20261002-v1 (Theorem FL, Corollaries FL.5-FL.6, Proposition FL.4, Lemmas FL.1-FL.3; +PROOF.md). + +Standard library only; exact rationals except the floating-point margins printed by F4 (a fixed, deterministic +construction of the model certificates on three parameter points). Output: +RESULTS.json (sorted keys), byte-identical under -O. Usage: + python3 -B -S fold_check.py # exit 0, prints RESULTS.json + python3 -B -S fold_check.py --mutant M3 # exit 1 (only its control fails; M1..M7) + unknown mutant label -> exit 2 + +Controls + F1 the model at the pins and the weight identity (3.5): det Hess G_w(M) = (1 + beta/2 - phi)/(4 phi), + d_z^2 G_w(M) = -(1 + beta/2), det Hess G_w(S) = -(1 - beta/2 + phi)/(4 phi) (exact central differences, exact for + cubics); the typed region {phi > 0, |phi - beta/2| < 1} = (2.4) of #242; the unnormalized determinants 6 lt + Y and + -6 lt + Y (Y = 3kB - gamma^2/4); the normalization (lt/gamma^2) G_k(X, (gamma/lt) z) = G_w(X, z); and the window + determinant identity |det Hess f| = k^(m-1) r^2 |det Hess F| for f = b + k r^3 F(L^-1 .), L = diag(r, rk, r^(3/2), ...), + m = 1, 2, 3, with r = t^2 (exact) + F2 Lemma FL.2: det d_w(grad G, G) = z^4 (X + 1/2)/(96 phi^2) and det d_w(grad G, G - L_S) = z^4 (X - 1/2)/(96 phi^2) as + polynomial identities (psi = 1/phi, derivatives by exact polynomial differentiation); the critical values + -z^2 (2 + beta)/12 on X = -1/2 and -z^2 (2 - beta)/12 on X = 1/2; A0 - L_S = (X - 1/2)^2 (X + 1)/(12 phi) and + A0(2) = 25/(48 phi) (the axial path of Lemma FL.2(c)); the double + root z = a/chi and the inflection value A0 + a^3/(6 chi^2) on D = 0, and the phi-scaling of A0 - L_S; the asymptotic + ridge coefficients kappa_+- (2.2) (leading-order slice equation, and R(X)/X^3 at X = +-10^6 in floating point); + kappa_+ = 0 for the witness; the Jacobian k^3 gamma^2/(12 mu^4) of (gamma, B, C3) -> w (exact differentiation) + F3 Lemma FL.1: the exponent table (1.2) for every monomial of degree <= 4 (d = 2, 3, 4), the pinned/exact list, and the + order-5 remainder; on exactly pinned degree-6 fields (d = 2 and d = 3, rational coefficients, r = t^2) the C^2 error + of the soft window against G_k^(d) at 4 points drops by about 100 (d = 2) and 10 (d = 3) when r drops by 100 + F4 Proposition FL.4: the exact D' witness G_(3/2, 8/3, 20/3)(X, z) = (1/9) G_(1/6, 0, 0)(X + 2z, 3z) (polynomial identity); + the factorization showing that G_(1/6, 0, 0) has exactly the critical points (+-1/2, 0) and (-3, 35/8), hence the + witness exactly three, with values 0, -1/36, -125/384; the crossing forms at S on rational points (case (D): vertical + descending -a(1/2) < 0, ridge tangent ascending R''(1/2) = det/G_zz > 0; case (D'): vertical ascending, valley tangent + descending); the two facts used in Lemma FL.2(c) (beta > 2, chi < 0: the slice at 1/beta is open; chi > 0: + D > a^2 on |X| < 1/2), on rational points; and the floating-point certificate margins on three fixed points, one per + case (R), (D), (D') + F5 Theorem FL: the change of variables 36 mu^2 - Y^2 = (gamma^4/16)(phi^-2 - (1 - t)^2), dmu/dphi = -gamma^2/(24 phi^2), + (384/gamma^6)(gamma^4/16)(gamma^2/24) = 1, and chi = k^2 C3 gamma/mu^2 = chi0 phi^2, on random rationals + F6 Corollaries FL.5-FL.6: the factorization (4.2) R - L_S = (X - 1/2)^2 [2(X + 1) + 3 phi (X + 1/2)^2]/(24 phi) at + w = (phi, 0, 0) and its discriminant 4 - 12 phi; the concave slice (G = R at z_r = p/2); for phi in {2/5, 1/2, 3/4, 9/10}, + exact Sturm counts showing R > L_S + 1/200 on [-3, -1/2] and R(-3) > 0 (the (R) certificate along the ridge); the + pushforward ell^(-1/3) (ell/k^2)/(3 k^(2/3)) = ell^(2/3) k^(-8/3)/3, the cumulative factor 3/5 and the cusp-cutoff + exponent 2/3 - 5/12 = 1/4 +""" +import json +import math +import sys +from fractions import Fraction as Fr + +MUTANTS = ('M1', 'M2', 'M3', 'M4', 'M5', 'M6', 'M7') +MUT = None +FACT = [math.factorial(n) for n in range(16)] +H = Fr(1, 2) + + +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 + + +# --------------------------------------------------------------------------------- sparse polynomials over Q +class P: + """Polynomial in NV variables: dict exponent-tuple -> Fraction.""" + NV = 5 + + def __init__(self, terms=None): + self.t = {} + for e, c in (terms or {}).items(): + c = Fr(c) + if c != 0: + self.t[e] = self.t.get(e, Fr(0)) + c + self.t = {e: c for e, c in self.t.items() if c != 0} + + @staticmethod + def const(c): + return P({(0,) * P.NV: c}) + + @staticmethod + def var(i): + e = [0] * P.NV; e[i] = 1 + return P({tuple(e): 1}) + + def __add__(self, o): + o = o if isinstance(o, P) else P.const(o) + d = dict(self.t) + for e, c in o.t.items(): + d[e] = d.get(e, Fr(0)) + c + return P(d) + + __radd__ = __add__ + + def __neg__(self): + return P({e: -c for e, c in self.t.items()}) + + def __sub__(self, o): + return self + (-(o if isinstance(o, P) else P.const(o))) + + def __rsub__(self, o): + return P.const(o) - self + + def __mul__(self, o): + if not isinstance(o, P): + return P({e: c * Fr(o) for e, c in self.t.items()}) + d = {} + for e1, c1 in self.t.items(): + for e2, c2 in o.t.items(): + e = tuple(a + b for a, b in zip(e1, e2)) + d[e] = d.get(e, Fr(0)) + c1 * c2 + return P(d) + + __rmul__ = __mul__ + + def __pow__(self, n): + out = P.const(1) + for _ in range(n): + out = out * self + return out + + def d(self, i): + d = {} + for e, c in self.t.items(): + if e[i] > 0: + f = list(e); f[i] -= 1 + d[tuple(f)] = d.get(tuple(f), Fr(0)) + c * e[i] + return P(d) + + def __eq__(self, o): + o = o if isinstance(o, P) else P.const(o) + return (self - o).t == {} + + def __call__(self, *vals): + out = Fr(0) + for e, c in self.t.items(): + v = c + for x, n in zip(vals, e): + if n: + v *= Fr(x) ** n + out += v + return out + + +def det2(M): + return M[0][0] * M[1][1] - M[0][1] * M[1][0] + + +def det3(M): + return (M[0][0] * (M[1][1] * M[2][2] - M[1][2] * M[2][1]) - M[0][1] * (M[1][0] * M[2][2] - M[1][2] * M[2][0]) + + M[0][2] * (M[1][0] * M[2][1] - M[1][1] * M[2][0])) + + +def mdet(A): + A = [row[:] for row in A] + n = len(A); s = Fr(1) + for c in range(n): + piv = next((i for i in range(c, n) if A[i][c] != 0), None) + if piv is None: + return Fr(0) + if piv != c: + A[c], A[piv] = A[piv], A[c]; s = -s + s *= A[c][c] + for i in range(c + 1, n): + f = A[i][c] / A[c][c] + A[i] = [a - f * b for a, b in zip(A[i], A[c])] + return s + + +def solve(M, v): + 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)] + + +# variables for F2: X, z, psi (= 1/phi), beta, chi +VX, VZ, VPSI, VB, VC = (P.var(i) for i in range(5)) + + +def model_poly(): + A0 = (VX + H) ** 2 * (VX - 1) * VPSI * Fr(1, 12) + G = A0 + H * (VX * VX - Fr(1, 4)) * VZ - H * (1 - VB * VX) * VZ * VZ + VC * VZ ** 3 * Fr(1, 6) + return A0, G + + +def Gw(X, z, phi, beta, chi): + X, z, phi, beta, chi = (Fr(v) for v in (X, z, phi, beta, chi)) + return ((X + H) ** 2 * (X - 1) / (12 * phi) + H * (X * X - Fr(1, 4)) * z - H * (1 - beta * X) * z * z + + chi * z ** 3 / 6) + + +def Gk(X, ze, g, lt, k, B, C3): + return (2 * (X + H) ** 2 * (X - 1) + H * (X * X - Fr(1, 4)) * g * ze - H * lt * ze * ze + H * k * B * X * ze * ze + + k * k * C3 * ze ** 3 / 6) + + +def hess_num(f, x, y, h=Fr(1, 7)): + """exact Hessian of a polynomial of degree <= 3 by central differences""" + fxx = (f(x + h, y) - 2 * f(x, y) + f(x - h, y)) / (h * h) + fyy = (f(x, y + h) - 2 * f(x, y) + f(x, y - h)) / (h * h) + fxy = (f(x + h, y + h) - f(x + h, y - h) - f(x - h, y + h) + f(x - h, y - h)) / (4 * h * h) + return [[fxx, fxy], [fxy, fyy]] + + +# ------------------------------------------------------------------------------------------------ F1 +def control_F1(): + ok = True + rng = LCG(901) + n = 0 + for _ in range(40): + phi = Fr(1 + rng.nxt() % 50, 1 + rng.nxt() % 20); beta = rng.rat(9, 4); chi = rng.rat(9, 4) + f = lambda X, z: Gw(X, z, phi, beta, chi) + HM = hess_num(f, -H, Fr(0)); HS = hess_num(f, H, Fr(0)) + detS_expected = -(1 - beta / 2 + phi) / (4 * phi) + if MUT == 'M1': + detS_expected = (1 - beta / 2 + phi) / (4 * phi) + ok &= det2(HM) == (1 + beta / 2 - phi) / (4 * phi) and HM[1][1] == -(1 + beta / 2) + ok &= det2(HS) == detS_expected + typed = phi > 0 and abs(phi - beta / 2) < 1 + ok &= typed == (det2(HM) > 0 and HM[1][1] < 0 and det2(HS) < 0) + # unnormalized blocks and the normalization + k = Fr(1 + rng.nxt() % 9, 1 + rng.nxt() % 5); g = rng.rat(7, 3, nonzero=True); lt = Fr(1 + rng.nxt() % 9, 1 + rng.nxt() % 4) + B = rng.rat(7, 3); C3 = rng.rat(7, 3); Y = 3 * k * B - g * g / 4 + fk = lambda X, ze: Gk(X, ze, g, lt, k, B, C3) + ok &= det2(hess_num(fk, -H, Fr(0))) == 6 * lt + Y and det2(hess_num(fk, H, Fr(0))) == -6 * lt + Y + ph, be, ch = g * g / (24 * lt), k * B / lt, k * k * C3 * g / lt ** 2 + for X, z in ((Fr(-3, 2), Fr(1, 3)), (Fr(1, 5), Fr(-2)), (Fr(2), Fr(5, 7))): + ok &= lt / g ** 2 * Gk(X, g / lt * z, g, lt, k, B, C3) == Gw(X, z, ph, be, ch) + n += 1 + # the window determinant identity: f = b + k r^3 F(L^-1 x), L = diag(r, r k, r^(3/2), ...), r = t^2 + wd = [] + for m in (1, 2, 3): + for t in (Fr(1, 3), Fr(2, 7)): + r = t * t; k = Fr(5, 4) + L = [r, r * k] + [t ** 3] * (m - 1) + HF = [[rng.rat(5, 3) for _ in range(m + 1)] for _ in range(m + 1)] + HF = [[(HF[i][j] + HF[j][i]) / 2 for j in range(m + 1)] for i in range(m + 1)] + # Hess f = k r^3 L^-1 HF L^-1 (L diagonal) + Hf = [[k * r ** 3 * HF[i][j] / (L[i] * L[j]) for j in range(m + 1)] for i in range(m + 1)] + ok &= mdet(Hf) == k ** (m - 1) * r ** 2 * mdet(HF) + wd.append(m) + return ok, {'random_parameter_points': n, 'window_identity_dims_checked': sorted(set(wd))} + + +# ------------------------------------------------------------------------------------------------ F2 +def control_F2(): + ok = True + A0, G = model_poly() + GX, Gz = G.d(0), G.d(1) + LS = VPSI * Fr(-1, 24) + dphi = lambda F: -(VPSI * VPSI) * F.d(2) # d/dphi = -psi^2 d/dpsi + dbeta = lambda F: F.d(3) + dchi = lambda F: F.d(4) + # (b) the level conditions + rows0 = [[dphi(F), dbeta(F), dchi(F)] for F in (GX, Gz, G)] + rowsS = [[dphi(F), dbeta(F), dchi(F)] for F in (GX, Gz, G - LS)] + c96 = Fr(1, 96) if MUT != 'M2' else Fr(1, 95) + ok &= det3(rows0) == VZ ** 4 * (VX + H) * VPSI ** 2 * c96 + ok &= det3(rowsS) == VZ ** 4 * (VX - H) * VPSI ** 2 * c96 + # (c) the lines X = -1/2 and X = 1/2 + rng = LCG(902) + for _ in range(12): + beta = rng.rat(9, 4, nonzero=True); phi = Fr(1 + rng.nxt() % 30, 1 + rng.nxt() % 9) + z = 1 / beta; chi = beta * (2 + beta) # X = -1/2: z = (2 + beta)/chi = 1/beta + vals = (-H, z, 1 / phi, beta, chi) + ok &= GX(*vals) == 0 and Gz(*vals) == 0 and G(*vals) == -z * z * (2 + beta) / 12 + z = -1 / beta; chi = beta * beta - 2 * beta # X = 1/2: z = (2 - beta)/chi = -1/beta + if chi != 0: + vals = (H, z, 1 / phi, beta, chi) + ok &= GX(*vals) == 0 and Gz(*vals) == 0 and G(*vals) - LS(*vals) == -z * z * (2 - beta) / 12 + # (d) A0 - L_S, and the end of the axial path: A0(2) = 25/(48 phi) + ok &= A0 - LS == (VX - H) ** 2 * (VX + 1) * VPSI * Fr(1, 12) + for phi in (Fr(1, 7), Fr(2, 3), Fr(5, 2)): + ok &= A0(Fr(2), Fr(0), 1 / phi, Fr(0), Fr(0)) == Fr(25, 48) / phi + # (e) the inflection value on D = 0 and its phi-derivative + for _ in range(12): + X = rng.rat(9, 4); beta = rng.rat(5, 3); phi = Fr(1 + rng.nxt() % 30, 1 + rng.nxt() % 9) + p = X * X - Fr(1, 4); a = 1 - beta * X + if p == 0 or a == 0: + continue + chi = a * a / p # D = a^2 - chi p = 0 + zi = a / chi + vals = (X, zi, 1 / phi, beta, chi) + ok &= Gz(*vals) == 0 and Gz.d(1)(*vals) == 0 # double critical point of the slice + ok &= G(*vals) == A0(*vals) + a ** 3 / (6 * chi * chi) + # d/dphi (A0 + a^3/(6 chi^2) - L_S) = -(A0 - L_S)/phi: A0 - L_S is phi^-1 times a function of X alone + for _ in range(6): + X = rng.rat(9, 4); ph1 = Fr(1 + rng.nxt() % 30, 1 + rng.nxt() % 9); ph2 = ph1 + Fr(1, 3) + AL = lambda ph: (X + H) ** 2 * (X - 1) / (12 * ph) + Fr(1, 24) / ph + ok &= AL(ph1) * ph1 == AL(ph2) * ph2 + # (f) kappa_+- : series of z_r and float check of R(X)/X^3 + kap = [] + for (phi, beta, chi) in ((Fr(1, 2), Fr(1, 3), Fr(-2)), (Fr(3, 2), Fr(8, 3), Fr(20, 3)), (Fr(1, 5), Fr(-2), Fr(3)), + (Fr(2, 3), Fr(5, 2), Fr(-1, 4))): + om2 = beta * beta - chi + if om2 <= 0: + continue + omf = math.sqrt(float(om2)) + kp = 1 / (12 * float(phi)) + (2 * omf - float(beta)) / (6 * (omf - float(beta)) ** 2) + km = 1 / (12 * float(phi)) - (2 * omf + float(beta)) / (6 * (omf + float(beta)) ** 2) + # zeta0 = 1/(omega - beta) solves (chi/2) z^2 + beta z + 1/2 = 0 (leading order of the slice equation) + z0 = 1 / (omf - float(beta)) + ok &= abs(float(chi) / 2 * z0 * z0 + float(beta) * z0 + 0.5) < 1e-12 + ok &= abs(kp - (1 / (12 * float(phi)) + z0 / 3 + float(beta) * z0 * z0 / 6)) < 1e-12 + for X, kexp in ((1e6, kp), (-1e6, km)): + p = X * X - 0.25; a = 1 - float(beta) * X; Dd = a * a - float(chi) * p + sq = math.sqrt(Dd) + zr = p / (a + sq) if a >= 0 else (a - sq) / float(chi) + R = (X + 0.5) ** 2 * (X - 1) / (12 * float(phi)) + p * zr / 3 - a * zr * zr / 6 + ok &= abs(R / X ** 3 - kexp) < 1e-5 * (1 + abs(kexp)) + kap.append([str(phi), str(beta), str(chi), round(kp, 9), round(km, 9)]) + # the D' witness has kappa_+ = 0 exactly (beta^2 - chi = 4/9, omega = 2/3): R = O(X^2) on the right + om = Fr(2, 3); phi, beta = Fr(3, 2), Fr(8, 3) + ok &= 1 / (12 * phi) + (2 * om - beta) / (6 * (om - beta) ** 2) == 0 + # (g) the Jacobian of (gamma, B, C3) -> (phi, beta, chi) at fixed mu, k, by exact differentiation + VG, VBB, VC3 = P.var(0), P.var(1), P.var(2) + for _ in range(8): + mu = Fr(1 + rng.nxt() % 9, 1 + rng.nxt() % 4); k = Fr(1 + rng.nxt() % 9, 1 + rng.nxt() % 4) + w = [VG * VG * (1 / (24 * mu)), VBB * (k / mu), VC3 * VG * (k * k / mu ** 2)] + J = [[w[i].d(j) for j in range(3)] for i in range(3)] + ok &= det3(J) == VG * VG * (k ** 3 / (12 * mu ** 4)) + return ok, {'kappa_pm_float': kap, 'witness_kappa_plus': '0'} + + +# ------------------------------------------------------------------------------------------------ F3 +def control_F3(): + ok = True + table = {} + for d in (2, 3, 4): + mdim = d - 1 + listed = 0; checked = 0 + for i in range(5): + for j in range(5 - i): + for L in range(5 - i - j): + if mdim == 1 and L > 0: + continue + e = Fr(i + j) + Fr(3 * L, 2) - 3 + if MUT == 'M3': + e = Fr(i + j + L) - 3 + deg = i + j + L + pinned_or_exact = ((L == 0 and i + j <= 3) or (L == 2 and i == j == 0) or (L == 1 and i + j <= 1)) + if pinned_or_exact: + listed += 1 + continue + checked += 1 + bound = Fr(1, 2) if L == 1 else Fr(1) + ok &= e >= bound and deg >= 3 + # Taylor remainder of order 5: e >= 2 for every (i, j, L) with i + j + L = 5 + for i in range(6): + for j in range(6 - i): + L = 5 - i - j + if mdim == 1 and L > 0: + continue + ok &= Fr(i + j) + Fr(3 * L, 2) - 3 >= 2 + table[str(d)] = {'listed_pinned_or_exact': listed, 'checked_monomials': checked} + # exactly pinned degree-6 fields: d = 2 and d = 3 + rates = {} + for d in (2, 3): + rng = LCG(903 + d) + if d == 2: + mono = [(i, n - i, 0) for n in range(7) for i in range(n, -1, -1)] + pinned = [(0, 0, 0), (1, 0, 0), (2, 0, 0), (3, 0, 0), (0, 1, 0), (1, 1, 0)] + else: + mono = [(i, j, n - i - j) for n in range(7) for i in range(n + 1) for j in range(n + 1 - i)] + 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)] + + 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 + k = Fr(2, 3); lt = Fr(3, 2); b = Fr(1, 5); mu = Fr(-5, 4) # stiff A_22 = mu < 0 + free = {mm: rng.rat(5, 3) for mm in mono if mm not in pinned} + if d == 3: + free[(0, 1, 1)] = Fr(0); free[(0, 0, 2)] = mu + errs = [] + ts = (Fr(1, 10), Fr(1, 100)) if d == 2 else (Fr(1, 100), Fr(1, 1000)) + for t in ts: + r = t * t + c = dict(free); c[(0, 2, 0)] = -lt * r / k + rows, rhs = [], [] + pins = [((-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))] + if d == 3: + pins += [(-r / 2, 0, (0, 0, 1)), (r / 2, 0, (0, 0, 1))] + for (x, target, kind) in pins: + 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)] + sx, sy, sw = r, r * k, t ** 3 # x = r X, y = r k zeta, w = r^(3/2) eta + 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))): + if d == 2: + et = Fr(0) + x, y, w = sx * X, sy * ze, sw * et + s = k * r ** 3 + # window field and its derivatives (chain rule), minus the model G_k - (1/(2k)) lambda_2 eta^2 + F0 = (val(c, x, y, w) - b) / s + lam2 = -mu + Gm = Gk(X, ze, g, lt, k, B, C3) - lam2 * et * et / (2 * k) + err = [abs(F0 - Gm)] + # first and second derivatives of the model (exact, by hand) + GX = 2 * (2 * (X + H) * (X - 1) + (X + H) ** 2) + X * g * ze + H * k * B * ze * ze + Gz = H * (X * X - Fr(1, 4)) * g - lt * ze + k * B * X * ze + k * k * C3 * ze * ze / 2 + Ge = -lam2 * et / k + GXX = 2 * (2 * (X - 1) + 4 * (X + H)) + g * ze + GXz = X * g + k * B * ze + Gzz = -lt + k * B * X + k * k * C3 * ze + Gee = -lam2 / k + derivs = [((1, 0, 0), sx, GX), ((0, 1, 0), sy, Gz), ((2, 0, 0), sx * sx, GXX), ((1, 1, 0), sx * sy, GXz), + ((0, 2, 0), sy * sy, Gzz)] + if d == 3: + derivs += [((0, 0, 1), sw, Ge), ((0, 0, 2), sw * sw, Gee), ((1, 0, 1), sx * sw, Fr(0)), + ((0, 1, 1), sy * sw, Fr(0))] + for (dx, dy, dw), fac, gm in derivs: + err.append(abs(val(c, x, y, w, dx, dy, dw) * fac / s - gm)) + e.append(max(err)) + errs.append(max(e)) + ratio = errs[0] / errs[1] + expected = Fr(100) if d == 2 else Fr(10) # r drops by 100: rate r (d=2), r^(1/2) (d=3) + ok &= expected / 3 <= ratio <= 3 * expected + rates[str(d)] = {'C2_error_at_two_r': [str(float(v)) for v in errs], 'ratio': str(float(ratio)), + 'expected_ratio': str(expected)} + return ok, {'exponent_table': table, 'pinned_fields': rates} + + +# ------------------------------------------------------------------------------------------------ F4 +def slice_crit(X, phi, beta, chi): + """float: (z_r, z_v, R, V) of the slice, or None if open""" + p = X * X - 0.25; a = 1 - beta * X; Dd = a * a - chi * p + if Dd <= 0: + return None + sq = math.sqrt(Dd) + if a >= 0: + zr = p / (a + sq); zv = (a + sq) / chi + else: + zr = (a - sq) / chi; zv = p / (a - sq) + A0 = (X + 0.5) ** 2 * (X - 1) / (12 * phi) + crit = lambda z: A0 + p * z / 3 - a * z * z / 6 + return zr, zv, crit(zr), crit(zv) + + +def gw_f(X, z, phi, beta, chi): + return ((X + 0.5) ** 2 * (X - 1) / (12 * phi) + 0.5 * (X * X - 0.25) * z - 0.5 * (1 - beta * X) * z * z + chi * z ** 3 / 6) + + +def certificate_margins(case, phi, beta, chi, n=4000): + """float margins of the constructions of Proposition FL.4, Step 3, on the model (fixed grids).""" + LS = -1.0 / (24 * phi) + sgn = 1.0 if chi > 0 else -1.0 + out = {} + if case == 'R': + # path: the ridge from -1/2 leftwards until R > 0 (for this point M dies on the left ridge) + vals = [] + for i in range(n + 1): + X = -0.5 - 6.0 * i / n + c = slice_crit(X, phi, beta, chi) + vals.append(c[2]) + if c[2] > 0.05: + break + out['path_min_minus_LS'] = min(vals) - LS + out['endpoint_value'] = vals[-1] + return out + + def scan(direction, stop_at_half): + X = -0.5 + while True: + Xn = X + direction * 1e-3 + c = slice_crit(Xn, phi, beta, chi) + if c is None or c[2] <= LS: + return Xn + if stop_at_half and Xn >= 0.5: + return 0.5 + X = Xn + XL = scan(-1, False) + XR = scan(+1, case == 'D') + tau = 0.02 + while any(slice_crit(XL - tau * i / 20, phi, beta, chi) is None for i in range(21)) or ( + case != 'D' and any(slice_crit(XR + tau * i / 20, phi, beta, chi) is None for i in range(21))): + tau /= 2 + J = (XL - tau, (0.5 if case == 'D' else XR + tau)) + Z = 30.0 + bmax = -1e9; imax_off = -1e9 + rho = 0.08 + for i in range(n + 1): + X = J[0] + (J[1] - J[0]) * i / n + zr, zv, R, V = slice_crit(X, phi, beta, chi) + zd = zr - Z * sgn + if not (case == "D'" and abs(X - 0.5) < rho): + bmax = max(bmax, V - LS) # the valley boundary (off the crossing arc) + bmax = max(bmax, gw_f(X, zd, phi, beta, chi) - LS) # the dead-end boundary + if abs(X + 0.5) > rho: + imax_off = max(imax_off, R) # interior: G <= R(X) on each closed slice segment + if i == 0 or (i == n and case != 'D'): + bmax = max(bmax, R - LS) # vertical sides + out['I_M'] = [round(XL, 3), round(XR, 3)] + out['tau'] = tau + out['boundary_max_minus_LS'] = bmax + out['interior_max_off_M'] = imax_off + if case == 'D': + # the right side: the slice at 1/2 between z_d and z_v, off |z| < rho; and the ridge path margins + zr, zv, R, V = slice_crit(0.5, phi, beta, chi) + zs = [zr - Z * sgn + (zv - zr + Z * sgn) * i / n for i in range(n + 1)] + out['right_side_max_minus_LS_off_S'] = max(gw_f(0.5, z, phi, beta, chi) - LS for z in zs if abs(z) > rho) + out['ridge_path_min_minus_LS_off_S'] = min(slice_crit(-0.5 + (1 - rho) * i / n, phi, beta, chi)[2] - LS + for i in range(n + 1)) + X1 = 0.5 + rho + out['A0_minus_LS_at_X1'] = (X1 + 0.5) ** 2 * (X1 - 1) / (12 * phi) - LS + if case == "D'": + zr, zv, R, V = slice_crit(0.5, phi, beta, chi) + far = -1.0 if zr > 0 else 1.0 + zs = [zr + (far * 40 - zr) * i / n for i in range(n + 1)] + vs = [gw_f(0.5, z, phi, beta, chi) for z in zs] + out['slice_path_min_minus_LS_off_S'] = min(v - LS for z, v in zip(zs, vs) if abs(z) > rho) + out['slice_path_end_value'] = max(vs) + out['ridge_path_min_minus_LS'] = min(slice_crit(-0.5 + i / n, phi, beta, chi)[2] - LS for i in range(n + 1)) + return out + + +def control_F4(): + ok = True + # (a) the exact D' witness + A0, G = model_poly() + Xp, zp = VX + 2 * VZ, 3 * VZ + def subst(F, phi, beta, chi, Xs, zs): + out = P.const(0) + for e, c in F.t.items(): + out = out + (Xs ** e[0]) * (zs ** e[1]) * (c * (1 / Fr(phi)) ** e[2] * Fr(beta) ** e[3] * Fr(chi) ** e[4]) + return out + lhs = subst(G, Fr(3, 2), Fr(8, 3), Fr(20, 3), VX, VZ) + rhs = subst(G, Fr(1, 6), Fr(0), Fr(0), Xp, zp) * Fr(1, 9) + ok &= lhs == rhs + # critical points of G_(1/6,0,0): (+-1/2, 0) and (-3, 35/8); images under the inverse shear (X, z) = (x - 2w/3, w/3) + g0 = lambda x, w: Gw(x, w, Fr(1, 6), 0, 0) + crit0 = [(-H, Fr(0)), (H, Fr(0)), (Fr(-3), Fr(35, 8))] + vals = [] + GXp, Gzp = G.d(0), G.d(1) + for x, w in crit0: + X, z = x - 2 * w / 3, w / 3 + args = (X, z, Fr(2, 3), Fr(8, 3), Fr(20, 3)) # psi = 1/phi = 2/3 + ok &= GXp(*args) == 0 and Gzp(*args) == 0 + vals.append(G(*args)) + ok &= vals[-1] == g0(x, w) / 9 + ok &= vals == [Fr(0), Fr(-1, 36), Fr(-125, 384)] + # exactly three critical points of G_(1/6,0,0): d_z G = (x^2 - 1/4)/2 - z vanishes iff z = (x^2 - 1/4)/2, and then + # d_x G(x, (x^2 - 1/4)/2) = (x^2 - 1/4)(3/2 + x/2) = (x + 1/2)(x - 1/2)(x + 3)/2 as a polynomial in x + G0 = subst(G, Fr(1, 6), Fr(0), Fr(0), VX, VZ) + ok &= G0.d(1) == (VX * VX - Fr(1, 4)) * H - VZ + zr0 = (VX * VX - Fr(1, 4)) * H + gx_on = subst(G0.d(0), Fr(1), Fr(0), Fr(0), VX, zr0) + ok &= gx_on == (VX + H) * (VX - H) * (VX + 3) * H + # (b) crossing forms at S on rational points + rng = LCG(904) + nD = nDp = 0 + for _ in range(60): + phi = Fr(1 + rng.nxt() % 40, 1 + rng.nxt() % 20); beta = rng.rat(12, 3); chi = rng.rat(9, 4, nonzero=True) + if not (abs(phi - beta / 2) < 1) or beta == 2: + continue + HS = hess_num(lambda X, z: Gw(X, z, phi, beta, chi), H, Fr(0)) + a_half = 1 - beta / 2 + ok &= HS[1][1] == -a_half + schur = det2(HS) / HS[1][1] # R''(1/2) in (D), V''(1/2) in (D') + tangent = (Fr(1), -HS[0][1] / HS[1][1]) + form = (HS[0][0] * tangent[0] ** 2 + 2 * HS[0][1] * tangent[0] * tangent[1] + HS[1][1] * tangent[1] ** 2) + ok &= form == schur + if MUT == 'M4': + schur = -schur + if a_half > 0: + ok &= HS[1][1] < 0 and schur > 0 # (D): vertical descending, ridge tangent ascending + nD += 1 + else: + ok &= HS[1][1] > 0 and schur < 0 # (D'): vertical ascending, valley tangent descending + nDp += 1 + # the witness: (D') forms + HSw = hess_num(lambda X, z: Gw(X, z, Fr(3, 2), Fr(8, 3), Fr(20, 3)), H, Fr(0)) + ok &= HSw[1][1] == Fr(1, 3) and det2(HSw) / HSw[1][1] < 0 + # (c) float certificate margins on four fixed points + pts = {'R': (0.5, 0.0, 0.001), 'D': (0.2, 0.2, 0.1), "D'": (1.5, 8 / 3, 20 / 3)} + marg = {} + for case, (phi, beta, chi) in pts.items(): + m = certificate_margins(case, phi, beta, chi) + marg[case] = {k: (round(v, 6) if isinstance(v, float) else v) for k, v in m.items()} + if case == 'R': + ok &= m['path_min_minus_LS'] > 0 and m['endpoint_value'] > 0 + else: + ok &= m['boundary_max_minus_LS'] < 0 and m['interior_max_off_M'] < 0 + if case == 'D': + ok &= (m['right_side_max_minus_LS_off_S'] < 0 and m['ridge_path_min_minus_LS_off_S'] > 0 + and m['A0_minus_LS_at_X1'] > 0) + if case == "D'": + ok &= (m['slice_path_min_minus_LS_off_S'] > 0 and m['slice_path_end_value'] > 0 + and m['ridge_path_min_minus_LS'] > 0) + # Lemma FL.2(c): the inequalities behind the trichotomy, exactly on rational points + tri = {'beta>2,chi<0 open at 1/beta': 0, 'chi>0: D > a^2 on |X| < 1/2': 0} + for _ in range(40): + beta = 2 + Fr(1 + rng.nxt() % 40, 1 + rng.nxt() % 9); chi = -Fr(1 + rng.nxt() % 50, 1 + rng.nxt() % 9) + X = 1 / beta; p = X * X - Fr(1, 4); a = 1 - beta * X + ok &= a == 0 and p < 0 and a * a - chi * p < 0 + tri['beta>2,chi<0 open at 1/beta'] += 1 + chi2 = -chi; X2 = rng.rat(9, 20) + if abs(X2) < H: + p2 = X2 * X2 - Fr(1, 4); a2 = 1 - beta * X2; D2 = a2 * a2 - chi2 * p2 + ok &= (D2 > a2 * a2) if MUT != 'M7' else (D2 < a2 * a2) # so sqrt(D) > |a|: z_r < 0 < z_v + tri['chi>0: D > a^2 on |X| < 1/2'] += 1 + return ok, {'witness_critical_values': [str(v) for v in vals], 'crossing_forms_points': {'D': nD, "D'": nDp}, + 'certificate_margins_float': marg, 'trichotomy_points': tri} + + +# ------------------------------------------------------------------------------------------------ F5 +def control_F5(): + ok = True + rng = LCG(905) + for _ in range(30): + g = rng.rat(9, 4, nonzero=True); k = Fr(1 + rng.nxt() % 9, 1 + rng.nxt() % 4); B = rng.rat(9, 4) + C3 = rng.rat(9, 4); phi = Fr(1 + rng.nxt() % 50, 1 + rng.nxt() % 20) + mu = g * g / (24 * phi); t = 12 * k * B / g ** 2; Y = 3 * k * B - g * g / 4 + ok &= 36 * mu * mu - Y * Y == g ** 4 / 16 * (phi ** -2 - (1 - t) ** 2) + # dmu/dphi exactly via the difference quotient of the rational function g^2/(24 phi) + e = Fr(1, 10 ** 5) + ok &= ((g * g / (24 * (phi + e))) - mu) / e == -g * g / (24 * phi * (phi + e)) + ok &= k * k * C3 * g / mu ** 2 == 576 * k * k * C3 / g ** 3 * phi ** 2 + c384 = Fr(384) if MUT != 'M5' else Fr(385) + ok &= c384 * Fr(1, 16) * Fr(1, 24) == 1 + return ok, {'random_points': 30} + + +# ------------------------------------------------------------------------------------------------ F6 +def sturm_count(poly, lo, hi): + """number of distinct real roots of poly (list of Fraction coefficients, low to high) in (lo, hi]""" + def trim(p): + while len(p) > 1 and p[-1] == 0: + p = p[:-1] + return p + def deriv(p): + return [i * p[i] for i in range(1, len(p))] or [Fr(0)] + def rem(a, b): + a = a[:] + while len(a) >= len(b) and any(a): + f = a[-1] / b[-1]; s = len(a) - len(b) + for i in range(len(b)): + a[s + i] -= f * b[i] + a = trim(a[:-1]) if a[-1] == 0 else trim(a) + if len(a) < len(b): + break + return trim(a) + seq = [trim(poly), trim(deriv(poly))] + while len(seq[-1]) > 1 or seq[-1][0] != 0: + r = rem(seq[-2], seq[-1]) + if r == [Fr(0)] or not any(r): + break + seq.append([-c for c in r]) + def ev(p, x): + out = Fr(0) + for c in reversed(p): + out = out * x + c + return out + def changes(x): + vals = [ev(p, x) for p in seq] + vals = [v for v in vals if v != 0] + return sum(1 for u, v in zip(vals, vals[1:]) if (u > 0) != (v > 0)) + return changes(lo) - changes(hi) + + +def polymul(a, b): + out = [Fr(0)] * (len(a) + len(b) - 1) + for i, u in enumerate(a): + for j, v in enumerate(b): + out[i + j] += u * v + return out + + +def control_F6(): + ok = True + # the factorization (4.2): with psi = 1/phi, 24 (R - L_S) = psi (X - 1/2)^2 2 (X + 1) + 3 (X - 1/2)^2 (X + 1/2)^2, + # i.e. R - L_S = (X - 1/2)^2 [2 (X + 1) + 3 phi (X + 1/2)^2]/(24 phi); discriminant of the bracket in X: 4 - 12 phi + Rpoly = (VX + H) ** 2 * (VX - 1) * VPSI * Fr(1, 12) + (VX * VX - Fr(1, 4)) ** 2 * Fr(1, 8) + LSp = VPSI * Fr(-1, 24) + rhs = (VX - H) ** 2 * ((VX + 1) * 2 * VPSI + (VX + H) ** 2 * 3) * Fr(1, 24) + ok &= Rpoly - LSp == rhs + for phi in (Fr(1, 5), Fr(1, 3), Fr(2, 5), Fr(7, 9)): + a2, a1, a0 = 3 * phi, 3 * phi + 2, 3 * phi / 4 + 2 # 2(X + 1) + 3 phi (X + 1/2)^2 = a2 X^2 + a1 X + a0 + ok &= a1 * a1 - 4 * a2 * a0 == 4 - 12 * phi + # R(X) at w = (phi, 0, 0) equals g/(24 phi kappa), g = kappa [2 (X+1/2)^2 (X-1) + 3 phi (X^2-1/4)^2] + rng = LCG(906) + for _ in range(10): + phi = Fr(1 + rng.nxt() % 50, 1 + rng.nxt() % 20); X = rng.rat(9, 4) + R = (X + H) ** 2 * ((X - 1) / (12 * phi) + (X - H) ** 2 / 8) + g_over_kappa = 2 * (X + H) ** 2 * (X - 1) + 3 * phi * (X * X - Fr(1, 4)) ** 2 + ok &= R == g_over_kappa / (24 * phi) + # the concave slice: G(X, z) <= R(X) with equality at z_r = p/2 (a = 1) + zr = (X * X - Fr(1, 4)) / 2 + ok &= Gw(X, zr, phi, 0, 0) == R and Gw(X, zr + Fr(1, 3), phi, 0, 0) < R + # (R) certificates on the concave ridge, exactly: R - L_S - delta has no root on [-3, -1/2] and R(-3) > 0 + cert = {} + for phi in (Fr(2, 5), Fr(1, 2), Fr(3, 4), Fr(9, 10)): + LS = Fr(-1, 24) / phi + # R(X) = (X + 1/2)^2 [(X - 1)/(12 phi) + (X - 1/2)^2 / 8] as coefficients in X (low to high) + sq = [Fr(1, 4), Fr(1), Fr(1)] # (X + 1/2)^2 + inner = [Fr(-1) / (12 * phi) + Fr(1, 32), Fr(1) / (12 * phi) - Fr(1, 8), Fr(1, 8)] + Rc = polymul(sq, inner) + delta = Fr(1, 200) if MUT != 'M6' else Fr(10) + poly = Rc[:]; poly[0] -= LS + delta + roots = sturm_count(poly, Fr(-3), -H) + Rm3 = sum(c * Fr(-3) ** i for i, c in enumerate(Rc)) + ok &= roots == 0 and Rm3 > 0 + ok &= sum(c * (-H) ** i for i, c in enumerate(poly)) > 0 # sign at -1/2 (then on all of [-3, -1/2]) + cert[str(phi)] = {'roots_of_R_minus_LS_minus_delta_on_[-3,-1/2]': roots, 'R(-3)': str(Rm3)} + # corollary bookkeeping + for _ in range(6): + k = Fr(1 + rng.nxt() % 9, 1 + rng.nxt() % 4); s = Fr(1 + rng.nxt() % 9, 1 + rng.nxt() % 7) + ell = k * s ** 3; r = s # r = (ell/k)^(1/3) + ok &= r ** 3 == ell / k + # r dr/dell = (1/3) k^(-2/3) ell^(-1/3): with r = (ell/k)^(1/3), dr/dell = r/(3 ell) + ok &= r * (r / (3 * ell)) == Fr(1, 3) * (r * r / ell) # = (1/3) k^(-2/3) ell^(-1/3) + ok &= (r * r / ell) ** 3 == 1 / (k * k * ell) + e7 = Fr(-1, 3) + 1 - Fr(2, 3) # ell^(-1/3) * ell * k^-2 * k^(-2/3) + ok &= e7 == 0 and Fr(-1, 3) + 1 == Fr(2, 3) and -2 - Fr(2, 3) == Fr(-8, 3) + ok &= Fr(2, 3) + 1 == Fr(5, 3) and 1 / Fr(5, 3) == Fr(3, 5) + ok &= Fr(-8, 3) + 1 == Fr(-5, 3) and Fr(1, 4) * Fr(-5, 3) == Fr(-5, 12) and Fr(2, 3) - Fr(5, 12) == Fr(1, 4) + return ok, {'R_certificates_phi_0_0': cert} + + +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: fold_check.py [--mutant M1..M7]', file=sys.stderr) + return 2 + f1, f1d = control_F1() + f2, f2d = control_F2() + f3, f3d = control_F3() + f4, f4d = control_F4() + f5, f5d = control_F5() + f6, f6d = control_F6() + res = {'object': 'CL-FOLD-LIMIT-20261002-v1', 'scientific_effect': 'NONE', + 'controls': {'F1_model_pins_and_weight': f1, 'F2_lemma_FL2_generic': f2, 'F3_lemma_FL1_window': f3, + 'F4_proposition_FL4_certificates': f4, 'F5_theorem_FL_bookkeeping': f5, 'F6_corollaries': f6}, + 'F1': f1d, 'F2': f2d, 'F3': f3d, 'F4': f4d, 'F5': f5d, 'F6': f6d} + 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 + + +if __name__ == '__main__': + sys.exit(main(sys.argv[1:])) From 555ef295070ba5bb74d795e136a12bced5c034a9 Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 11:31:38 +0000 Subject: [PATCH 2/6] Fold-scale limit v1.1: record #170/#175 as prior work; add Proposition FL.7 AUTH-243-01 (author finding). v1 did not cite the merged OpenAI packets Math- #170 (d = 2, Theorem S, section 9) and #175 (fixed d >= 3, Theorem F, section 7). Those packets already prove the failure law r^-3 (1 - p_r) -> alpha_1 + alpha_2, hence the existence of the fold-scale limit and the compact-window l^(2/3) coefficient. They also prove the window-to-global decision transfer (#170 Theorem E, #175 Theorem H) and the critical-chord bound d(M) >= L_S (#170 section 3). v1.1: - adds a prior-work statement and narrows "What is new" to the identification with #242's F, continuity and local uniformity, and a partly different route (the cap-region domination is shared); - adds Proposition FL.7. #242's model is #170's cubic, G_k(X, zeta) = P_theta(X, k zeta)/k, with the same typed domain, weight and maximin decision. So F = k A_* a_fail: directly in d = 2, and by uniqueness of limits in d >= 3; - relabels Corollaries FL.5-FL.6 as recovering #170/#175; - credits #170/#175 for Proposition FL.4's transfer and Lemma FL.2(c)'s chord; - adds control F7 and mutant M8, and records #170 and #175 in SOURCES.json and the workflow triggers. Sections 1-3 are unchanged. A clean-context same-family referee on the change returned ACCEPT WITH FIXES; all fixes are applied. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .github/workflows/soft-fold-limit.yml | 2 + frontiers/soft_fold_limit_20261002/PROOF.md | 254 ++++++++++++++---- frontiers/soft_fold_limit_20261002/README.md | 51 +++- .../soft_fold_limit_20261002/RESULTS.json | 9 +- .../soft_fold_limit_20261002/SOURCES.json | 43 ++- .../soft_fold_limit_20261002/fold_check.py | 78 +++++- 6 files changed, 353 insertions(+), 84 deletions(-) diff --git a/.github/workflows/soft-fold-limit.yml b/.github/workflows/soft-fold-limit.yml index b06c0de43..1762c0fad 100644 --- a/.github/workflows/soft-fold-limit.yml +++ b/.github/workflows/soft-fold-limit.yml @@ -16,6 +16,8 @@ name: Soft fold limit exact controls - 'frontiers/c7_total_bounded_20260929/PROOF.md' - 'frontiers/elder_dimension_lift_20260928/PROOF.md' - 'frontiers/third_order_rate_20261001/PROOF.md' + - 'frontiers/local_elder_geometry_20260930/PROOF.md' + - 'frontiers/concave_fibre_elder_20260930/PROOF.md' # the consumed unmerged #242 PROOF is bound through the API (recorded head, blob and the current PR head) - 'frontiers/soft_rejected_pairs_20261002/PROOF.md' workflow_dispatch: diff --git a/frontiers/soft_fold_limit_20261002/PROOF.md b/frontiers/soft_fold_limit_20261002/PROOF.md index c491b6e0f..caedaef6c 100644 --- a/frontiers/soft_fold_limit_20261002/PROOF.md +++ b/frontiers/soft_fold_limit_20261002/PROOF.md @@ -1,8 +1,9 @@ -# The fold-scale rejection limit in every dimension: Conjecture 6 of Math- #242, the constant in `1 − p_r ~ r³`, and the compact-window `ℓ^{2/3}` law +# The fold-scale rejection limit through #242's soft model: Conjecture 6 of Math- #242 in every dimension, and its identification with the failure law of #170/#175 -Object: CL-FOLD-LIMIT-20261002-v1. +Object: CL-FOLD-LIMIT-20261002-v1.1 (v1: `008b299`; v1.1 adds the prior-work record AUTH-243-01 and Proposition FL.7). Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 2 October 2026. -Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem FL, Corollaries FL.5 and FL.6, Proposition FL.4, Lemmas FL.1–FL.3). +Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem FL, Propositions FL.4 and FL.7, Lemmas FL.1–FL.3; Corollaries FL.5 and FL.6 +recover merged results of #170 and #175). 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. @@ -12,33 +13,72 @@ as **Conjecture 6** that the actual rejected kernel at a fixed gap `k = ℓ/r³` lim_{r↓0} (k/r)·r^{−2}A_r^{rej}(b, k, u) = F(k; b, u). (0.0) -This note proves (0.0) in every `d ≥ 2`. It is the fold-scale counterpart of #207's Theorem CU.4 (the cusp kernels at a fixed -`κ`). The new points are that the decision is #242's Lemma 2 (slices are cubic, and in one case the saddle is a slice -minimum), that the transverse curvature must be zoomed at the scale `r` (the soft layer), and that the domination comes from -the parent's deterministic cap region ([P] §7), which confines rejection to that layer. +**Prior work (recorded in v1.1, AUTH-243-01).** That the left side of (0.0) converges is not new. Two OpenAI packets are +merged on `main` as author-side candidates, each at its stated conditional scope: +- Math- #170 (`d = 2`), conditional on its Theorem E and its source interfaces; +- Math- #175 (fixed `d ≥ 3`), conditional on its source interfaces and its Theorem H. + +Their Theorems S and F prove, at fixed marks, `r^{−3}(1 − p_r) → a_fail` (`= α₁ + α₂` in #170, `a1 + a2` in #175). This is an +identified Gaussian integral over the window-saddle classifier of #170's typed cubic. Their §§9 and 7 give the compact-window +`ℓ^{2/3}` coefficient. Since `(k/r)r^{−2}A_r^{rej} = kA_r·r^{−3}(1 − p_r)`, the left side of (0.0) tends to `kA_∗a_fail`. Two +further statements here are also theirs: +- the decision transfer from a window model to the global field (#170 Theorem E(2)–(3), #175 Theorem H); +- the lower bound `d(M̂) ≥ L_S` by a critical chord through `S` (#170 §3). + +#242 and v1 of this note did not cite them. What Conjecture 6 adds is the identification of the limit with #242's `F`. + +This note proves (0.0) in every `d ≥ 2` along a partly different route (§§1–3). It identifies the two limits directly in +`d = 2`, and by uniqueness of limits in `d ≥ 3` (Proposition FL.7). Theorem FL is the fold-scale counterpart of #207's +Theorem CU.4 (the cusp kernels at a fixed `κ`). Its decision step is #242's Lemma 2: the slices are cubic, and in one case +the saddle is a slice minimum. The transverse curvature is zoomed at the scale `r` (the soft layer), and the domination +comes from the parent's deterministic cap region ([P] §7), which confines rejection to that layer. The zoom and the +domination are as in #170 §8 and #175 §§2 and 4. **What is new.