From f8a0ee3979f82791b5d70eb624722ce003c5637b Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 1 Oct 2026 00:39:30 +0000 Subject: [PATCH 1/2] Add Theorem CU: cusp crossover and second-order term of the elder lifetime law nu_eld(ell) = c ell^{-1/3} + c_1 ell^{1/4} + o(ell^{1/4}) with c_1 = -(192/7) 2^{1/4} int int pi_0 E_0[|Y|^{7/4}|Delta|^{1/4} 1{A<0} | b] < 0, via the cusp limit field at r ~ k (CU.1), the quartic-ridge model whose elder window is exactly |phi| < 1/3 inside the typed window |phi| < 1 (CU.2), stability of the decision by compact-window path and trap arguments (CU.3), kernel limits at fixed kappa (CU.4) and a window-probability bound for the typed weight (CU.5). Author-side proof candidate; scientific effect NONE. Consumes the unmerged #191 (blob 441152df) and #198 (blob abfb98ae), pinned by blob and verified by the workflow. Exact stdlib controls C1-C7 with mutants M1-M4; the exploration (not a control) gives c_1 = -0.26939883 (d = 2) and -0.21184835 (d = 3, c_1/c = -5.071062) by deterministic quadrature. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .github/workflows/cusp-second-order.yml | 99 +++ .../cusp_second_order_20261001/EXPLORE.json | 134 ++++ frontiers/cusp_second_order_20261001/PROOF.md | 441 ++++++++++++ .../cusp_second_order_20261001/README.md | 61 ++ .../cusp_second_order_20261001/RESULTS.json | 68 ++ .../cusp_second_order_20261001/SOURCES.json | 165 +++++ .../cusp_second_order_20261001/cusp_check.py | 627 ++++++++++++++++++ .../explore_cusp.py | 361 ++++++++++ 8 files changed, 1956 insertions(+) create mode 100644 .github/workflows/cusp-second-order.yml create mode 100644 frontiers/cusp_second_order_20261001/EXPLORE.json create mode 100644 frontiers/cusp_second_order_20261001/PROOF.md create mode 100644 frontiers/cusp_second_order_20261001/README.md create mode 100644 frontiers/cusp_second_order_20261001/RESULTS.json create mode 100644 frontiers/cusp_second_order_20261001/SOURCES.json create mode 100644 frontiers/cusp_second_order_20261001/cusp_check.py create mode 100644 frontiers/cusp_second_order_20261001/explore_cusp.py diff --git a/.github/workflows/cusp-second-order.yml b/.github/workflows/cusp-second-order.yml new file mode 100644 index 000000000..11d088378 --- /dev/null +++ b/.github/workflows/cusp-second-order.yml @@ -0,0 +1,99 @@ +name: Cusp second-order exact controls + +'on': + pull_request: + paths: + - 'frontiers/cusp_second_order_20261001/**' + - '.github/workflows/cusp-second-order.yml' + # bound merged sources (consumed in SOURCES.json) and the pinned unmerged dependencies (#191, #198; verified by blob) + - 'frontiers/remainder_vanishing_20260930/PROOF.md' + - 'frontiers/remainder_rate_20260930/PROOF.md' + - 'frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md' + - 'frontiers/c7_total_bounded_20260929/PROOF.md' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md' + - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md' + - 'reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md' + - 'frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md' + - 'frontiers/c7_zero_gap_limit_20260929/PROOF.md' + workflow_dispatch: + +permissions: + contents: 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.request + root = pathlib.Path('frontiers/cusp_second_order_20261001').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']) + for entry in manifest['consumed'] + manifest['cited_only']: + data = pathlib.Path(entry['path']).read_bytes() + framed = b'blob ' + str(len(data)).encode() + b'\0' + data + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or hashlib.sha1(framed).hexdigest() != entry['git_blob']: + raise ValueError('merged source drifted: ' + entry['path']) + # each consumed unmerged dependency (#191, #198) is verified by its pinned blob: from the tree if the dependency + # has landed (then it must be byte-identical to the pinned version, else this packet needs rebinding), + # otherwise fetched by blob id through the repository API; absence or drift fails the job + for entry in manifest.get('consumed_unmerged', []): + tree_path = pathlib.Path(entry['path']) + if tree_path.exists(): + data = tree_path.read_bytes() + where = 'tree' + else: + req = urllib.request.Request( + 'https://api.github.com/repos/' + os.environ['GH_REPOSITORY'] + '/git/blobs/' + entry['git_blob'], + headers={'Authorization': 'Bearer ' + os.environ['GH_TOKEN'], 'Accept': 'application/vnd.github+json'}) + with urllib.request.urlopen(req, timeout=60) as resp: + blob = json.load(resp) + if blob.get('encoding') != 'base64': + raise ValueError('unexpected blob encoding for ' + entry['path']) + data = base64.b64decode(blob['content']) + where = 'blob api' + framed = b'blob ' + str(len(data)).encode() + b'\0' + data + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or hashlib.sha1(framed).hexdigest() != entry['git_blob']: + raise ValueError('consumed unmerged dependency differs from the pinned blob (rebind required): ' + entry['path'] + ' via ' + where) + print(json.dumps({'consumed_unmerged_verified': entry['path'], 'git_blob': entry['git_blob'], 'via': where})) + script = str(root / 'cusp_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', 'M9'], 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/cusp_second_order_20261001/EXPLORE.json b/frontiers/cusp_second_order_20261001/EXPLORE.json new file mode 100644 index 000000000..ce941a438 --- /dev/null +++ b/frontiers/cusp_second_order_20261001/EXPLORE.json @@ -0,0 +1,134 @@ +{ + "X1": { + "agree_with_window_1_3": 32, + "disagreements": [], + "r": 0.01, + "random_trials": 40, + "targeted_r_0.003": [ + { + "elder": true, + "kappa": 0.799, + "phi_target": 0.3, + "typed": true + }, + { + "elder": true, + "kappa": 0.61, + "phi_target": 0.3, + "typed": true + }, + { + "elder": true, + "kappa": 1.483, + "phi_target": 0.32, + "typed": true + }, + { + "elder": true, + "kappa": 2.774, + "phi_target": 0.32, + "typed": true + }, + { + "elder": false, + "kappa": 0.419, + "phi_target": 0.345, + "typed": true + }, + { + "elder": false, + "kappa": 1.618, + "phi_target": 0.345, + "typed": true + }, + { + "elder": false, + "kappa": 1.494, + "phi_target": 0.36, + "typed": true + }, + { + "elder": false, + "kappa": 2.471, + "phi_target": 0.36, + "typed": true + }, + { + "elder": true, + "kappa": 1.776, + "phi_target": -0.3, + "typed": true + }, + { + "elder": true, + "kappa": 0.885, + "phi_target": -0.3, + "typed": true + }, + { + "elder": true, + "kappa": 1.088, + "phi_target": -0.32, + "typed": true + }, + { + "elder": true, + "kappa": 0.802, + "phi_target": -0.32, + "typed": true + }, + { + "elder": false, + "kappa": 0.932, + "phi_target": -0.345, + "typed": true + }, + { + "elder": false, + "kappa": 0.484, + "phi_target": -0.345, + "typed": true + }, + { + "elder": false, + "kappa": 0.482, + "phi_target": -0.36, + "typed": true + }, + { + "elder": false, + "kappa": 1.798, + "phi_target": -0.36, + "typed": true + } + ], + "typed": 32 + }, + "X2": { + "d2": { + "J": 0.5518754382, + "J_coarse": 0.5518754382, + "c": 0.073406919306, + "c1": -0.26939883, + "c1_over_c": -3.669938, + "method": "composite Gauss-Legendre, a = -w^4: 24x12 by 56x12 nodes (coarse: 12x12 by 28x12)" + }, + "d3": { + "J": 1.363390706, + "J_coarse": 1.363390722, + "c": 0.041775931841, + "c1": -0.21184835, + "c1_over_c": -5.071062, + "mc_check": { + "J": 1.36198, + "J_se": 0.00089, + "c1": -0.21163, + "c1_se": 0.00014, + "method": "Monte Carlo on the cone, 8000000 samples, seed 1001, 10 batches" + }, + "method": "cone quadrature: Gauss-Legendre 64 x 64 in (rho, u), trapezoid 9 in theta, Gauss-Laguerre 64 in s (coarse: Laguerre 32)" + } + }, + "not_a_control": true, + "object": "CL-CUSP-SECOND-ORDER-20261001-v1 exploration" +} diff --git a/frontiers/cusp_second_order_20261001/PROOF.md b/frontiers/cusp_second_order_20261001/PROOF.md new file mode 100644 index 000000000..e7d3c0ce0 --- /dev/null +++ b/frontiers/cusp_second_order_20261001/PROOF.md @@ -0,0 +1,441 @@ +# The cusp crossover: the elder rule at `r ≍ k` and the second-order term `ν_eld(ℓ) = cℓ^{−1/3} + c₁ℓ^{1/4} + o(ℓ^{1/4})` + +Object: CL-CUSP-SECOND-ORDER-20261001-v1. +Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 1 October 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR REVIEW REQUIRED. Scientific effect: NONE — no register, +graph, STATUS, PROOF_INDEX, prize or Boolean change; the numerical values of §7 are exploration, not certified. +Same GitHub account as every lane; zero organizational independence. +**Dependencies (unmerged, consumed):** Math- #191 (`frontiers/remainder_vanishing_20260930/PROOF.md`, v1.1 blob +`441152df`: Lemma E (E.1) and Steps 1–3 with its dominating variable `T`, §2 (2.1), the admissible radius `r_0^*`) and +Math- #198 (`frontiers/remainder_rate_20260930/PROOF.md`, v1.1 blob `abfb98ae`: §1 the elder-density identity and +Lemma B (B.2), §2 Lemma F′, §3 (3.1)–(3.2) and Lemma W). This note cannot be integrated before both and must be rebound +if either changes. Merged inputs: [R], [C7-K], [P] with [E1]/[E2]/[REC] directly; [182] (the deterministic barrier, +inside #198 Lemmas B and F′; its §3 has a Slice-A acceptance from this same author only) and [Z] ((Z10), (Z13)–(Z14) and +`r_0^{[Z]}`, inside #198 and #191) transitively. + +## 0. Statement + +Setting and notation are those of [R] (`frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, blob `247b3ecf`), +#191 §0 and #198 §0: fixed `d ≥ 2`, `m = d − 1`, `L > 0`, the [P] field on `X = R^d/(LZ^d)`; near pins `M = −ru/2`, +`S = ru/2` at heights `b`, `b − kr³` with zero gradients; the regression law `Q = Q_{r,b,k}` and the coupling +`(F_r, F_0)` of [R] (R3)–(R4); `P = 1 + |b| + k`; the typed weight `W_r` with `W_r/r² = F_d(K_M)F_{d−1}(K_S)`; the elder +mark `e = 1{d_f(M) = f(S)}` for the maximin `d_f(M) = sup{min_t f(γ(t)) : γ(0) = M, f(γ(1)) > f(M)}` ([P] §8, Borel by +[E2] §9); the kernels + + A_r = 12π_r(v_r)E_Q[W_r/r²], A_r^{eld} = 12π_r(v_r)E_Q[(W_r/r²)e], A_0 = 12π_0(v_0)z_0, + +so that ([P] (10.2), #191 (2.1), #198 §1) `ν_eld(ℓ) = ∫_0^{r_0^*}∫_R∫_{S^{d−1}} r^{−2}A_r^{eld}(b, ℓ/r³, u) dσ db dr + ν_eld^{far,r_0^*}(ℓ)` +and `cℓ^{−1/3} = ∫_0^∞∫∫ r^{−2}A_0(b, ℓ/r³, u) dσ db dr`, `c = c_{d,L}` of [P] (15.2), `r_0^*` the admissible radius of #191 §0. + +**Cusp variables.** `κ := k/r = ℓ/r⁴` and `s := rℓ^{−1/4} = κ^{−1/4}`. The gap is `ℓ = kr³ = κr⁴`: the scale `r ≍ k` +is the scale at which the height gap is quartic in the separation. + +**Jets.** For a `C⁵` function `f`, in the frame `(u, Θ)` of the pair: `f₄ := ∂_u⁴f(0)`, `γ := ∇_Θ∂_u²f(0) ∈ R^m`, +`A := D_Θ²f(0) ∈ Sym(m)`, `Δ := det A`, + + Y := (f₄/12)Δ − γᵀadj(A)γ/4, φ := Y/(6κΔ) = (f₄ − 3γᵀA^{−1}γ)/(72κ) (A invertible). (0.1) + +(`Y` is #191's `Y` at `k = 0`; it is a scalar.) `E₀[· | b]` denotes expectation over the jets of `F₀` under the contact +law at the zero-gap target `v₀(b, 0)`, i.e. given `f(0) = b`, `∂_uf(0) = ∂_u²f(0) = ∂_u³f(0) = 0`, +`∇_Θf(0) = ∇_Θ∂_uf(0) = 0`. + +**Theorem CU (second-order term of the elder lifetime law).** For every `d ≥ 2`, `L > 0`, for the canonical marked +Kac–Rice version of the elder density ([P] §9 read as [E2]), + + ν_eld(ℓ) = c ℓ^{−1/3} + c₁ ℓ^{1/4} + o(ℓ^{1/4}) (ℓ ↓ 0), (CU.1) + c₁ = −(192/7)·2^{1/4} ∫_{S^{d−1}}∫_R π₀(u; v₀(b,0)) E₀[|Y|^{7/4}|Δ|^{1/4}1{A < 0} | b] db dσ(u) ∈ (−∞, 0). (CU.2) + +Equivalently, with the parity factorization of [P] §15 (odd and even jets are independent), +`c₁ = −(192/7)2^{1/4}∫_{S^{d−1}} p_G(0)p_{V_u}(0)(2π)^{−1/2}τ_u^{−1} E[|Y|^{7/4}|Δ|^{1/4}1{A<0} | V_u = 0, G = 0, t_u = 0] dσ(u)`, +`G = ∇f(0)`, `V_u = (∂_u²f, ∇_Θ∂_uf)(0)`, `t_u = ∂_u³f(0)`, `τ_u² = Var(t_u | G = 0)`. So +`ν_eld(ℓ) = cℓ^{−1/3}(1 + (c₁/c)ℓ^{7/12} + o(ℓ^{7/12}))`; for the manuscript's field (`d = 3`, `L = 24`) the exploration of +§7 gives `c₁ ≈ −0.21185`, `c₁/c ≈ −5.071`. + +The mechanism is a **cusp crossover**: rescaled at `x = rX` (axis), `y = r²Ξ` (transverse, `Ξ ∈ R^m`), heights `r⁴`, +the pinned field converges to the random polynomial + + 𝔓(X, Ξ) = 2κ(X + ½)²(X − 1) + (f₄/24)(X² − ¼)² + ½(X² − ¼)γ·Ξ + ½ΞᵀAΞ (0.2) + +(Theorem CU.1), whose superlevel topology reduces exactly to the quartic ridge +`g(X) = κ[2(X + ½)²(X − 1) + 3φ(X² − ¼)²]` (Theorem CU.2). For `𝔓`, the pair is typed iff `|φ| < 1` — the sign window +of #198 Lemma W — and **`S` is `M`'s elder partner iff `|φ| < 1/3`**: the elder window is exactly one third of the typed +window, its boundary being the configurations in which the two competing saddles (`φ = 1/3`) or maxima (`φ = −1/3`) +have equal heights, `g(X₃) + κ = κ(φ + 1)³(3φ − 1)/(16φ³)` and `g(X₃) = κ(φ − 1)³(3φ + 1)/(16φ³)`. The model's decision +is stable (Proposition CU.3), so the elder kernel at fixed `κ` converges (Theorem CU.4); a window-probability bound +(Lemma CU.5) controls the intermediate separations; and the cusp integral gives (CU.2), with +`∫_0^∞ loss ds = (16/7)2^{1/4}|Y|^{7/4}|Δ|^{1/4}`. + +**What is not claimed.** No rate for the `o(ℓ^{1/4})`; no certified numerical value of `c₁` (§7 is exploration); no +expansion of the candidate or rejected densities (Remark 2 records the formal ones); no uniformity in `d`, `L`; no +finite-radius band; nothing beyond the existential scope of Theorem R, #191 and #198. Conditional on #191, #198 and +this note's review, the exponent question left in #198 Remark 1 is settled: `1/4` is the true order and the +coefficient is negative. + +## 1. The cusp limit field + +**Theorem CU.1.** Let `f ∈ C⁵(X)`, `0 < r ≤ 1`, `κ > 0`, and let `M = −ru/2`, `S = ru/2` be critical points of `f` with +`f(M) = b`, `f(S) = b − κr⁴`. For `R ≥ 1` with `2Rr ≤ L/4`, put `𝒲_R := [−R, R] × {|Ξ| ≤ R}` and +`𝔉(X, Ξ) := r^{−4}[f(rXu + r²ΘΞ) − b]`. Let `𝔓` be (0.2) with the jets `(f₄, γ, A)` of `f` at `0`. Then + + ‖𝔉 − 𝔓‖_{C²(𝒲_R)} ≤ C(d, R) r ‖f‖_{C⁵}. (1.1) + +*Proof.* Write `ϑ_j := ∂_u^jf(0)`, `ψ_j := ∂_u^j∇_Θf(0)` (so `ϑ₄ = f₄`, `ψ₂ = γ`), `h := r/2`, `N := ‖f‖_{C⁵}`. +Taylor's formula along the axis, `ϑ(x) := f(xu) = Σ_{j≤4}ϑ_jx^j/j! + ρ(x)` with `|ρ| ≤ N|x|⁵/120`, `|ρ'| ≤ N|x|⁴/24`, and +the pins `ϑ'(±h) = 0`, `ϑ(−h) = b`, `ϑ(h) = b − 16κh⁴` give successively +`ϑ'(h) − ϑ'(−h) = 2ϑ₂h + ϑ₄h³/3 + O(Nh⁴) = 0`, `ϑ'(h) + ϑ'(−h) = 2ϑ₁ + ϑ₃h² + O(Nh⁴) = 0`, +`ϑ(h) − ϑ(−h) = 2ϑ₁h + ϑ₃h³/3 + O(Nh⁵) = −16κh⁴` and `ϑ(h) + ϑ(−h) = 2ϑ₀ + ϑ₂h² + ϑ₄h⁴/12 + O(Nh⁵) = 2b − 16κh⁴`, hence + + ϑ₂ = −(r²/24)f₄ + O(Nr³), ϑ₃ = 12κr + O(Nr²), ϑ₁ = −(3/2)κr³ + O(Nr⁴), ϑ₀ − b = (−κ/2 + f₄/384)r⁴ + O(Nr⁵). + +The transverse pins `ψ(±h) = 0` for `ψ(x) := ∇_Θf(xu) = ψ₀ + ψ₁x + γx²/2 + ψ₃x³/6 + O(N|x|⁴)` give +`ψ₀ = −(r²/8)γ + O(Nr⁴)` and `ψ₁ = O(Nr²)`. Expand `f` at `0` to total order 4 in `(x, y)`: the remainder `ρ₅` satisfies +`|∂^αρ₅(x, y)| ≤ CN(|x| + |y|)^{5−|α|}` (`|α| ≤ 2`), and at `(x, y) = (rX, r²Ξ)`, `(X, Ξ) ∈ 𝒲_R`, `|x| + |y| ≤ 2Rr`; with +`∂_X = r∂_x`, `∂_Ξ = r²∂_y`, every derivative of order `≤ 2` of `r^{−4}ρ₅(rX, r²Ξ)` is `O(C_R N r)`. The Taylor monomials +`x^iy^j` with weighted degree `i + 2|j| ≥ 5` contribute `O(C_R N r)` in `C²(𝒲_R)` likewise. The monomials of weighted +degree `≤ 4` are `(i, |j|) ∈ {(0,0), (1,0), (2,0), (3,0), (4,0), (0,1), (1,1), (2,1), (0,2)}`; inserting the pinned values, + + r^{−4}[ϑ₀ − b + ϑ₁rX + ϑ₂r²X²/2 + ϑ₃r³X³/6 + f₄r⁴X⁴/24 + ψ₀·r²Ξ + ψ₁·r³XΞ + γ·r⁴X²Ξ/2 + r⁴ΞᵀAΞ/2] + = [−κ/2 − (3/2)κX + 2κX³] + (f₄/24)[1/16 − X²/2 + X⁴] + [−γ·Ξ/8 + γ·X²Ξ/2] + ½ΞᵀAΞ + O(C_R N r), + +and `−κ/2 − (3/2)κX + 2κX³ = 2κ(X + ½)²(X − 1)`, `1/16 − X²/2 + X⁴ = (X² − ¼)²`, `−γ·Ξ/8 + γ·X²Ξ/2 = ½(X² − ¼)γ·Ξ`. ∎ + +(Checker C4 verifies, exactly, that for pinned polynomials of degree 6 in `d = 2, 3` the Laurent polynomial +`r^{−4}[f(rX, r²Ξ) − b]` has no negative powers of `r` and constant term exactly `𝔓(X, Ξ)`. A referee pass also checked +the `O(r)` rate on a non-polynomial field.) + +## 2. The model: fiber reduction and the elder window + +**Theorem CU.2.** Let `κ > 0`, `f₄ ∈ R`, `γ ∈ R^m`, `A ∈ Sym(m)` negative definite; `𝔓` as in (0.2); `φ`, `Y` as in +(0.1); `X₃ := −1/(2φ)` for `φ ≠ 0`. + +(a) *(Fiber reduction.)* `𝔓(X, Ξ) = g(X) + ½(Ξ − Ξ*(X))ᵀA(Ξ − Ξ*(X))`, with `Ξ*(X) := −½(X² − ¼)A^{−1}γ` and + + g(X) = κ[2(X + ½)²(X − 1) + 3φ(X² − ¼)²], g'(X) = 6κ(X² − ¼)(1 + 2φX). (2.1) + +(b) *(Critical points, types.)* The critical points of `𝔓` are `(X, Ξ*(X))` with `g'(X) = 0`: `M = (−½, 0)`, `S = (½, 0)` +and, for `φ ∉ {0, ±1}`, `(X₃, Ξ*(X₃))`. `𝔓(M) = 0`, `𝔓(S) = −κ`; `det Hess 𝔓(M) = 6κ(φ − 1)Δ = Y − 6κΔ`, +`det Hess 𝔓(S) = 6κ(φ + 1)Δ = Y + 6κΔ`; `M` is a nondegenerate maximum and `S` a nondegenerate critical point of index +`m` iff `|φ| < 1`. + +(c) *(Elder window.)* Let `|φ| < 1` and let `d_𝔓(M)` be the maximin connection level of `M` for `𝔓` on `R^{1+m}` (paths +to points with `𝔓 > 0`; `−∞` if there are none). Then + + d_𝔓(M) = −κ = 𝔓(S) if |φ| < 1/3; d_𝔓(M) = g(X₃) > −κ if 1/3 < φ < 1; d_𝔓(M) = −∞ if −1 < φ < −1/3. + +(d) `g(X₃) + κ = κ(φ + 1)³(3φ − 1)/(16φ³)`, `g(X₃) = κ(φ − 1)³(3φ + 1)/(16φ³)`; at `φ = 1/3`, +`g − g(½) = (κ/16)(2X − 1)²(2X + 3)²` (the two saddles `½`, `−3/2` at equal height); at `φ = −1/3`, +`g = −(κ/16)(2X − 3)²(2X + 1)²` (the two maxima `−½`, `3/2` at equal height). + +*Proof.