** - **Theorem FL** (§3). (0.0) holds for every `d ≥ 2`, `L > 0`, `b ∈ R`, `k > 0` and `u ∈ S^{d−1}`, uniformly for `(b, k)` in compact subsets of `R × (0, ∞)`. The limit defining `F` in #242 (3.1) exists, and `F` is continuous and positive. -- **Corollary FL.5** (§4, the selection loss). `r^{−3}(1 − p_r(b, k, u)) → F(k; b, u)/(k·A_∗(b, k, u)) ∈ (0, ∞)`. [P]'s - Theorem A gives `1 − p_r ≤ Cr³` and the merged dimension lift gives `1 − p_r ≥ cr³`; this identifies the constant. -- **Corollary FL.6** (§4, the compact-window density difference). For the compact mark windows of [P]'s Theorem B, - `ν_cand(ℓ) − ν_eld(ℓ) = d_{𝐁,𝐊}ℓ^{2/3} + o(ℓ^{2/3})` with `d_{𝐁,𝐊} = (1/3)∫∫∫_{S^{d−1}×𝐁×𝐊} k^{−8/3}F(k; b, u) > 0`. This - sharpens [P] (1.2) from `O(ℓ^{2/3})` to an asymptotic law; it is the compact-window analogue of #242's Conjecture 7. -- **Proposition FL.4** (§2, stability of #242's Lemma 2). For every parameter `ϖ` in a full-measure set `𝒦`, every - `C²`-small perturbation of the model with the two pins as critical points has the elder decision of the model, for the - global superlevel filtration of any field that realizes it in a window. The certificates are explicit paths and traps - crossing the saddle along model curves. In `d ≥ 3` the stiff directions enter as a negative definite form. -- **Lemma FL.2** (§2). `𝒦` has full measure; the proof gives explicit Jacobians, `z⁴(X ± ½)/(96φ²)` for the level - conditions. In the model a rejected maximum always dies strictly above the saddle level (by an axial path, an observation - of OpenAI Codex on #242), and #242's case (D′) requires `χ > 0`: for `β > 2`, `χ < 0` the pair is always rejected. + - *Relative to #170/#175,* which are pointwise at fixed marks, the identification with `F`, the continuity and the local + uniformity are new. + - *The decision and convergence steps take a different route.* They condition on the contact Hessian with a Weyl zoom + and use #242's slice decision with Proposition FL.4. They do not consume [CUB], [TWO], [SC], [LOCAL] or #175's + hard-fibre lemma. + - *The domination step is the same cap-region mechanism as theirs.* +- **Proposition FL.7** (§4, the identification). #242's model and #170's cubic are the same function, + `G_k(X, ζ) = P_θ(X, kζ)/k` with `θ = (−λ̃/k, γ, B, C_3)`, and their typed domains, limiting weights and maximin decisions + agree. + - In `d = 2`, `F = kA_∗a_fail` is a direct identity of model integrals. With #170 Theorem S it gives a second proof of + (0.0) at each fixed `(b, k, u)`. + - In `d ≥ 3` the identity follows from Theorem FL and #175 Theorem F by uniqueness of limits. It is therefore conditional + on #175's interfaces. +- **Corollaries FL.5 and FL.6** (§4) are not new. They are #170 Theorem S and §9 (`d = 2`) and #175 Theorem F and §7 + (`d ≥ 3`), recovered with the constants in #242's form: + - `r^{−3}(1 − p_r) → F/(k·A_∗)`, now locally uniformly in `(b, k)`; + - `ν_cand − ν_eld = d_{𝐁,𝐊}ℓ^{2/3} + o(ℓ^{2/3})`, with `d_{𝐁,𝐊} = C_fail^{𝐁,𝐊}` of #170 §9 in `d = 2`. +- **Proposition FL.4** (§2) re-proves, in #242's normalization, the decision transfer of #170 Theorem E(2)–(3) and #175 + Theorem H. For every parameter `ϖ` in a full-measure set `𝒦`, every `C²`-small perturbation of the model with the two pins + as critical points has the model's elder decision. This holds for the global superlevel filtration of any field that + realizes the perturbation in a window. + - *What differs is the method:* explicit paths and traps crossing the saddle along model curves (Lemma FL.3), an `ε`-form + for a merely continuous global field, and the generic set `𝒦` adapted to #242's slice decision. + - In `d ≥ 3` the stiff directions enter as a negative definite form; this is #175's hard tube. +- **Lemma FL.2** (§2). `𝒦` has full measure, with explicit Jacobians `z⁴(X ± ½)/(96φ²)` for the level conditions. + - In the model a rejected maximum always dies strictly above the saddle level. This is #170 §3's critical chord through + `S`; OpenAI Codex observed it on #242 as an axial path. + - #242's case (D′) requires `χ > 0`: for `β > 2`, `χ < 0` the pair is always rejected. - **Lemma FL.1** (§1) makes #242's expansions (2.1) and (2.6) quantitative: the soft window of a pinned `C⁵` field is `C²`-close to the model, at rate `r` (`d = 2`) or `r^{1/2}` (`d ≥ 3`), with a constant linear in `‖f‖_{C⁵}` and independent of the transverse eigenvalues. -**What is not claimed.** Conjecture 7 of #242 (the `ℓ^{2/3}` term of the unrestricted `ρ_rej`) is not proved: it needs -(0.0) uniformly from the cusp scale to the fold scale with integrable errors, and the intermediate separations (#242 §4, -inputs (i)–(iii)). Theorem FL is a limit without rate, at fixed `k` (locally uniformly). No numerical value is certified. +**What is not claimed.** +- Conjecture 7 of #242 (the `ℓ^{2/3}` term of the unrestricted `ρ_rej`) is not proved. It needs (0.0) uniformly from the + cusp scale to the fold scale with integrable errors, and the intermediate separations (#242 §4, inputs (i)–(iii)). +- Theorem FL is a limit without rate, at fixed `k` (locally uniformly). No numerical value is certified. +- No priority is claimed for #170's and #175's results: + - the existence of the limit in (0.0); + - the constant in `1 − p_r ~ r³`; + - the compact-window `ℓ^{2/3}` coefficient; + - the transfer of the window decision to the global field; + - the critical-chord bound `d(M̂) ≥ L_S`. **Dependencies.** - Consumed (merged): @@ -57,6 +97,22 @@ inputs (i)–(iii)). Theorem FL is a limit without rate, at fixed `k` (locally u - Consumed through #207 §0: [E2] (`reviews/d1_section9_borel_repair_20260925/REPAIR.md`, blob `fe9b9ce4`), for the Borel measurability of the elder mark. Reading rules for [P]: [E1] (`imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`, blob `213594d6`) and [REC] (`reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md`, blob `75da2597`). +- Compared (merged, author-side candidates at their stated conditional scope), and consumed by Proposition FL.7 only: + - #170 (`frontiers/local_elder_geometry_20260930/PROOF.md`, blob `ef2aa579`): + - §2: the typed cubic `P_θ`, `B′`, `C(P)`, the null sets, and Theorem E(1); + - §§7 and 8.1: the classifier `n(θ)`, (S1) and the weight `w_+`; + - §8.5 (S11): the weight limit; + - §8.6: the mass `α₁ + α₂`; + - Theorem S, (S2); + - §9: `C_fail`. + - #175 (`frontiers/concave_fibre_elder_20260930/PROOF.md`, blob `923d3236`): + - §§1–2: the model and the soft cubic, `n ∈ {0, 1, 2}`; + - §3: Theorem H; + - §5: Theorem F, (S1)–(S3); + - §7. + + Through them, FL.7 depends transitively on #170's [CUB] classifier (via Theorem E(1)). In `d ≥ 3` it also depends on + #175's source interfaces ([SC], [LOCAL], and the hard-fibre lemma). - Cited only: OpenAI Codex's #242 Slice B review (comment 5948352439, the axial path in Lemma FL.2(c)); OA dimension lift (`frontiers/elder_dimension_lift_20260928/PROOF.md`, blob `7303bd79`): (A3); #229 (`frontiers/third_order_rate_20261001/PROOF.md`, blob `110ed33a`): (R⁺.1); #242 Theorem 1 and Proposition 4 (same blob as @@ -199,8 +255,10 @@ null set. At `X = −1` the conditions force `a(−1) = 0` and then `χ = 0`; at inflection value, which is therefore `≥ L_S`; equality is excluded by (G3). *Proof of (c).* The axial segment `{(X, 0) : −½ ≤ X ≤ 2}` carries `G_ϖ = A₀`, with `A₀ − L_S = (X − ½)²(X + 1)/(12φ) ≥ 0` -(equality only at `Ŝ`), and it ends at `A₀(2) = 25/(48φ) > 0`. So `d(M̂) ≥ L_S` for every typed `ϖ`. This observation is -OpenAI Codex's, in its review of #242 Lemma 2 (comment 5948352439). Hence `e(G_ϖ) = 1` iff `d(M̂) ≤ L_S`, i.e. iff for every +(equality only at `Ŝ`), and it ends at `A₀(2) = 25/(48φ) > 0`. So `d(M̂) ≥ L_S` for every typed `ϖ`. This is #170 §3's +critical-chord bound through `S` (for a cubic, the restriction to a chord between two critical points is +`P(M) + (P(Y) − P(M))(3t² − 2t³)`). OpenAI Codex observed it for #242's model in its review of Lemma 2 +(comment 5948352439). Hence `e(G_ϖ) = 1` iff `d(M̂) ≤ L_S`, i.e. iff for every `c > L_S` the point `M` is the highest point of its component of `{G_ϖ > c}`. Assume this, with `ϖ ∈ 𝒦`. As in Step 1 of the proof of Lemma 2: - (i) `R < 0` on `I_M ∖ {−½}`. A point `X*` with `R(X*) > 0` would put a point above `0` into `M`'s component at the level @@ -511,10 +569,61 @@ continuously. All bounds above are uniform on `𝒞`. A diagonal argument then g uniform convergence. Positivity is proved in Corollary FL.5. (Uniformity in `u` would need `u_n → u` in this argument as well; it is not claimed.) ∎ -## 4. The selection loss and the compact-window density difference - -**Corollary FL.5 (the constant in `1 − p_r ~ r³`).** For fixed `u ∈ S^{d−1}`, uniformly for `(b, k)` in compact subsets of -`R × (0, ∞)`, +## 4. The identification with #170/#175, the selection loss, and the compact-window density difference + +**Proposition FL.7 (the model of #170 and #175).** For the jets `(λ̃, γ, B, C_3)` of (0.2), put +`θ := (s, a, β′, c) := (−λ̃/k, γ, B, C_3)`. In `d = 2` these are #170's coordinates `(f_zz(0)/r, f_xxz(0), f_xzz(0), f_zzz(0))` +(§8.1 there); in `d ≥ 3` they are #175's soft coordinates (`μ_1 = −rs`, §2 there). Let + + P_θ(X, Z) := 2kX³ − 3kX/2 − k/2 + sZ²/2 + (a/2)(X² − ¼)Z + (β′/2)XZ² + (c/6)Z³, B′ := β′ − a²/(12k), (4.0) + +be #170's typed cubic. Let `n(θ)` be its number of additional critical points with values in `(−k, 0)`, that is, the +number of points of `C(P_θ)` (#170 §2; the classifier of #170 §§7 and 8.1, with `n ∈ {0, 1, 2}` by #175 §2). Then: +- (i) `G_k(X, ζ) = P_θ(X, kζ)/k` identically. +- (ii) `Y = 3kB′`. So the typed domains agree (`s < −|B′|/2` iff `|Y| < 6λ̃`), and so do the limiting weights: + `9k²(4s² − B′²) = 36λ̃² − Y²`. +- (iii) For typed `θ` with `γ ≠ 0`, off #170's null sets `Σ_k ∪ Δ_k`, `1 − e(G_ϖ) = 1{n(θ) > 0}`. +- (iv) `F(k; b, u) = kA_∗(b, k, u)·a_fail(b, k, u)` and `d_{𝐁,𝐊} = C_fail^{𝐁,𝐊}`. + - In `d = 2`, `a_fail = α₁ + α₂` is #170's (S2), and `C_fail` is #170 §9. This case is an identity of model integrals. + - In `d ≥ 3`, `a_fail = a1 + a2` is #175's (S2)–(S3), and `C_fail` is #175 §7 (L1). This case uses uniqueness of limits + and is conditional on #175 Theorem F. + +*Proof.* +- (i) This is a polynomial identity (control F7). `2(X + ½)²(X − 1) = 2X³ − 3X/2 − ½`, and the terms of #242 (2.1) in `ζ`, + `ζ²` and `ζ³` are `(γ/2)(X² − ¼)ζ`, `−(λ̃/2)ζ² + (kB/2)Xζ²` and `(k²C_3/6)ζ³`. These are the terms of `P_θ(X, kζ)/k`. +- (ii) `Y = 3kB − γ²/4 = 3kB′`, and `λ̃ = −ks`. +- (iii) By (i) and (0.4), `G_ϖ(X, z) = (λ̃/(kγ²))P_θ(X, k(γ/λ̃)z)`. This is an invertible linear change of variables + fixing `M̂ = M` and `Ŝ = S`, followed by a positive factor; it preserves maximin levels up to that factor, and it maps + `L_S` to `−k`. By #170 Theorem E(1), the maximin level of `P_θ` is `h_*(θ) = max({−k} ∪ P_θ(C(P_θ))) ≥ −k`. So + `e(G_ϖ) = 1` iff `h_*(θ) = −k` iff `n(θ) = 0`. +- (iv) In `d = 2`, Step 5 of Theorem FL and Fubini give + `F = 12π_0(u; v_0)∫_0^∞dμ̃∫(36μ̃² − Y²)1{|Y| < 6μ̃}(1 − e(G_ϖ))h_0(0, γ, B, C_3)dγ dB dC_3`. Here `h_0` is the joint + density of `(𝔸, γ, B, C_3)` under the contact law, with `e_1` the transverse unit vector of #170's frame. Flipping `e_1` + changes neither `ϖ` nor `w_+` nor `n`. Put `μ̃ = −ks`, so that `dμ̃/ds = −k` and `∫_0^∞dμ̃ = k∫_{−∞}^0 ds`, and use + (ii)–(iii). This becomes `k·12π_0∫w_+(θ)h_0(0, a, β′, c)1{n(θ) > 0}dθ`, which is `k·A_∗·a_fail` by #170 (S1)–(S2), (S11) + and `A_∗ = 12π_0z_0`. + - *Every `d ≥ 2`.* Uniqueness of limits gives the same identity. `(k/r)r^{−2}A_r^{rej} = kA_r·r^{−3}(1 − p_r)` tends to `F` + by Theorem FL, and to `kA_∗a_fail` by #170 Theorem S or #175 Theorem F. + - *In `d ≥ 3` the integrands also agree, up to the normalization `c_m dO` of the Weyl formula.* + `𝒱(0, μ_2, …, μ_m)(μ_2⋯μ_m)² = Π_{j≥2}μ_j³Π_{2≤i 0`, `π_0 > 0` and `𝒱 > 0` almost everywhere. Hence `F > 0`. ∎ -[P]'s Theorem A gives `1 − p_r ≤ Cr³`, and the OA dimension lift (A3) gives `1 − p_r ≥ cr³`, both uniformly. (4.1) -identifies the constant at each `(b, k, u)`. +[P]'s Theorem A gives `1 − p_r ≤ Cr³`, and the OA dimension lift (A3) gives `1 − p_r ≥ cr³`, both uniformly. The constant +was first identified by #170 Theorem S (`d = 2`) and #175 Theorem F (`d ≥ 3`) as `α₁ + α₂`. (4.1) recovers it in #242's form, +and the two forms agree by Proposition FL.7. The positivity argument above is independent of theirs. -**Corollary FL.6 (the compact-window density difference).** Let `𝐁` (births) and `𝐊 = [k_−, k_+] ⊂ (0, ∞)` (gap marks) be -compact intervals of positive length, and let `ν_cand`, `ν_eld` be the compact-window densities of [P]'s Theorem B. Then +**Corollary FL.6 (the compact-window density difference; #170 §9, #175 §7).** Let `𝐁` (births) and +`𝐊 = [k_−, k_+] ⊂ (0, ∞)` (gap marks) be compact intervals of positive length, and let `ν_cand`, `ν_eld` be the compact-window +densities of [P]'s Theorem B. Then ν_cand(ℓ) − ν_eld(ℓ) = d_{𝐁,𝐊}ℓ^{2/3} + o(ℓ^{2/3}), d_{𝐁,𝐊} := (1/3)∫_{S^{d−1}}∫_𝐁∫_𝐊 k^{−8/3}F(k; b, u)dk db dσ(u) ∈ (0, ∞). (4.3) Consequently `E[N_cand(0, t] − N_eld(0, t]] = (3/5)d_{𝐁,𝐊}t^{5/3} + o(t^{5/3})`. And for every real `q > −5/3`, -`E Σ_{candidate, not selected, ℓ ≤ t}ℓ^q = d_{𝐁,𝐊}t^{q+5/3}/(q + 5/3) + o(t^{q+5/3})`. These sharpen [P] (1.2), (12.1) -and (12.3) from upper bounds to asymptotic laws. +`E Σ_{candidate, not selected, ℓ ≤ t}ℓ^q = d_{𝐁,𝐊}t^{q+5/3}/(q + 5/3) + o(t^{q+5/3})`. #170 §9 and #175 §7 already sharpen +[P] (1.2) and (12.1) from upper bounds to asymptotic laws, with `C_fail^{𝐁,𝐊} = d_{𝐁,𝐊}` (Proposition FL.7). The `q`-moment +form is the same integration. *Proof.* By [P] (11.1)–(11.2), at `r = (ℓ/k)^{1/3}`, `ν_cand(ℓ) − ν_eld(ℓ) = ℓ^{−1/3}∫A_r(1 − p_r)/(3k^{2/3}) db dk dσ = ℓ^{−1/3}∫A_r^{rej}/(3k^{2/3})`. Since `r³ = ℓ/k`, @@ -560,10 +672,11 @@ by integrating (4.3) against `ℓ^q dℓ` on `(0, t]`, as in [P] §12. ∎ 1. #242 Proposition 4 computes `F = F_0(b)H(k)` for the Gaussian kernel `e^{−|x|²/2}` on `R^d`, the model case, not for the torus field of Theorem FL. For that model the formula reads `d_{𝐁,𝐊} = (1/3)|S^{d−1}|∫_𝐁F_0 db·∫_𝐊 k^{−8/3}H(k)dk`; for the periodized torus field the jets' laws differ by `O(e^{−L²/8})` (#242 §0). -2. Without the lower cutoff `k_−`, the integral in (4.3) diverges at `k → 0`, where `F(k) → F_0 > 0`. Cutting it at the cusp - scale `k ≍ ℓ^{1/4}` produces `ℓ^{2/3}·ℓ^{−5/12} = ℓ^{1/4}`, the order of #229's term `(I^{cand} − c_1)ℓ^{1/4}`. This is why - the unrestricted `ℓ^{2/3}` coefficient (#242 Conjecture 7) needs the two-scale composite and the intermediate separations, - while the compact-window coefficient (4.3) does not. +2. Without the lower cutoff `k_−`, the integral in (4.3) diverges at `k → 0` whenever `F` stays bounded below there, as for + the Gaussian kernel, where `F(k) → F_0 > 0` (#242 Proposition 4). Cutting it at the cusp scale `k ≍ ℓ^{1/4}` produces + `ℓ^{2/3}·ℓ^{−5/12} = ℓ^{1/4}`, the order of #229's term `(I^{cand} − c_1)ℓ^{1/4}`. This is why the unrestricted `ℓ^{2/3}` + coefficient (#242 Conjecture 7) needs the two-scale composite and the intermediate separations, while the compact-window + coefficient (4.3) does not. ## 5. Numerical evidence (exploration; not part of the proofs) @@ -618,22 +731,47 @@ The scripts and records are in the project archive (`exploration/`; numpy and sc exception: #242's near-degenerate (D′) point flips at amplitudes `≥ 3·10⁻³`. This is consistent with `ε` depending on the margins. +**Proposition FL.7(iii) (v1.1 referee, independent).** The comparison was made on 1,800 typed points: 1,500 broad, and 300 +in the (D′) family, 141 of them elder. The parameters were `k ∈ [0.1, 5]` and `λ̃ ∈ [0.1, 10]`, with both signs of `γ`. +- *Method.* #170's classifier (the critical points of `P_θ` with values in `(−k, 0)`, found from a resultant) was compared + with an independent one-dimensional implementation of #242 Lemma 2 on `G_ϖ`. +- *Result.* There were no disagreements once the scanner's grid was refined near `X = ½`, where `I_M`'s right end can lie + very close to `½`. All 243 points with `β > 2`, `χ < 0` have `n > 0`. +- *The identity (iv).* `(γ⁶/384)I(t, χ_0) = k∫w_+1{n > 0}ds` was checked on four jet triples, to relative error + `≤ 1.2·10⁻⁶`. + ## 6. Scope and relation to other packets - **#242.** Conjecture 6 is Theorem FL. The definition (3.1) of `F` is the actual fold-scale limit of the rejected kernel in - every dimension. For the Gaussian kernel on `R^d` (#242's model case), #242 Proposition 4 describes `F`: `F = F_0H`, - `H(k) = 1 + (12/25)k² + O(k³)`. + every dimension. For the Gaussian kernel on `R^d` (#242's model case), #242 Proposition 4 describes `F` as `F = F_0H`, with + `H(k) = 1 + (12/25)k² + O(k³)`. #170/#175 define `a_fail` for the torus field only. In `d = 2`, Proposition FL.7(iv) is an + identity of model integrals, so `F_0H/k` is the integral `A_∗a_fail` evaluated with the `R^d` kernel's jets. The jets of the + periodized torus field differ from these by `O(e^{−L²/8})`. Proposition 2′ of #242 (the stiff directions) is transferred to the field here, as part of Proposition FL.4. +- **#170 and #175 (merged author-side candidates, at their stated conditional scope).** Their Theorems S and F prove the + failure law `r^{−3}(1 − p_r) → a_fail` at fixed marks. + - *Their route.* #170's Theorem E decides the window cubic by critical chords and isolating caps, with [CUB]'s classifier, + and transfers it to the global field. #175's Theorem H adds, for `d ≥ 3`, a hard-fibre barrier of width `r^{3/2}`. + - *Their domination* is the parent's cap implication on `{λ_1 > (4/(3k))rM_3², rM_4 ≤ 3k/10}`, with [P] (6.2) and (7.7). + - *Their compact-window coefficient* is in their §§9 and 7. + - *Theorem FL's route.* It reaches the same limit through a different decision and convergence step: conditioning on the + contact Hessian with a Weyl zoom, #242's slice decision, and the path/trap stability of Proposition FL.4. It does not + consume [CUB], [TWO], [SC], [LOCAL] or the hard-fibre lemma. + - *Shared steps.* Theorem FL's domination step uses the same cap-region mechanism and the same `m = 1` near/far split + (#170 §8.3). The stiff scale `r^{3/2}` of Proposition FL.4 is #175's hard tube. + - *Proposition FL.7* shows that the two limits are the same Gaussian integral in `d = 2`. In `d ≥ 3` they are equal by + uniqueness of limits. - **#242 Conjecture 7** is not proved. It needs inputs (i)–(ii) of #242 §4. Input (i) asks for (3.1) with an error that, integrated against `dr` from the cusp scale to the fold scale, is `O(ℓ^{3/4})` after the composite is subtracted. That needs quantitative margins in Proposition FL.4 near the decision boundary, which the present pointwise argument does not give. Input (ii) (the intermediate separations) is untouched. -- **[P], the dimension lift, and [C7-K].** Corollaries FL.5 and FL.6 identify the constants in the `Θ(r³)` selection loss - and the `Θ(ℓ^{2/3})` compact-window density difference. Their orders were known from [P] Theorems A and B, the OA lower - bound (A3), and [C7-K] (K2). -- **Matching.** #242 Theorem 1 gives `κ𝒜^{rej} → F_0` as `κ → ∞` at the cusp scale. Theorem FL gives - `(k/r)r^{−2}A_r^{rej} → F(k)`, with `F(k) → F_0` as `k → 0` (#242 §3). The two limits agree in the overlap, which is what - the composite of #242 Lemma 5 assumes. +- **[P], the dimension lift, and [C7-K].** Corollaries FL.5 and FL.6 restate, in #242's form, the constants of the `Θ(r³)` + selection loss and of the `Θ(ℓ^{2/3})` compact-window density difference, which #170/#175 identified first. Their orders + were known from [P] Theorems A and B, the OA lower bound (A3), and [C7-K] (K2). +- **Matching.** #242 Theorem 1 gives `κ𝒜^{rej} → F_0` as `κ → ∞` at the cusp scale, and Theorem FL gives the fold-scale + limit `F(k)`. That `F(k) → F_0` as `k → 0` is proved in #242 only for the Gaussian kernel on `R^d` (Proposition 4, + `H → 1`). It is what the composite of #242 Lemma 5 assumes. Neither limit is uniform in the overlap, and that is not + claimed here. - **Not claimed:** any rate in (3.1); uniformity as `k → 0` or `k → ∞`; any statement about `ν_cand`'s own corrections. ## 7. Controls @@ -675,14 +813,23 @@ The scripts and records are in the project archive (`exploration/`; numpy and sc - exact Sturm certificates of `(R)` along the ridge for `φ ∈ {2/5, 1/2, 3/4, 9/10}`: no root of `R − L_S − 1/200` on `[−3, −½]`, and `R(−3) > 0`; - the exponents of the pushforward (4.4) and of Remark 2. +- **F7.** Proposition FL.7: + - `G_k(X, ζ) = P_θ(X, kζ)/k` as a polynomial identity in `(X, ζ)`, on random rational jets with `λ̃` of either sign; + - `Y = 3kB′`, the weight identity `9k²(4s² − B′²) = 36λ̃² − Y²`, and the equivalence of the typed domains; + - the hard-factor identity `𝒱(0, μ_2, …, μ_m)(μ_2⋯μ_m)² = Π μ_j³Π(μ_j − μ_i)` for `m = 2, 3, 4`; + - the exponent `−8/3 + 1 = −5/3` of `d_{𝐁,𝐊} = C_fail^{𝐁,𝐊}`, and `dμ̃/ds = −k`. -Mutants `M1`–`M7` each fail only their own control (`M1` F1, `M2` F2, `M3` F3, `M4` and `M7` F4, `M5` F5, `M6` F6); the checker +Mutants `M1`–`M8` each fail only their own control (`M1` F1, `M2` F2, `M3` F3, `M4` and `M7` F4, `M5` F5, `M6` F6, `M8` F7); the checker names the failing control on stderr, and an unknown label exits 2. -**What the controls do not test.** Proposition FL.4 for non-polynomial perturbations, the transversality and Sard arguments -themselves (only their Jacobians), Lemma FL.2(c) beyond its two elementary facts (it is tested numerically in §5), and the -probabilistic steps of Theorem FL (coupling, conditioning, domination). These are argued in the text and are the subject of -review slices B–D. +**What the controls do not test.** +- Proposition FL.4 for non-polynomial perturbations. +- The transversality and Sard arguments themselves (only their Jacobians). +- Lemma FL.2(c) beyond its two elementary facts (it is tested numerically in §5). +- The probabilistic steps of Theorem FL (coupling, conditioning, domination). +- The use of #170 Theorem E(1) in Proposition FL.7(iii). The referee's numerical comparison in §5 tests it. + +These are argued in the text and are the subject of review slices B–E. ## 8. Sources (exact identities in `SOURCES.json`) @@ -698,6 +845,8 @@ review slices B–D. | #242 | `frontiers/soft_rejected_pairs_20261002/PROOF.md` at `87912bd` (`8b2f5fae`), unmerged | §0, (2.1)–(2.6), Lemmas 2–3, Proposition 2′, (3.1) — consumed | | OA lift | `frontiers/elder_dimension_lift_20260928/PROOF.md` (`7303bd79`) | (A3) — cited | | #229 | `frontiers/third_order_rate_20261001/PROOF.md` (`110ed33a`) | (R⁺.1) — cited | +| #170 | `frontiers/local_elder_geometry_20260930/PROOF.md` (`ef2aa579`) | §2 (Theorem E(1)), §§7–8 (the classifier, (S1), (S11), the mass `α₁ + α₂`), Theorem S, §9 — compared; consumed by Proposition FL.7 only | +| #175 | `frontiers/concave_fibre_elder_20260930/PROOF.md` (`923d3236`) | §§1–2, §3 (Theorem H), §5 (Theorem F, (S1)–(S3)), §7 — compared; consumed by Proposition FL.7 only | ## 9. Review slices @@ -709,6 +858,7 @@ review slices B–D. - **C** — §3, Steps 1–3: the coupling and the conditioning on `𝔸`, the Weyl zoom (3.4), the window identity (3.5), and the almost-everywhere convergence (full measure of `ϖ* ∈ 𝒦`). - **D** — §3, Steps 4–5: the domination (cap region, (3.6), the `m = 1` far branch), and the identification with #242 (3.1). -- **E** — §4: Corollaries FL.5 and FL.6 (the positivity of `F`, and the use of [P] (11.2) and [C7-K] (K2)), with §§5–6. +- **E** — §4: Proposition FL.7 (the identification with #170/#175) and Corollaries FL.5 and FL.6 (the positivity of `F`, and + the use of [P] (11.2) and [C7-K] (K2)), with §§5–6. A reviewer should record, per slice: ACCEPT, ACCEPT WITH FIXES (list), or REJECT (with the failing step). diff --git a/frontiers/soft_fold_limit_20261002/README.md b/frontiers/soft_fold_limit_20261002/README.md index 5c178fa11..aea7e3a9c 100644 --- a/frontiers/soft_fold_limit_20261002/README.md +++ b/frontiers/soft_fold_limit_20261002/README.md @@ -1,22 +1,34 @@ -# The fold-scale rejection limit in every dimension +# The fold-scale rejection limit through #242's soft model **Author-side proof candidate (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** -Object `CL-FOLD-LIMIT-20261002-v1`. Full text: [`PROOF.md`](PROOF.md). +Object `CL-FOLD-LIMIT-20261002-v1.1`; v1 was `008b299`. Full text: [`PROOF.md`](PROOF.md). + +## Prior work (AUTH-243-01) + +Two OpenAI packets are merged on `main` as author-side candidates, each at its stated conditional scope: Math- #170 +(`d = 2`) and Math- #175 (fixed `d ≥ 3`). At fixed marks they already prove: +- the failure law `r^{−3}(1 − p_r) → α₁ + α₂` (Theorems S and F), and hence the existence of the fold-scale limit of the + rejected kernel, `kA_∗(α₁ + α₂)`; +- the compact-window `ℓ^{2/3}` coefficient (their §§9 and 7); +- the transfer of the window decision to the global field (#170 Theorem E(2)–(3), #175 Theorem H); +- the critical-chord bound `d(M̂) ≥ L_S` (#170 §3). + +v1 of this packet did not cite them. v1.1 records this and claims no priority for those statements. ## Result Math- #242 (open) defines the fold-scale rejection rate `F(k; b, u)` through an explicit soft model and the elder decision -of its Lemma 2, and states as Conjecture 6 that the rejected kernel at a fixed gap `k = ℓ/r³` converges to it. This note -proves that conjecture in every dimension and draws two consequences for the parent's Theorems A and B. +of its Lemma 2. Its Conjecture 6 states that the rejected kernel at a fixed gap `k = ℓ/r³` converges to `F`. | | Statement | Status | |---|---|---| -| **Theorem FL** | `(k/r)r^{−2}A_r^{rej}(b, k, u) → F(k; b, u)` as `r → 0`, every `d ≥ 2`, uniformly for `(b, k)` in compacts; the limit in #242 (3.1) exists, and `F` is continuous and positive | proof | -| **Corollary FL.5** | `r^{−3}(1 − p_r) → F/(k·A_∗) ∈ (0, ∞)` (fixed `u`, uniformly for `(b, k)` in compacts): the constant in the `Θ(r³)` selection loss of [P] Theorem A and the OA lower bound (A3) | proof | -| **Corollary FL.6** | compact mark windows: `ν_cand − ν_eld = d_{𝐁,𝐊}ℓ^{2/3} + o(ℓ^{2/3})`, `d_{𝐁,𝐊} = (1/3)∫∫∫k^{−8/3}F > 0`; so also `E[N_cand − N_eld](0, t] ~ (3/5)d_{𝐁,𝐊}t^{5/3}` | proof | -| **Proposition FL.4** | stability of #242 Lemma 2's decision: every `C²`-small perturbation with the pins as critical points has the model's decision, for the global superlevel filtration of any field realizing it in a window; stiff directions included | proof | -| **Lemma FL.2** | the generic set `𝒦` has full measure; a rejected model maximum always dies strictly above the saddle level (an axial path, observed by OpenAI Codex on #242), and case (D′) needs `χ > 0` | proof | +| **Theorem FL** | `(k/r)r^{−2}A_r^{rej}(b, k, u) → F(k; b, u)` as `r → 0`, for every `d ≥ 2`, uniformly for `(b, k)` in compacts. The limit in #242 (3.1) exists, and `F` is continuous and positive. This is Conjecture 6. The identification with `F`, the continuity and the local uniformity are new. The decision and convergence steps take a different route from #170/#175; the cap-region domination is shared. | proof | +| **Proposition FL.7** | #242's model is #170's cubic: `G_k(X, ζ) = P_θ(X, kζ)/k`, with `θ = (−λ̃/k, γ, B, C_3)`. The typed domain, weight and maximin decision are the same, so `F = kA_∗a_fail`. In `d = 2` this is an identity of model integrals, which with #170 Theorem S gives a second proof of Conjecture 6. In `d ≥ 3` it follows by uniqueness of limits, conditional on #175. | proof | +| **Corollary FL.5** | `r^{−3}(1 − p_r) → F/(k·A_∗)`: #170 Theorem S and #175 Theorem F, in #242's form | recovers merged results | +| **Corollary FL.6** | compact mark windows: `ν_cand − ν_eld = d_{𝐁,𝐊}ℓ^{2/3} + o(ℓ^{2/3})`, with `d_{𝐁,𝐊} = C_fail^{𝐁,𝐊}` (#170 §9, #175 §7) | recovers merged results | +| **Proposition FL.4** | the decision transfer of #170 Theorem E(2)–(3) and #175 Theorem H, re-proved in #242's normalization by paths and traps (an `ε`-form for a continuous global field), with the generic set adapted to #242's slice decision; stiff directions included | proof (method new) | +| **Lemma FL.2** | the generic set `𝒦` has full measure; a rejected model maximum always dies strictly above the saddle level (#170 §3's critical chord; observed on #242 by OpenAI Codex); case (D′) needs `χ > 0` | proof | | **Lemma FL.3** | a nondegenerate saddle can be crossed along a model curve, robustly under `C²`-small perturbation | proof | | **Lemma FL.1** | the soft window of a pinned `C⁵` field is `C²`-close to the model: rate `r` (`d = 2`), `r^{1/2}` (`d ≥ 3`), constant linear in `‖f‖_{C⁵}` | proof | @@ -57,8 +69,9 @@ on three fixed points. Its output is `RESULTS.json`, byte-identical under `-O`. | F4 | Proposition FL.4: the exact (D′) witness `G_{(3/2, 8/3, 20/3)}(X, z) = (1/9)G_{(1/6, 0, 0)}(X + 2z, 3z)`, a factorization proving it has exactly three critical points, the crossing forms at `Ŝ`, the two facts used in Lemma FL.2(c), and floating-point certificate margins on three fixed points | | F5 | Theorem FL: the change of variables `μ̃ ↔ φ`, the constant `384`, and `χ = χ_0φ²` | | F6 | the corollaries: the factorization (4.2) of `R − L_S` at `ϖ = (φ, 0, 0)` and its discriminant, exact Sturm certificates of rejection for `φ ∈ {2/5, 1/2, 3/4, 9/10}`, and the exponents of the pushforward | +| F7 | Proposition FL.7: `G_k(X, ζ) = P_θ(X, kζ)/k` as a polynomial identity, `Y = 3kB′`, the weight identity `9k²(4s² − B′²) = 36λ̃² − Y²`, the typed domains, the hard factor `𝒱(0, μ)(Πμ_j)² = Πμ_j³Π(μ_j − μ_i)` (`m = 2, 3, 4`), and the exponent of `d_{𝐁,𝐊} = C_fail`. The decision equivalence (iii) is tested numerically by the referee (PROOF §5), not by F7 | -Mutants `M1`–`M7` each fail only their own control (the checker names it on stderr), and an unknown label exits 2: +Mutants `M1`–`M8` each fail only their own control (the checker names it on stderr), and an unknown label exits 2: python3 -B -S fold_check.py # exit 0, output = RESULTS.json python3 -B -S fold_check.py --mutant M1 # exit 1 @@ -68,7 +81,9 @@ Mutants `M1`–`M7` each fail only their own control (the checker names it on st - No rate in Theorem FL, and no uniformity as `k → 0` or `k → ∞`. - #242 Conjecture 7 (the unrestricted `ℓ^{2/3}` term) is not proved; it still needs #242 §4's inputs (i)–(ii). - No numerical constant is certified. -- No change to #242, [P] or any other packet; #242 is consumed at a pinned blob. +- No priority for the existence of the limit, the constant in `1 − p_r ~ r³`, the compact-window `ℓ^{2/3}` coefficient, the + window-to-global decision transfer, or the critical-chord bound (#170, #175). +- No change to #242, [P], #170, #175 or any other packet; #242 is consumed at a pinned blob. ## Review record @@ -89,5 +104,19 @@ The same referee then checked the fixes: **ACCEPT WITH MINOR FIXES**. Every fix description of [CAP]. All seven were applied. After that check, Step 4 of Proposition FL.4 was reordered so that `W` and `R_η` are fixed after `ε` and visibly contain the balls of Lemma FL.3; the argument is unchanged. +**v1.1 (author, after opening).** The author found that #170/#175 had not been cited. This is recorded as AUTH-243-01 on the +PR. +- *Changes.* v1.1 adds the prior-work statement, Proposition FL.7, control F7 and mutant M8. It relabels Corollaries + FL.5–FL.6, credits #170/#175 for the decision transfer and the critical chord, and updates §§4–9. The arguments of + §§1–3 are unchanged. +- *Referee pass on the change.* A third clean-context same-family referee checked Proposition FL.7 and the prior-work + record: **ACCEPT WITH FIXES** for both. + - It re-derived (i) and (ii) with sympy, and checked the normalizations of (iv) end to end. It compared (iii) on 1,800 + typed points, with no disagreement. + - It found three MAJOR wording findings: the decision-transfer and critical-chord credits; the conditional status of the + `d ≥ 3` identity; and the description of #170/#175's route and of what is shared. + - It also found four MINOR findings and six NITs. + - All are applied. + Same GitHub account and same provider as the author: zero organizational-independence credit. Nonauthor review is required for every slice (`PROOF.md` §9). diff --git a/frontiers/soft_fold_limit_20261002/RESULTS.json b/frontiers/soft_fold_limit_20261002/RESULTS.json index 2df93a56e..e0c003f40 100644 --- a/frontiers/soft_fold_limit_20261002/RESULTS.json +++ b/frontiers/soft_fold_limit_20261002/RESULTS.json @@ -142,14 +142,19 @@ } } }, + "F7": { + "hard_factor_cases": 14, + "random_jets": 40 + }, "controls": { "F1_model_pins_and_weight": true, "F2_lemma_FL2_generic": true, "F3_lemma_FL1_window": true, "F4_proposition_FL4_certificates": true, "F5_theorem_FL_bookkeeping": true, - "F6_corollaries": true + "F6_corollaries": true, + "F7_proposition_FL7_identification": true }, - "object": "CL-FOLD-LIMIT-20261002-v1", + "object": "CL-FOLD-LIMIT-20261002-v1.1", "scientific_effect": "NONE" } diff --git a/frontiers/soft_fold_limit_20261002/SOURCES.json b/frontiers/soft_fold_limit_20261002/SOURCES.json index 80b88dd82..f990eb4d3 100644 --- a/frontiers/soft_fold_limit_20261002/SOURCES.json +++ b/frontiers/soft_fold_limit_20261002/SOURCES.json @@ -1,8 +1,12 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-FOLD-LIMIT-20261002-v1", - "kind": "author-side proof candidate, every d >= 2: Theorem FL (the fold-scale limit (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u), Conjecture 6 of Math- #242, uniformly on compacts of (b, k); the limit defining F exists, F is continuous and positive); Corollary FL.5 (r^-3 (1 - p_r) -> F/(k A_*), A_* the limit of A_r); Corollary FL.6 (compact mark windows BB x KK: nu_cand - nu_eld = d l^(2/3) + o(l^(2/3)), d = (1/3) int k^(-8/3) F > 0); Proposition FL.4 (stability of the model decision of #242 Lemma 2 under C^2-small perturbations, stiff directions included); Lemma FL.2 (the generic set has full measure; a rejected model maximum dies strictly above the saddle (an axial path, observed by OpenAI Codex on #242); case (D-prime) needs chi > 0); Lemma FL.3 (crossing a nondegenerate saddle); Lemma FL.1 (the soft window, rate r in d = 2 and r^(1/2) in d >= 3)", + "object": "CL-FOLD-LIMIT-20261002-v1.1", + "supersedes": { + "object": "CL-FOLD-LIMIT-20261002-v1", + "head": "008b2997936ae8a677a05cae00db010006ae319f" + }, + "kind": "author-side proof candidate, every d >= 2: Theorem FL (Conjecture 6 of Math- #242: the fold-scale limit (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u), uniformly on compacts of (b, k); the limit defining F exists, F is continuous and positive; the existence of the limit, as r^-3 (1 - p_r) -> alpha_1 + alpha_2, was proved earlier by the merged #170 (d = 2) and #175 (d >= 3); the decision and convergence steps here take a different route, the cap-region domination is shared); Proposition FL.7 (G_k(X, zeta) = P_theta(X, k zeta)/k with theta = (-lt/k, gamma, B, C3): same typed domain, weight and maximin decision, so F = k A_* a_fail, directly in d = 2 and by uniqueness of limits in d >= 3); Corollaries FL.5-FL.6 (recover #170 Theorem S and section 9, #175 Theorem F and section 7, in #242 form; not new); Proposition FL.4 (re-proves the window-to-global decision transfer of #170 Theorem E(2)-(3) and #175 Theorem H in #242 normalization, by paths and traps); Lemma FL.2 (the generic set has full measure; d(M) >= L_S by the critical chord of #170 section 3, observed on #242 by OpenAI Codex; case (D-prime) needs chi > 0); Lemma FL.3 (crossing a nondegenerate saddle); Lemma FL.1 (the soft window, rate r in d = 2 and r^(1/2) in d >= 3)", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { @@ -72,6 +76,22 @@ "git_blob": "75da2597971510f843f8d90c743950cb8c177342", "tag": "REC", "use": "reading rule for [P]" + }, + { + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "bytes": 39722, + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "tag": "170", + "use": "Math- #170 (merged author-side candidate, OpenAI, at its stated conditional scope; prior work on the failure law in d = 2 and on the window-to-global decision transfer, recorded in v1.1 as AUTH-243-01): section 2 (the typed cubic P_theta, B', C(P), the null sets, Theorem E(1)), section 3 (the critical chord, d(M) >= P(S)), sections 7-8 (the classifier n, (S1), (S11), the mass alpha_1 + alpha_2), Theorem S ((S2): r^-3 (1 - p_r) -> alpha_1 + alpha_2), section 9 (C_fail); consumed by Proposition FL.7 only" + }, + { + "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 on the failure law in fixed d >= 3 and on the hard-fibre decision transfer, recorded in v1.1 as AUTH-243-01): sections 1-2 (the model and the soft cubic, n in {0, 1, 2}), section 3 (Theorem H), section 5 (Theorem F, (S1)-(S3)), section 7 (L1); consumed by Proposition FL.7 only" } ], "consumed_unmerged": [ @@ -113,7 +133,8 @@ "M4": "F4", "M5": "F5", "M6": "F6", - "M7": "F4" + "M7": "F4", + "M8": "F7" }, "replayed": [ "fold_check.py" @@ -123,23 +144,23 @@ "files": [ { "path": "PROOF.md", - "bytes": 59974, - "sha256": "5f5a411d1333c8f1eb83ec02794e154d6793a41f7d822cf8faca82cba020ea9f" + "bytes": 72241, + "sha256": "9047cc30a37af0aed43ed289d6f94ce31a2a3607bd67c71195158390a48d67d2" }, { "path": "README.md", - "bytes": 7764, - "sha256": "2c0a3dc154b973f42aa7f9326ea6d354362d227d6fc31e15e56af2db86a28dac" + "bytes": 10568, + "sha256": "f8981defe4468c9edc04ef0101dc7c60fe45a7a37514a70c55c831ecbe61c299" }, { "path": "fold_check.py", - "bytes": 37203, - "sha256": "680442d62ba7b146e9a55ad99797e7520ffe97a96943fc0a4982cba604e8d7cf" + "bytes": 40386, + "sha256": "425ac7ac63efa7e1fdfa2d53d16fd9f2138216f9630e500487e55d4907339ba7" }, { "path": "RESULTS.json", - "bytes": 2796, - "sha256": "61ca77e1eb57e22d60eb37bd8c97cd388c1814245406856c379c4ca51966c762" + "bytes": 2903, + "sha256": "168118193b1f7fc9b93416541e93237a82fbe5935306ad2f71cb013a6c7dd43d" } ] } diff --git a/frontiers/soft_fold_limit_20261002/fold_check.py b/frontiers/soft_fold_limit_20261002/fold_check.py index af3384bd7..85f7af3af 100644 --- a/frontiers/soft_fold_limit_20261002/fold_check.py +++ b/frontiers/soft_fold_limit_20261002/fold_check.py @@ -1,12 +1,12 @@ #!