* (a) `½(Ξ − Ξ*)ᵀA(Ξ − Ξ*) = ½ΞᵀAΞ + ½(X² − ¼)γ·Ξ + (1/8)(X² − ¼)²γᵀA^{−1}γ`; subtracting from (0.2) leaves +`2κ(X + ½)²(X − 1) + ((f₄ − 3γᵀA^{−1}γ)/24)(X² − ¼)²`, and `(f₄ − 3γᵀA^{−1}γ)/24 = 3κφ`. (b) In the coordinates +`(X, Z)`, `Z := Ξ − Ξ*(X)` (a `C^∞` diffeomorphism with unit Jacobian), `𝔓 = g(X) + ½ZᵀAZ`: critical points are +`{Z = 0, g' = 0}`, and at a critical point the Hessian changes by a congruence of determinant one, so +`det Hess_{(X,Ξ)}𝔓 = g''·Δ` and the inertia is that of `diag(g'', A)`. From (2.1), `g''(−½) = 6κ(φ − 1)`, +`g''(½) = 6κ(φ + 1)`, and `6κφΔ = Y`. (c) For every `t`, `{𝔓 > t} = {(X, Ξ*(X) + Z) : g(X) > t, Zᵀ(−A)Z < 2(g(X) − t)}` — +over each `X` with `g(X) > t` a nonempty open ellipsoid, empty otherwise. Hence a point above `0` exists over `X` iff +`g(X) > 0`; every path satisfies `min 𝔓 ≤ min g(X(·))`, and the lifted path `X ↦ (X, Ξ*(X))` has `𝔓 = g`. So `d_𝔓(M)` is +the one-dimensional maximin `max(d₋, d₊)`, `d₋` (resp. `d₊`) the supremum over `x < −½` (resp. `x > −½`) with `g(x) > 0` +of `min g` on the segment between `−½` and `x` (`sup ∅ = −∞`). On `(−½, ½)`, `g' < 0` (`1 + 2φX ≥ 1 − |φ| > 0`), so `g` +falls from `0` to `−κ`. *`0 ≤ φ < 1/3`:* on `(½, ∞)`, `g' > 0` and `g → +∞`, so `d₊ = −κ`; on `(−∞, −½)`, either `φ = 0` +and `g' > 0` (so `g < 0` there and `d₋ = −∞`), or `X₃ < −3/2` is a local minimum with `g(X₃) < −κ` by (d) and `g → +∞` as +`X → −∞`, so `d₋ = g(X₃) < −κ`. *`1/3 < φ < 1`:* `d₊ = −κ` as before; `X₃ ∈ (−3/2, −½)` is a local minimum, `g → +∞` to +the left, and `g(X₃) > −κ` by (d), so `d₋ = g(X₃)`. *`−1/3 < φ < 0`:* on `(−∞, −½)`, `g' > 0`, so `d₋ = −∞`; on `(½, ∞)`, +`g` rises to the local maximum `X₃ > 3/2` with `g(X₃) > 0` by (d) (`(φ−1)³ < 0`, `3φ + 1 > 0`, `φ³ < 0`), so `d₊ = −κ`. +*`−1 < φ < −1/3`:* `d₋ = −∞`; now `g(X₃) < 0` with `X₃ ∈ (½, 3/2)`, `g < 0` on `(−½, ∞)`, so `d₊ = −∞` and `M` is the unique +global maximum. (d) is algebra (checker C1). ∎ + +The typed condition `|φ| < 1` is `|Y| < 6κ|Δ|`, the sign window of #198 Lemma W at leading order; the elder condition +is `|Y| < 2κ|Δ|`. Checker C2 recomputes (c) for 117 rational `φ` by an exact one-dimensional union-find maximin on a +grid containing every critical point (hence exact), and C5 the determinants and inertias of (b). A referee pass +confirmed (c) by an independent two-dimensional maximin of the full model (no fiber reduction) in 34 cases. + +## 3. Stability of the decision + +**Proposition CU.3.** Fix `δ ∈ (0, 1/20)`, put `R_δ := 30δ^{−2}` and `𝒲 := [−3, 2] × {|Ξ| ≤ R_δ}`, and let `K_δ` be the +set of `(κ, f₄, γ, A)` with `κ ∈ [δ, δ^{−1}]`, `A ≤ −δI`, `‖A‖ + |γ| + |f₄| ≤ δ^{−1}` and +`|φ| ∈ [0, 1/3 − δ] ∪ [1/3 + δ, 1 − δ]`. There is `ε = ε(δ, m) > 0` such that: if `f ∈ C²(X)`, `M = −ru/2` and `S = ru/2` +are critical points of `f` with `f(M) = b`, `f(S) = b − κr⁴`, the map `Φ : (X, Ξ) ↦ rXu + r²ΘΞ` is injective on `𝒲` +(so that, being a local diffeomorphism, it maps the interior and boundary of every subregion of `𝒲` onto the interior +and boundary of its image), `(κ, f₄, γ, A) ∈ K_δ` and `‖𝔉 − 𝔓‖_{C²(𝒲)} ≤ ε` (`𝔉 = r^{−4}(f∘Φ − b)`, `𝔓 = 𝔓_{κ,f₄,γ,A}`, +`C²` norms with operator norms on the derivatives), then `M` is a nondegenerate local maximum, `S` a nondegenerate +critical point of index `m`, and + + e(f) = 1{d_f(M) = f(S)} = 1{|φ| < 1/3}. (3.1) + +*Proof.* Constants `C_δ, μ_δ, c_δ, η_δ, …` below depend only on `δ, m` (continuity on the compact `K_δ`). + +*Step 1 (the ridge of `𝔉`).* On `𝒲`, `∂_Ξ²𝔉 ≤ A + εI ≤ −(δ/2)I` for `ε ≤ δ/2`. For `X ∈ [−3, 2]`, +`|Ξ*(X)| ≤ (35/8)δ^{−2}`. Since `𝔉(X, ·) ≤ g(X) − (δ/2)|· − Ξ*(X)|² + ε` and `𝔉(X, Ξ*(X)) ≥ g(X) − ε`, every maximizer of +`𝔉(X, ·)` over `|Ξ| ≤ R_δ` lies within `2(ε/δ)^{1/2} < 1` of `Ξ*(X)`, hence in the interior; by strict concavity it is +unique, `Ξ_𝔉(X)`, and the first-order identity `A(Ξ_𝔉 − Ξ*) = ∂_Ξ𝔓(X, Ξ_𝔉) = ∂_Ξ(𝔓 − 𝔉)(X, Ξ_𝔉)` improves this to +`|Ξ_𝔉 − Ξ*| ≤ ε/δ`. By the implicit function theorem `Ξ_𝔉 ∈ C¹`, and `g_𝔉(X) := 𝔉(X, Ξ_𝔉(X))` is `C²` with +`g_𝔉' = ∂_X𝔉(·, Ξ_𝔉)`, `g_𝔉'' = [∂_X²𝔉 − ∂_{XΞ}𝔉(∂_Ξ²𝔉)^{−1}∂_{ΞX}𝔉](·, Ξ_𝔉)`; the same formulas hold for `g` with `(𝔓, Ξ*)`. +Hence `‖g_𝔉 − g‖_{C²([−3,2])} ≤ C_δε`. + +*Step 2 (connectivity reduces to one dimension).* For every `t`, each nonempty `X`-fiber of `𝒲 ∩ {𝔉 > t}` is convex +(a ball cut by a superlevel set of a concave function) and contains `Ξ_𝔉(X)`, and it is nonempty iff `g_𝔉(X) > t`. So the +path components of `𝒲 ∩ {𝔉 > t}` are the parts over the components of `{X ∈ [−3, 2] : g_𝔉(X) > t}`, and the supremum of +`𝔉` over such a part is the supremum of `g_𝔉` over the interval. + +*Step 3 (transverse boundary).* On `|Ξ| = R_δ`, `|Ξ − Ξ*| ≥ 25δ^{−2}`, so `𝔉 ≤ max_{[−3,2]}g − 312δ^{−3} + ε`; since +`|g| ≤ 280κ ≤ 280δ^{−1}` on `[−3, 2]` and `δ^{−2} > 400`, `𝔉 < −2κ − 1` there. + +*Step 4 (types).* `M ↔ (−½, 0)` and `S ↔ (½, 0)` are critical points of `𝔉`, so `Ξ_𝔉(±½) = 0`, `g_𝔉'(±½) = 0`, +`g_𝔉(−½) = 0` and `g_𝔉(½) = −κ` exactly. `g''(−½) = 6κ(φ − 1) ≤ −6δ²` and `g''(½) ≥ 6δ²`, so for `C_δε < 3δ²` the +Schur complement `g_𝔉''` of `∂_Ξ²𝔉 < 0` in `Hess 𝔉` is negative at `M` and positive at `S` (Haynsworth): `M` is a +nondegenerate maximum and `S` has index `m`, for `𝔉` and hence for `f`. + +*Step 5 (the decision).* Let `τ := δ/10` and let `Z_g` be the set of **all** real zeros of `g'`: `±½` and, for `φ ≠ 0`, +`X₃` — wherever it lies, inside or outside `[−3, 2]`. All are nondegenerate (`g''(X₃) = 3κ(1 − φ²)/φ`), `|X₃ ∓ ½| ≥ δ/2`, +`|g''| ≥ c_δ > 0` on the `τ`-neighbourhoods of the points of `Z_g` (referee computation: `c_δ ≈ 3.6κδ`), and `|g'| ≥ μ_δ > 0` +on the points of `[−3, 2]` at distance `≥ τ` from `Z_g` (compactness of `K_δ`). For `C_δε < ½min(μ_δ, c_δτ)`, `g_𝔉'` has +exactly the zeros `±½` and, when `X₃ ∈ [−3 + τ, 2 − τ]`, one more zero `X₃^𝔉` within `τ` of `X₃` (nondegenerate), with the +same sign pattern as `g'` elsewhere on `[−3, 2]` away from the `τ`-neighbourhoods of `Z_g`. + +*Case E, `|φ| ≤ 1/3 − δ`.* By §2, `g` increases on `[−3/2, −½]` (for `φ > 0` because `X₃ < −3/2 − 4δ`), decreases on +`[−½, ½]` and increases on `[½, X_R]`, where `X_R := 2` if `φ ≥ 0` and `X_R := min(2, X₃ − τ)` if `φ < 0`; and +`g(−3/2) = κ(−5 + 12φ) ≤ −κ(1 + 12δ)`, `g(X_R) ≥ η_δ > 0` (`g(2) = κ(12.5 + 42.1875φ) ≥ 1.9κ` for `φ ≥ −¼`, and for +`φ < −¼`, `X₃ ≤ 2` and `g(X₃ − τ) ≥ g(X₃) − Cκτ² ≥ 11κδ` since `g(X₃) ≥ 12κδ` by (d); numerically `g(X_R) ≥ 0.8κ` at +`δ = 1/21`). The same holds for `g_𝔉`, with `−κ(1 + 11δ)` and `η_δ/2`. *Path:* the ridge curve `X ↦ Φ(X, Ξ_𝔉(X))`, +`X ∈ [−½, X_R]`, runs from `M` through `S` to a point where `f > b`, and along it `f − b = r⁴g_𝔉 ≥ −κr⁴ = f(S) − b`. So +`d_f(M) ≥ f(S)`. *Trap:* let `D := (−3/2, ½) × {|Ξ| < R_δ}` and `D_phys := Φ(D)`, `∂D_phys = Φ(∂D)`. On `∂D`: +`𝔉 ≤ g_𝔉(−3/2) < −κ` at `X = −3/2`; `𝔉 < −2κ − 1` at `|Ξ| = R_δ`; at `X = ½`, `𝔉(½, Ξ) ≤ g_𝔉(½) = −κ` with equality only at +`Ξ = 0`, i.e. at `S`. So `f ≤ f(S)` on `∂D_phys`. The path component `C` of `{f > f(S)}` containing `M` does not meet +`∂D_phys` (the first exit of a path in `C` from `D_phys` would land on it), hence `C ⊂ D_phys`, and by Step 2 its supremum is +`b + r⁴ sup g_𝔉` over the component of `{g_𝔉 > −κ} ∩ (−3/2, ½)` containing `−½`, which is `b + r⁴g_𝔉(−½) = b`. So `C` +contains no point above `b`, and every path from `M` to a point above `b` leaves `C` and meets `{f ≤ f(S)}`: +`d_f(M) ≤ f(S)`. Hence `d_f(M) = f(S)` and `e = 1`. + +*Case R⁺, `φ ∈ [1/3 + δ, 1 − δ]`.* `X₃ ∈ (−3/2, −½ − δ/2)` is a nondegenerate local minimum of `g`; `g` decreases on +`[−3, X₃]` and increases on `[X₃, −½]`; `g(X₃) ≥ −κ + 0.44δκ` by (d); `g(−3) = κ(−50 + 229.6875φ) ≥ 26κ`. For `g_𝔉` the +same holds with `g_𝔉(X₃^𝔉) ≥ −κ + 0.4δκ`. The ridge curve from `M` to `X = −3` ends where `f > b` and has minimum +`b + r⁴g_𝔉(X₃^𝔉) > f(S)`: `d_f(M) > f(S)`, `e = 0`. + +*Case R⁻, `φ ∈ [−1 + δ, −1/3 − δ]`.* `X₃ ∈ (½ + δ/2, 3/2)`; `g` increases on `[−3/2, −½]`, decreases on `[−½, ½]`, +increases on `[½, X₃]`, decreases on `[X₃, 2]`; `g(−3/2) ≤ −9κ`, `g(2) ≤ −1.5κ`, `g(X₃) ≤ −1.5δκ < 0`. Put +`t_* := −(5/4)κ` and `D' := (−3/2, 2) × {|Ξ| < R_δ}`: `𝔉 < t_*` on `∂D'` (at the ends by the values of `g_𝔉`, on the +transverse boundary by Step 3), the component `C'` of `{f > b + r⁴t_*}` containing `M` lies in `Φ(D')`, and its supremum is +at most `b + r⁴max(g_𝔉(−½), g_𝔉(X₃^𝔉)) = b`. So `d_f(M) ≤ b + r⁴t_* < f(S)` (or `M` is the global maximum, `d_f(M) = −∞`), +and `e = 0`. ∎ + +Only window information is used: every conclusion is either an explicit path inside `Φ(𝒲)` or a trap whose boundary +lies inside it, so the exterior landscape cannot change the decision. No Morse–Smale, separatrix or genericity +assumption is made; `e` is computed exactly. + +## 4. The kernels at the cusp scale + +**Theorem CU.4.** For fixed `b ∈ R`, `u ∈ S^{d−1}` and `κ > 0`, as `r ↓ 0`, + + r^{−2}A_r(b, κr, u) → 𝒜^{cand}, r^{−2}A_r^{eld}(b, κr, u) → 𝒜^{eld}, r^{−2}A₀(b, κr, u) → 𝒜^{con}, (4.1) + +`𝒜^• := 12π₀(u; v₀(b, 0))E₀[w^• | b]` with `w^{cand} := (36κ²Δ² − Y²)1{A < 0, |Y| < 6κ|Δ|}`, +`w^{eld} := (36κ²Δ² − Y²)1{A < 0, |Y| < 2κ|Δ|}` and `w^{con} := 36κ²Δ²1{A < 0}`. + +*Proof.* Work under the coupling (R3) with `k = κr`; `T` dominates `1 + k + ‖F_r‖_{C⁷} + ‖F₀‖_{C⁷} + r^{−2}‖F_r − F₀‖_{C²}` +with `‖T‖_p ≤ C_pP` ([R] (R4); #191 §1). The contact law of `F₀` has target `v₀(b, κr)`, which differs from `v₀(b, 0)` +only in the coordinate `12k`; the regression mean is affine in the target and the covariance does not depend on it, so +`F₀ = F₀⁰ + 12k·w₀` with `F₀⁰` of target `v₀(b, 0)` and `w₀` a fixed smooth function: the `E₀`-variables are the jets of +`F₀⁰`, and those of `F₀` differ from them by deterministic `O(κr)` vectors. + +(i) *Weights.* By #198 (3.1), `det K_M = r(Y_r − 6κΔ₀) + R_M`, `det K_S = r(Y_r + 6κΔ₀) + R_S`, `|R_i| ≤ Cr²(1 + k)T^{N₁}`, +with `Y_r = (f₄/12)Δ₀ + 3k tr(adj(A₀)B) − q` (#191's `Y`). As `r → 0`, `Y_r → Y` and `Δ₀ → Δ` (the `E₀`-variables) in every +`L^p`, so `det K_i/r → Y ∓ 6κΔ` in every `L^p`. + +(ii) *Types.* `{W_r > 0} = {index K_M = d, index K_S = d − 1}`. On `{A < 0, |Y| ≠ 6κ|Δ|}`, for small `r`, `A_M, A_S < 0` +and, by Haynsworth, the indices are `m + 1{σ_i < 0}` with `σ_i = det K_i/det A_i`; with `sign Δ = (−1)^m` this gives +`typed ⟺ |Y| < 6κ|Δ|` in the limit. On `{A < 0}^c`, a typed configuration needs `A_M < 0`, hence `λ_max(A₀) ≤ CrT`, of +probability `→ 0` (the deterministic `O(κr)` shift between `A₀` and `A` included). So +`1{W_r > 0} → 1{A < 0, |Y| < 6κ|Δ|}` in probability. + +(iii) *Marks.* By Theorem CU.1, `‖𝔉_r − 𝔓_r‖_{C²(𝒲)} ≤ C_δrT` for the rescaled `F_r` and `𝔓_r` built from the jets of +`F_r`. On `{(κ, jets of F_r) ∈ K_δ, C_δrT ≤ ε(δ)}`, Proposition CU.3 gives `e = 1{|φ_r| < 1/3}`, `φ_r` built from the jets of +`F_r`, and `φ_r → φ = Y/(6κΔ)` in probability. The typed event minus this good event has probability at most +`P(C_δrT > ε(δ)) + P(|jets(F_r) − E₀-jets| > η_δ) + P((κ, E₀-jets) ∉ K_{2δ}, A < 0, |Y| < 6κ|Δ|) + o(1)`, with `η_δ` so small +that `η_δ`-perturbations of points of `K_{2δ}` stay in `K_δ`; the first two terms `→ 0` as `r → 0`, and the third `→ 0` as +`δ → 0`, because the `E₀`-jets are a nondegenerate Gaussian vector ([P] §2 finite-jet rank) and, given `(A, γ)`, `f₄` has a +nondegenerate Gaussian law — so `φ` has a continuous distribution and `P(|φ| ∈ (1/3 − 2δ, 1/3 + 2δ) ∪ (1 − 2δ, 1))`, +`P(λ_max(A) ∈ (−2δ, 0))`, `P(large jets)` all vanish as `δ → 0`. + +(iv) *Expectations.* On the typed event `W_r/r² = −det K_M det K_S`, and by #198 (3.2) +`W_r/r² ≤ C(κ²r²Δ₀² + r⁴(1 + k)²T^{2N₁})`, so `(W_r/r²)/r²` is bounded in `L²` and uniformly integrable. With (i)–(iii), +`E_Q[W_r/r²]/r² → E₀[w^{cand} | b]` and `E_Q[(W_r/r²)e]/r² → E₀[w^{eld} | b]` (note `|φ| < 1/3 ⟺ |Y| < 2κ|Δ|`). + +(v) *Densities.* `π_r(v_r) → π₀(v₀(b, 0))` by [R] (R5) and the continuity of `π₀` in the target; +`A₀(b, κr, u)/r² = 12π₀(v₀(b, κr))·36κ²E[Δ²1{A < 0} | v₀(b, κr)] → 𝒜^{con}`. ∎ + +## 5. The typed window has small probability + +**Lemma CU.5.** There are `C, N` and, for `m = d − 1`, the exponents `θ = 1/2`, `β = min(1/8, 1/(4m))` such that for +`0 < r ≤ r₁`, `b ∈ R`, `0 ≤ k ≤ 1`, + + E_Q[W_r/r²] ≤ C (k² + r⁴)(min(1, k/r)^{θ} + r^{β}) P^N. (5.1) + +(#198 Lemma W is (5.1) without the factor in parentheses.) + +*Proof.* Let `J` be the vector of all partial derivatives of `f = F_r` at `0` of orders `≤ 8` in the frame `(u, Θ)`, +`J = (J_pin, f₄, J')`, where `J_pin := (f, ∂_uf, ∂_u²f, ∂_u³f, ∇_Θf, ∇_Θ∂_uf)(0)` and `J'` is the rest. (`J'` must exclude the +pinned jets: with `∂_u²f(0)` in `J'`, the pin row `U₃` would determine `(r²/24)f₄` up to `O(r⁷)`.) + +*Conditional density of `f₄`.* Under `Q`, `(f₄, J')` is Gaussian with the law of `(f₄, J')` given `U_r = v_r`, so +`Var_Q(f₄ | J') = Var(f₄ | J', U_r)`. As `r → 0`, `U_r → U₀ = J_pin` in `L²` of the field, and `(J', J_pin)` and +`(f₄, J', J_pin)` are families of distinct partial derivatives at one point, hence nondegenerate ([P] §2). The Gram +matrix of `(J', U_r)` therefore stays uniformly invertible for `r ≤ r₁` (uniformly over frames), and +`Var(f₄ | J', U_r) → Var(f₄ | J', J_pin) > 0`. So the conditional density of `f₄` given `J'` under `Q` is bounded by a +constant `C₀`. + +*Affine structure.* Expanding `α_i = ∂_u²f(x_iu)/r`, `A_i = D_Θ²f(x_iu)`, `β_i = ∇_Θ∂_uf(x_iu)/r` (`x_i = ∓r/2`) to order 8 +in `x_i` and inserting the pin identities to the same order (`∂_u²f(0) = −(r²/24)f₄ + (terms in J') + …`, +`∂_u³f(0) = 12k − (r²/40)∂_u⁵f(0) + …`, `∇_Θ∂_uf(0) = (terms in J')`; the transverse pins involve no `f₄`), +`det K_i = α_i det A_i − rβ_iᵀadj(A_i)β_i = d_i(J')f₄ + c_i(J') + ρ_i`, with `d_i(J') = (r/12)det A_i(J')` and a Taylor +remainder `|ρ_i| ≤ Cr⁶(1 + ‖f‖_{C⁹})^{N₂}`. (A referee pass verified exactly, on a general pinned polynomial of degree 8 +in `d = 2`, that `A_i, β_i` contain no `f₄` and `det K_i` is of degree 1 in `f₄` with slope `(r/12)det A_i`.) Let +`E₁ := {‖F_r‖_{C⁹} + T ≤ r^{−η}}` with `η := 1/(8max(N₁, N₂))`; on `E₁`, `|ρ_i| ≤ Cr⁵`, and `P(E₁^c) ≤ C_q r^qP^{N_q}` for every +`q` ([R] (R4)). + +*The window.* Let `Δ := det D_Θ²f(0)` (a coordinate of `J'`) and `B₁ := {|Δ| ≥ r^{1/2}}`. On `B₁ ∩ E₁` and for small `r`, +`det A_M(J')`, `det A_S(J')` have the sign of `Δ` and modulus `≥ r^{1/2}/2`, so `|d_i| ≥ r^{3/2}/24`. On the typed event the +two determinants have opposite signs (#198 §3), so `f₄` lies between the roots `−(c_i + ρ_i)/d_i`; hence +`{W_r > 0} ∩ B₁ ∩ E₁ ⊂ {f₄ ∈ I(J')}`, `I(J')` the `J'`-measurable interval spanned by `−c_M/d_M`, `−c_S/d_S`, enlarged by +`Cr⁵/min|d_i|`. To bound its length on this event, write `ℓ_i := d_if₄ + c_i` and evaluate at the actual `f₄ ∈ I(J')` +(`|f₄| ≤ r^{−η}` on `E₁`): by #198 (3.1), `ℓ_S − ℓ_M = det K_S − det K_M − (ρ_S − ρ_M) = 12kΔ₀ + R_S − R_M − (ρ_S − ρ_M)`, so +`|ℓ_S(f₄) − ℓ_M(f₄)| ≤ 12k|Δ| + Cr²r^{−N₁η}` on `E₁` (the swap `Δ₀ → Δ` costs `Cr²T ≤ Cr^{2−η}`); and +`|d_S − d_M| ≤ Cr²r^{−N₂η}`. Since `−c_S/d_S + c_M/d_M = [ℓ_M(f₄) − ℓ_S(f₄) + (d_S − d_M)(f₄ + c_M/d_M)]/d_S`, this gives +`|I(J')| ≤ 2(12k|Δ| + Cr^{2−N₁η} + Cr⁵)/|d_S| ≤ λ := C(k/r + r^{1/4})`. (Alternatively, the intrinsic polynomial identity +`det K_S − det K_M = 12k·det D_Θ²f(0) + r²Π(f₄, J', k)` for pinned jets — verified exactly by the referee pass on a +general pinned degree-8 polynomial in `d = 2` — gives the same bound without `T`.) The event `{|I(J')| ≤ λ}` is +`J'`-measurable and contains `{W_r > 0} ∩ B₁ ∩ E₁`, so +`P_Q({W_r > 0} ∩ B₁ ∩ E₁) ≤ E[1{|I(J')| ≤ λ}P_Q(f₄ ∈ I(J') | J')] ≤ C₀λ`. + +*Small determinant.* Under `Q`, `D_Θ²f(0)` is a nondegenerate Gaussian symmetric matrix and `Δ` a polynomial of degree `m` +in it. `E_Q[Δ²] ≥ c > 0` uniformly in `(r, b, k, frame)`: the top (`m`-th) Wiener-chaos component of `det(μ + G)` is the +Wick product of `det G`, which does not depend on the mean `μ`. So by the Carbery–Wright inequality +`P_Q(|Δ| < r^{1/2}) ≤ Cr^{1/(2m)}`. + +*Weights.* On the typed event, `W_r/r² ≤ C(k²Δ₀² + r⁴(1 + k)²T^{2N₁})` (#198 (3.2)), whose `L²` norm is `≤ C(k² + r⁴)P^N`. +By Cauchy–Schwarz, `E_Q[(W_r/r²)1_{B₁∩E₁}] ≤ C(k² + r⁴)P^N·min(1, C₀λ)^{1/2} ≤ C(k² + r⁴)P^N(min(1, k/r)^{1/2} + r^{1/8})`, +`E_Q[(W_r/r²)1_{B₁^c}] ≤ C(k² + r⁴)P^N r^{1/(4m)}` and `E_Q[(W_r/r²)1_{E₁^c}] ≤ C(k² + r⁴)P^N r^q`. Summing gives (5.1). ∎ + +## 6. Proof of Theorem CU + +Fix `S ≥ 1` and `ε₀ ∈ (0, β/(12(3 + β)))` with `β` of Lemma CU.5, put `ρ₁ := ℓ^{1/12 − ε₀}` and take `ℓ` so small that +`Sℓ^{1/4} < ρ₁ ≤ min(r₁, r_0^*)` and `Sℓ^{1/4} ≤ r_0^{[K]}`. Split + + ν_eld(ℓ) − cℓ^{−1/3} = I₁ + I₂ + I₃ + I₄ + I₅, + +`I₁ := ∫_{r 1/4` by the choice of `ε₀`. So `|I₂| ≤ Cℓ^{1/4}S^{−7} + o(ℓ^{1/4})`. + +*`I₃, I₄, I₅`.* #198 Lemma B (B.2) gives `0 ≤ I₃ ≤ Cℓ^{1/3}(ρ₁^{−1} + ℓρ₁^{−5}) = O(ℓ^{1/4+ε₀})`; `|I₄| ≤ Cℓ²ρ₁^{−7} = o(ℓ^{1/4})`; +#198 Lemma F′ gives `0 ≤ I₅ ≤ Cℓ^{1/3}`. + +*Conclusion.* `ℓ^{−1/4}(ν_eld(ℓ) − cℓ^{−1/3})` has `limsup` and `liminf` within `CS^{−7}` of +`∫∫∫_0^S(𝒜^{eld} − 𝒜^{con}) ds db dσ`. The integrand equals `−12π₀E₀[loss(s) | b]` with +`loss(s) := Y²1{|Y| < 2s^{−4}|Δ|} + 36s^{−8}Δ²1{|Y| ≥ 2s^{−4}|Δ|}` on `{A < 0}` (and `0` off it); `loss ≥ 0` and, with +`s₁ := (2|Δ|/|Y|)^{1/4}`, + + ∫_0^∞ loss(s) ds = Y²s₁ + (36/7)Δ²s₁^{−7} = (16/7)Y²s₁ = (16/7)·2^{1/4}|Y|^{7/4}|Δ|^{1/4} (6.1) + +(since `36Δ²s₁^{−8} = 9Y²`; checker C6). By Tonelli the `s`-integral over `(0, ∞)` converges absolutely; letting +`S → ∞` gives (CU.1) with `c₁ = −12∫∫π₀E₀[(16/7)2^{1/4}|Y|^{7/4}|Δ|^{1/4}1{A < 0} | b] db dσ`, which is (CU.2). The +expectation is finite (Gaussian moments; `π₀` has Gaussian decay in `b`) and strictly positive because `Y ≠ 0`, +`A < 0` has positive probability under the nondegenerate contact law; so `c₁ < 0`. ∎ + +## 7. Remarks + +1. **The crossover.** At fixed `κ` the elder share of the typed weight is the weight of `{|φ| < 1/3}` inside + `{|φ| < 1}`, with weight `36κ²Δ²(1 − φ²)`. As `κ → ∞` (the cap regime `r ≪ k`) the rejected weighted mass concentrates + on `|Δ| ≲ |Y|/κ` and is `≍ 1/κ = r/k`: the scaling `r³/k` of [C7-K] (K2) for the mass, i.e. a rejected *fraction* + `≍ (r/k)³`, the `Θ(r³)` of [P] Theorem A and Math-#149 at fixed `k` (a referee pass found rejected fraction `× κ³ → 0.0106` + in `d = 2`). As `κ → 0` the typed weight itself vanishes relative to the contact weight (it is `O(κ)` of it), and + within it the elder share tends to `13/27`: since `∫_{|y| 0`. A proof needs the rate of the + `B_{d,L}`-part (the candidate kernel at `k → 0` over intermediate separations and the far candidate kernel), which + is not supplied here. +3. **Numerical values (exploration, not controls).