/usr/bin/env python3 -"""Exact controls for CL-FOLD-LIMIT-20261002-v1 (Theorem FL, Corollaries FL.5-FL.6, Proposition FL.4, Lemmas FL.1-FL.3; -PROOF.md). +"""Exact controls for CL-FOLD-LIMIT-20261002-v1.1 (Theorem FL, Propositions FL.4 and FL.7, Corollaries FL.5-FL.6, +Lemmas FL.1-FL.3; PROOF.md). Standard library only; exact rationals except the floating-point margins printed by F4 (a fixed, deterministic construction of the model certificates on three parameter points). Output: RESULTS.json (sorted keys), byte-identical under -O. Usage: python3 -B -S fold_check.py # exit 0, prints RESULTS.json - python3 -B -S fold_check.py --mutant M3 # exit 1 (only its control fails; M1..M7) + python3 -B -S fold_check.py --mutant M3 # exit 1 (only its control fails; M1..M8) unknown mutant label -> exit 2 Controls @@ -40,13 +40,18 @@ exact Sturm counts showing R > L_S + 1/200 on [-3, -1/2] and R(-3) > 0 (the (R) certificate along the ridge); the pushforward ell^(-1/3) (ell/k^2)/(3 k^(2/3)) = ell^(2/3) k^(-8/3)/3, the cumulative factor 3/5 and the cusp-cutoff exponent 2/3 - 5/12 = 1/4 + F7 Proposition FL.7 (the model of #170/#175): G_k(X, zeta) = P_theta(X, k zeta)/k as a polynomial identity in + (X, zeta) with theta = (s, a, beta', c) = (-lt/k, gamma, B, C3), on random rational jets (lt of either sign); + Y = 3k B' (B' = beta' - a^2/(12k)); 9k^2 (4 s^2 - B'^2) = 36 lt^2 - Y^2; typed domains s < -|B'|/2 iff |Y| < 6 lt; + the hard factor V(0, mu_2..mu_m) (mu_2...mu_m)^2 = prod mu_j^3 prod (mu_j - mu_i) for m = 2, 3, 4; d mu~/ds = -k; + and the exponent -8/3 + 1 = -5/3 of d_BK = C_fail """ import json import math import sys from fractions import Fraction as Fr -MUTANTS = ('M1', 'M2', 'M3', 'M4', 'M5', 'M6', 'M7') +MUTANTS = ('M1', 'M2', 'M3', 'M4', 'M5', 'M6', 'M7', 'M8') MUT = None FACT = [math.factorial(n) for n in range(16)] H = Fr(1, 2) @@ -737,6 +742,61 @@ def control_F6(): return ok, {'R_certificates_phi_0_0': cert} +# ------------------------------------------------------------------------------------------------ F7 +def control_F7(): + ok = True + rng = LCG(907) + n = 0 + for _ in range(40): + k = Fr(1 + rng.nxt() % 9, 1 + rng.nxt() % 5); g = rng.rat(7, 3, nonzero=True) + lt = Fr(1 + rng.nxt() % 9, 1 + rng.nxt() % 4) * (1 if rng.nxt() % 3 else -1) + B = rng.rat(7, 3); C3 = rng.rat(7, 3) + s, a, bp, c = -lt / k, g, B, C3 + # G_k of #242 (2.1), as a polynomial in (X, zeta) = (VX, VZ) + Gk_poly = (2 * (VX + H) ** 2 * (VX - 1) + H * g * (VX * VX - Fr(1, 4)) * VZ - H * lt * VZ * VZ + + H * k * B * VX * VZ * VZ + k * k * C3 * Fr(1, 6) * VZ ** 3) + # #170's typed cubic (4.0) at Z = k zeta, divided by k + Zk = VZ * k + P170 = (2 * k * VX ** 3 - Fr(3, 2) * k * VX - k * H + s * H * Zk * Zk + a * H * (VX * VX - Fr(1, 4)) * Zk + + bp * H * VX * Zk * Zk + c * Fr(1, 6) * Zk ** 3) * (1 / k) + if MUT == 'M8': + P170 = P170 + Fr(1, 1000) * VX * VZ * VZ + ok &= Gk_poly == P170 + for X, ze in ((Fr(-3, 2), Fr(1, 3)), (Fr(1, 5), Fr(-2)), (Fr(2), Fr(5, 7))): + ok &= Gk(X, ze, g, lt, k, B, C3) == Gk_poly(X, ze, 0, 0, 0) + Bp = bp - a * a / (12 * k); Y = 3 * k * B - g * g / 4 + ok &= Y == 3 * k * Bp + ok &= 9 * k * k * (4 * s * s - Bp * Bp) == 36 * lt * lt - Y * Y + ok &= (s < -abs(Bp) / 2) == (abs(Y) < 6 * lt) + ok &= (-k * (s + Fr(1, 10 ** 6)) - (-k * s)) / Fr(1, 10 ** 6) == -k # mu~ = -k s: d mu~/ds = -k + n += 1 + # the hard factor: Vandermonde(0, mu_2..mu_m) (mu_2...mu_m)^2 = prod mu_j^3 prod_{i Date: Fri, 2 Oct 2026 12:14:57 +0000 Subject: [PATCH 3/6] Fold-scale limit v1.2: rebind #242 v1.2; cite Codex's direct d >= 3 cross-check - Rebinds the consumed #242 to v1.2 (head 1f86fea, PROOF blob ad4beb84). Bounded comparison of the consumed slices (section 0, (2.1)-(2.6), Lemmas 2-3, Proposition 2-prime, (3.1)): the model, its normalization, Lemma 3 and (3.1) are unchanged. Lemma 2's proof is rewritten at the exact level, and its statement now says that beta > 2, chi <= 0 is rejected, which is Lemma FL.2(c) here. The other changes are wording, citation and the B-03 absolute Jacobian. No statement here changes. - Proposition FL.7(iv) cites OpenAI Codex's antecedent cross-check (5951210964), which argues the d >= 3 identity directly, as root-authored support. - Rebased onto main f534754 (every consumed and cited merged blob is unchanged). Object v1.2, supersedes v1.1 (0740b2c). Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- frontiers/soft_fold_limit_20261002/PROOF.md | 15 +++++++--- frontiers/soft_fold_limit_20261002/README.md | 3 +- .../soft_fold_limit_20261002/RESULTS.json | 2 +- .../soft_fold_limit_20261002/SOURCES.json | 30 +++++++++---------- .../soft_fold_limit_20261002/fold_check.py | 4 +-- 5 files changed, 31 insertions(+), 23 deletions(-) diff --git a/frontiers/soft_fold_limit_20261002/PROOF.md b/frontiers/soft_fold_limit_20261002/PROOF.md index caedaef6c..3d9faa87d 100644 --- a/frontiers/soft_fold_limit_20261002/PROOF.md +++ b/frontiers/soft_fold_limit_20261002/PROOF.md @@ -1,6 +1,9 @@ # The fold-scale rejection limit through #242's soft model: Conjecture 6 of Math- #242 in every dimension, and its identification with the failure law of #170/#175 -Object: CL-FOLD-LIMIT-20261002-v1.1 (v1: `008b299`; v1.1 adds the prior-work record AUTH-243-01 and Proposition FL.7). +Object: CL-FOLD-LIMIT-20261002-v1.2. Versions: +- v1, `008b299`. +- v1.1, `0740b2c`: the prior-work record AUTH-243-01, and Proposition FL.7. +- v1.2: rebinds #242 v1.2, and cites Codex's direct `d ≥ 3` cross-check of FL.7. Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 2 October 2026. Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem FL, Propositions FL.4 and FL.7, Lemmas FL.1–FL.3; Corollaries FL.5 and FL.6 recover merged results of #170 and #175). @@ -92,7 +95,7 @@ domination are as in #170 §8 and #175 §§2 and 4. - #207 (`frontiers/cusp_second_order_20261001/PROOF.md`, blob `f6df5a73`): §0 (the elder mark and the maximin), Theorem CU.1 (the pin identities), and the structure of Proposition CU.3 and Theorem CU.4. - [C7-K] (`frontiers/c7_total_bounded_20260929/PROOF.md`, blob `28748b08`): (K2). -- Consumed (unmerged): #242 (`frontiers/soft_rejected_pairs_20261002/PROOF.md` at `87912bd`, blob `8b2f5fae`): §0, +- Consumed (unmerged): #242 (`frontiers/soft_rejected_pairs_20261002/PROOF.md` at `1f86fea`, blob `ad4beb84`; v1.2): §0, (2.1)–(2.5), Lemmas 2 and 3, Proposition 2′ ((2.6)), (3.1). This note must be rebound if that blob changes. - Consumed through #207 §0: [E2] (`reviews/d1_section9_borel_repair_20260925/REPAIR.md`, blob `fe9b9ce4`), for the Borel measurability of the elder mark. Reading rules for [P]: [E1] (`imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`, @@ -607,6 +610,9 @@ number of points of `C(P_θ)` (#170 §2; the classifier of #170 §§7 and 8.1, w - *In `d ≥ 3` the integrands also agree, up to the normalization `c_m dO` of the Weyl formula.* `𝒱(0, μ_2, …, μ_m)(μ_2⋯μ_m)² = Π_{j≥2}μ_j³Π_{2≤i= 2: Theorem FL (Conjecture 6 of Math- #242: the fold-scale limit (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u), uniformly on compacts of (b, k); the limit defining F exists, F is continuous and positive; the existence of the limit, as r^-3 (1 - p_r) -> alpha_1 + alpha_2, was proved earlier by the merged #170 (d = 2) and #175 (d >= 3); the decision and convergence steps here take a different route, the cap-region domination is shared); Proposition FL.7 (G_k(X, zeta) = P_theta(X, k zeta)/k with theta = (-lt/k, gamma, B, C3): same typed domain, weight and maximin decision, so F = k A_* a_fail, directly in d = 2 and by uniqueness of limits in d >= 3); Corollaries FL.5-FL.6 (recover #170 Theorem S and section 9, #175 Theorem F and section 7, in #242 form; not new); Proposition FL.4 (re-proves the window-to-global decision transfer of #170 Theorem E(2)-(3) and #175 Theorem H in #242 normalization, by paths and traps); Lemma FL.2 (the generic set has full measure; d(M) >= L_S by the critical chord of #170 section 3, observed on #242 by OpenAI Codex; case (D-prime) needs chi > 0); Lemma FL.3 (crossing a nondegenerate saddle); Lemma FL.1 (the soft window, rate r in d = 2 and r^(1/2) in d >= 3)", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { - "branch_base": "7636cae5c386f9725f9eec127dfd4ea5ec630435" + "branch_base": "f5347541a6ef7b84de96120dcce6137b816b8033" }, "consumed": [ { @@ -97,12 +97,12 @@ "consumed_unmerged": [ { "path": "frontiers/soft_rejected_pairs_20261002/PROOF.md", - "bytes": 63690, - "sha256": "c312e423990442cfaad44459a2a20b231029828cae97018ba68724c20613331b", - "git_blob": "8b2f5fae2a0014a401003463d7c2b2221e7124f5", + "bytes": 79394, + "sha256": "ab952b2a2c7648beb5f7473b3f028cdc495c8317c52522bee144d26eecc8e2bb", + "git_blob": "ad4beb8467a775c7c76f85d5584c13338d0e66be", "tag": "242", - "use": "Math- #242 (open): section 0, (2.1)-(2.6) (the soft model and its normalization), Lemma 2 (the model decision), Lemma 3 (the a priori elder region, used for domination in the identification of the limit), Proposition 2-prime (the stiff directions), (3.1) (the definition of F); Theorem 1 and Proposition 4 are cited", - "head": "87912bd8f28fe0d31fcb525670d1fd71a00e0db1", + "use": "Math- #242 (open; v1.2): section 0, (2.1)-(2.6) (the soft model and its normalization), Lemma 2 (the model decision; v1.2 rewrites its proof at the exact level and states that beta > 2, chi <= 0 is rejected, as Lemma FL.2(c) here), Lemma 3 (the a priori elder region, used for domination in the identification of the limit), Proposition 2-prime (the stiff directions), (3.1) (the definition of F); Theorem 1 and Proposition 4 are cited", + "head": "1f86fea035f838cd3957b20bea58fc2082a4e3a4", "pr": "Math- #242" } ], @@ -144,23 +144,23 @@ "files": [ { "path": "PROOF.md", - "bytes": 72241, - "sha256": "9047cc30a37af0aed43ed289d6f94ce31a2a3607bd67c71195158390a48d67d2" + "bytes": 72742, + "sha256": "7716a113bcf8c1725f9ec771d58ef033767f0a7bb50d592b841ca0ffe86f31af" }, { "path": "README.md", - "bytes": 10568, - "sha256": "f8981defe4468c9edc04ef0101dc7c60fe45a7a37514a70c55c831ecbe61c299" + "bytes": 10687, + "sha256": "0be812e8a4785462587ac0e572dd42a04d3d1789a03d948ebb220649d5f5da6a" }, { "path": "fold_check.py", "bytes": 40386, - "sha256": "425ac7ac63efa7e1fdfa2d53d16fd9f2138216f9630e500487e55d4907339ba7" + "sha256": "762695347e6b46aab81759c299cbc2a5dd3860e53017fc1ddbda1b43994850ae" }, { "path": "RESULTS.json", "bytes": 2903, - "sha256": "168118193b1f7fc9b93416541e93237a82fbe5935306ad2f71cb013a6c7dd43d" + "sha256": "459bbdf3c84b77548e53824df889b13a8ebd0fe6b0d7d7cadf8a4d73821641e9" } ] } diff --git a/frontiers/soft_fold_limit_20261002/fold_check.py b/frontiers/soft_fold_limit_20261002/fold_check.py index 85f7af3af..c2e1e77e9 100644 --- a/frontiers/soft_fold_limit_20261002/fold_check.py +++ b/frontiers/soft_fold_limit_20261002/fold_check.py @@ -1,5 +1,5 @@ #!/usr/bin/env python3 -"""Exact controls for CL-FOLD-LIMIT-20261002-v1.1 (Theorem FL, Propositions FL.4 and FL.7, Corollaries FL.5-FL.6, +"""Exact controls for CL-FOLD-LIMIT-20261002-v1.2 (Theorem FL, Propositions FL.4 and FL.7, Corollaries FL.5-FL.6, Lemmas FL.1-FL.3; PROOF.md). Standard library only; exact rationals except the floating-point margins printed by F4 (a fixed, deterministic @@ -814,7 +814,7 @@ def main(argv): f5, f5d = control_F5() f6, f6d = control_F6() f7, f7d = control_F7() - res = {'object': 'CL-FOLD-LIMIT-20261002-v1.1', 'scientific_effect': 'NONE', + res = {'object': 'CL-FOLD-LIMIT-20261002-v1.2', 'scientific_effect': 'NONE', 'controls': {'F1_model_pins_and_weight': f1, 'F2_lemma_FL2_generic': f2, 'F3_lemma_FL1_window': f3, 'F4_proposition_FL4_certificates': f4, 'F5_theorem_FL_bookkeeping': f5, 'F6_corollaries': f6, 'F7_proposition_FL7_identification': f7}, From 483556485cf15cc8662425e2195699029c713ff5 Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 15:49:34 +0000 Subject: [PATCH 4/6] Fold-scale limit v1.3: apply OA-243-B-01/02/03 and D-01; record D.1; rebind #242 v1.3 Applies the four minor findings of the nonauthor slice reviews of v1.2 (11a4cd4): - OA-243-B-01: Proposition FL.4 states the m = 1 convention (no stiff variables; the eta, q_-, R_eta clauses are vacuous). - OA-243-B-02: the (Trap) fact splits off the trivial case c' >= G(M_0), where d_f(M) <= f(M) <= b + s c'. - OA-243-B-03: Step 4 fixes the planar rectangle W = W_varpi first (containing the certificates and r_varpi-discs about both pins), then shrinks the Lemma FL.3 radii to fit; only rho_3, eps_0, rho, delta, eps and R_eta depend on Q. - OA-243-D-01: Step 5's Weyl equality keeps the ordering lambda_1 < lambda_2 (upper limit min(eps, lambda_2); eps for m = 1). Records Codex's direct identity D.1 (F = k A_* a_fail in every fixed d, reviewed in 5954510413) in a Remark after Proposition FL.7, and rebinds #242 v1.3 (d504cdb, blob 5fd29e63), whose consumed sections 0-3 are byte-identical to v1.2 (blob ad4beb84). PROOF section 9 lists the changed bytes. No statement changes in substance; fold_check.py and RESULTS.json are unchanged. Scientific effect: NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- frontiers/soft_fold_limit_20261002/PROOF.md | 109 +++++++++++++----- frontiers/soft_fold_limit_20261002/README.md | 31 ++++- .../soft_fold_limit_20261002/SOURCES.json | 88 +++++++++++--- 3 files changed, 181 insertions(+), 47 deletions(-) diff --git a/frontiers/soft_fold_limit_20261002/PROOF.md b/frontiers/soft_fold_limit_20261002/PROOF.md index 3d9faa87d..2652db872 100644 --- a/frontiers/soft_fold_limit_20261002/PROOF.md +++ b/frontiers/soft_fold_limit_20261002/PROOF.md @@ -1,9 +1,13 @@ # The fold-scale rejection limit through #242's soft model: Conjecture 6 of Math- #242 in every dimension, and its identification with the failure law of #170/#175 -Object: CL-FOLD-LIMIT-20261002-v1.2. Versions: +Object: CL-FOLD-LIMIT-20261002-v1.3. Versions: - v1, `008b299`. - v1.1, `0740b2c`: the prior-work record AUTH-243-01, and Proposition FL.7. -- v1.2: rebinds #242 v1.2, and cites Codex's direct `d ≥ 3` cross-check of FL.7. +- v1.2, `11a4cd4`: rebinds #242 v1.2, and cites Codex's direct `d ≥ 3` cross-check of FL.7. Nonauthor slices A–E + reviewed this head. +- v1.3: applies the four minor findings of those reviews, OA-243-B-01, B-02, B-03 (Proposition FL.4) and D-01 (Step 5 of + Theorem FL). It records Codex's reviewed direct identity D.1 after Proposition FL.7, and rebinds #242 v1.3, whose + §§0–3 are byte-identical to v1.2. No statement changes; §9 lists the changed bytes. Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 2 October 2026. Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem FL, Propositions FL.4 and FL.7, Lemmas FL.1–FL.3; Corollaries FL.5 and FL.6 recover merged results of #170 and #175). @@ -52,7 +56,8 @@ domination are as in #170 §8 and #175 §§2 and 4. - In `d = 2`, `F = kA_∗a_fail` is a direct identity of model integrals. With #170 Theorem S it gives a second proof of (0.0) at each fixed `(b, k, u)`. - In `d ≥ 3` the identity follows from Theorem FL and #175 Theorem F by uniqueness of limits. It is therefore conditional - on #175's interfaces. + on #175's interfaces. Codex's supplement D.1, reviewed by another Codex agent, gives it directly as an identity of + model integrals in every fixed `d`, without #175 Theorem F (Remark after Proposition FL.7). - **Corollaries FL.5 and FL.6** (§4) are not new. They are #170 Theorem S and §9 (`d = 2`) and #175 Theorem F and §7 (`d ≥ 3`), recovered with the constants in #242's form: - `r^{−3}(1 − p_r) → F/(k·A_∗)`, now locally uniformly in `(b, k)`; @@ -95,8 +100,10 @@ domination are as in #170 §8 and #175 §§2 and 4. - #207 (`frontiers/cusp_second_order_20261001/PROOF.md`, blob `f6df5a73`): §0 (the elder mark and the maximin), Theorem CU.1 (the pin identities), and the structure of Proposition CU.3 and Theorem CU.4. - [C7-K] (`frontiers/c7_total_bounded_20260929/PROOF.md`, blob `28748b08`): (K2). -- Consumed (unmerged): #242 (`frontiers/soft_rejected_pairs_20261002/PROOF.md` at `1f86fea`, blob `ad4beb84`; v1.2): §0, - (2.1)–(2.5), Lemmas 2 and 3, Proposition 2′ ((2.6)), (3.1). This note must be rebound if that blob changes. +- Consumed (unmerged): #242 (`frontiers/soft_rejected_pairs_20261002/PROOF.md` at `d504cdb`, blob `5fd29e63`; v1.3): §0, + (2.1)–(2.5), Lemmas 2 and 3, Proposition 2′ ((2.6)), (3.1). #242's §§0–3 are byte-identical to its v1.2 (blob + `ad4beb84`), which the slice reviews of v1.2 of this note read; v1.3 changed only #242's header, §4's numerical + sentences and §§5–9. This note must be rebound if that blob changes. - Consumed through #207 §0: [E2] (`reviews/d1_section9_borel_repair_20260925/REPAIR.md`, blob `fe9b9ce4`), for the Borel measurability of the elder mark. Reading rules for [P]: [E1] (`imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`, blob `213594d6`) and [REC] (`reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md`, blob `75da2597`). @@ -306,7 +313,13 @@ on `|w| ≤ ρ_3` when `ρ_3` and `ε_0` are small. Taylor's formula with integr ### 2.3 The stability proposition **Proposition FL.4 (stability of the model decision).** Let `ϖ ∈ 𝒦` and let `𝒬` be a positive definite -`(m − 1) × (m − 1)` matrix (nothing if `m = 1`), `𝔮(η) := ½ηᵀ𝒬η`, `q_− := λ_min(𝒬)`. There are a compact rectangle +`(m − 1) × (m − 1)` matrix (nothing if `m = 1`), `𝔮(η) := ½ηᵀ𝒬η`, `q_− := λ_min(𝒬)`. + +*Convention for `m = 1` (OA-243-B-01).* There are then no stiff variables `η`. Read `W × B̄_{R_η}` as `W`, `𝔮` as `0`, and +`M̂_0`, `Ŝ_0` as `M̂`, `Ŝ`. Every clause below that involves `η`, `𝒬`, `q_−` or `R_η` is vacuous and is omitted, including the +term `q_−ρ²/8` in the choice of `ε` and the choice of `R_η` in Step 4. + +There are a compact rectangle `W = W_ϖ ⊂ R²` containing `M̂` and `Ŝ` in its interior, a radius `R_η = R_η(ϖ, 𝒬)` and an `ε = ε(ϖ, 𝒬) > 0` with the following property. Let `𝔊 ∈ C²(W × B̄_{R_η})` satisfy @@ -321,7 +334,8 @@ Let `f` be continuous on `X`, and let `Ψ : W × B̄_{R_η} → X` be continuous If moreover `f` is `C²` near `M` and `S` and `Ψ` is a `C²` diffeomorphism near `M̂_0` and `Ŝ_0`, then `M` is a nondegenerate local maximum of `f` and `S` a nondegenerate critical point of index `m`. -*Proof.* Constants `δ, ρ, τ` below depend on `ϖ` and `𝒬` only. +*Proof.* The curves `σ_±`, the path `Γ`, the set `𝒟_2`, the constants `τ`, `Z`, `ρ_1`, `r_ϖ` and the rectangle `W` below +depend on `ϖ` only. The radius `ρ`, the margin `δ`, and `ε` and `R_η` depend on `ϖ` and `𝒬` (OA-243-B-03). *Step 1 (types).* By (S2), `Hess 𝔊(M̂_0)` is within `ε` of `diag(Hess G_ϖ(M̂), −𝒬)`, which is negative definite (on `𝒯`, `det Hess G_ϖ(M̂) = (1 + β/2 − φ)/(4φ) > 0` and `∂_z²G_ϖ(M̂) = −(1 + β/2) < 0`). Likewise `Hess 𝔊(Ŝ_0)` is within `ε` of @@ -334,13 +348,15 @@ persists (Weyl), and it transfers to `f` through `Ψ`. - (Trap) Let `𝒟` be open in `R^d`, with `𝒟 ⊂ int(W × B̄_{R_η})` and `𝒟̄ ⊂ W × B̄_{R_η}`. Suppose `M̂_0 ∈ 𝒟`, `𝔊 ≤ c′` on `∂𝒟`, and `𝔊 < 𝔊(M̂_0)` on `𝒟 ∖ {M̂_0}`. Then `d_f(M) ≤ b + sc′`. - Indeed, by invariance of domain `Ψ(𝒟)` is open in `X`. Its closure lies in the compact set `Ψ(𝒟̄)`, so its boundary lies in - `Ψ(∂𝒟)`, where `f ≤ b + sc′`. The path component `C` of `{f > b + sc′}` containing `M` therefore lies in `Ψ(𝒟)`, so - `f < f(M)` on `C ∖ {M}`. A path from `M` to a point above `f(M)` must leave `C`, and so it meets `{f ≤ b + sc′}`. + Indeed, if `c′ ≥ 𝔊(M̂_0)`, the bound is trivial: every admissible path starts at `M`, so + `d_f(M) ≤ f(M) = b + s𝔊(M̂_0) ≤ b + sc′` (OA-243-B-02). Let now `c′ < 𝔊(M̂_0)`, so that `M ∈ {f > b + sc′}`. By invariance + of domain `Ψ(𝒟)` is open in `X`. Its closure lies in the compact set `Ψ(𝒟̄)`, so its boundary lies in `Ψ(∂𝒟)`, where + `f ≤ b + sc′`. The path component `C` of `{f > b + sc′}` containing `M` therefore lies in `Ψ(𝒟)`, so `f < f(M)` on + `C ∖ {M}`. A path from `M` to a point above `f(M)` must leave `C`, and so it meets `{f ≤ b + sc′}`. -*Step 3 (model certificates).