** For the Gaussian kernel `e^{−|z|²/2}` on `R^d` (the periodized + SIDE24 covariance differs by factors `1 + O(e^{−288})`), the parity form of (CU.2) reduces to + `c₁ = −(16/7)2^{1/4}π^{−(d−1/2)}J_d`, `J_d := E[|Y|^{7/4}|Δ|^{1/4}1{A<0} | V_u = 0, G = 0, t_u = 0]` (`d = 2, 3`), with the + conditional laws of checker C7. `d = 2`: `c = 0.0734069193`, `c₁ = −0.26939883`, `c₁/c = −3.669938` (composite + Gauss–Legendre quadrature, two resolutions agreeing to ten digits; confirmed by an independent referee quadrature); + `d = 3`: `c = 0.0417759318`, `c₁ = −0.21184835`, `c₁/c = −5.071062` (deterministic quadrature over the negative-definite + cone, two resolutions agreeing to `10^{−8}` relative), cross-checked by the shipped fixed-seed Monte Carlo over the same + law (`−0.21163 ± 0.00014`, 1.6 standard errors), a development Monte Carlo (`−0.21188 ± 0.00010`, `2·10⁸` samples) and + an independent referee conditional Monte Carlo (`−0.211848 ± 0.000024`, `5·10⁸` samples). So + `ν_{3,24}(ℓ) ≈ 0.041776ℓ^{−1/3} − 0.21185ℓ^{1/4}`: the relative correction `5.071ℓ^{7/12}` is 1% at `ℓ ≈ 2.3·10^{−5}` and + 10% at `ℓ ≈ 1.2·10^{−3}`. The exploration also + tests (3.1) against the actual global elder rule of random pinned plane quintics (graded-grid union-find maximin of + the rescaled field): at `r = 0.01` all 32 typed random cases agree with `|φ| < 1/3`, and in targeted cases at + `r = 0.003` the boundary lies between `|φ| = 0.32` and `0.345`. (Development runs at `r = 0.05`, not shipped, disagreed + only within `O(r/κ)` of `|φ| = 1/3` or `|φ| = 1`.) +4. **For the manuscript.** Once #191, #198 and this note are reviewed, the abstract can read + `ν_{3,24}(ℓ) = c_{3,24}ℓ^{−1/3} + c₁ℓ^{1/4} + o(ℓ^{1/4})` with `c₁ < 0` the explicit Gaussian integral (CU.2), numerically + `c₁ ≈ −0.2118`. Until then V3 edit E10 (`(1 + O(ℓ^{1/3}))`) is the safe statement; E12/E13 are strengthened by this + note at candidate status. +5. **The collision field.** `𝔓` is the zoom-in limit at the scale where a maximum and a saddle are about to annihilate + with gap comparable to the fourth power of their separation. Theorems CU.1–CU.3 are deterministic statements about + `C⁵` functions: any field whose rescaled pinned landscape is `C²`-close to `𝔓` has the elder window `1/3` there, and + only the law of `(f₄, γ, A)` — hence `c₁` — depends on the field. Theorem CU.4, Lemma CU.5 and §6 are proved here only + for the [P] torus field. +6. **Consistency.** (i) #198 Remark 1's formal candidate computation is reproduced exactly (window `|Y| < 6κ|Δ|`, loss + `min(Y², 36κ²Δ²)`), and its elder inequality `I^{eld} ≤ I^{cand}` is sharpened to `c₁ = 2·3^{−1/4}I^{cand}`. (ii) The + typed window `|φ| < 1` is #198 Lemma W's sign window. (iii) Theorem CU is consistent with #198's `O(ℓ^{1/4})` and #191's + `o(1)` (it uses the same inputs, so this is not an independent confirmation of them). + +## 8. Sources (exact identities in `SOURCES.json`) + +| Tag | Path | Role | +|---|---|---| +| [R] | `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (blob `247b3ecf`) | (R3)–(R5), §4 scaled Hessians, Lemma R3.2 — consumed | +| [C7-K] | `frontiers/c7_total_bounded_20260929/PROOF.md` (blob `28748b08`) | (K2); §4 `1 − e ≤ 1_{G_r^c}` — consumed | +| [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d`) | §2 finite-jet rank, §4 regression, §8 maximin, §§9–10 as [E2], (10.2), §15 (15.2) — consumed | +| [E1] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` (blob `213594d6`) | reading rule | +| [E2] | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` (blob `fe9b9ce4`) | §9 replacement | +| [REC] | `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` (blob `75da2597`) | reading rule | +| [182] | `frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md` (blob `0d401877`) | §3 barrier — consumed through #198 Lemmas B, F′ | +| [Z] | `frontiers/c7_zero_gap_limit_20260929/PROOF.md` (blob `5b6328ea`) | (Z10), (Z13)–(Z14), `r_0^{[Z]}` — consumed through #198, #191 | +| #191 | `frontiers/remainder_vanishing_20260930/PROOF.md` (unmerged; v1.1 blob `441152df`) | (E.1), Lemma E Steps 1–3 and `T`, §2 (2.1), `r_0^*` — **consumed, unmerged** | +| #198 | `frontiers/remainder_rate_20260930/PROOF.md` (unmerged; v1.1 blob `abfb98ae`) | §1 elder-density identity and Lemma B (B.2), §2 Lemma F′, §3 (3.1)–(3.2) and Lemma W — **consumed, unmerged** | +| [CW] | A. Carbery, J. Wright, *Distributional and L^q norm inequalities for polynomials over convex bodies in R^n*, Math. Res. Lett. 8 (2001) 233–248 | small-ball bound for `det` (Lemma CU.5) — external, classical | + +## 9. Exact controls and exploration + +`cusp_check.py` (stdlib, exact rationals; `RESULTS.json` its output, byte-identical under `-O`): C1 the quartic +ridge identities and factorizations; C2 the elder window by exact 1D maximin for 117 rational `φ`; C3 the fiber +reduction; C4 the cusp limit field for pinned polynomials of degree 6 in `d = 2, 3`; C5 the model's endpoint +determinants and typed window; C6 the cusp integrals (6.1) and the candidate analogue; C7 the Gaussian-kernel +conditional covariances used by the exploration. Mutants M1 (threshold 1/2), M2 (constant 8/7), M3 (drop `γ·Ξ`), +M4 (coefficient 2 in `e₄`) exit 1; an unknown label exits 2. + +**What the controls do not test.** Theorem CU.1 for non-polynomial fields, Proposition CU.3, Theorem CU.4, Lemma CU.5 +and §6 are proved in prose only. `explore_cusp.py` (stdlib, floating point, not replayed by the workflow; output in +`EXPLORE.json`) is exploration: the actual-landscape test of the elder window and the numerical `c₁`. Neither is +acceptance. + +## 10. Review slices + +A: Theorems CU.1–CU.2 — the pinned Taylor bookkeeping, the fiber reduction and the one-dimensional maximin (§§1–2). +B: Proposition CU.3 — the ridge of a perturbed field, the reduction of connectivity, the trap and path arguments, the + margins (§3). +C: Theorem CU.4 and Lemma CU.5 — the kernel limits and the window probability (§§4–5), including the conditional + density of `f₄` and the small-determinant bound. +D: §6 — the decomposition, the dominations, the choice of `ρ₁`, the cusp integral (6.1), the sign of `c₁`; §7. diff --git a/frontiers/cusp_second_order_20261001/README.md b/frontiers/cusp_second_order_20261001/README.md new file mode 100644 index 000000000..b0580bb16 --- /dev/null +++ b/frontiers/cusp_second_order_20261001/README.md @@ -0,0 +1,61 @@ +# The cusp crossover and the second-order term of the elder lifetime law (CL-CUSP-SECOND-ORDER-20261001-v1) + +Author-side proof candidate, Anthropic Claude, 1 October 2026. Scientific effect NONE. Nonauthor review required. +**Depends on Math- #191** (Lemma E, (E.1), §2; v1.1 blob `441152df`) **and Math- #198** (§1, Lemma B (B.2), Lemma F′, +§3 (3.1)–(3.2), Lemma W; v1.1 blob `abfb98ae`), both unmerged author-side candidates of this session: this packet cannot +be integrated before them and must be rebound if either changes. Everything else consumed is merged: [R], [C7-K], the +reconciled [P] chain directly; [182] and [Z] transitively, through #191 and #198. + +**Statement (Theorem CU).** For the periodized Gaussian field on the fixed torus (every `d ≥ 2`, `L > 0`) and the +canonical marked Kac–Rice version of the elder lifetime density, + + ν_eld(ℓ) = c ℓ^{−1/3} + c₁ ℓ^{1/4} + o(ℓ^{1/4}), + c₁ = −(192/7)·2^{1/4} ∫_{S^{d−1}}∫_R π₀(u; v₀(b,0)) E₀[|Y|^{7/4}|Δ|^{1/4}1{A < 0} | b] db dσ(u) < 0, + +`Y = (f₄/12)Δ − γᵀadj(A)γ/4`, `Δ = det A`, built from the jets `f₄ = ∂_u⁴f(0)`, `γ = ∇_Θ∂_u²f(0)`, `A = D_Θ²f(0)` under the +zero-gap contact law. #198 had the order `O(ℓ^{1/4})`; this packet identifies the term and its sign. For the manuscript's +field (`d = 3`, `L = 24`) the exploration gives `c₁ ≈ −0.21185`, `c₁/c ≈ −5.071`: +`ν_{3,24}(ℓ) = c_{3,24}ℓ^{−1/3}(1 − 5.071ℓ^{7/12} + o(ℓ^{7/12}))`, a 1% correction at `ℓ ≈ 2.3·10⁻⁵`. + +**Mechanism — the cusp crossover.** At the scale `r ≍ k` (gap `ℓ = kr³` comparable to `r⁴`; `κ := k/r = ℓ/r⁴`), rescaled by +`x = rX` (axis), `y = r²Ξ` (transverse), heights `r⁴`, the pinned field converges to a universal random polynomial +`𝔓 = 2κ(X + ½)²(X − 1) + (f₄/24)(X² − ¼)² + ½(X² − ¼)γ·Ξ + ½ΞᵀAΞ` (Theorem CU.1). Maximizing over the concave transverse +fibers reduces its superlevel topology exactly to the quartic ridge `g(X) = κ[2(X + ½)²(X − 1) + 3φ(X² − ¼)²]`, +`φ = Y/(6κΔ)` (Theorem CU.2). The pair is typed iff `|φ| < 1` (the sign window of #198 Lemma W) and **`S` is `M`'s elder +partner iff `|φ| < 1/3`**: the critical values factor as `g(X₃) + κ = κ(φ + 1)³(3φ − 1)/(16φ³)` and +`g(X₃) = κ(φ − 1)³(3φ + 1)/(16φ³)`, and at `φ = ±1/3` the two competing saddles (resp. maxima) have equal heights. The +decision is stable under `C²`-small perturbations in a fixed window, by compact-window path-and-trap arguments only +(Proposition CU.3), so the typed and elder kernels converge at fixed `κ` (Theorem CU.4); a window-probability bound +(Lemma CU.5: `E[W_r/r²] ≤ C(k² + r⁴)(min(1, k/r)^{1/2} + r^β)P^N`) controls the intermediate separations; and the cusp +integral `∫₀^∞ loss ds = (16/7)2^{1/4}|Y|^{7/4}|Δ|^{1/4}` gives `c₁`. The scale `r ≍ k` interpolates between the cap +regime (`κ → ∞`: rejected fraction `≍ (r/k)³`, the `Θ(r³)` of [P] Theorem A) and the small-`κ` end, where the typed +weight is `O(κ)` of the contact weight and the elder share of it tends to `13/27` (Remark 1; Theorem Z's +all-rejected statement is the different, non-commuting limit `k → 0` at fixed `r`). + +**Not claimed:** a rate for the `o(ℓ^{1/4})`; certified numerical values (the numbers are exploration); the candidate +and rejected expansions (formal, Remark 2); uniformity in `d, L`. + +**Files.** `PROOF.md`. `cusp_check.py` (stdlib exact controls; `RESULTS.json` its output, byte-identical under `-O`; +mutants M1–M4 exit 1, an unknown label exits 2): C1 the quartic-ridge identities and boundary factorizations; C2 the +elder window by exact 1D maximin for 117 rational `φ`; C3 the fiber reduction; C4 the cusp limit field for pinned +polynomials of degree 6 in `d = 2, 3`; C5 the model's endpoint determinants and typed window; C6 the cusp integrals; +C7 the Gaussian-kernel conditional covariances used by the exploration. `explore_cusp.py` (stdlib, floating point, +**not a control**, not replayed by the workflow, about a minute; `EXPLORE.json` its output): X1 the actual global elder +rule of random pinned plane quintics at `r = 0.01` agrees with `|φ| < 1/3` in 32/32 typed cases, and in targeted cases at +`r = 0.003` the boundary lies between `|φ| = 0.32` and `0.345`; X2 `c₁` for the Gaussian kernel — `d = 2`: `−0.26939883`, +`c₁/c = −3.669938` (Gauss–Legendre quadrature, two resolutions agree to ten digits); `d = 3`: `−0.21184835`, +`c₁/c = −5.071062` (deterministic quadrature over the negative-definite cone, two resolutions agree to `10⁻⁸` +relative; a fixed-seed Monte Carlo over the same law gives `−0.21163 ± 0.00014`). `SOURCES.json` (exact identities; +#191/#198 recorded as consumed-unmerged and verified by the workflow by blob). + +**Pre-submission referee record (same author family; not acceptance).** Two clean-context Claude referee passes read +v1 before this PR. Slices A/B confirmed CU.1–CU.2 symbolically and by an independent two-dimensional maximin of the +full model (34 cases) and found closeable gaps in Proposition CU.3 (all real zeros of `g'` in Step 5, the margin +`−2κ − 1` in Step 3, `X_R := min(2, X₃ − τ)`); slices C/D found closeable gaps in Lemma CU.5 (`E₁` must control `T`; the +`J'`-measurable window event; the Carbery–Wright uniformity) and a false remark (the small-`κ` elder share is `13/27`, +not `0`), and independently reproduced `c₁` (`d = 2` quadrature; `d = 3` Monte Carlo `−0.211848 ± 0.000024`). Every +finding is applied in this version. + +**What the controls do not test.** Theorem CU.1 for non-polynomial fields, Proposition CU.3, Theorem CU.4, +Lemma CU.5 and the assembly (§6) are proved in prose only. Review slices (PROOF.md §10): A the cusp field and the +model; B stability; C the kernel limits and the window probability; D the assembly and the remarks. diff --git a/frontiers/cusp_second_order_20261001/RESULTS.json b/frontiers/cusp_second_order_20261001/RESULTS.json new file mode 100644 index 000000000..d56f88be9 --- /dev/null +++ b/frontiers/cusp_second_order_20261001/RESULTS.json @@ -0,0 +1,68 @@ +{ + "checks": { + "C1_quartic_ridge_identities": { + "info": { + "boundary_identities": 6, + "parameter_points": 21 + }, + "passed": true + }, + "C2_elder_window_exact_maximin": { + "info": { + "elder": 39, + "grid": "1/8 on [-40,40] plus critical points", + "phi_values": 117, + "rejected": 78 + }, + "passed": true + }, + "C3_fiber_reduction": { + "info": { + "points": 144 + }, + "passed": true + }, + "C4_cusp_limit_field_pinned_polynomials": { + "info": { + "degree": 6, + "dimensions": [ + 2, + 3 + ], + "points": 24 + }, + "passed": true + }, + "C5_model_endpoint_determinants_and_typed_window": { + "info": { + "cases": 60 + }, + "passed": true + }, + "C6_cusp_integrals": { + "info": { + "candidate_constant": "(8/7) 6^(1/4)", + "elder_constant": "(16/7) 2^(1/4)", + "points": 3 + }, + "passed": true + }, + "C7_gaussian_kernel_covariances": { + "info": { + "D1": "4/3", + "d2_even_block": "[[8/3, 2], [2, 30]]", + "d2_gamma_var": "2", + "d3_gamma_cov": "2 I", + "d3_regression": "3/5, 3/5, 0", + "d3_residual": "138/5", + "d3_transverse_block": "[[8/3, 2/3, 0], [2/3, 8/3, 0], [0, 0, 1]]", + "tau2": "6" + }, + "passed": true + } + }, + "mutant": null, + "object": "CL-CUSP-SECOND-ORDER-20261001-v1", + "passed": true, + "scientific_effect": "NONE" +} diff --git a/frontiers/cusp_second_order_20261001/SOURCES.json b/frontiers/cusp_second_order_20261001/SOURCES.json new file mode 100644 index 000000000..67eaa3f3c --- /dev/null +++ b/frontiers/cusp_second_order_20261001/SOURCES.json @@ -0,0 +1,165 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "object": "CL-CUSP-SECOND-ORDER-20261001-v1", + "kind": "author-side proof candidate: Theorem CU, nu_eld = c ell^-1/3 + c_1 ell^1/4 + o(ell^1/4) with c_1 = -(192/7) 2^(1/4) J < 0, via the cusp limit field (CU.1), the quartic-ridge model and its elder window |phi| < 1/3 (CU.2), stability (CU.3), kernel limits (CU.4) and the window probability (CU.5); 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Lett. 8 (2001) 233-248", + "use": "small-ball bound for det (Lemma CU.5)" + } + ], + "mutants": [ + "M1", + "M2", + "M3", + "M4" + ], + "replayed": [ + "cusp_check.py" + ], + "not_replayed": [ + "explore_cusp.py" + ], + "referee": { + "kind": "pre-submission clean-context referee passes, same author family (Anthropic Claude subagents); NOT acceptance, NOT independent", + "passes": [ + { + "slices": "A, B (Theorems CU.1-CU.2, Proposition CU.3)", + "result": "CU.1-CU.2 confirmed symbolically and by an independent two-dimensional maximin of the full model (34 cases); closeable gaps in CU.3 Step 5 (all real zeros of g-prime, X_R := min(2, X_3 - tau), c_delta on the tau-neighbourhoods) and Step 3 (margin -2 kappa - 1); notation and wording; checker docstring count 117" + }, + { + "slices": "C, D (Theorem CU.4, Lemma CU.5, section 6, remarks)", + "result": "closeable gaps in CU.5 (E_1 must control T; the J-prime-measurable window event; Carbery-Wright uniformity via the top Wiener chaos); section 4 details (K_{2 delta}, the C r T bound); a false remark (the small-kappa elder share is 13/27, not 0); transitive dependencies [182], [Z]; c_1 reproduced independently (d = 2 quadrature; d = 3 Monte Carlo -0.211848 +- 0.000024)" + } + ], + "disposition": "every finding applied in v1 as submitted" + }, + "files": [ + { + "path": "PROOF.md", + "bytes": 40307, + "sha256": "28632a87d651785eb1f24e1b92908b8872c3a732987dde34af85199ae0f3ea9f" + }, + { + "path": "cusp_check.py", + "bytes": 26196, + "sha256": "841f375606d68fc68b9e590064990e0846cd062ea31370355800a01c56138932" + }, + { + "path": "RESULTS.json", + "bytes": 1328, + "sha256": "8f2b5a3a5d3556468897c3c51f6935fa3c98a9c3fb515263b60a1de19bc6d2ad" + }, + { + "path": "README.md", + "bytes": 5903, + "sha256": "a50ff18f6eb66d36857d5f92bc27eb9ed10c42cb1e2c28544161200cd58d054b" + }, + { + "path": "explore_cusp.py", + "bytes": 14953, + "sha256": "8c2072c9f23760f63ecf1b4c420d1272f12dad3cc77711c48dcdacd53a2733e4" + }, + { + "path": "EXPLORE.json", + "bytes": 2422, + "sha256": "ba77febb6b44c779363ecd4ce06c4bbd45df2b350513c931e05d649aca719032" + } + ] +} diff --git a/frontiers/cusp_second_order_20261001/cusp_check.py b/frontiers/cusp_second_order_20261001/cusp_check.py new file mode 100644 index 000000000..0792e9fdd --- /dev/null +++ b/frontiers/cusp_second_order_20261001/cusp_check.py @@ -0,0 +1,627 @@ +#!