* By Lemma FL.2(c), `ϖ ∈ 𝒦` falls in exactly one of three cases. Each comes with a margin -`δ > 0` and a radius `ρ > 0`. Let `ρ_3` be the smallest of the radii that Lemma FL.3 gives for the three maps of Step 4; we -take `2ρ ≤ ρ_3` (`ρ` may be decreased at the cost of `δ`). +*Step 3 (model certificates).* By Lemma FL.2(c), `ϖ ∈ 𝒦` falls in exactly one of three cases. Each comes with a +certificate built from `ϖ` alone and, for every small radius `ρ > 0`, a margin `δ = δ(ρ) > 0`; decreasing `ρ` decreases +`δ`. The radius `ρ` is fixed in Step 4, with `2ρ ≤ ρ_3`. - **(R)** `e(G_ϖ) = 0`, so `M` dies above `L_S`. Then there is a compact path `Γ` from `M̂` to a point `𝔭` with `G_ϖ(𝔭) > 2δ`, along which `G_ϖ ≥ L_S + 2δ`. (By the definition of the maximin, and because components of open subsets of @@ -388,16 +404,25 @@ just outside `I_M`. (Here `R′` is the derivative of `R`.) `Ŝ` to a point `𝔭 = (½, z_𝔭)` on the far side of `0`, with `G_ϖ(𝔭) > 2δ`. The slice at `½` has its only critical points at `z_r(½)` (maximum) and `0` (minimum, value `L_S`), and it tends to `+∞` on the far side. -In both cases compactness gives `δ` and `ρ`. +In both cases the curves `σ_±`, the path `Γ` and the set `𝒟_2` do not depend on `ρ`, and for every small `ρ > 0` +compactness gives `δ(ρ) > 0`. In case (R), `δ` does not depend on `ρ` at all. + +*Step 4 (persistence).* First fix a radius `r_ϖ > 0` and a compact rectangle `W = W_ϖ` whose interior contains `Γ`, +`𝒟̄_2` and the closed `r_ϖ`-discs about `M̂` and `Ŝ`. Both depend on `ϖ` only (OA-243-B-03). + +The three maps are `t ↦ (σ_+(t), 0)` and `(t, η) ↦ (σ_−(t), η)` at `Ŝ_0`, and the identity at `M̂_0`. The forms of +`H_0 := G_ϖ − 𝔮` on their tangent spaces are `σ_+′ᵀ𝐇_Sσ_+′ > 0`, `diag(σ_−′ᵀ𝐇_Sσ_−′, −𝒬) < 0` and +`diag(Hess G_ϖ(M̂), −𝒬) < 0`, with `𝐇_S := Hess G_ϖ(Ŝ)`. +- Let `ε_0` and `ρ_3` be the smallest tolerance and radius that Lemma FL.3 gives for the three maps, and `ρ_c` the largest + of their radii `ρ_3 sup|Dc|`. +- Decrease `ρ_3` until `2ρ_c ≤ r_ϖ`. Lemma FL.3 still holds with the same `ε_0`: its proof uses `H` only on + `c(B̄_{ρ_3})`, and its margins improve as `ρ_3` decreases. +- Fix `ρ ∈ (0, ρ_3/2]` small enough for Step 3, and put `δ := δ(ρ)`. +- Choose `ε ≤ min(δ/2, ε_0, q_−ρ²/8)`, with `ε < 1`, so small that Step 1 applies. Choose a radius `R_η ≥ 2ρ_c` with + `q_−R_η²/2 ≥ max_W G_ϖ − L_S + 3`. -*Step 4 (persistence).* The three maps are `t ↦ (σ_+(t), 0)` and `(t, η) ↦ (σ_−(t), η)` at `Ŝ_0`, and the identity at -`M̂_0`. The forms of `H_0 := G_ϖ − 𝔮` on their tangent spaces are `σ_+′ᵀ𝐇_Sσ_+′ > 0`, `diag(σ_−′ᵀ𝐇_Sσ_−′, −𝒬) < 0` and -`diag(Hess G_ϖ(M̂), −𝒬) < 0`, with `𝐇_S := Hess G_ϖ(Ŝ)`. Let `ρ_3` and `ε_0` be the smallest radius and tolerance that -Lemma FL.3 gives for the three maps (with `2ρ ≤ ρ_3` as in Step 3), and `ρ_c` the largest of their radii `ρ_3 sup|Dc|`. -Choose `ε ≤ min(δ/2, ε_0, q_−ρ²/8)`, with `ε < 1`, so small that Step 1 applies. Then fix a compact rectangle `W` whose -interior contains `Γ`, `𝒟̄_2` and the closed `2ρ_c`-discs about `M̂` and `Ŝ`, and a radius `R_η ≥ 2ρ_c` with -`q_−R_η²/2 ≥ max_W G_ϖ − L_S + 3`. So `W × B̄_{R_η}` contains the balls of Lemma FL.3, and `δ, ρ, ρ_3, ε_0, ε` do not -depend on `W` or `R_η`. By (S2), `|𝔊(M̂_0)| ≤ ε` and `|𝔊(Ŝ_0) − L_S| ≤ ε`. +Then `W × B̄_{R_η}` contains the balls `B(s, 2ρ_c)` of Lemma FL.3. Only `ρ_3`, `ε_0`, `ρ`, `δ`, `ε` and `R_η` depend on +`𝒬`, as the statement says. By (S2), `|𝔊(M̂_0)| ≤ ε` and `|𝔊(Ŝ_0) − L_S| ≤ ε`. - **(R).** On `Γ × {0}`, `𝔊 ≥ L_S + 2δ − ε > L_S + ε ≥ 𝔊(Ŝ_0)`, and `𝔊(𝔭, 0) > 2δ − ε > 𝔊(M̂_0)`. By (Path), `d_f(M) ≥ b + s(L_S + 2δ − ε) > f(S)`. So `1{d_f(M) = f(S)} = 0`. @@ -554,9 +579,15 @@ By #242 §2, the change of variables `μ̃ = γ*²/(24φ)` turns `36μ̃² − Y `(γ*²/(24φ²))dφ`. So by Fubini, `∫_0^∞h dμ̃ = (μ_2⋯μ_m)²E[(γ*⁶/384)I(t*, χ_0*)]`, with `t*`, `χ_0*` as in (0.3) and `I` as in #242 (2.5). -It remains to identify (3.7) with #242 (3.1). By Weyl's formula, -`ε^{−1}E_{v_0}[Υ1{𝔸 < 0, λ_1 < ε}] = c_m∫∫ε^{−1}∫_0^ε p_𝔸(a)𝒱(λ)E[Υ(F_0^a)]dλ_1 dλ_2⋯dO`, where -`Υ := (γ_1⁶/384)(λ_2⋯λ_m)²I(t, χ_0)`. The inner average tends to its value at `λ_1 = 0`, for three reasons: +It remains to identify (3.7) with #242 (3.1). By Weyl's formula, with `0 < λ_1 < ⋯ < λ_m` the eigenvalues of `−a`, + + ε^{−1}E_{v_0}[Υ1{𝔸 < 0, λ_1 < ε}] + = c_m∫_{0<λ_2<⋯<λ_m}∫_O ε^{−1}∫_0^{min(ε, λ_2)} p_𝔸(a)𝒱(λ)E[Υ(F_0^a)] dλ_1 dλ_2⋯dλ_m dO, + +where `Υ := (γ_1⁶/384)(λ_2⋯λ_m)²I(t, χ_0)`. For `m = 1` the upper limit is `ε` and there are no hard eigenvalues. The +ordering `λ_1 < λ_2` must be kept (OA-243-D-01). Write the inner average as `∫_0^1 1{εs < λ_2}(⋯)|_{λ_1 = εs} ds`. For +every fixed ordered hard spectrum the indicator tends to `1`, and the bounds below dominate in `s`, the hard spectrum and +`O`. The inner average tends to its value at `λ_1 = 0`, for three reasons: - by #242 Lemma 3, `γ⁶I ≤ C(γ⁶ + k³|B|³ + k⁴C_3²)`, which is integrable with Gaussian weights; - `I` is continuous at almost every `(t, χ_0)`. By Remark 4 of §2, the actual decision is constant near each point of `𝒦`; for almost every `(t, χ_0)` the set `{φ : (φ, 2tφ, χ_0φ²) ∉ 𝒦}` is null, so `1{φ ∈ 𝓡(t′, χ_0′)} → 1{φ ∈ 𝓡(t, χ_0)}` for @@ -615,6 +646,16 @@ number of points of `C(P_θ)` (#170 §2; the classifier of #170 §§7 and 8.1, w or factor-of-two discrepancy. This is root-authored support and is not reviewed here. - *The last claim.* Integrating `(1/3)k^{−8/3}F = A_∗a_fail/(3k^{5/3})` gives `d_{𝐁,𝐊} = C_fail^{𝐁,𝐊}`. ∎ +*Remark (D.1: a direct identity in every `d`).* OpenAI Codex's supplement D.1 is §3 of its review 5391986303 on #242. It +is bound to #242 v1.2's PROOF, whose §§0–3 are unchanged in v1.3. +- D.1 proves `F = kA_∗a_fail` directly, as an identity of model integrals in every fixed `d`, under its stated + contact-density and deterministic-classifier interfaces (#170, #175 and [SC]). It uses neither Theorem FL nor #175 + Theorem F. +- A different Codex agent reviewed it (5954510413, on this PR): PASS_TECHNICAL_SCOPED, with no finding. The review notes + that an author may adopt the identity in FL.7 while keeping every hypothesis needed for an actual-field conclusion. +- With D.1, (iv) holds in `d ≥ 3` without #175 Theorem F. This note records D.1 as reviewed support and does not reproduce + it. D.1, its review and this note are on the same account (organizational independence 0). + **A second proof of (0.0) in `d = 2`.** At each fixed `(b, k, u)`, conditional on #170's interfaces, (0.0) follows without the field analysis of §§1–3. The ingredients are: - #170 Theorem S; @@ -627,7 +668,7 @@ dominated convergence with #242 Lemma 3's bound; Step 5 of Theorem FL proves it Theorem FL is then a second proof with a partly different route, and it also gives continuity and local uniformity. In `d ≥ 3` the direct computation would also need the Weyl normalization of #175 §2 matched with (3.4). This note does not do it; -Codex's cross-check 5951210964 argues it. +Codex's cross-check 5951210964 argues it, and D.1 (the Remark above) proves it. **Corollary FL.5 (the constant in `1 − p_r ~ r³`; #170 Theorem S, #175 Theorem F).** For fixed `u ∈ S^{d−1}`, uniformly for `(b, k)` in compact subsets of `R × (0, ∞)`, @@ -767,7 +808,7 @@ in the (D′) family, 141 of them elder. The parameters were `k ∈ [0.1, 5]` an - *Shared steps.* Theorem FL's domination step uses the same cap-region mechanism and the same `m = 1` near/far split (#170 §8.3). The stiff scale `r^{3/2}` of Proposition FL.4 is #175's hard tube. - *Proposition FL.7* shows that the two limits are the same Gaussian integral in `d = 2`. In `d ≥ 3` they are equal by - uniqueness of limits. + uniqueness of limits, and directly by Codex's reviewed D.1 (Remark after FL.7). - **#242 Conjecture 7** is not proved. It needs inputs (i)–(ii) of #242 §4. Input (i) asks for (3.1) with an error that, integrated against `dr` from the cusp scale to the fold scale, is `O(ℓ^{3/4})` after the composite is subtracted. That needs quantitative margins in Proposition FL.4 near the decision boundary, which the present pointwise argument does not @@ -849,7 +890,7 @@ These are argued in the text and are the subject of review slices B–E. | [CAP] | `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` (`0633aca3`) | §§1, 5: the deterministic cap theorem — consumed (as in [P] §7) | | [E1], [REC] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` (`213594d6`), `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` (`75da2597`) | reading rules for [P] | | [E2] | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` (`fe9b9ce4`) | the elder mark is Borel — consumed | -| #242 | `frontiers/soft_rejected_pairs_20261002/PROOF.md` at `1f86fea` (`ad4beb84`, v1.2), unmerged | §0, (2.1)–(2.6), Lemmas 2–3, Proposition 2′, (3.1) — consumed | +| #242 | `frontiers/soft_rejected_pairs_20261002/PROOF.md` at `d504cdb` (`5fd29e63`, v1.3; §§0–3 as in v1.2's `ad4beb84`), unmerged | §0, (2.1)–(2.6), Lemmas 2–3, Proposition 2′, (3.1) — consumed | | OA lift | `frontiers/elder_dimension_lift_20260928/PROOF.md` (`7303bd79`) | (A3) — cited | | #229 | `frontiers/third_order_rate_20261001/PROOF.md` (`110ed33a`) | (R⁺.1) — cited | | #170 | `frontiers/local_elder_geometry_20260930/PROOF.md` (`ef2aa579`) | §2 (Theorem E(1)), §§7–8 (the classifier, (S1), (S11), the mass `α₁ + α₂`), Theorem S, §9 — compared; consumed by Proposition FL.7 only | @@ -869,3 +910,15 @@ These are argued in the text and are the subject of review slices B–E. the use of [P] (11.2) and [C7-K] (K2)), with §§5–6. A reviewer should record, per slice: ACCEPT, ACCEPT WITH FIXES (list), or REJECT (with the failing step). + +**Changed bytes in v1.3** (for delta checks against the v1.2 slice reviews at `11a4cd4`): +- *Header:* the object label and versions; the FL.7 item of "What is new" (D.1); the #242 dependency (rebound to v1.3). +- *§2 (slice B):* the `m = 1` convention after Proposition FL.4's first sentence (B-01); the trivial case in the proof of + (Trap) (B-02); the proof's first sentence, Step 3's first paragraph, the sentence after the certificates, and Step 4's + first paragraph (B-03). +- *§3 (slice D):* the Weyl equality in Step 5, with the ordered upper limit `min(ε, λ_2)` and the sentence after it (D-01). +- *§4 (slice E):* the Remark on D.1 after Proposition FL.7, and the last clause of the paragraph after "A second proof". +- *§§6, 8–9:* the FL.7 item of §6, the #242 source row, and this list. + +No statement changes in substance; Proposition FL.4's statement gains only the `m = 1` convention. §§0–1, §§2.1–2.2, +§3's Steps 1–4, the proofs of FL.5–FL.7, §§5 and 7, `fold_check.py` and `RESULTS.json` are unchanged. diff --git a/frontiers/soft_fold_limit_20261002/README.md b/frontiers/soft_fold_limit_20261002/README.md index 25caa403d..30340250a 100644 --- a/frontiers/soft_fold_limit_20261002/README.md +++ b/frontiers/soft_fold_limit_20261002/README.md @@ -2,8 +2,18 @@ **Author-side proof candidate (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** -Object `CL-FOLD-LIMIT-20261002-v1.2`. Versions: v1 `008b299`; v1.1 `0740b2c` (prior work, Proposition FL.7); v1.2 (rebinds -#242 v1.2 and cites Codex's `d ≥ 3` cross-check). Full text: [`PROOF.md`](PROOF.md). +Object `CL-FOLD-LIMIT-20261002-v1.3`. Full text: [`PROOF.md`](PROOF.md). Versions: +- v1 `008b299`; +- v1.1 `0740b2c`: prior work, Proposition FL.7; +- v1.2 `11a4cd4`: rebinds #242 v1.2 and cites Codex's `d ≥ 3` cross-check. Nonauthor slices A–E reviewed this head. +- v1.3 applies those reviews' four minor findings: + - OA-243-B-01, the `m = 1` convention in Proposition FL.4; + - OA-243-B-02, the trivial case of the (Trap) fact; + - OA-243-B-03, the rectangle `W` chosen before the `𝒬`-dependent radii; + - OA-243-D-01, the ordered domain `λ_1 < min(ε, λ_2)` in Step 5. + + It also records Codex's reviewed direct identity D.1 after FL.7, and rebinds #242 v1.3 (`d504cdb`, blob `5fd29e63`), + whose consumed §§0–3 are byte-identical to v1.2. No statement changes in substance. PROOF §9 lists the changed bytes. ## Prior work (AUTH-243-01) @@ -25,7 +35,7 @@ of its Lemma 2. Its Conjecture 6 states that the rejected kernel at a fixed gap | | Statement | Status | |---|---|---| | **Theorem FL** | `(k/r)r^{−2}A_r^{rej}(b, k, u) → F(k; b, u)` as `r → 0`, for every `d ≥ 2`, uniformly for `(b, k)` in compacts. The limit in #242 (3.1) exists, and `F` is continuous and positive. This is Conjecture 6. The identification with `F`, the continuity and the local uniformity are new. The decision and convergence steps take a different route from #170/#175; the cap-region domination is shared. | proof | -| **Proposition FL.7** | #242's model is #170's cubic: `G_k(X, ζ) = P_θ(X, kζ)/k`, with `θ = (−λ̃/k, γ, B, C_3)`. The typed domain, weight and maximin decision are the same, so `F = kA_∗a_fail`. In `d = 2` this is an identity of model integrals, which with #170 Theorem S gives a second proof of Conjecture 6. In `d ≥ 3` it follows by uniqueness of limits, conditional on #175. | proof | +| **Proposition FL.7** | #242's model is #170's cubic: `G_k(X, ζ) = P_θ(X, kζ)/k`, with `θ = (−λ̃/k, γ, B, C_3)`. The typed domain, weight and maximin decision are the same, so `F = kA_∗a_fail`. In `d = 2` this is an identity of model integrals, which with #170 Theorem S gives a second proof of Conjecture 6. In `d ≥ 3` it follows by uniqueness of limits, conditional on #175; Codex's reviewed supplement D.1 gives it directly as a model identity (Remark after FL.7). | proof | | **Corollary FL.5** | `r^{−3}(1 − p_r) → F/(k·A_∗)`: #170 Theorem S and #175 Theorem F, in #242's form | recovers merged results | | **Corollary FL.6** | compact mark windows: `ν_cand − ν_eld = d_{𝐁,𝐊}ℓ^{2/3} + o(ℓ^{2/3})`, with `d_{𝐁,𝐊} = C_fail^{𝐁,𝐊}` (#170 §9, #175 §7) | recovers merged results | | **Proposition FL.4** | the decision transfer of #170 Theorem E(2)–(3) and #175 Theorem H, re-proved in #242's normalization by paths and traps (an `ε`-form for a continuous global field), with the generic set adapted to #242's slice decision; stiff directions included | proof (method new) | @@ -119,5 +129,16 @@ PR. - It also found four MINOR findings and six NITs. - All are applied. -Same GitHub account and same provider as the author: zero organizational-independence credit. Nonauthor review is -required for every slice (`PROOF.md` §9). +**Nonauthor reviews (OpenAI Codex; same GitHub account, organizational independence 0; Dylan's personal reading pending).** +- Slice A at v1.1 `0740b2c` (review 5391585764): ACCEPT / PASS_TECHNICAL_SCOPED, no finding. +- At v1.2 `11a4cd4`, PROOF blob `3d9faa87`: + - Slice B (5392212174): ACCEPT WITH THREE MINOR FIXES, OA-243-B-01, B-02 and B-03; + - Slice C (5392145839): ACCEPT / PASS_TECHNICAL_SCOPED; + - Slice D (5392277487): ACCEPT WITH ONE MINOR CLARIFICATION, OA-243-D-01; + - Slice E (5392675391): ACCEPT / PASS_TECHNICAL_SCOPED, no new finding. +- D.1, Codex's root-authored direct identity `F = kA_∗a_fail` (§3 of review 5391986303 on #242), was reviewed by a different + Codex agent (5954510413): PASS_TECHNICAL_SCOPED, no finding. +- v1.3 applies the four findings and asks the B, D and E reviewers for bounded delta checks of the changed bytes. + +The same-family referees are the same GitHub account and the same provider as the author, so they carry zero +organizational-independence credit. The nonauthor reviews are listed above; the slices are in `PROOF.md` §9. diff --git a/frontiers/soft_fold_limit_20261002/SOURCES.json b/frontiers/soft_fold_limit_20261002/SOURCES.json index 5f542b190..c51e31b69 100644 --- a/frontiers/soft_fold_limit_20261002/SOURCES.json +++ b/frontiers/soft_fold_limit_20261002/SOURCES.json @@ -1,16 +1,17 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-FOLD-LIMIT-20261002-v1.2", + "object": "CL-FOLD-LIMIT-20261002-v1.3", "supersedes": { - "object": "CL-FOLD-LIMIT-20261002-v1.1", - "head": "0740b2cae546866c5f5f27e770ba023bde64bbe8" + "object": "CL-FOLD-LIMIT-20261002-v1.2", + "head": "11a4cd4e3b3f927eb9d4a743d8a43e387f1a66da", + "proof_blob": "3d9faa87d717fbd68b784b2ae408505fb2cc56dc" }, - "kind": "author-side proof candidate, every d >= 2: Theorem FL (Conjecture 6 of Math- #242: the fold-scale limit (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u), uniformly on compacts of (b, k); the limit defining F exists, F is continuous and positive; the existence of the limit, as r^-3 (1 - p_r) -> alpha_1 + alpha_2, was proved earlier by the merged #170 (d = 2) and #175 (d >= 3); the decision and convergence steps here take a different route, the cap-region domination is shared); Proposition FL.7 (G_k(X, zeta) = P_theta(X, k zeta)/k with theta = (-lt/k, gamma, B, C3): same typed domain, weight and maximin decision, so F = k A_* a_fail, directly in d = 2 and by uniqueness of limits in d >= 3); Corollaries FL.5-FL.6 (recover #170 Theorem S and section 9, #175 Theorem F and section 7, in #242 form; not new); Proposition FL.4 (re-proves the window-to-global decision transfer of #170 Theorem E(2)-(3) and #175 Theorem H in #242 normalization, by paths and traps); Lemma FL.2 (the generic set has full measure; d(M) >= L_S by the critical chord of #170 section 3, observed on #242 by OpenAI Codex; case (D-prime) needs chi > 0); Lemma FL.3 (crossing a nondegenerate saddle); Lemma FL.1 (the soft window, rate r in d = 2 and r^(1/2) in d >= 3)", + "kind": "author-side proof candidate, every d >= 2 (v1.3: applies the minor findings OA-243-B-01 (the m = 1 convention in Proposition FL.4), B-02 (the trivial case of the Trap fact), B-03 (the rectangle W fixed before the Q-dependent radii) and D-01 (the ordered domain lambda_1 < min(eps, lambda_2) in Step 5) of the nonauthor slice reviews of v1.2; records Codex's reviewed direct identity D.1 (F = k A_* a_fail in every fixed d); rebinds #242 v1.3, whose consumed sections 0-3 are byte-identical to v1.2; no statement changes in substance): Theorem FL (Conjecture 6 of Math- #242: the fold-scale limit (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u), uniformly on compacts of (b, k); the limit defining F exists, F is continuous and positive; the existence of the limit, as r^-3 (1 - p_r) -> alpha_1 + alpha_2, was proved earlier by the merged #170 (d = 2) and #175 (d >= 3); the decision and convergence steps here take a different route, the cap-region domination is shared); Proposition FL.7 (G_k(X, zeta) = P_theta(X, k zeta)/k