/usr/bin/env python3 +"""Exact finite controls for PROOF.md (CL-CUSP-SECOND-ORDER-20261001-v1). Standard library only; exact rationals. + + C1 the quartic ridge of Theorem CU.2: g(X) = kappa[2(X+1/2)^2(X-1) + 3 phi (X^2-1/4)^2] has + g'(X) = 6 kappa (X^2-1/4)(1+2 phi X), g(-1/2) = 0, g(1/2) = -kappa, g''(-1/2) = 6 kappa (phi-1), + g''(1/2) = 6 kappa (phi+1); at X3 = -1/(2 phi): g(X3) + kappa = kappa (phi+1)^3 (3 phi-1)/(16 phi^3) and + g(X3) = kappa (phi-1)^3 (3 phi+1)/(16 phi^3); at phi = 1/3, g - g(1/2) = (kappa/16)(2X-1)^2(2X+3)^2 and at + phi = -1/3, g = -(kappa/16)(2X-3)^2(2X+1)^2 -- all as exact polynomial identities at rational kappa, phi; + C2 the elder window: for the 117 rational phi = n/60, |n| <= 59, n != +-20 (the two boundaries phi = +-1/3 excluded), + the maximin connection level of M = -1/2 for g, computed by an exact one-dimensional union-find on a rational + grid that contains every critical point (so it is the exact maximin, not an approximation), equals g(1/2) = + -kappa iff |phi| < 1/3; typed (g''(-1/2) < 0 < g''(1/2)) iff |phi| < 1; mutant M1 asserts the threshold 1/2; + C3 the fiber reduction: P(X,Y) = g(X) + (1/2)(Y - Y*)^T A (Y - Y*) with Y* = -(1/2)(X^2-1/4) A^{-1} gamma and + e4 = f4 - 3 gamma^T A^{-1} gamma, exactly at rational points for m = 1, 2 (A negative definite rational); + mutant M4 uses the coefficient 2 in place of 3; + C4 the cusp limit field (Theorem CU.1): for pinned polynomials of degree <= 6 in d = 2 and d = 3 (the + 2(d+1) pinned coefficients solved exactly as polynomials in r from f(M) = b, f(S) = b - kappa r^4, + grad f(M) = grad f(S) = 0), r^-4 [f(r X, r^2 Y) - b] is a polynomial in r whose constant term is exactly + P(X,Y) = 2 kappa (X+1/2)^2(X-1) + (f4/24)(X^2-1/4)^2 + (1/2)(X^2-1/4) gamma.Y + (1/2) Y^T A Y, with (f4, gamma, A) + the polynomial's jets at 0; mutant M3 drops the gamma.Y term; + C5 the endpoint determinants of the model: det Hess P(M) = Y - 6 kappa Delta, det Hess P(S) = Y + 6 kappa Delta, + Y = (f4/12) Delta - gamma^T adj(A) gamma/4, and the inertia of Hess P at M and S (exact congruence) is + (d negative) and (d-1 negative) iff |phi| < 1 with phi = Y/(6 kappa Delta), m = 1, 2; + C6 the cusp integrals: int_0^oo loss_eld ds = (16/7) Y^2 s1, s1 = (2|Delta|/|Y|)^(1/4), equal to + (16/7) 2^(1/4) |Y|^(7/4) |Delta|^(1/4) (checked through the exact fourth-power identity), and the candidate + analogue (8/7) 6^(1/4) |Y|^(7/4) |Delta|^(1/4), at rational points where the fourth roots are rational; + mutant M2 uses 8/7 for the elder constant; + C7 the Gaussian-kernel conditional covariances used by the exploration (exact Schur complements): d = 2: + (f_yy, f_xxxx) | (f_xx, f_xy) has covariance [[8/3, 2], [2, 30]], f_xxy | (f_x, f_y, f_xxx) variance 2, + Var(f_xxx | grad f) = 6, D_1 = 4/3; d = 3: (f_yy, f_zz, f_yz) | (f_xx, f_xy, f_xz) has covariance + [[8/3, 2/3, 0], [2/3, 8/3, 0], [0, 0, 1]], f_xxxx | (f_yy, f_zz, f_yz, f_xx, f_xy, f_xz) has regression + coefficient 3/5 on each of f_yy, f_zz and residual variance 138/5, and (f_xxy, f_xxz) | (grad f, f_xxx) has + covariance 2 I. + M mutants (--mutant M1|M2|M3|M4) exit 1; an unknown label exits 2. + Scope: C1-C7 check exact identities and finitely many rational instances. They do NOT test Theorem CU.1 for + non-polynomial fields, Proposition CU.3 (stability), Theorem CU.4 (Gaussian kernel limits), Lemma CU.5 (window + probability) or the assembly of Theorem CU; those are proved in prose only. The controls are not acceptance. +""" +import argparse +import itertools +import json +import sys +from fractions import Fraction as F + +HALF = F(1, 2) +QUARTER = F(1, 4) + + +# ------------------------------------------------------------------------------------------ small exact helpers + +def det(M): + n = len(M) + A = [row[:] for row in M] + d = F(1) + for i in range(n): + p = next((r for r in range(i, n) if A[r][i] != 0), None) + if p is None: + return F(0) + if p != i: + A[i], A[p] = A[p], A[i] + d = -d + d *= A[i][i] + for r in range(i + 1, n): + fct = A[r][i] / A[i][i] + for c in range(i, n): + A[r][c] -= fct * A[i][c] + return d + + +def inverse(M): + n = len(M) + A = [M[i][:] + [F(int(i == j)) for j in range(n)] for i in range(n)] + for i in range(n): + p = next(r for r in range(i, n) if A[r][i] != 0) + A[i], A[p] = A[p], A[i] + piv = A[i][i] + A[i] = [x / piv for x in A[i]] + for r in range(n): + if r != i and A[r][i] != 0: + fct = A[r][i] + A[r] = [x - fct * y for x, y in zip(A[r], A[i])] + return [row[n:] for row in A] + + +def adj(M): + n = len(M) + if n == 1: + return [[F(1)]] + dM = det(M) + if dM != 0: + inv = inverse(M) + return [[dM * inv[i][j] for j in range(n)] for i in range(n)] + out = [[F(0)] * n for _ in range(n)] + for i in range(n): + for j in range(n): + minor = [[M[a][b] for b in range(n) if b != i] for a in range(n) if a != j] + out[i][j] = (-1) ** (i + j) * det(minor) + return out + + +def quad(v, M, w): + return sum(v[i] * M[i][j] * w[j] for i in range(len(v)) for j in range(len(w))) + + +def inertia(M): + """(negative, zero, positive) counts by exact symmetric congruence diagonalisation.""" + n = len(M) + A = [row[:] for row in M] + neg = zero = pos = 0 + idx = list(range(n)) + while idx: + piv = next((i for i in idx if A[i][i] != 0), None) + if piv is None: + pair = next(((i, j) for i in idx for j in idx if i < j and A[i][j] != 0), None) + if pair is None: + zero += len(idx) + break + i, j = pair + for c in range(n): + A[i][c] += A[j][c] + for rr in range(n): + A[rr][i] += A[rr][j] + piv = i + p = A[piv][piv] + if p < 0: + neg += 1 + else: + pos += 1 + rest = [i for i in idx if i != piv] + for i in rest: + fct = A[i][piv] / p + for c in range(n): + A[i][c] -= fct * A[piv][c] + for i in rest: + A[i][piv] = A[piv][i] = F(0) + idx = rest + return neg, zero, pos + + +def iroot(nn, k): + if nn < 0: + return None + lo, hi = 0, 1 + while hi ** k <= nn: + hi *= 2 + while lo < hi: + mid = (lo + hi) // 2 + if mid ** k < nn: + lo = mid + 1 + else: + hi = mid + return lo if lo ** k == nn else None + + +def rational_root(x, k): + a, b = iroot(x.numerator, k), iroot(x.denominator, k) + return None if a is None or b is None else F(a, b) + + +# ------------------------------------------------------------------------------ polynomials in one variable + +def padd(p, q): + n = max(len(p), len(q)) + return [(p[i] if i < len(p) else F(0)) + (q[i] if i < len(q) else F(0)) for i in range(n)] + + +def pmul(p, q): + out = [F(0)] * (len(p) + len(q) - 1) + for i, a in enumerate(p): + if a: + for j, b in enumerate(q): + out[i + j] += a * b + return out + + +def pscale(p, c): + return [c * a for a in p] + + +def peval(p, x): + out = F(0) + for a in reversed(p): + out = out * x + a + return out + + +def pderiv(p): + return [i * p[i] for i in range(1, len(p))] or [F(0)] + + +def ppow(p, k): + out = [F(1)] + for _ in range(k): + out = pmul(out, p) + return out + + +def ptrim(p): + p = list(p) + while len(p) > 1 and p[-1] == 0: + p.pop() + return p + + +# ------------------------------------------------------------------------------------------------ the model + +def ridge_poly(kappa, phi): + """g(X)/1 as coefficients in X: kappa [2 (X+1/2)^2 (X-1) + 3 phi (X^2-1/4)^2].""" + a = pmul(ppow([HALF, F(1)], 2), [F(-1), F(1)]) + b = ppow([-QUARTER, F(0), F(1)], 2) + return ptrim(pscale(padd(pscale(a, F(2)), pscale(b, 3 * phi)), kappa)) + + +def check_C1(mutant): + pts = 0 + for kappa in (F(1), F(3, 7), F(5, 2)): + for phi in (F(1, 5), F(-2, 7), F(3, 4), F(-9, 10), F(1, 3), F(-1, 3), F(1, 11)): + g = ridge_poly(kappa, phi) + gp = pderiv(g) + target = pscale(pmul([-QUARTER, F(0), F(1)], [F(1), 2 * phi]), 6 * kappa) + if ptrim(gp) != ptrim(target): + return False, "g' != 6 kappa (X^2-1/4)(1+2 phi X)" + if peval(g, -HALF) != 0 or peval(g, HALF) != -kappa: + return False, 'pinned heights' + gpp = pderiv(gp) + if peval(gpp, -HALF) != 6 * kappa * (phi - 1) or peval(gpp, HALF) != 6 * kappa * (phi + 1): + return False, 'endpoint curvatures' + X3 = -1 / (2 * phi) + if peval(gp, X3) != 0: + return False, 'third critical point' + if peval(g, X3) + kappa != kappa * (phi + 1) ** 3 * (3 * phi - 1) / (16 * phi ** 3): + return False, 'factorisation (phi+1)^3 (3 phi - 1)' + if peval(g, X3) != kappa * (phi - 1) ** 3 * (3 * phi + 1) / (16 * phi ** 3): + return False, 'factorisation (phi-1)^3 (3 phi + 1)' + pts += 1 + # boundary configurations: double roots + gp3 = ridge_poly(kappa, F(1, 3)) + gm3 = ridge_poly(kappa, F(-1, 3)) + dbl_p = pscale(pmul(ppow([F(-1), F(2)], 2), ppow([F(3), F(2)], 2)), kappa / 16) + dbl_m = pscale(pmul(ppow([F(-3), F(2)], 2), ppow([F(1), F(2)], 2)), -kappa / 16) + if ptrim(padd(gp3, [kappa])) != ptrim(dbl_p) or ptrim(gm3) != ptrim(dbl_m): + return False, 'boundary double-root identities' + return True, {'parameter_points': pts, 'boundary_identities': 6} + + +def maximin_1d(xs, vals, iM): + """Exact 1D union-find maximin: death level of the component born at index iM (the level at which it first + merges with a component whose maximum exceeds vals[iM]); -inf if it never does.""" + n = len(xs) + order = sorted(range(n), key=lambda i: -vals[i]) + parent = [-1] * n + top = [None] * n + + def find(a): + while parent[a] != a: + parent[a] = parent[parent[a]] + a = parent[a] + return a + for p in order: + parent[p] = p + top[p] = vals[p] + for q in (p - 1, p + 1): + if 0 <= q < n and parent[q] >= 0: + a, b = find(p), find(q) + if a != b: + if top[a] < top[b]: + a, b = b, a + if parent[iM] >= 0 and find(iM) == b and top[a] > vals[iM]: + return vals[p] + parent[b] = a + return None + + +def check_C2(mutant): + thr = HALF if mutant == 'M1' else F(1, 3) + kappa = F(1) + R = F(40) + base = [F(n, 8) for n in range(-320, 321)] # step 1/8 on [-40, 40] + n_elder = n_rej = 0 + for n in range(-59, 60): + phi = F(n, 60) + if abs(phi) == F(1, 3): + continue + g = ridge_poly(kappa, phi) + pts = set(base) | {-HALF, HALF} + if phi != 0: + X3 = -1 / (2 * phi) + if abs(X3) < R: + pts.add(X3) + xs = sorted(pts) + vals = [peval(g, x) for x in xs] + iM = xs.index(-HALF) + dl = maximin_1d(xs, vals, iM) + elder = (dl is not None and dl == -kappa) + if elder != (abs(phi) < thr): + return False, 'elder window fails at phi = %s (death level %s)' % (phi, dl) + gpp = pderiv(pderiv(g)) + typed = peval(gpp, -HALF) < 0 < peval(gpp, HALF) + if not typed: + return False, 'typed fails inside |phi| < 1 at phi = %s' % phi + n_elder += elder + n_rej += (not elder) + for phi in (F(11, 10), F(-11, 10), F(1), F(-1)): + gpp = pderiv(pderiv(ridge_poly(kappa, phi))) + if peval(gpp, -HALF) < 0 < peval(gpp, HALF): + return False, 'typed holds outside |phi| < 1' + return True, {'phi_values': n_elder + n_rej, 'elder': n_elder, 'rejected': n_rej, 'grid': '1/8 on [-40,40] plus critical points'} + + +def model_P(kappa, f4, gamma, A, X, Y, drop_gamma=False): + m = len(A) + val = 2 * kappa * (X + HALF) ** 2 * (X - 1) + (f4 / 24) * (X * X - QUARTER) ** 2 + if not drop_gamma: + val += HALF * (X * X - QUARTER) * sum(gamma[j] * Y[j] for j in range(m)) + val += HALF * quad(Y, A, Y) + return val + + +def neg_def(m, seed): + """A rational negative definite m x m matrix: -(B B^T + I/2).""" + vals = [F(1, 3), F(-2, 5), F(3, 7), F(1, 2), F(-1, 4), F(2, 9)] + B = [[vals[(seed + i * m + j) % len(vals)] for j in range(m)] for i in range(m)] + return [[-(sum(B[i][k] * B[j][k] for k in range(m)) + (HALF if i == j else 0)) for j in range(m)] for i in range(m)] + + +def check_C3(mutant): + coef = 2 if mutant == 'M4' else 3 + pts = 0 + for m in (1, 2): + for seed in range(3): + A = neg_def(m, seed) + Ainv = inverse(A) + gamma = [F(2, 3), F(-5, 4)][:m] + for kappa, f4 in ((F(1), F(7, 3)), (F(2, 5), F(-11, 2))): + e4 = f4 - coef * quad(gamma, Ainv, gamma) + for X in (F(-2), F(-1, 2), F(1, 3), F(5, 4)): + Ys = [(X * X - QUARTER) * (-HALF) * sum(Ainv[j][k] * gamma[k] for k in range(m)) for j in range(m)] + gX = 2 * kappa * (X + HALF) ** 2 * (X - 1) + (e4 / 24) * (X * X - QUARTER) ** 2 + for Y in ([F(1, 2), F(-3, 2)][:m], [F(0), F(2)][:m], [F(-7, 5), F(1, 9)][:m]): + Z = [Y[j] - Ys[j] for j in range(m)] + if model_P(kappa, f4, gamma, A, X, Y) != gX + HALF * quad(Z, A, Z): + return False, 'fiber identity fails (m = %d)' % m + pts += 1 + return True, {'points': pts} + + +# ------------------------------------------------------------ C4: pinned polynomials and the cusp limit field + +def monomials(m, deg): + out = [] + for total in range(deg + 1): + for i in range(total + 1): + for js in itertools.product(range(total - i + 1), repeat=m): + if sum(js) == total - i: + out.append((i, js)) + return out + + +def pinned_poly_in_r(m, kappa, b, free): + """Coefficients c[(i, js)] as polynomials in r (lists), with the 2(d+1) pinned ones solved exactly. + M = (-r/2, 0), S = (r/2, 0); f(M) = b, f(S) = b - kappa r^4, grad f(M) = grad f(S) = 0. h = r/2.""" + zero_js = tuple([0] * m) + c = {key: [val] for key, val in free.items()} + hpow = lambda k: [F(0)] * k + [F(1, 2 ** k)] # h^k = r^k / 2^k as a polynomial in r + ax = lambda i: c.get((i, zero_js), [F(0)]) + # transverse gradient pins: psi_j(x) = sum_i c[(i, e_j)] x^i vanishes at x = +-h + for j in range(m): + ej = tuple(1 if t == j else 0 for t in range(m)) + c0 = [F(0)] + c1 = [F(0)] + for i in range(2, 7): + ci = c.get((i, ej), [F(0)]) + if i % 2 == 0: + c0 = padd(c0, pscale(pmul(ci, hpow(i)), F(-1))) + else: + c1 = padd(c1, pscale(pmul(ci, hpow(i - 1)), F(-1))) + c[(0, ej)] = c0 + c[(1, ej)] = c1 + # axial pins + c2 = [F(0)] + for i in range(4, 7): + if i % 2 == 0: + c2 = padd(c2, pscale(pmul(ax(i), hpow(i - 2)), F(-i, 2))) + c3 = pscale(hpow(1), 4 * kappa) # 4 kappa h + for i in range(5, 7, 2): + c3 = padd(c3, pscale(pmul(ax(i), hpow(i - 3)), F(1 - i, 2))) + c1 = pscale(hpow(3), -8 * kappa) + for i in range(5, 7, 2): + c1 = padd(c1, pscale(pmul(ax(i), hpow(i - 1)), F(-1))) + c1 = padd(c1, pscale(pmul(c3, hpow(2)), F(-1))) + c0 = padd([b], pscale(hpow(4), -8 * kappa)) # b - kappa r^4 / 2 = b - 8 kappa h^4 + c0 = padd(c0, pscale(pmul(c2, hpow(2)), F(-1))) + for i in range(4, 7, 2): + c0 = padd(c0, pscale(pmul(ax(i), hpow(i)), F(-1))) + c[(0, zero_js)] = c0 + c[(1, zero_js)] = c1 + c[(2, zero_js)] = c2 + c[(3, zero_js)] = c3 + return c + + +def eval_poly_in_r(c, m, x_poly, y_polys): + total = [F(0)] + for (i, js), coef in c.items(): + term = pmul(coef, ppow(x_poly, i)) + for j in range(m): + term = pmul(term, ppow(y_polys[j], js[j])) + total = padd(total, term) + return total + + +def check_C4(mutant): + checked = 0 + vals = [F(1, 3), F(-2, 5), F(3, 7), F(-1, 2), F(5, 4), F(-3, 8), F(2, 9), F(-7, 6), F(1, 5), F(4, 11)] + for m in (1, 2): + mons = monomials(m, 6) + zero_js = tuple([0] * m) + pinned = {(0, zero_js), (1, zero_js), (2, zero_js), (3, zero_js)} + for j in range(m): + ej = tuple(1 if t == j else 0 for t in range(m)) + pinned |= {(0, ej), (1, ej)} + for trial in range(2): + free = {} + for idx, key in enumerate(mons): + if key in pinned: + continue + free[key] = vals[(idx * 7 + trial * 3) % len(vals)] + # make the transverse quadratic negative definite + for j in range(m): + ejj = (0, tuple(2 if t == j else 0 for t in range(m))) + free[ejj] = -abs(free.get(ejj, F(1))) - 1 + if m == 2: + free[(0, (1, 1))] = F(1, 3) # keep the transverse block negative definite + kappa = [F(3, 5), F(7, 4)][trial] + b = F(1, 7) + c = pinned_poly_in_r(m, kappa, b, free) + # jets at 0 + f4 = 24 * free[(4, zero_js)] + gamma = [] + A = [[F(0)] * m for _ in range(m)] + for j in range(m): + ej = tuple(1 if t == j else 0 for t in range(m)) + gamma.append(2 * free.get((2, ej), F(0))) + for k in range(m): + if j == k: + A[j][j] = 2 * free.get((0, tuple(2 if t == j else 0 for t in range(m))), F(0)) + else: + A[j][k] = free.get((0, tuple(1 if t in (j, k) else 0 for t in range(m))), F(0)) + for X in (F(-1), F(1, 3), F(3, 2)): + for Y in ([F(1, 2), F(-1)][:m], [F(-2, 3), F(1, 4)][:m]): + x_poly = [F(0), X] # r X + y_polys = [[F(0), F(0), Y[j]] for j in range(m)] # r^2 Y_j + val = eval_poly_in_r(c, m, x_poly, y_polys) + val = padd(val, [-b]) + val = val + [F(0)] * 6 + if any(val[i] != 0 for i in range(4)): + return False, 'negative powers of r in r^-4 [f(rX, r^2 Y) - b] (m = %d)' % m + model = model_P(kappa, f4, gamma, A, X, Y, drop_gamma=(mutant == 'M3')) + if mutant != 'M3': + # the pinned -gamma.Y/8 and the other low-order terms are inside model_P already + pass + if val[4] != model: + return False, 'cusp limit field mismatch (m = %d): %s vs %s' % (m, val[4], model) + checked += 1 + return True, {'points': checked, 'dimensions': [2, 3], 'degree': 6} + + +# ------------------------------------------------------------------------------------ C5: model Hessians + +def check_C5(mutant): + cases = 0 + for m in (1, 2): + for seed in range(3): + A = neg_def(m, seed) + Delta = det(A) + adjA = adj(A) + gamma = [F(3, 4), F(-1, 3)][:m] + for kappa in (F(1, 2), F(2)): + for f4 in (F(-40), F(-3), F(0), F(5), F(60)): + Y = (f4 / 12) * Delta - quad(gamma, adjA, gamma) / 4 + n = m + 1 + HM = [[F(0)] * n for _ in range(n)] + HS = [[F(0)] * n for _ in range(n)] + HM[0][0] = -6 * kappa + f4 / 12 + HS[0][0] = 6 * kappa + f4 / 12 + for j in range(m): + HM[0][j + 1] = HM[j + 1][0] = -gamma[j] / 2 + HS[0][j + 1] = HS[j + 1][0] = gamma[j] / 2 + for k in range(m): + HM[j + 1][k + 1] = HS[j + 1][k + 1] = A[j][k] + if det(HM) != Y - 6 * kappa * Delta or det(HS) != Y + 6 * kappa * Delta: + return False, 'endpoint determinants' + phi = Y / (6 * kappa * Delta) + typed = inertia(HM)[0] == m + 1 and inertia(HS)[0] == m and inertia(HS)[1] == 0 + if typed != (abs(phi) < 1): + return False, 'typed window != |phi| < 1 at phi = %s' % phi + cases += 1 + return True, {'cases': cases} + + +# -------------------------------------------------------------------------------------- C6: cusp integrals + +def check_C6(mutant): + const_e = F(8, 7) if mutant == 'M2' else F(16, 7) + pts = 0 + for Yv, Dv in ((F(1), F(8)), (F(1, 2), F(81, 32)), (F(2), F(1)), (F(3), F(3, 2))): + s1 = rational_root(2 * Dv / Yv, 4) + sc = rational_root(6 * Dv / Yv, 4) + if s1 is not None: + integral = Yv * Yv * s1 + F(36, 7) * Dv * Dv * s1 ** -7 + if integral != const_e * Yv * Yv * s1: + return False, 'elder cusp integral' + if (const_e * Yv * Yv * s1) ** 4 != const_e ** 4 * 2 * Yv ** 7 * Dv: + return False, 'elder fourth-power identity' + # the loss at a few s, and its continuity structure (jump at s1 from Y^2 to 9 Y^2) + if 36 * Dv * Dv * s1 ** -8 != 9 * Yv * Yv: + return False, 'jump at s1' + pts += 1 + if sc is not None: + integral = Yv * Yv * sc + F(36, 7) * Dv * Dv * sc ** -7 + if integral != F(8, 7) * Yv * Yv * sc or (F(8, 7) * Yv * Yv * sc) ** 4 != F(8, 7) ** 4 * 6 * Yv ** 7 * Dv: + return False, 'candidate cusp integral' + if 36 * Dv * Dv * sc ** -8 != Yv * Yv: + return False, 'candidate loss continuity at s*' + pts += 1 + if pts < 3: + return False, 'too few exact points' + # ratio of the elder and candidate constants: (16/7) 2^(1/4) / ((8/7) 6^(1/4)) = 2 * 3^(-1/4): check its 4th power + if (F(16, 7) / F(8, 7)) ** 4 * F(2, 6) != F(16, 3): + return False, 'ratio' + return True, {'points': pts, 'elder_constant': '(16/7) 2^(1/4)', 'candidate_constant': '(8/7) 6^(1/4)'} + + +# -------------------------------------------------------------------- C7: Gaussian-kernel covariances + +def kap_n(n): + if n % 2: + return F(0) + dfact = 1 + for t in range(n - 1, 0, -2): + dfact *= t + return F((-1) ** (n // 2) * dfact) + + +def cov(alpha, beta): + """E[d^alpha f(0) d^beta f(0)] for the Gaussian kernel exp(-|z|^2/2).""" + out = F((-1) ** sum(beta)) + for a, bb in zip(alpha, beta): + out *= kap_n(a + bb) + return out + + +def