with theta = (-lt/k, gamma, B, C3): same typed domain, weight and maximin decision, so F = k A_* a_fail, directly in d = 2 and by uniqueness of limits in d >= 3); Corollaries FL.5-FL.6 (recover #170 Theorem S and section 9, #175 Theorem F and section 7, in #242 form; not new); Proposition FL.4 (re-proves the window-to-global decision transfer of #170 Theorem E(2)-(3) and #175 Theorem H in #242 normalization, by paths and traps); Lemma FL.2 (the generic set has full measure; d(M) >= L_S by the critical chord of #170 section 3, observed on #242 by OpenAI Codex; case (D-prime) needs chi > 0); Lemma FL.3 (crossing a nondegenerate saddle); Lemma FL.1 (the soft window, rate r in d = 2 and r^(1/2) in d >= 3)", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { - "branch_base": "f5347541a6ef7b84de96120dcce6137b816b8033" + "branch_base": "a808ba25de9db22110db8b7555a64852f38e031d" }, "consumed": [ { @@ -97,12 +98,12 @@ "consumed_unmerged": [ { "path": "frontiers/soft_rejected_pairs_20261002/PROOF.md", - "bytes": 79394, - "sha256": "ab952b2a2c7648beb5f7473b3f028cdc495c8317c52522bee144d26eecc8e2bb", - "git_blob": "ad4beb8467a775c7c76f85d5584c13338d0e66be", + "bytes": 87672, + "sha256": "2d6981593fc32533fd3f3a25c8c424bff829c7b2baa47da148b9502520e62467", + "git_blob": "5fd29e63e81980a968009c7e8c3ba8e0af9e46a8", "tag": "242", - "use": "Math- #242 (open; v1.2): section 0, (2.1)-(2.6) (the soft model and its normalization), Lemma 2 (the model decision; v1.2 rewrites its proof at the exact level and states that beta > 2, chi <= 0 is rejected, as Lemma FL.2(c) here), Lemma 3 (the a priori elder region, used for domination in the identification of the limit), Proposition 2-prime (the stiff directions), (3.1) (the definition of F); Theorem 1 and Proposition 4 are cited", - "head": "1f86fea035f838cd3957b20bea58fc2082a4e3a4", + "use": "Math- #242 (open; v1.3, whose sections 0-3 are byte-identical to v1.2 blob ad4beb84, which the v1.2 slice reviews of this note read): section 0, (2.1)-(2.6) (the soft model and its normalization), Lemma 2 (the model decision; v1.2 rewrites its proof at the exact level and states that beta > 2, chi <= 0 is rejected, as Lemma FL.2(c) here), Lemma 3 (the a priori elder region, used for domination in the identification of the limit), Proposition 2-prime (the stiff directions), (3.1) (the definition of F); Theorem 1 and Proposition 4 are cited", + "head": "d504cdb15851b4a9140755fee6a295282c503ed4", "pr": "Math- #242" } ], @@ -144,13 +145,13 @@ "files": [ { "path": "PROOF.md", - "bytes": 72742, - "sha256": "7716a113bcf8c1725f9ec771d58ef033767f0a7bb50d592b841ca0ffe86f31af" + "bytes": 77255, + "sha256": "2c7202ea7dab1313ee57f682d241904aea62d340238ec824ad05f551320f2907" }, { "path": "README.md", - "bytes": 10687, - "sha256": "0be812e8a4785462587ac0e572dd42a04d3d1789a03d948ebb220649d5f5da6a" + "bytes": 12315, + "sha256": "13aa1c8b152a6f6b2cb81766c3e9b93fa621d63b6d073c3399429b8ebe30dbc5" }, { "path": "fold_check.py", @@ -162,5 +163,64 @@ "bytes": 2903, "sha256": "459bbdf3c84b77548e53824df889b13a8ebd0fe6b0d7d7cadf8a4d73821641e9" } + ], + "nonauthor_reviews": [ + { + "slice": "A", + "review": "pullrequestreview-5391585764 (supersedes 5391580522, transport-corrupted)", + "at": "0740b2c", + "verdict": "ACCEPT / PASS_TECHNICAL_SCOPED", + "findings": [], + "reviewer": "OpenAI Codex (delegated AI review; same GitHub account; organizational independence 0; Dylan personal reading pending)" + }, + { + "slice": "B", + "review": "pullrequestreview-5392212174", + "at": "11a4cd4", + "verdict": "ACCEPT WITH THREE MINOR FIXES / PASS_TECHNICAL_SCOPED", + "findings": [ + "OA-243-B-01", + "OA-243-B-02", + "OA-243-B-03" + ], + "applied_in": "v1.3", + "reviewer": "OpenAI Codex (delegated AI review; same GitHub account; organizational independence 0; Dylan personal reading pending)" + }, + { + "slice": "C", + "review": "pullrequestreview-5392145839", + "at": "11a4cd4", + "verdict": "ACCEPT / PASS_TECHNICAL_SCOPED", + "findings": [], + "reviewer": "OpenAI Codex (delegated AI review; same GitHub account; organizational independence 0; Dylan personal reading pending)" + }, + { + "slice": "D", + "review": "pullrequestreview-5392277487", + "at": "11a4cd4", + "verdict": "ACCEPT WITH ONE MINOR CLARIFICATION / PASS_TECHNICAL_SCOPED", + "findings": [ + "OA-243-D-01" + ], + "applied_in": "v1.3", + "reviewer": "OpenAI Codex (delegated AI review; same GitHub account; organizational independence 0; Dylan personal reading pending)" + }, + { + "slice": "E", + "review": "pullrequestreview-5392675391", + "at": "11a4cd4", + "verdict": "ACCEPT / PASS_TECHNICAL_SCOPED", + "findings": [], + "reviewer": "OpenAI Codex (delegated AI review; same GitHub account; organizational independence 0; Dylan personal reading pending)" + }, + { + "slice": "D.1 supplement (Codex root-authored; section 3 of #242 review 5391986303)", + "comment": "5954510413", + "at": "#242 PROOF ad4beb84", + "verdict": "PASS_TECHNICAL_SCOPED", + "findings": [], + "use_here": "recorded as reviewed support in the Remark after Proposition FL.7 (v1.3)", + "reviewer": "OpenAI Codex (delegated AI review; same GitHub account; organizational independence 0; Dylan personal reading pending)" + } ] } From 15870b638748eb3f4e152fbb77ff41dac264e923 Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 16:48:39 +0000 Subject: [PATCH 5/6] Fold-scale limit v1.4: rebind #242 v1.4; harden the binding of the consumed #242 - #242 v1.4 (ffc7005, PROOF blob c1592915) changes only its header, section 5 and section 9, after Codex's C78 review 5394316763. The consumed sections 0-3 are byte-identical to v1.3 and to v1.2's ad4beb84 (43,505 bytes, SHA-256 7548a21d...), which the slice reviews read. - Workflow: the #242 PR is resolved first. Once it has merged, its pinned regular-file blob must be present in this tree, with no fallback to the historical head (automated finding 4167121714 on #244; same pattern here). A forced-merged negative test fails as intended. No statement or proof changes; fold_check.py and RESULTS.json are unchanged. Scientific effect NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .github/workflows/soft-fold-limit.yml | 18 ++++++++---- frontiers/soft_fold_limit_20261002/PROOF.md | 23 +++++++++++---- frontiers/soft_fold_limit_20261002/README.md | 5 +++- .../soft_fold_limit_20261002/SOURCES.json | 28 +++++++++---------- 4 files changed, 48 insertions(+), 26 deletions(-) diff --git a/.github/workflows/soft-fold-limit.yml b/.github/workflows/soft-fold-limit.yml index 1762c0fad..e9cc7f822 100644 --- a/.github/workflows/soft-fold-limit.yml +++ b/.github/workflows/soft-fold-limit.yml @@ -115,7 +115,8 @@ jobs: # 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'): + merged = bool(pr.get('merged')) + if merged: ref = pr.get('merge_commit_sha') elif pr.get('state') == 'open': ref = pr['head']['sha'] @@ -125,12 +126,18 @@ jobs: 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 + return ref, merged 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 + # a consumed dependency's PR is resolved first (Codex review 4167121714 on #244). Once it has merged, + # its source must be the recorded regular-file blob in this tree: a deleted, renamed or changed landed + # source fails, and there is no fallback to the historical head. While it is open, the source is bound + # through the API. Cited sources are bound through the API. + now, merged = current(entry) if kind == 'consumed_unmerged' else (None, False) landed = tree_bytes(entry) if kind == 'consumed_unmerged' else None + if merged and landed is None: + raise ValueError('dependency ' + entry['pr'] + ' has merged but its pinned source is absent from this ' + 'tree: ' + entry['path']) if landed is not None: data, where = landed, 'tree' else: @@ -139,9 +146,8 @@ jobs: 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})) + 'via': where, 'current_ref_checked': now, 'dependency_merged': merged})) script = str(root / 'fold_check.py') expected = (root / 'RESULTS.json').read_bytes() for flags in (['-B', '-S'], ['-B', '-O', '-S']): diff --git a/frontiers/soft_fold_limit_20261002/PROOF.md b/frontiers/soft_fold_limit_20261002/PROOF.md index 2652db872..64091adf9 100644 --- a/frontiers/soft_fold_limit_20261002/PROOF.md +++ b/frontiers/soft_fold_limit_20261002/PROOF.md @@ -1,6 +1,6 @@ # The fold-scale rejection limit through #242's soft model: Conjecture 6 of Math- #242 in every dimension, and its identification with the failure law of #170/#175 -Object: CL-FOLD-LIMIT-20261002-v1.3. Versions: +Object: CL-FOLD-LIMIT-20261002-v1.4. Versions: - v1, `008b299`. - v1.1, `0740b2c`: the prior-work record AUTH-243-01, and Proposition FL.7. - v1.2, `11a4cd4`: rebinds #242 v1.2, and cites Codex's direct `d ≥ 3` cross-check of FL.7. Nonauthor slices A–E @@ -8,6 +8,11 @@ Object: CL-FOLD-LIMIT-20261002-v1.3. Versions: - v1.3: applies the four minor findings of those reviews, OA-243-B-01, B-02, B-03 (Proposition FL.4) and D-01 (Step 5 of Theorem FL). It records Codex's reviewed direct identity D.1 after Proposition FL.7, and rebinds #242 v1.3, whose §§0–3 are byte-identical to v1.2. No statement changes; §9 lists the changed bytes. +- v1.4: rebinds #242 v1.4 (`ffc7005`, blob `c1592915`). #242 v1.4 changes only its header and §5, after the C78 delta + review (5394316763), so the consumed §§0–3 are still byte-identical to the bytes the slice reviews read. The workflow now + resolves the #242 PR before reading the source. Once #242 has merged, its pinned blob must be in this tree, with no + fallback to the historical head. This applies the automated finding 4167121714 (raised on #244, same pattern). No + statement or proof changes; §9 lists the changed bytes. Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 2 October 2026. Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem FL, Propositions FL.4 and FL.7, Lemmas FL.1–FL.3; Corollaries FL.5 and FL.6 recover merged results of #170 and #175). @@ -100,10 +105,10 @@ domination are as in #170 §8 and #175 §§2 and 4. - #207 (`frontiers/cusp_second_order_20261001/PROOF.md`, blob `f6df5a73`): §0 (the elder mark and the maximin), Theorem CU.1 (the pin identities), and the structure of Proposition CU.3 and Theorem CU.4. - [C7-K] (`frontiers/c7_total_bounded_20260929/PROOF.md`, blob `28748b08`): (K2). -- Consumed (unmerged): #242 (`frontiers/soft_rejected_pairs_20261002/PROOF.md` at `d504cdb`, blob `5fd29e63`; v1.3): §0, +- Consumed (unmerged): #242 (`frontiers/soft_rejected_pairs_20261002/PROOF.md` at `ffc7005`, blob `c1592915`; v1.4): §0, (2.1)–(2.5), Lemmas 2 and 3, Proposition 2′ ((2.6)), (3.1). #242's §§0–3 are byte-identical to its v1.2 (blob - `ad4beb84`), which the slice reviews of v1.2 of this note read; v1.3 changed only #242's header, §4's numerical - sentences and §§5–9. This note must be rebound if that blob changes. + `ad4beb84`), which the slice reviews of v1.2 of this note read. v1.3 changed only #242's header, §4's numerical + sentences and §§5–9, and v1.4 only its header, §5 and §9. This note must be rebound if that blob changes. - Consumed through #207 §0: [E2] (`reviews/d1_section9_borel_repair_20260925/REPAIR.md`, blob `fe9b9ce4`), for the Borel measurability of the elder mark. Reading rules for [P]: [E1] (`imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`, blob `213594d6`) and [REC] (`reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md`, blob `75da2597`). @@ -890,7 +895,7 @@ These are argued in the text and are the subject of review slices B–E. | [CAP] | `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` (`0633aca3`) | §§1, 5: the deterministic cap theorem — consumed (as in [P] §7) | | [E1], [REC] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` (`213594d6`), `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` (`75da2597`) | reading rules for [P] | | [E2] | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` (`fe9b9ce4`) | the elder mark is Borel — consumed | -| #242 | `frontiers/soft_rejected_pairs_20261002/PROOF.md` at `d504cdb` (`5fd29e63`, v1.3; §§0–3 as in v1.2's `ad4beb84`), unmerged | §0, (2.1)–(2.6), Lemmas 2–3, Proposition 2′, (3.1) — consumed | +| #242 | `frontiers/soft_rejected_pairs_20261002/PROOF.md` at `ffc7005` (`c1592915`, v1.4; §§0–3 as in v1.2's `ad4beb84`), unmerged | §0, (2.1)–(2.6), Lemmas 2–3, Proposition 2′, (3.1) — consumed | | OA lift | `frontiers/elder_dimension_lift_20260928/PROOF.md` (`7303bd79`) | (A3) — cited | | #229 | `frontiers/third_order_rate_20261001/PROOF.md` (`110ed33a`) | (R⁺.1) — cited | | #170 | `frontiers/local_elder_geometry_20260930/PROOF.md` (`ef2aa579`) | §2 (Theorem E(1)), §§7–8 (the classifier, (S1), (S11), the mass `α₁ + α₂`), Theorem S, §9 — compared; consumed by Proposition FL.7 only | @@ -911,6 +916,14 @@ These are argued in the text and are the subject of review slices B–E. A reviewer should record, per slice: ACCEPT, ACCEPT WITH FIXES (list), or REJECT (with the failing step). +**Changed bytes in v1.4** (for a delta check against v1.3 at `4835564`): +- *Header:* the object label, the v1.4 entry, and the #242 dependency (rebound to v1.4). +- *§8:* the #242 source row. +- *§9:* this list. +- *Workflow:* the binding of the consumed #242 (finding 4167121714). + +Everything else in `PROOF.md`, and `fold_check.py` and `RESULTS.json`, is unchanged. + **Changed bytes in v1.3** (for delta checks against the v1.2 slice reviews at `11a4cd4`): - *Header:* the object label and versions; the FL.7 item of "What is new" (D.1); the #242 dependency (rebound to v1.3). - *§2 (slice B):* the `m = 1` convention after Proposition FL.4's first sentence (B-01); the trivial case in the proof of diff --git a/frontiers/soft_fold_limit_20261002/README.md b/frontiers/soft_fold_limit_20261002/README.md index 30340250a..8e6cab02a 100644 --- a/frontiers/soft_fold_limit_20261002/README.md +++ b/frontiers/soft_fold_limit_20261002/README.md @@ -2,7 +2,7 @@ **Author-side proof candidate (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** -Object `CL-FOLD-LIMIT-20261002-v1.3`. Full text: [`PROOF.md`](PROOF.md). Versions: +Object `CL-FOLD-LIMIT-20261002-v1.4`. Full text: [`PROOF.md`](PROOF.md). Versions: - v1 `008b299`; - v1.1 `0740b2c`: prior work, Proposition FL.7; - v1.2 `11a4cd4`: rebinds #242 v1.2 and cites Codex's `d ≥ 3` cross-check. Nonauthor slices A–E reviewed this head. @@ -14,6 +14,9 @@ Object `CL-FOLD-LIMIT-20261002-v1.3`. Full text: [`PROOF.md`](PROOF.md). Version It also records Codex's reviewed direct identity D.1 after FL.7, and rebinds #242 v1.3 (`d504cdb`, blob `5fd29e63`), whose consumed §§0–3 are byte-identical to v1.2. No statement changes in substance. PROOF §9 lists the changed bytes. +- v1.4 rebinds #242 v1.4 (`ffc7005`, blob `c1592915`). #242 v1.4 changes only its header and §5, after the C78 review + 5394316763, so the consumed §§0–3 are unchanged. The workflow now requires #242's pinned blob in the tree once #242 has + merged (automated finding 4167121714 on #244; same pattern here). No statement or proof changes. ## Prior work (AUTH-243-01) diff --git a/frontiers/soft_fold_limit_20261002/SOURCES.json b/frontiers/soft_fold_limit_20261002/SOURCES.json index c51e31b69..7ada57697 100644 --- a/frontiers/soft_fold_limit_20261002/SOURCES.json +++ b/frontiers/soft_fold_limit_20261002/SOURCES.json @@ -1,13 +1,13 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-FOLD-LIMIT-20261002-v1.3", + "object": "CL-FOLD-LIMIT-20261002-v1.4", "supersedes": { - "object": "CL-FOLD-LIMIT-20261002-v1.2", - "head": "11a4cd4e3b3f927eb9d4a743d8a43e387f1a66da", - "proof_blob": "3d9faa87d717fbd68b784b2ae408505fb2cc56dc" + "object": "CL-FOLD-LIMIT-20261002-v1.3", + "head": "483556485cf15cc8662425e2195699029c713ff5", + "proof_blob": "2652db8721d654b47763b18eea6b97ba9a0dc281" }, - "kind": "author-side proof candidate, every d >= 2 (v1.3: applies the minor findings OA-243-B-01 (the m = 1 convention in Proposition FL.4), B-02 (the trivial case of the Trap fact), B-03 (the rectangle W fixed before the Q-dependent radii) and D-01 (the ordered domain lambda_1 < min(eps, lambda_2) in Step 5) of the nonauthor slice reviews of v1.2; records Codex's reviewed direct identity D.1 (F = k A_* a_fail in every fixed d); rebinds #242 v1.3, whose consumed sections 0-3 are byte-identical to v1.2; no statement changes in substance): Theorem FL (Conjecture 6 of Math- #242: the fold-scale limit (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u), uniformly on compacts of (b, k); the limit defining F exists, F is continuous and positive; the existence of the limit, as r^-3 (1 - p_r) -> alpha_1 + alpha_2, was proved earlier by the merged #170 (d = 2) and #175 (d >= 3); the decision and convergence steps here take a different route, the cap-region domination is shared); Proposition FL.7 (G_k(X, zeta) = P_theta(X, k zeta)/k with theta = (-lt/k, gamma, B, C3): same typed domain, weight and maximin decision, so F = k A_* a_fail, directly in d = 2 and by uniqueness of limits in d >= 3); Corollaries FL.5-FL.6 (recover #170 Theorem S and section 9, #175 Theorem F and section 7, in #242 form; not new); Proposition FL.4 (re-proves the window-to-global decision transfer of #170 Theorem E(2)-(3) and #175 Theorem H in #242 normalization, by paths and traps); Lemma FL.2 (the generic set has full measure; d(M) >= L_S by the critical chord of #170 section 3, observed on #242 by OpenAI Codex; case (D-prime) needs chi > 0); Lemma FL.3 (crossing a nondegenerate saddle); Lemma FL.1 (the soft window, rate r in d = 2 and r^(1/2) in d >= 3)", + "kind": "author-side proof candidate, every d >= 2 (v1.4: rebinds #242 v1.4 (head ffc7005, blob c1592915), whose consumed sections 0-3 are byte-identical to v1.2 and v1.3; the workflow requires the pinned #242 blob in the tree once #242 has merged (automated finding 4167121714 on #244, same pattern); no statement or proof changes; v1.3: applies the minor findings OA-243-B-01 (the m = 1 convention in Proposition FL.4), B-02 (the trivial case of the Trap fact), B-03 (the rectangle W fixed before the Q-dependent radii) and D-01 (the ordered domain lambda_1 < min(eps, lambda_2) in Step 5) of the nonauthor slice reviews of v1.2; records Codex's reviewed direct identity D.1 (F = k A_* a_fail in every fixed d); rebinds #242 v1.3, whose consumed sections 0-3 are byte-identical to v1.2; no statement changes in substance): Theorem FL (Conjecture 6 of Math- #242: the fold-scale limit (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u), uniformly on compacts of (b, k); the limit defining F exists, F is continuous and positive; the existence of the limit, as r^-3 (1 - p_r) -> alpha_1 + alpha_2, was proved earlier by the merged #170 (d = 2) and #175 (d >= 3); the decision and convergence steps here take a different route, the cap-region domination is shared); Proposition FL.7 (G_k(X, zeta) = P_theta(X, k zeta)/k with theta = (-lt/k, gamma, B, C3): same typed domain, weight and maximin decision, so F = k A_* a_fail, directly in d = 2 and by uniqueness of limits in d >= 3); Corollaries FL.5-FL.6 (recover #170 Theorem S and section 9, #175 Theorem F and section 7, in #242 form; not new); Proposition FL.4 (re-proves the window-to-global decision transfer of #170 Theorem E(2)-(3) and #175 Theorem H in #242 normalization, by paths and traps); Lemma FL.2 (the generic set has full measure; d(M) >= L_S by the critical chord of #170 section 3, observed on #242 by OpenAI Codex; case (D-prime) needs chi > 0); Lemma FL.3 (crossing a nondegenerate saddle); Lemma FL.1 (the soft window, rate r in d = 2 and r^(1/2) in d >= 3)", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { @@ -98,12 +98,12 @@ "consumed_unmerged": [ { "path": "frontiers/soft_rejected_pairs_20261002/PROOF.md", - "bytes": 87672, - "sha256": "2d6981593fc32533fd3f3a25c8c424bff829c7b2baa47da148b9502520e62467", - "git_blob": "5fd29e63e81980a968009c7e8c3ba8e0af9e46a8", + "bytes": 90777, + "sha256": "f45ce06a0a78169fb5003cb91705f344c5e5867f6bd8271210253b423b53f9a6", + "git_blob": "c15929151e9f82813b7993a31b6735bcff8dc6fa", "tag": "242", - "use": "Math- #242 (open; v1.3, whose sections 0-3 are byte-identical to v1.2 blob ad4beb84, which the v1.2 slice reviews of this note read): section 0, (2.1)-(2.6) (the soft model and its normalization), Lemma 2 (the model decision; v1.2 rewrites its proof at the exact level and states that beta > 2, chi <= 0 is rejected, as Lemma FL.2(c) here), Lemma 3 (the a priori elder region, used for domination in the identification of the limit), Proposition 2-prime (the stiff directions), (3.1) (the definition of F); Theorem 1 and Proposition 4 are cited", - "head": "d504cdb15851b4a9140755fee6a295282c503ed4", + "use": "Math- #242 (open; v1.4, whose sections 0-3 are byte-identical to v1.3 and to v1.2 blob ad4beb84, which the v1.2 slice reviews of this note read): section 0, (2.1)-(2.6) (the soft model and its normalization), Lemma 2 (the model decision; v1.2 rewrites its proof at the exact level and states that beta > 2, chi <= 0 is rejected, as Lemma FL.2(c) here), Lemma 3 (the a priori elder region, used for domination in the identification of the limit), Proposition 2-prime (the stiff directions), (3.1) (the definition of F); Theorem 1 and Proposition 4 are cited", + "head": "ffc7005509554a378edd81136aa6e1f4fbd0eff8", "pr": "Math- #242" } ], @@ -145,13 +145,13 @@ "files": [ { "path": "PROOF.md", - "bytes": 77255, - "sha256": "2c7202ea7dab1313ee57f682d241904aea62d340238ec824ad05f551320f2907" + "bytes": 78202, + "sha256": "7d0d69a100c34188a950746fb6ca5afe3a865c0cc72f85269dcfcd6bdf05df8d" }, { "path": "README.md", - "bytes": 12315, - "sha256": "13aa1c8b152a6f6b2cb81766c3e9b93fa621d63b6d073c3399429b8ebe30dbc5" + "bytes": 12659, + "sha256": "b93c60afa625087bf355955d1aacb0e7b71e38e851556e83a9c701c754105db8" }, { "path": "fold_check.py", From 3a188bef39f5eca6fbb7c0cc43be86a52140895c Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 17:33:48 +0000 Subject: [PATCH 6/6] Fold-scale limit v1.5: rebind #242 v1.5 #242 v1.5 (ee56652, PROOF blob 271412db) changes only reporting in its header, section 5 and section 9 (C80 readback 5394603905), so the consumed sections 0-3 remain byte-identical. Only the status lines, section 8's #242 row, section 9's list, README and SOURCES.json change; SOURCES records the delivered v1.3/v1.4 reviews (5394499570, 5394480716, 5394576747). The workflow, fold_check.py and RESULTS.json are unchanged. Scientific effect NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- frontiers/soft_fold_limit_20261002/PROOF.md | 18 +++++-- frontiers/soft_fold_limit_20261002/README.md | 4 +- .../soft_fold_limit_20261002/SOURCES.json | 52 ++++++++++++++----- 3 files changed, 55 insertions(+), 19 deletions(-) diff --git a/frontiers/soft_fold_limit_20261002/PROOF.md b/frontiers/soft_fold_limit_20261002/PROOF.md index 64091adf9..6502cf7ba 100644 --- a/frontiers/soft_fold_limit_20261002/PROOF.md +++ b/frontiers/soft_fold_limit_20261002/PROOF.md @@ -1,6 +1,6 @@ # The fold-scale rejection limit through #242's soft model: Conjecture 6 of Math- #242 in every dimension, and its identification with the failure law of #170/#175 -Object: CL-FOLD-LIMIT-20261002-v1.4. Versions: +Object: CL-FOLD-LIMIT-20261002-v1.5. Versions: - v1, `008b299`. - v1.1, `0740b2c`: the prior-work record AUTH-243-01, and Proposition FL.7. - v1.2, `11a4cd4`: rebinds #242 v1.2, and cites Codex's direct `d ≥ 3` cross-check of FL.7. Nonauthor slices A–E @@ -13,6 +13,9 @@ Object: CL-FOLD-LIMIT-20261002-v1.4. Versions: resolves the #242 PR before reading the source. Once #242 has merged, its pinned blob must be in this tree, with no fallback to the historical head. This applies the automated finding 4167121714 (raised on #244, same pattern). No statement or proof changes; §9 lists the changed bytes. +- v1.5: rebinds #242 v1.5 (`ee56652`, blob `271412db`). #242 v1.5 changes only reporting in its header, §5 and §9, after + C80 (5394603905), so the consumed §§0–3 remain byte-identical. Nothing else changes, and §9 lists the changed bytes. + The v1.3 and v1.4 bytes were read by B/D/E (5394499570, 5394480716) and the workflow review (5394576747). Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 2 October 2026. Disposition: AUTHOR-SIDE PROOF CANDIDATE (Theorem FL, Propositions FL.4 and FL.7, Lemmas FL.1–FL.3; Corollaries FL.5 and FL.6 recover merged results of #170 and #175). @@ -105,10 +108,10 @@ domination are as in #170 §8 and #175 §§2 and 4. - #207 (`frontiers/cusp_second_order_20261001/PROOF.md`, blob `f6df5a73`): §0 (the elder mark and the maximin), Theorem CU.1 (the pin identities), and the structure of Proposition CU.3 and Theorem CU.4. - [C7-K] (`frontiers/c7_total_bounded_20260929/PROOF.md`, blob `28748b08`): (K2). -- Consumed (unmerged): #242 (`frontiers/soft_rejected_pairs_20261002/PROOF.md` at `ffc7005`, blob `c1592915`; v1.4): §0, +- Consumed (unmerged): #242 (`frontiers/soft_rejected_pairs_20261002/PROOF.md` at `ee56652`, blob `271412db`; v1.5): §0, (2.1)–(2.5), Lemmas 2 and 3, Proposition 2′ ((2.6)), (3.1). #242's §§0–3 are byte-identical to its v1.2 (blob `ad4beb84`), which the slice reviews of v1.2 of this note read. v1.3 changed only #242's header, §4's numerical - sentences and §§5–9, and v1.4 only its header, §5 and §9. This note must be rebound if that blob changes. + sentences and §§5–9, and v1.4 and v1.5 only its header, §5 and §9. This note must be rebound if that blob changes. - Consumed through #207 §0: [E2] (`reviews/d1_section9_borel_repair_20260925/REPAIR.md`, blob `fe9b9ce4`), for the Borel measurability of the elder mark. Reading rules for [P]: [E1] (`imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`, blob `213594d6`) and [REC] (`reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md`, blob `75da2597`). @@ -895,7 +898,7 @@ These are argued in the text and are the subject of review slices B–E. | [CAP] | `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` (`0633aca3`) | §§1, 5: the deterministic cap theorem — consumed (as in [P] §7) | | [E1], [REC] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` (`213594d6`), `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` (`75da2597`) | reading rules for [P] | | [E2] | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` (`fe9b9ce4`) | the elder mark is Borel — consumed | -| #242 | `frontiers/soft_rejected_pairs_20261002/PROOF.md` at `ffc7005` (`c1592915`, v1.4; §§0–3 as in v1.2's `ad4beb84`), unmerged | §0, (2.1)–(2.6), Lemmas 2–3, Proposition 2′, (3.1) — consumed | +| #242 | `frontiers/soft_rejected_pairs_20261002/PROOF.md` at `ee56652` (`271412db`, v1.5; §§0–3 as in v1.2's `ad4beb84`), unmerged | §0, (2.1)–(2.6), Lemmas 2–3, Proposition 2′, (3.1) — consumed | | OA lift | `frontiers/elder_dimension_lift_20260928/PROOF.md` (`7303bd79`) | (A3) — cited | | #229 | `frontiers/third_order_rate_20261001/PROOF.md` (`110ed33a`) | (R⁺.1) — cited | | #170 | `frontiers/local_elder_geometry_20260930/PROOF.md` (`ef2aa579`) | §2 (Theorem E(1)), §§7–8 (the classifier, (S1), (S11), the mass `α₁ + α₂`), Theorem S, §9 — compared; consumed by Proposition FL.7 only | @@ -916,6 +919,13 @@ These are argued in the text and are the subject of review slices B–E. A reviewer should record, per slice: ACCEPT, ACCEPT WITH FIXES (list), or REJECT (with the failing step). +**Changed bytes in v1.5** (for a delta check against v1.4 at `15870b6`): +- *Header:* the object label, the v1.5 entry, and the #242 dependency (rebound to v1.5). +- *§8:* the #242 source row. +- *§9:* this list. + +Everything else in `PROOF.md`, the workflow, `fold_check.py` and `RESULTS.json` is unchanged. + **Changed bytes in v1.4** (for a delta check against v1.3 at `4835564`): - *Header:* the object label, the v1.4 entry, and the #242 dependency (rebound to v1.4). - *§8:* the #242 source row. diff --git a/frontiers/soft_fold_limit_20261002/README.md b/frontiers/soft_fold_limit_20261002/README.md index 8e6cab02a..e1df8f396 100644 --- a/frontiers/soft_fold_limit_20261002/README.md +++ b/frontiers/soft_fold_limit_20261002/README.md @@ -2,7 +2,7 @@ **Author-side proof candidate (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** -Object `CL-FOLD-LIMIT-20261002-v1.4`. Full text: [`PROOF.md`](PROOF.md). Versions: +Object `CL-FOLD-LIMIT-20261002-v1.5`. Full text: [`PROOF.md`](PROOF.md). Versions: - v1 `008b299`; - v1.1 `0740b2c`: prior work, Proposition FL.7; - v1.2 `11a4cd4`: rebinds #242 v1.2 and cites Codex's `d ≥ 3` cross-check. Nonauthor slices A–E reviewed this head. @@ -17,6 +17,8 @@ Object `CL-FOLD-LIMIT-20261002-v1.4`. Full text: [`PROOF.md`](PROOF.md). Version - v1.4 rebinds #242 v1.4 (`ffc7005`, blob `c1592915`). #242 v1.4 changes only its header and §5, after the C78 review 5394316763, so the consumed §§0–3 are unchanged. The workflow now requires #242's pinned blob in the tree once #242 has merged (automated finding 4167121714 on #244; same pattern here). No statement or proof changes. +- v1.5 rebinds #242 v1.5 (`ee56652`, blob `271412db`; reporting-only changes after C80), with the consumed §§0–3 unchanged. + The v1.3/v1.4 deltas are reviewed: B/D/E in 5394499570 and 5394480716, and the workflow in 5394576747. ## Prior work (AUTH-243-01) diff --git a/frontiers/soft_fold_limit_20261002/SOURCES.json b/frontiers/soft_fold_limit_20261002/SOURCES.json index 7ada57697..37b8a38a4 100644 --- a/frontiers/soft_fold_limit_20261002/SOURCES.json +++ b/frontiers/soft_fold_limit_20261002/SOURCES.json @@ -1,13 +1,13 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-FOLD-LIMIT-20261002-v1.4", + "object": "CL-FOLD-LIMIT-20261002-v1.5", "supersedes": { - "object": "CL-FOLD-LIMIT-20261002-v1.3", - "head": "483556485cf15cc8662425e2195699029c713ff5", - "proof_blob": "2652db8721d654b47763b18eea6b97ba9a0dc281" + "object": "CL-FOLD-LIMIT-20261002-v1.4", + "head": "15870b638748eb3f4e152fbb77ff41dac264e923", + "proof_blob": "64091adf95bf21e4445c4effa70ee92f84ad2da6" }, - "kind": "author-side proof candidate, every d >= 2 (v1.4: rebinds #242 v1.4 (head ffc7005, blob c1592915), whose consumed sections 0-3 are byte-identical to v1.2 and v1.3; the workflow requires the pinned #242 blob in the tree once #242 has merged (automated finding 4167121714 on #244, same pattern); no statement or proof changes; v1.3: applies the minor findings OA-243-B-01 (the m = 1 convention in Proposition FL.4), B-02 (the trivial case of the Trap fact), B-03 (the rectangle W fixed before the Q-dependent radii) and D-01 (the ordered domain lambda_1 < min(eps, lambda_2) in Step 5) of the nonauthor slice reviews of v1.2; records Codex's reviewed direct identity D.1 (F = k A_* a_fail in every fixed d); rebinds #242 v1.3, whose consumed sections 0-3 are byte-identical to v1.2; no statement changes in substance): Theorem FL (Conjecture 6 of Math- #242: the fold-scale limit (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u), uniformly on compacts of (b, k); the limit defining F exists, F is continuous and positive; the existence of the limit, as r^-3 (1 - p_r) -> alpha_1 + alpha_2, was proved earlier by the merged #170 (d = 2) and #175 (d >= 3); the decision and convergence steps here take a different route, the cap-region domination is shared); Proposition FL.7 (G_k(X, zeta) = P_theta(X, k zeta)/k with theta = (-lt/k, gamma, B, C3): same typed domain, weight and maximin decision, so F = k A_* a_fail, directly in d = 2 and by uniqueness of limits in d >= 3); Corollaries FL.5-FL.6 (recover #170 Theorem S and section 9, #175 Theorem F and section 7, in #242 form; not new); Proposition FL.4 (re-proves the window-to-global decision transfer of #170 Theorem E(2)-(3) and #175 Theorem H in #242 normalization, by paths and traps); Lemma FL.2 (the generic set has full measure; d(M) >= L_S by the critical chord of #170 section 3, observed on #242 by OpenAI Codex; case (D-prime) needs chi > 0); Lemma FL.3 (crossing a nondegenerate saddle); Lemma FL.1 (the soft window, rate r in d = 2 and r^(1/2) in d >= 3)", + "kind": "author-side proof candidate, every d >= 2 (v1.5: rebinds #242 v1.5 (head ee56652, blob 271412db; reporting-only changes after C80), whose consumed sections 0-3 are byte-identical; no other change; v1.4: rebinds #242 v1.4 (head ffc7005, blob c1592915), whose consumed sections 0-3 are byte-identical to v1.2 and v1.3; the workflow requires the pinned #242 blob in the tree once #242 has merged (automated finding 4167121714 on #244, same pattern); no statement or proof changes; v1.3: applies the minor findings OA-243-B-01 (the m = 1 convention in Proposition FL.4), B-02 (the trivial case of the Trap fact), B-03 (the rectangle W fixed before the Q-dependent radii) and D-01 (the ordered domain lambda_1 < min(eps, lambda_2) in Step 5) of the nonauthor slice reviews of v1.2; records Codex's reviewed direct identity D.1 (F = k A_* a_fail in every fixed d); rebinds #242 v1.3, whose consumed sections 0-3 are byte-identical to v1.2; no statement changes in substance): Theorem FL (Conjecture 6 of Math- #242: the fold-scale limit (k/r) r^-2 A_r^rej(b, k, u) -> F(k; b, u), uniformly on compacts of (b, k); the limit defining F exists, F is continuous and positive; the existence of the limit, as r^-3 (1 - p_r) -> alpha_1 + alpha_2, was proved earlier by the merged #170 (d = 2) and #175 (d >= 3); the decision and convergence steps here take a different route, the cap-region domination is shared); Proposition FL.7 (G_k(X, zeta) = P_theta(X, k zeta)/k with theta = (-lt/k, gamma, B, C3): same typed domain, weight and maximin decision, so F = k A_* a_fail, directly in d = 2 and by uniqueness of limits in d >= 3); Corollaries FL.5-FL.6 (recover #170 Theorem S and section 9, #175 Theorem F and section 7, in #242 form; not new); Proposition FL.4 (re-proves the window-to-global decision transfer of #170 Theorem E(2)-(3) and #175 Theorem H in #242 normalization, by paths and traps); Lemma FL.2 (the generic set has full measure; d(M) >= L_S by the critical chord of #170 section 3, observed on #242 by OpenAI Codex; case (D-prime) needs chi > 0); Lemma FL.3 (crossing a nondegenerate saddle); Lemma FL.1 (the soft window, rate r in d = 2 and r^(1/2) in d >= 3)", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { @@ -98,12 +98,12 @@ "consumed_unmerged": [ { "path": "frontiers/soft_rejected_pairs_20261002/PROOF.md", - "bytes": 90777, - "sha256": "f45ce06a0a78169fb5003cb91705f344c5e5867f6bd8271210253b423b53f9a6", - "git_blob": "c15929151e9f82813b7993a31b6735bcff8dc6fa", + "bytes": 91967, + "sha256": "ee2930c1434bb765d11da0690a2abf3ab330d544ce7ede1ae076a4c29e206c75", + "git_blob": "271412dbc96a29590f5f805258e15c9e335cb430", "tag": "242", - "use": "Math- #242 (open; v1.4, whose sections 0-3 are byte-identical to v1.3 and to v1.2 blob ad4beb84, which the v1.2 slice reviews of this note read): section 0, (2.1)-(2.6) (the soft model and its normalization), Lemma 2 (the model decision; v1.2 rewrites its proof at the exact level and states that beta > 2, chi <= 0 is rejected, as Lemma FL.2(c) here), Lemma 3 (the a priori elder region, used for domination in the identification of the limit), Proposition 2-prime (the stiff directions), (3.1) (the definition of F); Theorem 1 and Proposition 4 are cited", - "head": "ffc7005509554a378edd81136aa6e1f4fbd0eff8", + "use": "Math- #242 (open; v1.5, whose sections 0-3 are byte-identical to v1.4, v1.3 and to v1.2 blob ad4beb84, which the v1.2 slice reviews of this note read): section 0, (2.1)-(2.6) (the soft model and its normalization), Lemma 2 (the model decision; v1.2 rewrites its proof at the exact level and states that beta > 2, chi <= 0 is rejected, as Lemma FL.2(c) here), Lemma 3 (the a priori elder region, used for domination in the identification of the limit), Proposition 2-prime (the stiff directions), (3.1) (the definition of F); Theorem 1 and Proposition 4 are cited", + "head": "ee56652e1b30f2bfbb476da269fab15eb24cecba", "pr": "Math- #242" } ], @@ -145,13 +145,13 @@ "files": [ { "path": "PROOF.md", - "bytes": 78202, - "sha256": "7d0d69a100c34188a950746fb6ca5afe3a865c0cc72f85269dcfcd6bdf05df8d" + "bytes": 78874, + "sha256": "f972f46d6b7348a4ff5d2eea022694364895a5273265818533dd01204b4c06a0" }, { "path": "README.md", - "bytes": 12659, - "sha256": "b93c60afa625087bf355955d1aacb0e7b71e38e851556e83a9c701c754105db8" + "bytes": 12892, + "sha256": "acc6421222ad925c23aea29c6a309c4343c54c21133fbdc27b320ebaee9bd1aa" }, { "path": "fold_check.py", @@ -221,6 +221,30 @@ "findings": [], "use_here": "recorded as reviewed support in the Remark after Proposition FL.7 (v1.3)", "reviewer": "OpenAI Codex (delegated AI review; same GitHub account; organizational independence 0; Dylan personal reading pending)" + }, + { + "slice": "B/D/E v1.3+v1.4 consolidated amendment (C79)", + "review": "pullrequestreview-5394499570", + "at": "15870b6 (PROOF blob 64091adf)", + "verdict": "ACCEPT / PASS_TECHNICAL_SCOPED for OA-243-B-01, B-02, B-03 and D-01, plus the D.1 adoption readback", + "findings": [], + "reviewer": "OpenAI Codex (delegated AI review; same GitHub account; organizational independence 0; Dylan personal reading pending)" + }, + { + "slice": "D/E changed-source review (D-01, D.1 adoption)", + "review": "pullrequestreview-5394480716", + "at": "15870b6 (PROOF blob 64091adf)", + "verdict": "ACCEPT / PASS_TECHNICAL_SCOPED", + "findings": [], + "reviewer": "OpenAI Codex (delegated AI review; same GitHub account; organizational independence 0; Dylan personal reading pending)" + }, + { + "slice": "v1.4 workflow/source-presence engineering verification", + "review": "pullrequestreview-5394576747", + "at": "15870b6 (PROOF blob 64091adf)", + "verdict": "PASS_TECHNICAL_SCOPED (historical fallback reproduced on v1.3 and rejected by v1.4)", + "findings": [], + "reviewer": "OpenAI Codex (delegated AI review; same GitHub account; organizational independence 0; Dylan personal reading pending)" } ] }