schur(targets, conds): + """Conditional covariance of targets given conds (lists of multi-indices).""" + Cc = [[cov(a, b) for b in conds] for a in conds] + Ct = [[cov(a, b) for b in targets] for a in targets] + X = [[cov(a, b) for b in conds] for a in targets] + if conds: + inv = inverse(Cc) + for i in range(len(targets)): + for j in range(len(targets)): + Ct[i][j] -= sum(X[i][p] * inv[p][q] * X[j][q] for p in range(len(conds)) for q in range(len(conds))) + return Ct + + +def check_C7(mutant): + # d = 2, coordinates (x, y) + fyy, f4, fxx, fxy = (0, 2), (4, 0), (2, 0), (1, 1) + C = schur([fyy, f4], [fxx, fxy]) + if C != [[F(8, 3), F(2)], [F(2), F(30)]]: + return False, 'd=2 even block %s' % C + g = schur([(2, 1)], [(1, 0), (0, 1), (3, 0)]) + if g != [[F(2)]]: + return False, 'd=2 gamma variance' + tau2 = schur([(3, 0)], [(1, 0), (0, 1)]) + if tau2 != [[F(6)]]: + return False, 'tau^2' + D1 = C[0][0] / 2 + if D1 != F(4, 3): + return False, 'D_1' + # d = 3, coordinates (x, y, z): regression of f_xxxx on (f_yy, f_zz, f_yz) after conditioning on (f_xx, f_xy, f_xz) + ev = [(0, 2, 0), (0, 0, 2), (0, 1, 1)] + Cy = schur(ev + [(4, 0, 0)], [(2, 0, 0), (1, 1, 0), (1, 0, 1)]) + Cyy = [row[:3] for row in Cy[:3]] + c4y = [Cy[3][j] for j in range(3)] + beta = [sum(inverse(Cyy)[i][j] * c4y[j] for j in range(3)) for i in range(3)] + resid = Cy[3][3] - sum(beta[i] * c4y[i] for i in range(3)) + if beta != [F(3, 5), F(3, 5), F(0)] or resid != F(138, 5): + return False, 'd=3 regression %s %s' % (beta, resid) + if Cyy != [[F(8, 3), F(2, 3), F(0)], [F(2, 3), F(8, 3), F(0)], [F(0), F(0), F(1)]]: + return False, 'd=3 transverse block' + g3 = schur([(2, 1, 0), (2, 0, 1)], [(1, 0, 0), (0, 1, 0), (0, 0, 1), (3, 0, 0)]) + if g3 != [[F(2), F(0)], [F(0), F(2)]]: + return False, 'd=3 gamma covariance' + return True, {'d2_even_block': '[[8/3, 2], [2, 30]]', 'd2_gamma_var': '2', 'tau2': '6', 'D1': '4/3', + 'd3_transverse_block': '[[8/3, 2/3, 0], [2/3, 8/3, 0], [0, 0, 1]]', 'd3_gamma_cov': '2 I', + 'd3_regression': '3/5, 3/5, 0', 'd3_residual': '138/5'} + + +CHECKS = [('C1_quartic_ridge_identities', check_C1), ('C2_elder_window_exact_maximin', check_C2), + ('C3_fiber_reduction', check_C3), ('C4_cusp_limit_field_pinned_polynomials', check_C4), + ('C5_model_endpoint_determinants_and_typed_window', check_C5), ('C6_cusp_integrals', check_C6), + ('C7_gaussian_kernel_covariances', check_C7)] +MUTANTS = ('M1', 'M2', 'M3', 'M4') + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument('--mutant', default=None) + args = ap.parse_args() + if args.mutant is not None and args.mutant not in MUTANTS: + sys.stderr.write('unknown mutant label\n') + return 2 + results = {} + ok_all = True + for name, fn in CHECKS: + ok, info = fn(args.mutant) + results[name] = {'passed': ok, 'info': info} + ok_all = ok_all and ok + out = {'object': 'CL-CUSP-SECOND-ORDER-20261001-v1', 'scientific_effect': 'NONE', 'passed': ok_all, + 'mutant': args.mutant, 'checks': results} + sys.stdout.write(json.dumps(out, indent=1, sort_keys=True) + '\n') + return 0 if ok_all else 1 + + +if __name__ == '__main__': + sys.exit(main()) diff --git a/frontiers/cusp_second_order_20261001/explore_cusp.py b/frontiers/cusp_second_order_20261001/explore_cusp.py new file mode 100644 index 000000000..40f35d7c5 --- /dev/null +++ b/frontiers/cusp_second_order_20261001/explore_cusp.py @@ -0,0 +1,361 @@ +#!/usr/bin/env python3 +"""EXPLORATION for PROOF.md (CL-CUSP-SECOND-ORDER-20261001-v1). Standard library only, floating point. +NOT a control, NOT evidence of acceptance, NOT replayed by the workflow (it takes about a minute). + + X1 the actual global elder decision of random pinned plane quintics at the cusp scale (r = 0.01, k = kappa r), + computed by a union-find maximin on a graded grid of the rescaled window, against the model window + |phi| < 1/3 (phi = Y/(6 kappa Delta)); random phi, and phi targeted at +-(0.30, 0.32, 0.345, 0.36); + X2 the second-order coefficient c_1 for the Gaussian kernel exp(-|z|^2/2) (L = infinity), with the leading + constants c_2, c_3 for the ratio c_1/c: d = 2 by composite Gauss-Legendre quadrature, d = 3 by a + deterministic quadrature over the negative-definite cone; each at two resolutions (the difference is the + convergence evidence), and d = 3 also by an independent fixed-seed Monte Carlo over the same law. +Output: EXPLORE.json (numbers rounded; seeds fixed). +""" +import json +import math +import random +import sys + +# ------------------------------------------------------------------------------------------------- X1 + +MONO = [(i, j) for i in range(6) for j in range(6) if i + j <= 5] +PINNED = [(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (1, 1)] + + +def solve4(A, b): + n = 4 + M = [A[i][:] + [b[i]] for i in range(n)] + for i in range(n): + p = max(range(i, n), key=lambda r: abs(M[r][i])) + M[i], M[p] = M[p], M[i] + for r in range(n): + if r != i: + f = M[r][i] / M[i][i] + M[r] = [x - f * y for x, y in zip(M[r], M[i])] + return [M[i][n] / M[i][i] for i in range(n)] + + +def build(r, kap, free): + c = dict(free) + c[(0, 1)] = -c[(2, 1)] * r ** 2 / 4 - c[(4, 1)] * r ** 4 / 16 + c[(1, 1)] = -c[(3, 1)] * r ** 2 / 4 + A, rhs = [], [] + for x0, val in ((-r / 2, 0.0), (r / 2, -kap * r ** 4)): + A.append([1, x0, x0 ** 2, x0 ** 3]) + rhs.append(val - c[(4, 0)] * x0 ** 4 - c[(5, 0)] * x0 ** 5) + for x0 in (-r / 2, r / 2): + A.append([0, 1, 2 * x0, 3 * x0 ** 2]) + rhs.append(-4 * c[(4, 0)] * x0 ** 3 - 5 * c[(5, 0)] * x0 ** 4) + c[(0, 0)], c[(1, 0)], c[(2, 0)], c[(3, 0)] = solve4(A, rhs) + return c + + +def feval(c, x, y): + return sum(v * x ** i * y ** j for (i, j), v in c.items()) + + +def hess(c, x, y): + fxx = sum(v * i * (i - 1) * x ** (i - 2) * y ** j for (i, j), v in c.items() if i >= 2) + fyy = sum(v * j * (j - 1) * x ** i * y ** (j - 2) for (i, j), v in c.items() if j >= 2) + fxy = sum(v * i * j * x ** (i - 1) * y ** (j - 1) for (i, j), v in c.items() if i >= 1 and j >= 1) + return fxx, fxy, fyy + + +def death_level(F, ny, nx, iM): + n = ny * nx + order = sorted(range(n), key=lambda p: -F[p]) + parent = [-1] * n + top = [0.0] * n + + def find(a): + while parent[a] != a: + parent[a] = parent[parent[a]] + a = parent[a] + return a + for p in order: + parent[p] = p + top[p] = F[p] + py, px = divmod(p, nx) + for dy in (-1, 0, 1): + for dx in (-1, 0, 1): + if dy == 0 and dx == 0: + continue + qy, qx = py + dy, px + dx + if 0 <= qy < ny and 0 <= qx < nx: + q = qy * nx + qx + if parent[q] >= 0: + a, b = find(p), find(q) + if a != b: + if top[a] < top[b]: + a, b = b, a + if parent[iM] >= 0 and find(iM) == b: + return F[p] + parent[b] = a + return -math.inf + + +def trial(rng, r, kap, phi_target=None, nX=241, nZ=121, X0=-3.5, X1=2.5): + free = {m: rng.gauss(0, 1) for m in MONO if m not in PINNED} + free[(0, 2)] = -abs(free[(0, 2)]) - 0.3 + if phi_target is not None: + a_, g_ = 2 * free[(0, 2)], 2 * free[(2, 1)] + free[(4, 0)] = (6 * kap * phi_target + g_ ** 2 / (4 * a_)) / 2 + c = build(r, kap, free) + fxxM, fxyM, fyyM = hess(c, -r / 2, 0.0) + fxxS, fxyS, fyyS = hess(c, r / 2, 0.0) + typed = (fyyM < 0 and fxxM * fyyM - fxyM ** 2 > 0) and (fxxS * fyyS - fxyS ** 2 < 0) + a, f4, g = 2 * c[(0, 2)], 24 * c[(4, 0)], 2 * c[(2, 1)] + phi = (f4 / 12 - g ** 2 / (4 * a)) / (6 * kap) + if not typed: + return {'typed': False, 'phi': phi} + Xs = [X0 + (X1 - X0) * i / (nX - 1) for i in range(nX)] + c02, c21 = c[(0, 2)], c[(2, 1)] + RZ = 3.0 * math.sqrt(max(1.0, abs(c[(4, 0)]) * 81 + 2 * kap * 27) / abs(c02)) + h0 = 0.01 * math.sqrt((1.0 + kap) / abs(c02)) + du = 2.0 / (nZ - 1) + lo, hi = 1e-6, 50.0 + for _ in range(100): + al = 0.5 * (lo + hi) + if RZ * al / math.sinh(al) * du > h0: + lo = al + else: + hi = al + Zs = [RZ * math.sinh(al * (-1 + du * k)) / math.sinh(al) for k in range(nZ)] + F = [] + for Z in Zs: + for X in Xs: + Ys = -c21 * X * X / (2 * c02) + F.append(feval(c, r * X, r * r * (Ys + Z)) / r ** 4) + ZM = c21 * 0.25 / (2 * c02) + iy = min(range(nZ), key=lambda k: abs(Zs[k] - ZM)) + ix = min(range(nX), key=lambda k: abs(Xs[k] + 0.5)) + while True: + best = (F[iy * nX + ix], iy, ix) + for dy in (-1, 0, 1): + for dx in (-1, 0, 1): + qy, qx = iy + dy, ix + dx + if 0 <= qy < nZ and 0 <= qx < nX and F[qy * nX + qx] > best[0]: + best = (F[qy * nX + qx], qy, qx) + if best[1] == iy and best[2] == ix: + break + iy, ix = best[1], best[2] + dl = death_level(F, nZ, nX, iy * nX + ix) + elder = math.isfinite(dl) and abs(dl + kap) < 0.02 * kap + 5e-3 + return {'typed': True, 'phi': phi, 'kappa': kap, 'death': dl, 'elder': elder, 'model': abs(phi) < 1 / 3} + + +def run_X1(): + rng = random.Random(20261001) + r = 0.01 + rand = [] + for _ in range(40): + kap = math.exp(rng.uniform(math.log(0.1), math.log(5.0))) + rand.append(trial(rng, r, kap)) + T = [x for x in rand if x['typed']] + agree = sum(1 for x in T if x['elder'] == x['model']) + tgt = [] + for phi in (0.30, 0.32, 0.345, 0.36, -0.30, -0.32, -0.345, -0.36): + for _ in range(2): + kap = math.exp(rng.uniform(math.log(0.3), math.log(3.0))) + res = trial(rng, 0.003, kap, phi_target=phi) + tgt.append({'phi_target': phi, 'kappa': round(kap, 3), 'typed': res['typed'], 'elder': res.get('elder')}) + return {'r': r, 'random_trials': len(rand), 'typed': len(T), 'agree_with_window_1_3': agree, + 'disagreements': [[round(x['phi'], 4), round(x['kappa'], 3)] for x in T if x['elder'] != x['model']], + 'targeted_r_0.003': tgt} + + +# ------------------------------------------------------------------------------------------------- X2 + +def hyp1f1_neg(a, b, x): + """1F1(a; b; -x) for x >= 0 via Kummer's transformation e^{-x} 1F1(b-a; b; x) (positive series), with the + large-x asymptotic expansion.""" + if x > 60.0: + # 1F1(a;b;-x) ~ Gamma(b)/Gamma(b-a) x^{-a} sum_n (a)_n (a-b+1)_n / n! x^{-n} + s, term = 1.0, 1.0 + for n in range(12): + term *= (a + n) * (a - b + 1 + n) / ((n + 1) * x) + s += term + return math.gamma(b) / math.gamma(b - a) * x ** (-a) * s + s, term, n = 1.0, 1.0, 0 + while True: + term *= (b - a + n) / ((b + n) * (n + 1)) * x + s += term + n += 1 + if term < 1e-17 * s and n > x: + break + return math.exp(-x) * s + + +P = 7 / 4 +GP = math.gamma((P + 1) / 2) / math.sqrt(math.pi) * 2 ** (P / 2) + + +def eabs_normal(m, v): + """E|X|^{7/4}, X ~ N(m, v).""" + return v ** (P / 2) * GP * hyp1f1_neg(-P / 2, 0.5, m * m / (2 * v)) + + +def gauss_legendre(n): + xs, ws = [], [] + for i in range(1, n + 1): + x = math.cos(math.pi * (i - 0.25) / (n + 0.5)) + for _ in range(100): + p0, p1 = 1.0, x + for k in range(2, n + 1): + p0, p1 = p1, ((2 * k - 1) * x * p1 - (k - 1) * p0) / k + dp = n * (x * p1 - p0) / (x * x - 1) + dx = p1 / dp + x -= dx + if abs(dx) < 1e-16: + break + xs.append(x) + ws.append(2 / ((1 - x * x) * dp * dp)) + return xs, ws + + +def composite(a, b, panels, xs, ws): + out = [] + h = (b - a) / panels + for k in range(panels): + lo = a + k * h + for x, w in zip(xs, ws): + out.append((lo + (x + 1) * h / 2, w * h / 2)) + return out + + +def laguerre(n): + """Gauss-Laguerre nodes and weights (weight e^{-s} on [0, oo)), Newton iteration on L_n.""" + xs, ws = [], [] + for i in range(1, n + 1): + if i == 1: + x = 3.0 / (1 + 2.4 * n) + elif i == 2: + x = xs[0] + 15.0 / (1 + 2.5 * n) + else: + x = xs[-1] + (1 + 2.55 * (i - 2)) / (1.9 * (i - 2)) * (xs[-1] - xs[-2]) + for _ in range(200): + p0, p1 = 1.0, 1.0 - x + for k in range(2, n + 1): + p0, p1 = p1, ((2 * k - 1 - x) * p1 - (k - 1) * p0) / k + dp = n * (p1 - p0) / x + dx = p1 / dp + x -= dx + if abs(dx) < 1e-15 * max(1.0, x): + break + p0, p1 = 1.0, 1.0 - x + for k in range(2, n + 2): + p0, p1 = p1, ((2 * k - 1 - x) * p1 - (k - 1) * p0) / k + xs.append(x) + ws.append(x / ((n + 1) ** 2 * p1 * p1)) + return xs, ws + + +def ey_d2(pw, pg): + """d = 2: (a, f4) ~ N(0, [[8/3, 2], [2, 30]]) given (f_xx, f_xy) = 0; gamma ~ N(0, 2); Y = (f4/12) a - gamma^2/4. + J = E[|Y|^{7/4} |a|^{1/4} 1{a < 0}] by composite Gauss-Legendre in a = -w^4 (pw panels) and gamma (pg panels).""" + Sa, Saf, Sf, Sg = 8 / 3, 2.0, 30.0, 2.0 + mu, s2 = Saf / Sa, Sf - Saf ** 2 / Sa + gx, gw = gauss_legendre(12) + wn = composite(0.0, (2 * Sa * 60) ** 0.125, pw, gx, gw) + gn = composite(-14.0, 14.0, pg, gx, gw) + EY = 0.0 + for w, ww in wn: + a = -w ** 4 + jac = 4 * w ** 3 + pa = math.exp(-a * a / (2 * Sa)) / math.sqrt(2 * math.pi * Sa) + v = (a / 12) ** 2 * s2 + inner = 0.0 + for g, gwt in gn: + m = mu * a * a / 12 - g * g / 4 + inner += gwt * eabs_normal(m, v) * math.exp(-g * g / (2 * Sg)) / math.sqrt(2 * math.pi * Sg) + EY += ww * jac * abs(a) ** 0.25 * pa * inner + return EY + + +def ey_d3(npan, nth, nlag): + """d = 3: the transverse block A given (f_xx, f_xy, f_xz) = 0 has eigenvalues t +- rho, t ~ N(0, 5/3) and + rho ~ Rayleigh(1) independent (checker C7: Var f_yy = Var f_zz = 8/3, Cov = 2/3, Var f_yz = 1); A < 0 iff + t < -rho. gamma ~ N(0, 2 I) is isotropic, so in the eigenbasis -gamma^T adj(A) gamma / 4 = s (|l2| cos^2 th + + |l1| sin^2 th) with s ~ Exp(1), th ~ U[0, pi/2]; f4 | A ~ N((3/5) tr A, 138/5) is integrated in closed form. + J by composite Gauss-Legendre in rho in [0, 8] and u in [0, 14^(1/4)] (t = -rho - u^4: the cone, with the + Delta^{1/4} endpoint singularity removed), trapezoid in th (spectral: a smooth function of cos 2th) and + Gauss-Laguerre in s.""" + gx, gw = gauss_legendre(8) + rn = composite(0.0, 8.0, npan, gx, gw) + un = composite(0.0, 14.0 ** 0.25, npan, gx, gw) + lx, lw = laguerre(nlag) + cs2 = [math.cos(k * (math.pi / 2) / nth) ** 2 for k in range(nth + 1)] + thw = [(0.5 if k in (0, nth) else 1.0) / nth for k in range(nth + 1)] + st2 = 5.0 / 3.0 + EY = 0.0 + for rho, rw in rn: + prho = rho * math.exp(-rho * rho / 2) + for u, uw in un: + w = u ** 4 + t = -rho - w + pt = math.exp(-t * t / (2 * st2)) / math.sqrt(2 * math.pi * st2) + a1, a2 = w, w + 2 * rho + Delta = a1 * a2 + m0 = 0.1 * t * Delta + v = (Delta / 12.0) ** 2 * 27.6 + inner = 0.0 + for c2, tw in zip(cs2, thw): + lam = a2 * c2 + a1 * (1 - c2) + inner += tw * sum(sw * eabs_normal(m0 + s * lam, v) for s, sw in zip(lx, lw)) + EY += rw * uw * 4 * u ** 3 * prho * pt * Delta ** 0.25 * inner + return EY + + +def mc_d3(N, seed=1001, batches=10): + """Independent Monte Carlo cross-check of ey_d3 (same law): rho ~ Rayleigh(1), t from the normal truncated to + the cone t < -rho, weight P(t < -rho | rho); returns (mean, batch standard error).""" + from statistics import NormalDist + nd = NormalDist() + rng = random.Random(seed) + sig_t = math.sqrt(5 / 3) + batch = N // batches + acc, means = 0.0, [] + for i in range(batch * batches): + rho = math.sqrt(-2.0 * math.log(1.0 - rng.random())) + w = nd.cdf(-rho / sig_t) + z = nd.inv_cdf(min(max(rng.random() * w, 1e-300), 1 - 1e-16)) + t = sig_t * z + l1, l2 = t + rho, t - rho + Delta = l1 * l2 + g1, g2 = rng.gauss(0, math.sqrt(2)), rng.gauss(0, math.sqrt(2)) + q = (g1 * g1 * l2 + g2 * g2 * l1) / 4 + acc += w * eabs_normal((0.6 * 2 * t) / 12 * Delta - q, (Delta / 12) ** 2 * 27.6) * Delta ** 0.25 + if (i + 1) % batch == 0: + means.append(acc / batch) + acc = 0.0 + m = sum(means) / len(means) + return m, math.sqrt(sum((x - m) ** 2 for x in means) / (len(means) - 1) / len(means)) + + +def run_X2(): + c2 = math.gamma(7 / 6) * 1.5 ** (1 / 3) * (4 / 3) / (2 * math.sqrt(3) * math.pi ** 1.5) + c3 = math.gamma(7 / 6) * 1.5 ** (1 / 3) * (29 / 6 - math.sqrt(6)) / (2 * math.sqrt(3) * math.pi ** 2.5) + k2 = -(16 / 7) * 2 ** 0.25 / math.pi ** 1.5 # c_1 = k_d J for the Gaussian kernel + k3 = -(16 / 7) * 2 ** 0.25 / math.pi ** 2.5 + J2c, J2 = ey_d2(12, 28), ey_d2(24, 56) + J3c, J3 = ey_d3(8, 8, 32), ey_d3(8, 8, 64) + M3, se3 = mc_d3(8000000) + return {'d2': {'J': round(J2, 10), 'J_coarse': round(J2c, 10), 'c': round(c2, 12), 'c1': round(k2 * J2, 8), + 'c1_over_c': round(k2 * J2 / c2, 6), + 'method': 'composite Gauss-Legendre, a = -w^4: 24x12 by 56x12 nodes (coarse: 12x12 by 28x12)'}, + 'd3': {'J': round(J3, 9), 'J_coarse': round(J3c, 9), 'c': round(c3, 12), 'c1': round(k3 * J3, 8), + 'c1_over_c': round(k3 * J3 / c3, 6), + 'method': 'cone quadrature: Gauss-Legendre 64 x 64 in (rho, u), trapezoid 9 in theta, ' + 'Gauss-Laguerre 64 in s (coarse: Laguerre 32)', + 'mc_check': {'J': round(M3, 5), 'J_se': round(se3, 5), 'c1': round(k3 * M3, 5), + 'c1_se': round(-k3 * se3, 5), + 'method': 'Monte Carlo on the cone, 8000000 samples, seed 1001, 10 batches'}}} + + +if __name__ == '__main__': + out = {'object': 'CL-CUSP-SECOND-ORDER-20261001-v1 exploration', 'not_a_control': True} + if len(sys.argv) < 2 or 'X2' in sys.argv[1:]: + out['X2'] = run_X2() + if len(sys.argv) < 2 or 'X1' in sys.argv[1:]: + out['X1'] = run_X1() + sys.stdout.write(json.dumps(out, indent=1, sort_keys=True) + '\n') From 826b8be512e86f2de73c33d6ac7a76908dc2e0e6 Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 1 Oct 2026 01:22:41 +0000 Subject: [PATCH 2/2] Theorem CU v1.1: candidate and rejected densities; repair Lemma CU.5 window Add Theorem CU' (section 7): nu_cand = c ell^{-1/3} + B_{d,L} + I_cand ell^{1/4} + o(ell^{1/4}) with I_cand = (3^{1/4}/2) c_1 < 0, and rho_rej = B_{d,L} + (I_cand - c_1) ell^{1/4} + o(ell^{1/4}) with I_cand - c_1 > 0, via the gap-Lipschitz Lemma L (the gap enters the pinned regression affinely, [R] (R3)); far pairs contribute O(ell). Checker C8 and mutant M5. Repair Lemma CU.5 after Codex P1 4150765060: the window length is now bounded through the symmetric root identity and the sign window; lambda and every downstream use are unchanged. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .../cusp_second_order_20261001/EXPLORE.json | 10 +- frontiers/cusp_second_order_20261001/PROOF.md | 216 ++++++++++++++---- .../cusp_second_order_20261001/README.md | 42 +++- .../cusp_second_order_20261001/RESULTS.json | 15 +- .../cusp_second_order_20261001/SOURCES.json | 50 ++-- .../cusp_second_order_20261001/cusp_check.py | 190 ++++++++++++++- .../explore_cusp.py | 14 +- 7 files changed, 445 insertions(+), 92 deletions(-) diff --git a/frontiers/cusp_second_order_20261001/EXPLORE.json b/frontiers/cusp_second_order_20261001/EXPLORE.json index ce941a438..0410db441 100644 --- a/frontiers/cusp_second_order_20261001/EXPLORE.json +++ b/frontiers/cusp_second_order_20261001/EXPLORE.json @@ -106,14 +106,17 @@ }, "X2": { "d2": { + "I_cand": -0.1772744, "J": 0.5518754382, "J_coarse": 0.5518754382, "c": 0.073406919306, "c1": -0.26939883, "c1_over_c": -3.669938, - "method": "composite Gauss-Legendre, a = -w^4: 24x12 by 56x12 nodes (coarse: 12x12 by 28x12)" + "method": "composite Gauss-Legendre, a = -w^4: 24x12 by 56x12 nodes (coarse: 12x12 by 28x12)", + "rejected_coefficient": 0.09212443 }, "d3": { + "I_cand": -0.13940406, "J": 1.363390706, "J_coarse": 1.363390722, "c": 0.041775931841, @@ -126,9 +129,10 @@ "c1_se": 0.00014, "method": "Monte Carlo on the cone, 8000000 samples, seed 1001, 10 batches" }, - "method": "cone quadrature: Gauss-Legendre 64 x 64 in (rho, u), trapezoid 9 in theta, Gauss-Laguerre 64 in s (coarse: Laguerre 32)" + "method": "cone quadrature: Gauss-Legendre 64 x 64 in (rho, u), trapezoid 9 in theta, Gauss-Laguerre 64 in s (coarse: Laguerre 32)", + "rejected_coefficient": 0.0724443 } }, "not_a_control": true, - "object": "CL-CUSP-SECOND-ORDER-20261001-v1 exploration" + "object": "CL-CUSP-SECOND-ORDER-20261001-v1.1 exploration" } diff --git a/frontiers/cusp_second_order_20261001/PROOF.md b/frontiers/cusp_second_order_20261001/PROOF.md index e7d3c0ce0..f6df5a735 100644 --- a/frontiers/cusp_second_order_20261001/PROOF.md +++ b/frontiers/cusp_second_order_20261001/PROOF.md @@ -1,17 +1,23 @@ # The cusp crossover: the elder rule at `r ≍ k` and the second-order term `ν_eld(ℓ) = cℓ^{−1/3} + c₁ℓ^{1/4} + o(ℓ^{1/4})` -Object: CL-CUSP-SECOND-ORDER-20261001-v1. +Object: CL-CUSP-SECOND-ORDER-20261001-v1.1 (v1 → v1.1, before any review: v1's formal Remark 2 is proved — Theorem CU′, +the candidate and rejected densities to order `ℓ^{1/4}`, §7, through the gap-Lipschitz Lemma L; a clean-context referee pass +on §7 found no error or gap and seven minor points, all applied; Lemma CU.5's window-length bound repaired after Codex +P1 4150765060 (the root difference is bounded symmetrically through the sign window); checker C8 and mutant M5 added; +§§1–6 otherwise unchanged except one clarifying clause in §4 (iv); remarks, sources, controls and slices renumbered +§§8–11). Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 1 October 2026. Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR REVIEW REQUIRED. Scientific effect: NONE — no register, -graph, STATUS, PROOF_INDEX, prize or Boolean change; the numerical values of §7 are exploration, not certified. +graph, STATUS, PROOF_INDEX, prize or Boolean change; the numerical values of §8 are exploration, not certified. Same GitHub account as every lane; zero organizational independence. **Dependencies (unmerged, consumed):** Math- #191 (`frontiers/remainder_vanishing_20260930/PROOF.md`, v1.1 blob `441152df`: Lemma E (E.1) and Steps 1–3 with its dominating variable `T`, §2 (2.1), the admissible radius `r_0^*`) and Math- #198 (`frontiers/remainder_rate_20260930/PROOF.md`, v1.1 blob `abfb98ae`: §1 the elder-density identity and Lemma B (B.2), §2 Lemma F′, §3 (3.1)–(3.2) and Lemma W). This note cannot be integrated before both and must be rebound -if either changes. Merged inputs: [R], [C7-K], [P] with [E1]/[E2]/[REC] directly; [182] (the deterministic barrier, -inside #198 Lemmas B and F′; its §3 has a Slice-A acceptance from this same author only) and [Z] ((Z10), (Z13)–(Z14) and -`r_0^{[Z]}`, inside #198 and #191) transitively. +if either changes. Merged inputs: [R], [C7-K], [P] with [E1]/[E2]/[REC] directly; [Z] directly in §7 ((Z2)–(Z3), (Z10), +the [P] §14 input of (Z16)) and transitively ((Z10), (Z13)–(Z14) and `r_0^{[Z]}`, inside #198 and #191); [182] (the +deterministic barrier, inside #198 Lemmas B and F′; its §3 has a Slice-A acceptance from this same author only) +transitively. ## 0. Statement @@ -49,7 +55,16 @@ Equivalently, with the parity factorization of [P] §15 (odd and even jets are i `c₁ = −(192/7)2^{1/4}∫_{S^{d−1}} p_G(0)p_{V_u}(0)(2π)^{−1/2}τ_u^{−1} E[|Y|^{7/4}|Δ|^{1/4}1{A<0} | V_u = 0, G = 0, t_u = 0] dσ(u)`, `G = ∇f(0)`, `V_u = (∂_u²f, ∇_Θ∂_uf)(0)`, `t_u = ∂_u³f(0)`, `τ_u² = Var(t_u | G = 0)`. So `ν_eld(ℓ) = cℓ^{−1/3}(1 + (c₁/c)ℓ^{7/12} + o(ℓ^{7/12}))`; for the manuscript's field (`d = 3`, `L = 24`) the exploration of -§7 gives `c₁ ≈ −0.21185`, `c₁/c ≈ −5.071`. +§8 gives `c₁ ≈ −0.21185`, `c₁/c ≈ −5.071`. + +**Theorem CU′ (the candidate and rejected densities; §7).** Under the same hypotheses, with `B_{d,L}` the equal-height +mass of [Z] (Z3), + + ν_cand(ℓ) = c ℓ^{−1/3} + B_{d,L} + I^{cand} ℓ^{1/4} + o(ℓ^{1/4}), I^{cand} = (3^{1/4}/2) c₁ < 0, (CU′.1) + ρ_rej(ℓ) = B_{d,L} + (I^{cand} − c₁) ℓ^{1/4} + o(ℓ^{1/4}), I^{cand} − c₁ = (1 − 3^{1/4}/2)|c₁| > 0. (CU′.2) + +The far pairs contribute `O(ℓ)`: the `ℓ^{1/4}` terms are carried by the cusp scale. The candidate density needs no elder +decision, and its non-cusp remainder is Lipschitz in the gap because the gap enters the pinned law affinely (Lemma L). The mechanism is a **cusp crossover**: rescaled at `x = rX` (axis), `y = r²Ξ` (transverse, `Ξ ∈ R^m`), heights `r⁴`, the pinned field converges to the random polynomial @@ -65,11 +80,10 @@ is stable (Proposition CU.3), so the elder kernel at fixed `κ` converges (Theor (Lemma CU.5) controls the intermediate separations; and the cusp integral gives (CU.2), with `∫_0^∞ loss ds = (16/7)2^{1/4}|Y|^{7/4}|Δ|^{1/4}`. -**What is not claimed.** No rate for the `o(ℓ^{1/4})`; no certified numerical value of `c₁` (§7 is exploration); no -expansion of the candidate or rejected densities (Remark 2 records the formal ones); no uniformity in `d`, `L`; no -finite-radius band; nothing beyond the existential scope of Theorem R, #191 and #198. Conditional on #191, #198 and -this note's review, the exponent question left in #198 Remark 1 is settled: `1/4` is the true order and the -coefficient is negative. +**What is not claimed.** No rate for the `o(ℓ^{1/4})` in (CU.1), (CU′.1) or (CU′.2); no certified numerical value of +`c₁`, `I^{cand}` or `B_{d,L}` (§8 is exploration); no uniformity in `d`, `L`; no finite-radius band; nothing beyond +the existential scope of Theorem R, [Z], #191 and #198. Conditional on #191, #198 and this note's review, the exponent +question left in #198 Remark 1 is settled: `1/4` is the true order and the coefficient is negative. ## 1. The cusp limit field @@ -257,7 +271,8 @@ nondegenerate Gaussian law — so `φ` has a continuous distribution and `P(|φ| `P(λ_max(A) ∈ (−2δ, 0))`, `P(large jets)` all vanish as `δ → 0`. (iv) *Expectations.* On the typed event `W_r/r² = −det K_M det K_S`, and by #198 (3.2) -`W_r/r² ≤ C(κ²r²Δ₀² + r⁴(1 + k)²T^{2N₁})`, so `(W_r/r²)/r²` is bounded in `L²` and uniformly integrable. With (i)–(iii), +`W_r/r² ≤ C(κ²r²Δ₀² + r⁴(1 + k)²T^{2N₁})`, so `(W_r/r²)/r²` is bounded in `L²` and uniformly integrable. With (i)–(ii) for the +candidate limit and (i)–(iii) for the elder limit, `E_Q[W_r/r²]/r² → E₀[w^{cand} | b]` and `E_Q[(W_r/r²)e]/r² → E₀[w^{eld} | b]` (note `|φ| < 1/3 ⟺ |Y| < 2κ|Δ|`). (v) *Densities.* `π_r(v_r) → π₀(v₀(b, 0))` by [R] (R5) and the continuity of `π₀` in the target; @@ -289,21 +304,27 @@ in `x_i` and inserting the pin identities to the same order (`∂_u²f(0) = −( `det K_i = α_i det A_i − rβ_iᵀadj(A_i)β_i = d_i(J')f₄ + c_i(J') + ρ_i`, with `d_i(J') = (r/12)det A_i(J')` and a Taylor remainder `|ρ_i| ≤ Cr⁶(1 + ‖f‖_{C⁹})^{N₂}`. (A referee pass verified exactly, on a general pinned polynomial of degree 8 in `d = 2`, that `A_i, β_i` contain no `f₄` and `det K_i` is of degree 1 in `f₄` with slope `(r/12)det A_i`.) Let -`E₁ := {‖F_r‖_{C⁹} + T ≤ r^{−η}}` with `η := 1/(8max(N₁, N₂))`; on `E₁`, `|ρ_i| ≤ Cr⁵`, and `P(E₁^c) ≤ C_q r^qP^{N_q}` for every +`E₁ := {‖F_r‖_{C⁹} + T ≤ r^{−η}}` with `η := 1/(8max(N₁, N₂))` (`N₁, N₂ ≥ m + 1`, enlarged if necessary, so `mη < 1/8`); on `E₁`, `|ρ_i| ≤ Cr⁵`, and `P(E₁^c) ≤ C_q r^qP^{N_q}` for every `q` ([R] (R4)). *The window.* Let `Δ := det D_Θ²f(0)` (a coordinate of `J'`) and `B₁ := {|Δ| ≥ r^{1/2}}`. On `B₁ ∩ E₁` and for small `r`, -`det A_M(J')`, `det A_S(J')` have the sign of `Δ` and modulus `≥ r^{1/2}/2`, so `|d_i| ≥ r^{3/2}/24`. On the typed event the -two determinants have opposite signs (#198 §3), so `f₄` lies between the roots `−(c_i + ρ_i)/d_i`; hence -`{W_r > 0} ∩ B₁ ∩ E₁ ⊂ {f₄ ∈ I(J')}`, `I(J')` the `J'`-measurable interval spanned by `−c_M/d_M`, `−c_S/d_S`, enlarged by -`Cr⁵/min|d_i|`. To bound its length on this event, write `ℓ_i := d_if₄ + c_i` and evaluate at the actual `f₄ ∈ I(J')` -(`|f₄| ≤ r^{−η}` on `E₁`): by #198 (3.1), `ℓ_S − ℓ_M = det K_S − det K_M − (ρ_S − ρ_M) = 12kΔ₀ + R_S − R_M − (ρ_S − ρ_M)`, so -`|ℓ_S(f₄) − ℓ_M(f₄)| ≤ 12k|Δ| + Cr²r^{−N₁η}` on `E₁` (the swap `Δ₀ → Δ` costs `Cr²T ≤ Cr^{2−η}`); and -`|d_S − d_M| ≤ Cr²r^{−N₂η}`. Since `−c_S/d_S + c_M/d_M = [ℓ_M(f₄) − ℓ_S(f₄) + (d_S − d_M)(f₄ + c_M/d_M)]/d_S`, this gives -`|I(J')| ≤ 2(12k|Δ| + Cr^{2−N₁η} + Cr⁵)/|d_S| ≤ λ := C(k/r + r^{1/4})`. (Alternatively, the intrinsic polynomial identity -`det K_S − det K_M = 12k·det D_Θ²f(0) + r²Π(f₄, J', k)` for pinned jets — verified exactly by the referee pass on a -general pinned degree-8 polynomial in `d = 2` — gives the same bound without `T`.) The event `{|I(J')| ≤ λ}` is -`J'`-measurable and contains `{W_r > 0} ∩ B₁ ∩ E₁`, so +`|det A_i(J') − Δ| ≤ Cr^{1−mη} ≤ |Δ|/2`, so `det A_M(J')`, `det A_S(J')` have the sign of `Δ` and `|d_i| ≥ (r/24)|Δ| ≥ r^{3/2}/24`. +Write `Λ_i(x) := d_ix + c_i` (`J'`-measurable coefficients), so that `det K_i = Λ_i(f₄) + ρ_i`, and `z_i := −c_i/d_i`. On the +typed event the two determinants have opposite signs (#198 §3) while `d_M`, `d_S` have the same sign, so `f₄` lies within +`max_i|ρ_i|/|d_i| ≤ Cr⁵/min|d_i|` of the segment between `z_M` and `z_S`; hence `{W_r > 0} ∩ B₁ ∩ E₁ ⊂ {f₄ ∈ I(J')}`, `I(J')` +the `J'`-measurable interval spanned by `z_M`, `z_S`, enlarged by `Cr⁵/min|d_i|`. Its length on this event: since +`Λ_i(f₄) = d_i(f₄ − z_i)`, `z_S − z_M = Λ_M(f₄)/d_M − Λ_S(f₄)/d_S`, and on the typed event the sign window (#198 Lemma W's +argument with (3.1)) gives `|det K_i| ≤ 12k|Δ₀| + 2max_j|R_j|`, so `|Λ_i(f₄)| ≤ 12k|Δ| + Cr^{2−N₁η}` on `E₁` (the swap +`Δ₀ → Δ` costs `12k|Δ₀ − Δ| ≤ Cr²T^m ≤ Cr^{2−mη}`, absorbed as `N₁ ≥ m + 1`, and `|ρ_i| ≤ Cr⁵`). With `|d_i| ≥ (r/24)|Δ|` and +`|Δ| ≥ r^{1/2}`, + + |I(J')| ≤ 48(12k|Δ| + Cr^{2−N₁η})/(r|Δ|) + Cr^{7/2} ≤ C(k/r + r^{1/2−N₁η}) ≤ λ := C(k/r + r^{1/4}). + +(v1 bounded `z_S − z_M` through an asymmetric identity containing `(d_S − d_M)(f₄ + c_M/d_M)` and did not bound that term — +Codex P1 4150765060 on Math- #207; the symmetric form needs no bound on `d_S − d_M`.) Put `I(J') := R` on the +`J'`-measurable set `{d_Md_S ≤ 0}`. The event `{|I(J')| ≤ λ}` is `J'`-measurable and `{W_r > 0} ∩ B₁ ∩ E₁` is contained in +`{f₄ ∈ I(J')} ∩ {|I(J')| ≤ λ}`; `E₁` and the sign window involve the coupling variables `T`, `Δ₀`, so this inclusion holds on +the coupling space, and the dominating event depends on `F_r` alone, so `P_Q({W_r > 0} ∩ B₁ ∩ E₁) ≤ E[1{|I(J')| ≤ λ}P_Q(f₄ ∈ I(J') | J')] ≤ C₀λ`. *Small determinant.* Under `Q`, `D_Θ²f(0)` is a nondegenerate Gaussian symmetric matrix and `Δ` a polynomial of degree `m` @@ -353,7 +374,106 @@ because `(1/12 − ε₀)(3 + β) > 1/4` by the choice of `ε₀`. So `|I₂| expectation is finite (Gaussian moments; `π₀` has Gaussian decay in `b`) and strictly positive because `Y ≠ 0`, `A < 0` has positive probability under the nondegenerate contact law; so `c₁ < 0`. ∎ -## 7. Remarks +## 7. The candidate and rejected densities + +For the candidate density no elder decision is needed, and the part of the remainder that is not cusp-scale — the +equal-height mass `B_{d,L}` of [Z] — is reached at a Lipschitz rate in the gap, because the gap enters the pinned law +affinely. Notation: `Ψ_ℓ^{cand}(b, y) = p_y(v_{b,ℓ})E_{Q_{y,b,ℓ}}[W]` and `Ψ_0` as in [Z] (Z2) (unmarked; #191 §0), so that +`r^{−2}A_r(b, ℓ/r³, u) = r^{d−1}Ψ_ℓ^{cand}(b, ru)` and `B_{d,L} = ∫_{X∖{0}}∫_R Ψ_0 db dy = ∫_0^{r₀}∫∫ r^{−2}A_r(b, 0, u) dσ db dr ++ ∫_{dist(0,y)≥r₀}∫Ψ_0 db dy` for every `r₀ ∈ (0, r_0^*]` (#191 (2.1)–(2.2), [Z] (Z3)). + +**Lemma L (the gap enters Lipschitz).** There are `C, c, N` such that for `0 < r ≤ r_0^*`, `b ∈ R`, `u ∈ S^{d−1}` and +`0 ≤ k ≤ 1`, + + |A_r(b, k, u) − A_r(b, 0, u)| ≤ C (k² + kr²)(1 + |b|)^N e^{−cb²}, (7.1) + +and for every `r₀ ∈ (0, r_0^*]` there is `C_{r₀}` such that for `dist(0, y) ≥ r₀` and `0 ≤ ℓ ≤ 1`, + + |Ψ_ℓ^{cand}(b, y) − Ψ_0(b, y)| ≤ C_{r₀} ℓ (1 + |b|)^N e^{−cb²}. (7.2) + +*Proof of (7.1).* *Affine structure.* [R] (R3) realizes `Q_{r,b,k}` as `F_r = F + C_rΣ_r^{−1}(v_r − U_r)` with one +unconditioned copy `F` for all `(b, k)`, and the target `v_r = (b − kr³/2, −kr², 0, 12k, 0, …, 0)` ([R] §2) is affine in `k`. +Hence `F_r^{(k)} = F_r^{(0)} + kω_r` with `ω_r := C_rΣ_r^{−1}δv_r`, `δv_r := (−r³/2, −r², 0, 12, 0, …, 0)`: deterministic, +pinned with `b = 0`, `k = 1` (`U_r(ω_r) = δv_r`), and `‖ω_r‖_{C⁷} ≤ ‖C_r‖_{C⁷}‖Σ_r^{−1}‖|δv_r| ≤ C` uniformly in `r ≤ r_0^*` and +frames by (R2). Put `T₀ := 1 + ‖F_r^{(0)}‖_{C⁷} + ‖ω_r‖_{C⁷}`; by (R3), `‖F_r^{(0)}‖_{C⁷} ≤ ‖F‖_{C⁷} + C(|b| + |U_r|)`, so +`‖T₀‖_p ≤ C_p(1 + |b|)`, and `‖F_r^{(k')}‖_{C⁷} ≤ T₀` for `0 ≤ k' ≤ 1`. (In the polynomial model of checker C8 the shift is +`ω_r = 2(x − r)(x + r/2)²`, with no transverse part, so `det K_i` is affine in `k'` with slope `∓6 det A_i`; for the actual +regression function `ω_r` has transverse parts, and only the bounds below are used.) + +*The typed product along the gap.* The scaled endpoint Hessians are linear in the field, so +`K_i(k') = K_i(0) + k'K_i[ω_r]`, with `‖K_i[ω_r]‖ ≤ C` (by #191 (1.1) for the pinned `ω_r`, `α_i[ω_r] = ∓6 + O(r)`) and +`‖K_i(k')‖ ≤ CT₀` (#191 (1.1), (1.3), (1.5)). Hence `det K_i(k')` is a polynomial in `k'` with +`|∂_{k'}det K_i| = |tr(adj K_i(k')·K_i[ω_r])| ≤ CT₀^{d−1}`. The typed product `w(k') := F_d(K_M(k'))F_{d−1}(K_S(k'))` is +continuous on `[0, k]`: an index changes only where an eigenvalue crosses zero, where the corresponding determinant, +hence `w`, vanishes. Between the finitely many zeros of the polynomial `p := det K_M det K_S` (if `p ≡ 0`, then `w ≡ 0`), +`w = ±p` or `w = 0`, so for every realization + + |w(k) − w(0)| ≤ ∫_0^k |∂_{k'}(det K_M det K_S)| 1{typed(k')} dk'. + +On `typed(k')` the two determinants have opposite signs, and the deterministic expansions of #191 §1 Steps 1–3 with +the `(1 + k)`-quantified remainder of #198 (3.1) — applied to the pinned `C⁷` function `F_r^{(k')}`, with `D_y²F_r^{(k')}(0)` +in place of `A₀` — give `det K_{M,S}(k') = ∓6k'Δ' + rY' + R_{M,S}` with `|R_i| ≤ Cr²T₀^{N₁}`; so, as in #198 Lemma W, +`|det K_i(k')| ≤ 12k'T₀^m + 2Cr²T₀^{N₁}`. Hence `|∂_{k'}(det K_M det K_S)| ≤ C(k' + r²)T₀^N` on `typed(k')`, so +`|w(k) − w(0)| ≤ C(k² + kr²)T₀^N`, and taking expectations, +`|Z_r/r²(b, k, u) − Z_r/r²(b, 0, u)| ≤ C(k² + kr²)(1 + |b|)^N`. + +*The density factor.* `π_r(v_r(b, ·))` is a Gaussian density whose covariance is uniformly invertible for `r ≤ r_0^*` +([R] (R2), (R5)), evaluated at a target affine in `k`: `|π_r(v_r(b, k)) − π_r(v_r(b, 0))| ≤ Ck(1 + |b|)e^{−cb²}`. With +`Z_r/r² ≤ C(k² + r⁴)(1 + |b|)^N` (#198 Lemma W) and `A_r = 12π_r(v_r)Z_r/r²`, (7.1) follows (`k³ ≤ k²`, `kr⁴ ≤ kr²`). + +*Proof of (7.2).* The same, with the two-site regression in place of the near one ([Z] (Z10): `f_ℓ = f_0 + ℓh_y`, +`f_0 ~ Q_{y,b,0}`, `h_y` deterministic). [Z] states (Z10) at fixed separation only; the uniform facts on the compact +`D_{r₀} = {dist(0, y) ≥ r₀}` are #198 §2 (a), (b), (2.2): `λ_min(Cov O_y) ≥ c₀ > 0`, the regression functions `ψ^y_j` are +bounded in `C³` uniformly (so `h_y = −ψ^y_{f(y)}` has `sup_y ‖h_y‖_{C²} ≤ C_{r₀}`), and the Hessian moments under `Q_{y,b,ℓ}` +are `≤ C_p(1 + |b|)^p` uniformly in `y ∈ D_{r₀}` and `ℓ ∈ [0, 1]` ((2.2) at `ℓ = 0` by the same proof); and +`|∂_ℓ p_y(v_{b,ℓ})| = p_y(v_{b,ℓ})|(Γ_y^{−1}v_{b,ℓ})_{f(y)}| ≤ C_{r₀}(1 + |b|)e^{−cb²}`. The typed product +`W(f_ℓ) = |det H_0 det H_y|1{H_0 < 0, index H_y = d − 1}` is continuous and piecewise polynomial in `ℓ` for the same reason, +with `|∂_ℓ(det H_0 det H_y)| ≤ C(‖f_0‖_{C²} + ‖h_y‖_{C²})^{2d−1}‖h_y‖_{C²}`. ∎ + +**Theorem CU′ (the candidate and rejected densities).** Under the hypotheses of Theorem CU, as `ℓ ↓ 0`, + + ν_cand(ℓ) = c ℓ^{−1/3} + B_{d,L} + I^{cand} ℓ^{1/4} + o(ℓ^{1/4}), + I^{cand} = −(96/7)·6^{1/4} ∫_{S^{d−1}}∫_R π₀(u; v₀(b,0)) E₀[|Y|^{7/4}|Δ|^{1/4}1{A < 0} | b] db dσ(u) = (3^{1/4}/2) c₁ < 0, (CU′.1) + + ρ_rej(ℓ) = ν_cand(ℓ) − ν_eld(ℓ) = B_{d,L} + (I^{cand} − c₁) ℓ^{1/4} + o(ℓ^{1/4}), I^{cand} − c₁ = (1 − 3^{1/4}/2)|c₁| > 0. (CU′.2) + +Moreover, for every `r₀ ∈ (0, r_0^*]`, `ν_cand^{far,r₀}(ℓ) = ∫_{dist(0,y)≥r₀}∫Ψ_0 db dy + O(ℓ)`: the `ℓ^{1/4}` terms are +near-diagonal, carried by the cusp scale. (CU′.1) uses only Lemma L, the candidate part of Theorem CU.4 (with CU.1), +#191 ((E.1), §§1–2), #198 ((3.1), Lemma W, §2), [R], [Z] and [P] §14 — not Proposition CU.3, Lemma CU.5, [C7-K] or #198 +Lemmas B, F′ — so it can be accepted separately; (CU′.2) = (CU′.1) − (CU.1) inherits everything Theorem CU uses. + +*Proof.* Fix `S ≥ 1` and `r₀ ∈ (0, r_0^*]`, and let `ℓ` be so small that `Sℓ^{1/4} < r₀`. By the separation-variable identity +(#191 (2.1)) and the decomposition of `B_{d,L}` above, + + ν_cand(ℓ) − cℓ^{−1/3} − B_{d,L} = J₁ − J₂ + J₃ − J₄ + J₅, + +`J₁ := ∫_{r 0`. A proof needs the rate of the - `B_{d,L}`-part (the candidate kernel at `k → 0` over intermediate separations and the far candidate kernel), which - is not supplied here. +2. **Candidate and rejected densities.** v1 recorded (CU′.1)–(CU′.2) as formal and said that a proof needs the rate of the + `B_{d,L}`-part (the candidate kernel at `k → 0` over intermediate separations and the far candidate kernel). §7 + supplies it in the simplest form — both kernels are Lipschitz in the gap (Lemma L), because the gap enters the + pinned regression affinely — and no elder information is needed for the candidate density. The rejected density + approaches `B_{d,L}` from above, the excess being the rejected part of the cusp-scale typed pairs (§7). 3. **Numerical values (exploration, not controls).** For the Gaussian kernel `e^{−|z|²/2}` on `R^d` (the periodized SIDE24 covariance differs by factors `1 + O(e^{−288})`), the parity form of (CU.2) reduces to `c₁ = −(16/7)2^{1/4}π^{−(d−1/2)}J_d`, `J_d := E[|Y|^{7/4}|Δ|^{1/4}1{A<0} | V_u = 0, G = 0, t_u = 0]` (`d = 2, 3`), with the @@ -382,7 +501,8 @@ expectation is finite (Gaussian moments; `π₀` has Gaussian decay in `b`) and law (`−0.21163 ± 0.00014`, 1.6 standard errors), a development Monte Carlo (`−0.21188 ± 0.00010`, `2·10⁸` samples) and an independent referee conditional Monte Carlo (`−0.211848 ± 0.000024`, `5·10⁸` samples). So `ν_{3,24}(ℓ) ≈ 0.041776ℓ^{−1/3} − 0.21185ℓ^{1/4}`: the relative correction `5.071ℓ^{7/12}` is 1% at `ℓ ≈ 2.3·10^{−5}` and - 10% at `ℓ ≈ 1.2·10^{−3}`. The exploration also + 10% at `ℓ ≈ 1.2·10^{−3}`. By (CU′.1)–(CU′.2), `I^{cand} = −0.1394041` and `I^{cand} − c₁ = +0.0724443` in `d = 3` + (`−0.1772744`, `+0.0921244` in `d = 2`). The exploration also tests (3.1) against the actual global elder rule of random pinned plane quintics (graded-grid union-find maximin of the rescaled field): at `r = 0.01` all 32 typed random cases agree with `|φ| < 1/3`, and in targeted cases at `r = 0.003` the boundary lies between `|φ| = 0.32` and `0.345`. (Development runs at `r = 0.05`, not shipped, disagreed @@ -401,41 +521,47 @@ expectation is finite (Gaussian moments; `π₀` has Gaussian decay in `b`) and typed window `|φ| < 1` is #198 Lemma W's sign window. (iii) Theorem CU is consistent with #198's `O(ℓ^{1/4})` and #191's `o(1)` (it uses the same inputs, so this is not an independent confirmation of them). -## 8. Sources (exact identities in `SOURCES.json`) +## 9. Sources (exact identities in `SOURCES.json`) | Tag | Path | Role | |---|---|---| | [R] | `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (blob `247b3ecf`) | (R3)–(R5), §4 scaled Hessians, Lemma R3.2 — consumed | | [C7-K] | `frontiers/c7_total_bounded_20260929/PROOF.md` (blob `28748b08`) | (K2); §4 `1 − e ≤ 1_{G_r^c}` — consumed | -| [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d`) | §2 finite-jet rank, §4 regression, §8 maximin, §§9–10 as [E2], (10.2), §15 (15.2) — consumed | +| [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d`) | §2 finite-jet rank, §4 regression, §8 maximin, §§9–10 as [E2], (10.2), §14 (far nondegeneracy), §15 (15.2) — consumed | | [E1] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` (blob `213594d6`) | reading rule | | [E2] | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` (blob `fe9b9ce4`) | §9 replacement | | [REC] | `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` (blob `75da2597`) | reading rule | | [182] | `frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md` (blob `0d401877`) | §3 barrier — consumed through #198 Lemmas B, F′ | -| [Z] | `frontiers/c7_zero_gap_limit_20260929/PROOF.md` (blob `5b6328ea`) | (Z10), (Z13)–(Z14), `r_0^{[Z]}` — consumed through #198, #191 | -| #191 | `frontiers/remainder_vanishing_20260930/PROOF.md` (unmerged; v1.1 blob `441152df`) | (E.1), Lemma E Steps 1–3 and `T`, §2 (2.1), `r_0^*` — **consumed, unmerged** | -| #198 | `frontiers/remainder_rate_20260930/PROOF.md` (unmerged; v1.1 blob `abfb98ae`) | §1 elder-density identity and Lemma B (B.2), §2 Lemma F′, §3 (3.1)–(3.2) and Lemma W — **consumed, unmerged** | +| [Z] | `frontiers/c7_zero_gap_limit_20260929/PROOF.md` (blob `5b6328ea`) | (Z2)–(Z3), (Z10), §2 (a.s. nonsingular endpoint Hessians), the [P] §14 input of (Z16) — consumed in §7; (Z10), (Z13)–(Z14), `r_0^{[Z]}` — consumed through #198, #191 | +| #191 | `frontiers/remainder_vanishing_20260930/PROOF.md` (unmerged; v1.1 blob `441152df`) | (E.1), Lemma E Steps 1–3 and `T`, §2 (2.1)–(2.2) and the unmarked kernel `Ψ_ℓ^{cand}`, `r_0^*` — **consumed, unmerged** | +| #198 | `frontiers/remainder_rate_20260930/PROOF.md` (unmerged; v1.1 blob `abfb98ae`) | §1 elder-density identity and Lemma B (B.2), §2 Lemma F′ and its uniform far facts (a), (b), (2.2), (2.1), §3 (3.1)–(3.2) and Lemma W — **consumed, unmerged** | | [CW] | A. Carbery, J. Wright, *Distributional and L^q norm inequalities for polynomials over convex bodies in R^n*, Math. Res. Lett. 8 (2001) 233–248 | small-ball bound for `det` (Lemma CU.5) — external, classical | -## 9. Exact controls and exploration +## 10. Exact controls and exploration `cusp_check.py` (stdlib, exact rationals; `RESULTS.json` its output, byte-identical under `-O`): C1 the quartic ridge identities and factorizations; C2 the elder window by exact 1D maximin for 117 rational `φ`; C3 the fiber reduction; C4 the cusp limit field for pinned polynomials of degree 6 in `d = 2, 3`; C5 the model's endpoint determinants and typed window; C6 the cusp integrals (6.1) and the candidate analogue; C7 the Gaussian-kernel -conditional covariances used by the exploration. Mutants M1 (threshold 1/2), M2 (constant 8/7), M3 (drop `γ·Ξ`), -M4 (coefficient 2 in `e₄`) exit 1; an unknown label exits 2. - -**What the controls do not test.** Theorem CU.1 for non-polynomial fields, Proposition CU.3, Theorem CU.4, Lemma CU.5 -and §6 are proved in prose only. `explore_cusp.py` (stdlib, floating point, not replayed by the workflow; output in +conditional covariances used by the exploration; C8 the affine gap structure behind Lemma L in the special case of pinned +polynomials (the `k`-shift `2(x − r)(x + r/2)²` has no transverse part there, so `∂_k det K_{M,S} = ∓6 det A_{M,S}` exactly; the +actual regression shift has transverse parts, covered only by the prose), the common `rY`, the constant `I^{cand}/c₁ = 3^{1/4}/2`, the +rejected cusp integral exactly in `Q(3^{1/4})` and the small-`κ` share `13/27`. Mutants M1 (threshold 1/2), M2 (constant +8/7), M3 (drop `γ·Ξ`), M4 (coefficient 2 in `e₄`), M5 (sign of `∂_k det K_M`) exit 1; an unknown label exits 2. + +**What the controls do not test.** Theorem CU.1 for non-polynomial fields, Proposition CU.3, Theorem CU.4, Lemma CU.5, +§6, Lemma L for random fields and the assembly of Theorem CU′ are proved in prose only. `explore_cusp.py` (stdlib, floating point, not replayed by the workflow; output in `EXPLORE.json`) is exploration: the actual-landscape test of the elder window and the numerical `c₁`. Neither is acceptance. -## 10. Review slices +## 11. Review slices A: Theorems CU.1–CU.2 — the pinned Taylor bookkeeping, the fiber reduction and the one-dimensional maximin (§§1–2). B: Proposition CU.3 — the ridge of a perturbed field, the reduction of connectivity, the trap and path arguments, the margins (§3). C: Theorem CU.4 and Lemma CU.5 — the kernel limits and the window probability (§§4–5), including the conditional density of `f₄` and the small-determinant bound. -D: §6 — the decomposition, the dominations, the choice of `ρ₁`, the cusp integral (6.1), the sign of `c₁`; §7. +D: §6 — the decomposition, the dominations, the choice of `ρ₁`, the cusp integral (6.1), the sign of `c₁`; §8. +E: §7 — Lemma L (the affine gap structure of the pinned regression, the continuity of the typed product along the gap, + the sign-window bound at intermediate gaps, the density factor; the far analogue through (Z10)) and the assembly of + Theorem CU′ (the decomposition `J₁`–`J₅`, the cusp integral (7.3), the constants). diff --git a/frontiers/cusp_second_order_20261001/README.md b/frontiers/cusp_second_order_20261001/README.md index b0580bb16..d41d62397 100644 --- a/frontiers/cusp_second_order_20261001/README.md +++ b/frontiers/cusp_second_order_20261001/README.md @@ -1,4 +1,4 @@ -# The cusp crossover and the second-order term of the elder lifetime law (CL-CUSP-SECOND-ORDER-20261001-v1) +# The cusp crossover and the second-order terms of the lifetime laws (CL-CUSP-SECOND-ORDER-20261001-v1.1) Author-side proof candidate, Anthropic Claude, 1 October 2026. Scientific effect NONE. Nonauthor review required. **Depends on Math- #191** (Lemma E, (E.1), §2; v1.1 blob `441152df`) **and Math- #198** (§1, Lemma B (B.2), Lemma F′, @@ -17,6 +17,17 @@ zero-gap contact law. #198 had the order `O(ℓ^{1/4})`; this packet identifies field (`d = 3`, `L = 24`) the exploration gives `c₁ ≈ −0.21185`, `c₁/c ≈ −5.071`: `ν_{3,24}(ℓ) = c_{3,24}ℓ^{−1/3}(1 − 5.071ℓ^{7/12} + o(ℓ^{7/12}))`, a 1% correction at `ℓ ≈ 2.3·10⁻⁵`. +**Theorem CU′ (v1.1, §7): the candidate and rejected densities.** With `B_{d,L}` the equal-height mass of Theorem Z, + + ν_cand(ℓ) = c ℓ^{−1/3} + B_{d,L} + I^{cand} ℓ^{1/4} + o(ℓ^{1/4}), I^{cand} = (3^{1/4}/2) c₁ < 0, + ρ_rej(ℓ) = B_{d,L} + (I^{cand} − c₁) ℓ^{1/4} + o(ℓ^{1/4}), I^{cand} − c₁ = (1 − 3^{1/4}/2)|c₁| > 0, + +so the rejected density approaches `B_{d,L}` from above (`I^{cand} − c₁ ≈ 0.0724` in `d = 3`), the excess being the rejected +part of the cusp-scale typed pairs (`1/3 ≤ |φ| < 1`). The candidate density needs no elder decision, and its non-cusp +remainder is Lipschitz in the gap — **Lemma L**: `|A_r(b, k, u) − A_r(b, 0, u)| ≤ C(k² + kr²)(1 + |b|)^Ne^{−cb²}` near the +diagonal and `O(ℓ)` for far pairs — because the gap enters the pinned regression affinely ([R] (R3)). (CU′.1) does not +use the elder machinery (Proposition CU.3, Lemma CU.5), so it can be accepted separately. + **Mechanism — the cusp crossover.** At the scale `r ≍ k` (gap `ℓ = kr³` comparable to `r⁴`; `κ := k/r = ℓ/r⁴`), rescaled by `x = rX` (axis), `y = r²Ξ` (transverse), heights `r⁴`, the pinned field converges to a universal random polynomial `𝔓 = 2κ(X + ½)²(X − 1) + (f₄/24)(X² − ¼)² + ½(X² − ¼)γ·Ξ + ½ΞᵀAΞ` (Theorem CU.1). Maximizing over the concave transverse @@ -32,30 +43,39 @@ regime (`κ → ∞`: rejected fraction `≍ (r/k)³`, the `Θ(r³)` of [P] Theo weight is `O(κ)` of the contact weight and the elder share of it tends to `13/27` (Remark 1; Theorem Z's all-rejected statement is the different, non-commuting limit `k → 0` at fixed `r`). -**Not claimed:** a rate for the `o(ℓ^{1/4})`; certified numerical values (the numbers are exploration); the candidate -and rejected expansions (formal, Remark 2); uniformity in `d, L`. +**Not claimed:** a rate for any of the `o(ℓ^{1/4})`; certified numerical values of `c₁`, `I^{cand}` or `B_{d,L}` (the +numbers are exploration); uniformity in `d, L`. **Files.** `PROOF.md`. `cusp_check.py` (stdlib exact controls; `RESULTS.json` its output, byte-identical under `-O`; mutants M1–M4 exit 1, an unknown label exits 2): C1 the quartic-ridge identities and boundary factorizations; C2 the elder window by exact 1D maximin for 117 rational `φ`; C3 the fiber reduction; C4 the cusp limit field for pinned polynomials of degree 6 in `d = 2, 3`; C5 the model's endpoint determinants and typed window; C6 the cusp integrals; -C7 the Gaussian-kernel conditional covariances used by the exploration. `explore_cusp.py` (stdlib, floating point, +C7 the Gaussian-kernel conditional covariances used by the exploration; C8 the affine gap structure behind Lemma L on +pinned polynomials (a special case: there the gap shift has no transverse part), `I^{cand}/c₁ = 3^{1/4}/2`, the rejected +cusp integral exactly in `Q(3^{1/4})`, the small-`κ` elder share `13/27` (mutant M5). `explore_cusp.py` (stdlib, floating point, **not a control**, not replayed by the workflow, about a minute; `EXPLORE.json` its output): X1 the actual global elder rule of random pinned plane quintics at `r = 0.01` agrees with `|φ| < 1/3` in 32/32 typed cases, and in targeted cases at `r = 0.003` the boundary lies between `|φ| = 0.32` and `0.345`; X2 `c₁` for the Gaussian kernel — `d = 2`: `−0.26939883`, `c₁/c = −3.669938` (Gauss–Legendre quadrature, two resolutions agree to ten digits); `d = 3`: `−0.21184835`, `c₁/c = −5.071062` (deterministic quadrature over the negative-definite cone, two resolutions agree to `10⁻⁸` -relative; a fixed-seed Monte Carlo over the same law gives `−0.21163 ± 0.00014`). `SOURCES.json` (exact identities; +relative; a fixed-seed Monte Carlo over the same law gives `−0.21163 ± 0.00014`); `I^{cand} = −0.1772744`, `−0.1394041` and +`I^{cand} − c₁ = 0.0921244`, `0.0724443` (`d = 2, 3`). `SOURCES.json` (exact identities; #191/#198 recorded as consumed-unmerged and verified by the workflow by blob). -**Pre-submission referee record (same author family; not acceptance).** Two clean-context Claude referee passes read -v1 before this PR. Slices A/B confirmed CU.1–CU.2 symbolically and by an independent two-dimensional maximin of the +**Referee record before any nonauthor review (same author family; not acceptance).** Two clean-context Claude referee +passes read v1 before this PR. Slices A/B confirmed CU.1–CU.2 symbolically and by an independent two-dimensional maximin of the full model (34 cases) and found closeable gaps in Proposition CU.3 (all real zeros of `g'` in Step 5, the margin `−2κ − 1` in Step 3, `X_R := min(2, X₃ − τ)`); slices C/D found closeable gaps in Lemma CU.5 (`E₁` must control `T`; the `J'`-measurable window event; the Carbery–Wright uniformity) and a false remark (the small-`κ` elder share is `13/27`, -not `0`), and independently reproduced `c₁` (`d = 2` quadrature; `d = 3` Monte Carlo `−0.211848 ± 0.000024`). Every -finding is applied in this version. +not `0`), and independently reproduced `c₁` (`d = 2` quadrature; `d = 3` Monte Carlo `−0.211848 ± 0.000024`). A third pass +read §7 of v1.1 (Lemma L, Theorem CU′): no error and no gap, seven minor points (attribution of the affine target to +[R] §2/(R3), the far uniform facts to #198 §2, the far Kac–Rice identity, scope of C8, the dependency split), and it +checked (7.1) by an exact-conditioning Monte Carlo in `d = 2`. Codex's automated review of v1 (P1 4150765060) found a +real gap in Lemma CU.5 — the window-length bound dropped an unbounded `(d_S − d_M)` term — repaired in v1.1 by bounding +the root difference symmetrically through the sign window (same `λ`, downstream unchanged); the repair was re-read by +the §7 referee with an exact randomized check. Every finding is applied. **What the controls do not test.** Theorem CU.1 for non-polynomial fields, Proposition CU.3, Theorem CU.4, -Lemma CU.5 and the assembly (§6) are proved in prose only. Review slices (PROOF.md §10): A the cusp field and the -model; B stability; C the kernel limits and the window probability; D the assembly and the remarks. +Lemma CU.5, the assembly (§6), Lemma L for random fields and the assembly of Theorem CU′ (§7) are proved in prose only. +Review slices (PROOF.md §11): A the cusp field and the model; B stability; C the kernel limits and the window +probability; D the assembly and the remarks; E Lemma L and Theorem CU′. diff --git a/frontiers/cusp_second_order_20261001/RESULTS.json b/frontiers/cusp_second_order_20261001/RESULTS.json index d56f88be9..c5cc87072 100644 --- a/frontiers/cusp_second_order_20261001/RESULTS.json +++ b/frontiers/cusp_second_order_20261001/RESULTS.json @@ -59,10 +59,23 @@ "tau2": "6" }, "passed": true + }, + "C8_gap_affine_structure_and_cand_constants": { + "info": { + "I_cand_over_c1": "3^(1/4)/2", + "dimensions": [ + 2, + 3 + ], + "pinned_cases": 8, + "rejected_integral_points": 2, + "small_kappa_elder_share": "13/27" + }, + "passed": true } }, "mutant": null, - "object": "CL-CUSP-SECOND-ORDER-20261001-v1", + "object": "CL-CUSP-SECOND-ORDER-20261001-v1.1", "passed": true, "scientific_effect": "NONE" } diff --git a/frontiers/cusp_second_order_20261001/SOURCES.json b/frontiers/cusp_second_order_20261001/SOURCES.json index 67eaa3f3c..5d7678b03 100644 --- a/frontiers/cusp_second_order_20261001/SOURCES.json +++ b/frontiers/cusp_second_order_20261001/SOURCES.json @@ -1,8 +1,8 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-CUSP-SECOND-ORDER-20261001-v1", - "kind": "author-side proof candidate: Theorem CU, nu_eld = c ell^-1/3 + c_1 ell^1/4 + o(ell^1/4) with c_1 = -(192/7) 2^(1/4) J < 0, via the cusp limit field (CU.1), the quartic-ridge model and its elder window |phi| < 1/3 (CU.2), stability (CU.3), kernel limits (CU.4) and the window probability (CU.5); consumes the unmerged #191 and #198; nonauthor analytic review required. explore_cusp.py / EXPLORE.json are exploration, not controls.", + "object": "CL-CUSP-SECOND-ORDER-20261001-v1.1", + "kind": "author-side proof candidate: Theorem CU, nu_eld = c ell^-1/3 + c_1 ell^1/4 + o(ell^1/4) with c_1 = -(192/7) 2^(1/4) J < 0, via the cusp limit field (CU.1), the quartic-ridge model and its elder window |phi| < 1/3 (CU.2), stability (CU.3), kernel limits (CU.4) and the window probability (CU.5); and Theorem CU-prime, nu_cand = c ell^-1/3 + B_{d,L} + I_cand ell^1/4 + o(ell^1/4) with I_cand = (3^(1/4)/2) c_1 and rho_rej = B_{d,L} + (I_cand - c_1) ell^1/4 + o(ell^1/4), via the gap-Lipschitz Lemma L; consumes the unmerged #191 and #198; nonauthor analytic review required. explore_cusp.py / EXPLORE.json are exploration, not controls.", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "read_at": { @@ -71,7 +71,7 @@ "sha256": "f0e563c45c41333f2bb6e5bf5cb98f0d3b592e2abf2af1c1b098c21ad77cb777", "git_blob": "5b6328ea3d6afcbeb731698b3834e536f72d1bcb", "tag": "Z", - "use": "(Z10), (Z13)-(Z14), the radius r_0^[Z]; consumed transitively through #198 and #191" + "use": "(Z2)-(Z3), (Z10), section 2 (a.s. nonsingular endpoint Hessians), the [P] section 14 input of (Z16): consumed directly in section 7 (Lemma L (7.2), Theorem CU-prime); (Z10), (Z13)-(Z14), the radius r_0^[Z]: consumed transitively through #198 and #191" } ], "cited_only": [], @@ -84,7 +84,7 @@ "bytes": 25353, "sha256": "0cb5b949261476f10e539677e2744e3288752474393af25ba787e67b1350f32a", "tag": "191", - "use": "CONSUMED: Lemma E (E.1) and Steps 1-3, section 2 (2.1), the admissible radius; unmerged author-side candidate of this session; this packet cannot be integrated before #191" + "use": "CONSUMED: Lemma E (E.1) and Steps 1-3, section 2 (2.1)-(2.2) and the unmarked kernel Psi_ell^cand, the far candidate identity, the admissible radius; unmerged author-side candidate of this session; this packet cannot be integrated before #191" }, { "path": "frontiers/remainder_rate_20260930/PROOF.md", @@ -94,7 +94,7 @@ "bytes": 33131, "sha256": "0b690df6ac40707ca06783f5641d8b480277c716df75b1031d84ee53a0575f72", "tag": "198", - "use": "CONSUMED: section 3 (3.1)-(3.2) and Lemma W, Lemma B (B.2), Lemma F-prime, the elder-density identity of section 1; unmerged author-side candidate of this session; this packet cannot be integrated before #198" + "use": "CONSUMED: section 3 (3.1)-(3.2) and Lemma W, Lemma B (B.2), Lemma F-prime with its uniform far facts (a), (b), (2.2) and (2.1), the elder-density identity of section 1; unmerged author-side candidate of this session; this packet cannot be integrated before #198" } ], "cited_external": [ @@ -108,7 +108,8 @@ "M1", "M2", "M3", - "M4" + "M4", + "M5" ], "replayed": [ "cusp_check.py" @@ -126,40 +127,51 @@ { "slices": "C, D (Theorem CU.4, Lemma CU.5, section 6, remarks)", "result": "closeable gaps in CU.5 (E_1 must control T; the J-prime-measurable window event; Carbery-Wright uniformity via the top Wiener chaos); section 4 details (K_{2 delta}, the C r T bound); a false remark (the small-kappa elder share is 13/27, not 0); transitive dependencies [182], [Z]; c_1 reproduced independently (d = 2 quadrature; d = 3 Monte Carlo -0.211848 +- 0.000024)" + }, + { + "slices": "E (section 7 of v1.1: Lemma L, Theorem CU-prime)", + "result": "no error and no gap; seven minor findings (the target moves in three rows, not one: cite [R] section 2 and (R3); bound omega_r from (R2); the a.s. appeal unnecessary; far uniform facts from #198 section 2 (a), (b), (2.2); the far Kac-Rice identity; J_1 dominated by (E.1) alone and the CU.4 (iv) split; C8 tests only the polynomial special case, and the separate-acceptance dependency split); constants re-verified symbolically; (7.1) checked by an exact-conditioning Monte Carlo in d = 2" + } + ], + "automated_nonauthor": [ + { + "source": "Codex review comment 4150765060 on Math- #207 (P1, v1 PROOF.md line 303)", + "finding": "Lemma CU.5: the window-length bound dropped the term (d_S - d_M)(f_4 + c_M/d_M) of an asymmetric root identity without bounding it", + "disposition": "repaired in v1.1: the root difference z_S - z_M = Lambda_M(f_4)/d_M - Lambda_S(f_4)/d_S is bounded symmetrically through the sign window with |d_i| >= (r/24)|Delta|; lambda and every downstream use unchanged; the repair re-read by the section-7 referee (no error or gap; three wording fixes applied; exact randomized check of the identity, the inclusion and the sign-window bounds, 2e5 rational draws)" } ], - "disposition": "every finding applied in v1 as submitted" + "disposition": "every finding of the three passes and of the automated review applied (v1 as submitted; v1.1 for section 7 and Lemma CU.5)" }, "files": [ { "path": "PROOF.md", - "bytes": 40307, - "sha256": "28632a87d651785eb1f24e1b92908b8872c3a732987dde34af85199ae0f3ea9f" + "bytes": 52605, + "sha256": "cffc1b83ac364da58bd650c3c301ea1babd38e19af54f74010bdc120edce38a8" }, { "path": "cusp_check.py", - "bytes": 26196, - "sha256": "841f375606d68fc68b9e590064990e0846cd062ea31370355800a01c56138932" + "bytes": 35169, + "sha256": "d6e6a841dd0c58144fb7a4daaf8a3cf391d92aa3bfbb1caf2c1162e4e439b69f" }, { "path": "RESULTS.json", - "bytes": 1328, - "sha256": "8f2b5a3a5d3556468897c3c51f6935fa3c98a9c3fb515263b60a1de19bc6d2ad" + "bytes": 1596, + "sha256": "038e0d0554a7575679d24492d110e1a823136dfb88dc556e45c58cc386c86339" }, { "path": "README.md", - "bytes": 5903, - "sha256": "a50ff18f6eb66d36857d5f92bc27eb9ed10c42cb1e2c28544161200cd58d054b" + "bytes": 7977, + "sha256": "0ba540b193294e240dacc3ad9a0595d559dfe2c2ff761d1dc472488c002eca1b" }, { "path": "explore_cusp.py", - "bytes": 14953, - "sha256": "8c2072c9f23760f63ecf1b4c420d1272f12dad3cc77711c48dcdacd53a2733e4" + "bytes": 15374, + "sha256": "b32efa718944ef93b5fe41266855878cbc500093ef376c6b07beb4a9567865ed" }, { "path": "EXPLORE.json", - "bytes": 2422, - "sha256": "ba77febb6b44c779363ecd4ce06c4bbd45df2b350513c931e05d649aca719032" + "bytes": 2552, + "sha256": "c8f631d66f413a4c9787e999114edc42a33cc9a65eef667b2f8de265f371ab74" } ] } diff --git a/frontiers/cusp_second_order_20261001/cusp_check.py b/frontiers/cusp_second_order_20261001/cusp_check.py index 0792e9fdd..98ac5070c 100644 --- a/frontiers/cusp_second_order_20261001/cusp_check.py +++ b/frontiers/cusp_second_order_20261001/cusp_check.py @@ -1,5 +1,5 @@ #!/usr/bin/env python3 -"""Exact finite controls for PROOF.md (CL-CUSP-SECOND-ORDER-20261001-v1). Standard library only; exact rationals. +"""Exact finite controls for PROOF.md (CL-CUSP-SECOND-ORDER-20261001-v1.1). Standard library only; exact rationals. C1 the quartic ridge of Theorem CU.2: g(X) = kappa[2(X+1/2)^2(X-1) + 3 phi (X^2-1/4)^2] has g'(X) = 6 kappa (X^2-1/4)(1+2 phi X), g(-1/2) = 0, g(1/2) = -kappa, g''(-1/2) = 6 kappa (phi-1), @@ -30,11 +30,18 @@ Var(f_xxx | grad f) = 6, D_1 = 4/3; d = 3: (f_yy, f_zz, f_yz) | (f_xx, f_xy, f_xz) has covariance [[8/3, 2/3, 0], [2/3, 8/3, 0], [0, 0, 1]], f_xxxx | (f_yy, f_zz, f_yz, f_xx, f_xy, f_xz) has regression coefficient 3/5 on each of f_yy, f_zz and residual variance 138/5, and (f_xxy, f_xxz) | (grad f, f_xxx) has - covariance 2 I. - M mutants (--mutant M1|M2|M3|M4) exit 1; an unknown label exits 2. - Scope: C1-C7 check exact identities and finitely many rational instances. They do NOT test Theorem CU.1 for + covariance 2 I; + C8 the gap enters affinely (Lemma L) and the constants of Theorem CU': for the pinned polynomials of C4 in d = 2, 3, + the family is affine in the gap with the k-shift omega = 2(x - r)(x + r/2)^2 exactly (transverse coefficients + gap-free), d det K_M/dk = -6 det A_M and d det K_S/dk = +6 det A_S exactly, det K_M + det K_S = 2rY + O(r^2) and + det K_S - det K_M = 12k det A(0) + O(r^2) (the common rY of the sign window); I_cand/c_1 = 3^(1/4)/2 (fourth power + 3/16); the rejected cusp integral int_{s1}^{3^(1/4)s1} (36 Delta^2 s^-8 - Y^2) ds = (16/7)Y^2 s1 - (8/7)Y^2 3^(1/4)s1 + exactly in Q(3^(1/4)); the small-kappa elder share 13/27; mutant M5 flips the sign of d det K_M/dk. + M mutants (--mutant M1|M2|M3|M4|M5) exit 1; an unknown label exits 2. + Scope: C1-C8 check exact identities and finitely many rational instances. They do NOT test Theorem CU.1 for non-polynomial fields, Proposition CU.3 (stability), Theorem CU.4 (Gaussian kernel limits), Lemma CU.5 (window - probability) or the assembly of Theorem CU; those are proved in prose only. The controls are not acceptance. + probability), Lemma L for random fields, or the assemblies of Theorems CU and CU'; those are proved in prose only. + The controls are not acceptance. """ import argparse import itertools @@ -531,6 +538,173 @@ def check_C6(mutant): return True, {'points': pts, 'elder_constant': '(16/7) 2^(1/4)', 'candidate_constant': '(8/7) 6^(1/4)'} +# ------------------------------------------------- C8: the gap enters affinely (Lemma L) and Theorem CU' + +def free_coefficients(m, trial): + """The free coefficients of check_C4 (degree 6, transverse quadratic negative definite).""" + vals = [F(1, 3), F(-2, 5), F(3, 7), F(-1, 2), F(5, 4), F(-3, 8), F(2, 9), F(-7, 6), F(1, 5), F(4, 11)] + zero_js = tuple([0] * m) + pinned = {(0, zero_js), (1, zero_js), (2, zero_js), (3, zero_js)} + for j in range(m): + ej = tuple(1 if t == j else 0 for t in range(m)) + pinned |= {(0, ej), (1, ej)} + free = {} + for idx, key in enumerate(monomials(m, 6)): + if key not in pinned: + free[key] = vals[(idx * 7 + trial * 3) % len(vals)] + for j in range(m): + ejj = (0, tuple(2 if t == j else 0 for t in range(m))) + free[ejj] = -abs(free.get(ejj, F(1))) - 1 + if m == 2: + free[(0, (1, 1))] = F(1, 3) + return free + + +def pdivr(p): + """p / r for a polynomial p in r with zero constant term.""" + if p[0] != 0: + raise ValueError('not divisible by r') + return p[1:] or [F(0)] + + +def scaled_hessian_dets(c, m, sign): + """det K at the pin x = sign r/2 as a polynomial in r, and det A there: K = [[alpha, sqrt(r) beta^T], + [sqrt(r) beta, A]], alpha = f_xx/r, beta = grad_y f_x / r, A = D_y^2 f, det K = alpha det A - r beta^T adj(A) beta.""" + zero_js = tuple([0] * m) + xp = [F(0), F(sign, 2)] + fxx = [F(0)] + for (i, js), coef in c.items(): + if js == zero_js and i >= 2: + fxx = padd(fxx, pscale(pmul(coef, ppow(xp, i - 2)), F(i * (i - 1)))) + alpha = pdivr(ptrim(fxx + [F(0)])) + beta, Am = [], [[[F(0)] for _ in range(m)] for _ in range(m)] + for j in range(m): + ej = tuple(1 if t == j else 0 for t in range(m)) + fxy = [F(0)] + for (i, js), coef in c.items(): + if js == ej and i >= 1: + fxy = padd(fxy, pscale(pmul(coef, ppow(xp, i - 1)), F(i))) + beta.append(pdivr(ptrim(fxy + [F(0)]))) + for l in range(m): + el = tuple((1 if t == j else 0) + (1 if t == l else 0) for t in range(m)) + mult = F(2) if j == l else F(1) + ent = [F(0)] + for (i, js), coef in c.items(): + if js == el: + ent = padd(ent, pscale(pmul(coef, ppow(xp, i)), mult)) + Am[j][l] = ent + if m == 1: + detA = Am[0][0] + badjb = pmul(beta[0], beta[0]) + else: + detA = padd(pmul(Am[0][0], Am[1][1]), pscale(pmul(Am[0][1], Am[1][0]), F(-1))) + badjb = padd(padd(pmul(pmul(beta[0], beta[0]), Am[1][1]), pmul(pmul(beta[1], beta[1]), Am[0][0])), + pscale(pmul(pmul(beta[0], beta[1]), Am[0][1]), F(-2))) + detK = padd(pmul(alpha, detA), pscale(pmul([F(0), F(1)], badjb), F(-1))) + return ptrim(detK), ptrim(detA) + + +class Q4: + """Exact arithmetic in Q(theta), theta^4 = 3 (theta = 3^(1/4)): a0 + a1 theta + a2 theta^2 + a3 theta^3.""" + def __init__(self, a): + self.a = [F(x) for x in a] + [F(0)] * (4 - len(a)) + + def __add__(self, o): + return Q4([x + y for x, y in zip(self.a, o.a)]) + + def __sub__(self, o): + return Q4([x - y for x, y in zip(self.a, o.a)]) + + def __mul__(self, o): + out = [F(0)] * 7 + for i, x in enumerate(self.a): + for j, y in enumerate(o.a): + out[i + j] += x * y + return Q4([out[k] + 3 * (out[k + 4] if k + 4 < 7 else 0) for k in range(4)]) + + def __eq__(self, o): + return self.a == o.a + + def power(self, n): + out = Q4([1]) + base = self if n >= 0 else Q4([0, 0, 0, F(1, 3)]) if self.a == [0, 1, 0, 0] else None + if base is None: + raise ValueError('negative powers only of theta') + for _ in range(abs(n)): + out = out * base + return out + + +def check_C8(mutant): + sgn = 1 if mutant == 'M5' else -1 + cases = 0 + for m in (1, 2): + for trial in range(2): + free = free_coefficients(m, trial) + zero_js = tuple([0] * m) + kappa0, b = [F(3, 5), F(7, 4)][trial], F(1, 7) + c0 = pinned_poly_in_r(m, kappa0, b, free) + c1 = pinned_poly_in_r(m, kappa0 + 1, b, free) + # (i) the pinned family is affine in kappa with kappa-part 2 kappa r (x - r)(x + r/2)^2, i.e. the k-shift + # omega = 2 (x - r)(x + r/2)^2 = 2x^3 - (3/2) r^2 x - r^3/2 per unit k = kappa r + diff = {key: ptrim(padd(c1.get(key, [F(0)]), pscale(c0.get(key, [F(0)]), F(-1)))) for key in set(c0) | set(c1)} + omega = {(3, zero_js): [F(0), F(2)], (1, zero_js): [F(0), F(0), F(0), F(-3, 2)], + (0, zero_js): [F(0), F(0), F(0), F(0), F(-1, 2)]} + for key, val in diff.items(): + if val != ptrim(omega.get(key, [F(0)])): + return False, 'pinned family not affine in the gap with the cubic shift (m = %d, %s)' % (m, key) + dM0, AM = scaled_hessian_dets(c0, m, -1) + dS0, AS = scaled_hessian_dets(c0, m, 1) + dM1, _ = scaled_hessian_dets(c1, m, -1) + dS1, _ = scaled_hessian_dets(c1, m, 1) + # (ii) d/dk det K_M = -6 det A_M and d/dk det K_S = +6 det A_S, exactly (d/dk = r^-1 d/dkappa) + if ptrim(padd(dM1, pscale(dM0, F(-1)))) != ptrim(pscale(pmul([F(0), F(1)], AM), F(6 * sgn))): + return False, 'd det K_M / dk != -6 det A_M (m = %d)' % m + if ptrim(padd(dS1, pscale(dS0, F(-1)))) != ptrim(pscale(pmul([F(0), F(1)], AS), F(6))): + return False, 'd det K_S / dk != 6 det A_S (m = %d)' % m + # (iii) the common rY: det K_M + det K_S = 2 r Y + O(r^2), det K_S - det K_M = 12 k det A(0) + O(r^2) + A0 = [[F(0)] * m for _ in range(m)] + gamma = [] + for j in range(m): + gamma.append(2 * free.get((2, tuple(1 if t == j else 0 for t in range(m))), F(0))) + for l in range(m): + el = tuple((1 if t == j else 0) + (1 if t == l else 0) for t in range(m)) + A0[j][l] = (2 if j == l else 1) * free.get((0, el), F(0)) + f4 = 24 * free[(4, zero_js)] + Y = (f4 / 12) * det(A0) - quad(gamma, adj(A0), gamma) / 4 + for kap, dM, dS in ((kappa0, dM0, dS0), (kappa0 + 1, dM1, dS1)): + tot = padd(dM, dS) + [F(0)] * 3 + dif = padd(padd(dS, pscale(dM, F(-1))), [F(0), -12 * kap * det(A0)]) + [F(0)] * 3 + if tot[0] != 0 or tot[1] != 2 * Y: + return False, 'det K_M + det K_S != 2 r Y + O(r^2) (m = %d)' % m + if dif[0] != 0 or dif[1] != 0: + return False, 'det K_S - det K_M != 12 k det A(0) + O(r^2) (m = %d)' % m + cases += 1 + # (iv) Theorem CU' constants: I_cand / c_1 = 3^(1/4)/2 (fourth power 3/16); the rejected cusp integral + # int_{s1}^{s2} (36 Delta^2 s^-8 - Y^2) ds = (16/7) Y^2 s1 - (8/7) Y^2 s2, s2 = 3^(1/4) s1, exactly in Q(3^(1/4)); + # the small-kappa elder share 13/27 + if ((F(8, 7) / F(16, 7)) ** 4) * F(6, 2) != F(3, 16): + return False, 'I_cand / c_1' + theta = Q4([0, 1]) + pts = 0 + for Yv, Dv in ((F(1), F(8)), (F(1, 2), F(81, 32)), (F(3), F(3, 2))): + s1 = rational_root(2 * Dv / Yv, 4) + if s1 is None: + continue + anti = lambda s: Q4([0]) - Q4([F(36, 7) * Dv * Dv]) * s.power(-7) * Q4([s1 ** -7]) - Q4([Yv * Yv * s1]) * s + # anti(t) evaluated at s = t * s1 with t in {1, theta}: F(s) = -(36/7) Delta^2 s^-7 - Y^2 s + val = anti(theta) - (Q4([0]) - Q4([F(36, 7) * Dv * Dv * s1 ** -7]) - Q4([Yv * Yv * s1])) + target = Q4([F(16, 7) * Yv * Yv * s1]) - Q4([F(8, 7) * Yv * Yv * s1]) * theta + if not (val == target): + return False, 'rejected cusp integral' + pts += 1 + share = (72 * 2 - F(2, 3) * 2 ** 3) / (72 * 6 - F(2, 3) * 6 ** 3) + if share != F(13, 27) or pts < 2: + return False, 'elder share 13/27 or too few points' + return True, {'pinned_cases': cases, 'dimensions': [2, 3], 'rejected_integral_points': pts, + 'I_cand_over_c1': '3^(1/4)/2', 'small_kappa_elder_share': '13/27'} + + # -------------------------------------------------------------------- C7: Gaussian-kernel covariances def kap_n(n): @@ -600,8 +774,8 @@ def check_C7(mutant): CHECKS = [('C1_quartic_ridge_identities', check_C1), ('C2_elder_window_exact_maximin', check_C2), ('C3_fiber_reduction', check_C3), ('C4_cusp_limit_field_pinned_polynomials', check_C4), ('C5_model_endpoint_determinants_and_typed_window', check_C5), ('C6_cusp_integrals', check_C6), - ('C7_gaussian_kernel_covariances', check_C7)] -MUTANTS = ('M1', 'M2', 'M3', 'M4') + ('C7_gaussian_kernel_covariances', check_C7), ('C8_gap_affine_structure_and_cand_constants', check_C8)] +MUTANTS = ('M1', 'M2', 'M3', 'M4', 'M5') def main(): @@ -617,7 +791,7 @@ def main(): ok, info = fn(args.mutant) results[name] = {'passed': ok, 'info': info} ok_all = ok_all and ok - out = {'object': 'CL-CUSP-SECOND-ORDER-20261001-v1', 'scientific_effect': 'NONE', 'passed': ok_all, + out = {'object': 'CL-CUSP-SECOND-ORDER-20261001-v1.1', 'scientific_effect': 'NONE', 'passed': ok_all, 'mutant': args.mutant, 'checks': results} sys.stdout.write(json.dumps(out, indent=1, sort_keys=True) + '\n') return 0 if ok_all else 1 diff --git a/frontiers/cusp_second_order_20261001/explore_cusp.py b/frontiers/cusp_second_order_20261001/explore_cusp.py index 40f35d7c5..772d75206 100644 --- a/frontiers/cusp_second_order_20261001/explore_cusp.py +++ b/frontiers/cusp_second_order_20261001/explore_cusp.py @@ -1,5 +1,5 @@ #!/usr/bin/env python3 -"""EXPLORATION for PROOF.md (CL-CUSP-SECOND-ORDER-20261001-v1). Standard library only, floating point. +"""EXPLORATION for PROOF.md (CL-CUSP-SECOND-ORDER-20261001-v1.1). Standard library only, floating point. NOT a control, NOT evidence of acceptance, NOT replayed by the workflow (it takes about a minute). X1 the actual global elder decision of random pinned plane quintics at the cusp scale (r = 0.01, k = kappa r), @@ -8,7 +8,8 @@ X2 the second-order coefficient c_1 for the Gaussian kernel exp(-|z|^2/2) (L = infinity), with the leading constants c_2, c_3 for the ratio c_1/c: d = 2 by composite Gauss-Legendre quadrature, d = 3 by a deterministic quadrature over the negative-definite cone; each at two resolutions (the difference is the - convergence evidence), and d = 3 also by an independent fixed-seed Monte Carlo over the same law. + convergence evidence), and d = 3 also by an independent fixed-seed Monte Carlo over the same law; with the + candidate coefficient I_cand = (3^(1/4)/2) c_1 and the rejected coefficient I_cand - c_1 of Theorem CU'. Output: EXPLORE.json (numbers rounded; seeds fixed). """ import json @@ -340,11 +341,14 @@ def run_X2(): J2c, J2 = ey_d2(12, 28), ey_d2(24, 56) J3c, J3 = ey_d3(8, 8, 32), ey_d3(8, 8, 64) M3, se3 = mc_d3(8000000) + q = 3 ** 0.25 / 2 # I_cand / c_1 (Theorem CU') return {'d2': {'J': round(J2, 10), 'J_coarse': round(J2c, 10), 'c': round(c2, 12), 'c1': round(k2 * J2, 8), - 'c1_over_c': round(k2 * J2 / c2, 6), + 'c1_over_c': round(k2 * J2 / c2, 6), 'I_cand': round(q * k2 * J2, 8), + 'rejected_coefficient': round((q - 1) * k2 * J2, 8), 'method': 'composite Gauss-Legendre, a = -w^4: 24x12 by 56x12 nodes (coarse: 12x12 by 28x12)'}, 'd3': {'J': round(J3, 9), 'J_coarse': round(J3c, 9), 'c': round(c3, 12), 'c1': round(k3 * J3, 8), - 'c1_over_c': round(k3 * J3 / c3, 6), + 'c1_over_c': round(k3 * J3 / c3, 6), 'I_cand': round(q * k3 * J3, 8), + 'rejected_coefficient': round((q - 1) * k3 * J3, 8), 'method': 'cone quadrature: Gauss-Legendre 64 x 64 in (rho, u), trapezoid 9 in theta, ' 'Gauss-Laguerre 64 in s (coarse: Laguerre 32)', 'mc_check': {'J': round(M3, 5), 'J_se': round(se3, 5), 'c1': round(k3 * M3, 5), @@ -353,7 +357,7 @@ def run_X2(): if __name__ == '__main__': - out = {'object': 'CL-CUSP-SECOND-ORDER-20261001-v1 exploration', 'not_a_control': True} + out = {'object': 'CL-CUSP-SECOND-ORDER-20261001-v1.1 exploration', 'not_a_control': True} if len(sys.argv) < 2 or 'X2' in sys.argv[1:]: out['X2'] = run_X2() if len(sys.argv) < 2 or 'X1' in sys.argv[1:]: