From ec968cb182de7b84a7a4300fbaca322ed52ea1e1 Mon Sep 17 00:00:00 2001 From: Claude Date: Wed, 30 Sep 2026 09:06:08 +0000 Subject: [PATCH 1/8] Add local-pairing packet: the local landscape decides the elder pairing (CL-LOCAL-PAIRING-20260930-v1) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Author-side proof candidate answering the 2026-09-30 question (which extra window critical points change the elder pairing, and whether the decision is local). Scientific effect NONE; nonauthor review required. * frontiers/local_pairing_20260930/LOCAL_PAIRING.md — Lemmas 2.1-2.2, Lemma P, Lemma P', Proposition 3.3 (deterministic, on the pinned planar cubic of [CUB]): off the null tie set, (M,S) is the elder pair iff n(theta) = 0; otherwise M dies at the highest extra window saddle. Theorem LP (conditional on [SC], [CUB], [P] at their scope): liminf r^-3 (1 - p_r) >= a_1 + a_2 = nu_1 + nu_2 - beta_far. Conjecture LP= with its obligation (O1)/(O1'). * pairing_check.py / RESULTS.json — stdlib exact controls (I0-I3, P1, P2, N1), byte-identical in -O; mutants M1-M4 exit 1, unknown label exit 2. * SOURCES.json — exact blob/sha256 identities of every consumed and cited source ([SC] and [TSL] consumed at frozen unmerged bytes). * exploration/ — numpy/scipy/sympy merge-tree evidence (not CI, not proof). * .github/workflows/local-pairing.yml — manifest, source-drift, replay and mutant gate. Reviewed twice by clean-context Claude subagents before landing (same provider and account: not a nonauthor review); all findings applied. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .github/workflows/local-pairing.yml | 60 + .../local_pairing_20260930/LOCAL_PAIRING.md | 443 ++ frontiers/local_pairing_20260930/README.md | 28 + frontiers/local_pairing_20260930/RESULTS.json | 40 + frontiers/local_pairing_20260930/SOURCES.json | 131 + .../exploration/RESULTS_EXPLORATION.json | 4735 +++++++++++++++++ .../exploration/elder_cubic.py | 205 + .../local_pairing_20260930/pairing_check.py | 273 + 8 files changed, 5915 insertions(+) create mode 100644 .github/workflows/local-pairing.yml create mode 100644 frontiers/local_pairing_20260930/LOCAL_PAIRING.md create mode 100644 frontiers/local_pairing_20260930/README.md create mode 100644 frontiers/local_pairing_20260930/RESULTS.json create mode 100644 frontiers/local_pairing_20260930/SOURCES.json create mode 100644 frontiers/local_pairing_20260930/exploration/RESULTS_EXPLORATION.json create mode 100644 frontiers/local_pairing_20260930/exploration/elder_cubic.py create mode 100644 frontiers/local_pairing_20260930/pairing_check.py diff --git a/.github/workflows/local-pairing.yml b/.github/workflows/local-pairing.yml new file mode 100644 index 000000000..9100dbfed --- /dev/null +++ b/.github/workflows/local-pairing.yml @@ -0,0 +1,60 @@ +name: Local pairing exact controls + +'on': + pull_request: + paths: + - 'frontiers/local_pairing_20260930/**' + - '.github/workflows/local-pairing.yml' + 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' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('frontiers/local_pairing_20260930').resolve() + manifest = json.loads((root / 'SOURCES.json').read_text()) + 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']) + script = str(root / 'pairing_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 ('M1', 'M2', 'M3', 'M4'): + 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/local_pairing_20260930/LOCAL_PAIRING.md b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md new file mode 100644 index 000000000..8d649da4e --- /dev/null +++ b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md @@ -0,0 +1,443 @@ +# Local pairing: the local landscape decides the elder pairing, and the selection loss is bounded below by the local cluster mass + +Object: CL-LOCAL-PAIRING-20260930-v1. +Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 30 September 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR REVIEW REQUIRED. Lemmas 2.1–2.2, Lemma P, Lemma P′ +and Proposition 3.3 are deterministic statements about the pinned cubic with complete written proofs +(reviewed twice by clean-context referee agents before landing; every blocking finding and amendment they +raised is applied below). Theorem LP (the lower bound with an identified functional) is conditional: it consumes exact +interfaces of [SC], [CUB] and [P] at their own review status. Conjecture LP= (equality) is stated with its +single remaining probabilistic obligation. Scientific effect: NONE — no register, graph, STATUS, prize or +Boolean changes; no numerical constant is claimed. + +## 0. What this packet answers, and what it does not + +The question put to the project on 2026-09-30 (input relayed by the owner) was: the account knows how many +extra window critical points there are near a short-lifetime pair (Theorem Q: `Θ(r³)`), but not *which +of them pair* and whether *the pairing is local*. Three sub-questions were asked. The answers, at the +scope stated below: + +1. **Mixed pairs are already local, conditionally.** For any cutoff `δ_r → 0` with `δ_r/r → ∞`, the + ordered-pair mass with a point outside the `δ_r` ball is `o(r³)` ([TSL] Theorem L, from [SC] (24) and + the [C6] moments; nonauthor-reviewed at conditional scope). Remote–remote pairs are `O_ρ(r⁵)` ([SC] (2)). + The only non-local piece of the *nonempty* event is a **remote singleton** of positive mass `β_far` + ([SC] (22); [TSL] Theorem T). So "the rare event is local" is true for pairs and false for singletons. + Nothing in this packet re-proves those statements. +2. **Which witnesses pair.** On the limiting local landscape — the pinned planar cubic `P_θ` of [CUB] — + the elder rule is decided exactly: off the null tie set `{s ≤ B, D² = T}`, `(M,S)` is the elder pair **iff** + the cubic has no extra strict-window critical point (`n(θ) = 0`) (Lemmas P, P′); if `n(θ) ≥ 1`, `M` dies at + the highest extra window saddle (Proposition 3.3). + Transferred to the field, this gives the lower bound + `liminf r⁻³ (1 − p_r) ≥ a₁ + a₂ = ν₁ + ν₂ − β_far` (Theorem LP), where `a₁, a₂` are the local coefficients + of [SC] (20) and `β_far` its remote coefficient. Equality — the statement that **remote witnesses never + pair** — is Conjecture LP= with one obligation (§5). This is the bridge from the count law to the + lifetime law: it bounds the existential `Θ(r³)` selection loss of [P] Theorem A / [ELDER] / [E_d] from + below by an explicit functional of the cluster measure, and identifies it under LP=. +3. **Cluster law.** Nothing new: [SC] (3) identifies `ν = ν₁δ₁ + ν₂δ₂`; this packet uses only its local part. + +What the bridge is worth for the lifetime density is stated without inflation in §6: it bounds from below +(and, under LP=, fixes) the coefficient of the `Θ(ℓ^{2/3})` compact-window selection loss — relative order +`ℓ` against the leading `ℓ^{−1/3}`; the leading law never needed it, and the unrestricted density's bounded +remainder is a different object. + +## 1. Setting and exact sources + +Fixed `d ≥ 2`, torus side `L`, birth `b`, gap mark `k > 0`, one orthonormal frame. Pins +`M = (−r/2, 0)`, `S = (r/2, 0)`, `f(M) = b`, `f(S) = b − k r³`, zero gradients; regression law `Q_r`; +typed determinant weight `W_r`, FULL normalizer `Z_r`, `Q_r^W = (W_r/Z_r) Q_r`. `p_r` is the `Q_r^W` +probability that the **global ordinary superlevel elder death partner** of `M` is `S` ([P] §1, [ELDER] §1). +On the generic locus its death height is the maximin + + d_f(M) = sup { min_t f(γ(t)) : γ(0) = M, f(γ(1)) > f(M) }, (E1) + +with the empty-supremum convention of [P] §8; `S` is the death partner iff `d_f(M) = f(S)` and `S` is the +critical point at that level ([ELDER] §1). A path from `M` to a point strictly above `b` whose values stay +strictly above `f(S)` therefore proves that `S` is **not** the death partner. Nothing here uses gradient +trajectories or separatrices. + +Sources (exact identities in `SOURCES.json`; none is revalidated here): + +| Tag | Source | What is consumed | +|---|---|---| +| [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | §1 model and `p_r`; Theorem A upper bound `1 − p_r ≤ C r³`; §8 genericity; §§9–11 radial ledger and (E9)-type intensity. | +| [ELDER] | `frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md` | (E1) maximin characterisation; §2 the stable-`C⁰`-connectivity transfer template; §4 (E9)–(E11) density gap in `d = 2`. | +| [CUB] | `frontiers/planar_cubic_cluster_20260929/PROOF.md` | (C1)–(C3) pinned cubic and shear; (C4) typed condition; Theorem C classifier `n(θ) ∈ {0,1,2}`; (C7)–(C13). Only the deterministic part. | +| [SC] | `frontiers/spectral_cluster_closure_20260929/PROOF.md` (Math-#162, frozen head `382c0f9`, proof blob `16c56821`) | §2 spectral coordinates and (10); §5 normal form: `[f(rX,rZ,0) − b]/r³ → P_θ` in `C²` on bounded sets, the limit `W_r/r⁴ → ∏h_j² w`, and the rescaled typed-jet measure (17); (18)–(20): `a₁, a₂`; (22): `β_far`; (2), (24) as cited in §0. Unmerged: consumed at its frozen bytes. | +| [TSL] | `frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md` (Math-#166, math head `7c82252`, blob `a32fd5f7`) | Theorem L, Theorem T — cited in §0 only; not consumed by any proof below. | +| [E_d], [EDL] | `frontiers/elder_lower_all_d_20260929/PROOF.md`, `frontiers/elder_dimension_lift_20260928/PROOF.md` | The existential lower bound `1 − p_r ≥ c r³` in every `d` that Theorem LP sharpens; not consumed. | +| [C7] | `frontiers/c7_total_bounded_20260929/PROOF.md` (Theorem K / Corollary T), `frontiers/c7_zero_gap_limit_20260929/PROOF.md` (Theorem Z), `frontiers/unrestricted_selection_difference_20260929/PROOF.md` (Theorem U) | Cited in §6.3 only for the order of the unrestricted remainder; not consumed. | + +A defect in a consumed interface blocks the corresponding statement here. In particular Theorem LP is +conditional on [SC] at exactly the same scope as [TSL]. + +## 2. The pinned cubic and the elder rule on it + +In the sheared coordinates `(u, Z)` of [CUB] (C2)–(C3), with `B = β − a²/(12k)` and +`D = (c − aβ/(4k) + a³/(72k²))/2`, + + P(u, Z) = A(u) + (Z²/2)(s + Bu) + (D/3) Z³, A(u) = 2k (u − 1)(u + 1/2)². (2.1) + +`M = (−1/2, 0)` with `P = 0`, `S = (1/2, 0)` with `P = −k`. The typed domain is `s < −|B|/2` ([CUB] (C4)). +Elementary facts about (2.1) (the identities among them, and the shear (2.1) itself against [CUB] (C1)–(C3), +are checked exactly by `pairing_check.py`, I0–I3; the monotonicity statements are one-line calculus): + +* **Axis.** `A′(u) = 6ku² − 3k/2`, so `A` is decreasing on `(−1/2, 1/2)` from `0` to `−k` and increasing for + `|u| > 1/2`; `A(u) + k = (k/2)(u + 1)(2u − 1)²`, so `A(u) > −k` iff `u > −1` and `u ≠ 1/2`; `A(u) > 0` iff + `u > 1`; `A(u) → +∞` as `u → +∞`. Because (2.1) has no term linear in `Z`, `∇P(u, 0) = (A′(u), 0)`: the + sheared axis `{Z = 0}` is an invariant line of `∇P`. +* **Vertical lines.** At fixed `u`, `∂_Z P = Z (s + Bu + DZ)`. If `s + Bu < 0` and `D ≠ 0`, `P(u, ·)` decreases + from `A(u)` at `Z = 0` to a minimum at `Z₁(u) = −(s + Bu)/D` (on the side `sign Z = sign D`) and then + increases to `+∞`. If `s + Bu = 0` and `D ≠ 0`, `P(u, Z) = A(u) + DZ³/3` is monotone to `+∞` on the `D` side +(level if `D = 0`). +* **Extra critical points.** `(u₁, Z₁)`, `Z₁ ≠ 0`, is critical iff `A′(u₁) + (B/2)Z₁² = 0` (conic) and + `s + Bu₁ + DZ₁ = 0` (line); then `P(u₁, Z₁) = A(u₁) + Z₁²(s + Bu₁)/6` ([CUB] (C8) in this form), so + `P(S₁) < A(u₁)` iff `s + Bu₁ < 0`, and `D ≠ 0` unless `s + Bu₁ = 0`. [CUB] Theorem C: every strict-window + extra critical point is a nondegenerate saddle, there are at most two, and their number `n(θ)` is (C6). For + `B > 0` they satisfy `−1/2 < u₁ < u_w < 1/2`; for `B < 0`, `u_w < u₁ < −1/2` with `u_w = −1/2 − B/(2s)`; + for `B = 0`, `u₁ = −1/2`. +* **`M` is a strict local maximum.** `Hess P(M) = diag(−6k, s − B/2)` and `s − B/2 < 0` on the typed domain. + +**Lemma 2.1 (growth at infinity; older components).** For every typed `θ`: (a) `|∇P(x)| → ∞` as `|x| → ∞`, +and `P` has finitely many critical points; (b) for every `h ≥ −k`, every unbounded component of `{P > h}` +contains points with `P > 0`. + +*Proof.* (a) Suppose `|∇P| ≤ C` along a sequence with `|x| → ∞`. If `|Z|` stayed bounded, `∂_u P = 6ku² − 3k/2 ++ (B/2)Z²` would diverge; so `|Z| → ∞` and `|s + Bu + DZ| ≤ C/|Z| → 0`. If `B = 0` this forces `D = 0` and then +`|s| ≤ C/|Z| → 0`, impossible. If `B ≠ 0`, `u = −(DZ + s)/B + o(1)` and +`∂_u P = [6kD²/B² + B/2] Z² + (12kDs/B²) Z + O(1)`; the `Z²` coefficient vanishes only when `12kD² + B³ = 0`, +which needs `B < 0` and `D ≠ 0`, and then the linear term `12kDs/B²` is nonzero because `s < 0`; either way +`∂_u P` diverges. Critical points are the intersections of the conic `A′(u) + (B/2)Z² = 0` (irreducible for +`B ≠ 0`; the two lines `u = ±1/2` for `B = 0`) with `Z = 0` or the line `s + Bu + DZ = 0`, finitely many. +(b) Write `P(ρ cos φ, ρ sin φ) = c₃(φ)ρ³ + c₂(φ)ρ² + c₁(φ)ρ + c₀` with `c₃(φ) = 2k cos³φ + (B/2) cos φ sin²φ ++ (D/3) sin³φ`, `c₂(φ) = (s/2) sin²φ` (the only quadratic term of (2.1) is `sZ²/2`), `c₁(φ) = −(3k/2) cos φ`, +`c₀ = −k/2`. At a zero `φ₀` of `c₃`, `sin φ₀ ≠ 0` (since `c₃(0), c₃(π) = ±2k`), so by continuity and compactness +there are `η ∈ (0, 2k)` and `σ > 0` with `|sin φ| ≥ σ` on `{|c₃(φ)| ≤ η}`; put `μ = |s|σ²/2`, so that `|c₂| ≥ μ` +there, and note `|c₂| ≤ |s|/2` everywhere. Choose `ρ₀` so large that for all `ρ′ ≥ ρ₀`: +`ηρ′³ > |s|ρ′²/2 + (3k/2)ρ′ + 3k/2`, `(μ/4)ρ′² > (3k/2)ρ′ + 3k/2`, `(μ/4)ρ′ > 3k/2`, and `3ηρ′² > |s|ρ′ + 3k/2`. +For `ρ ≥ ρ₀` and any `φ` one of four cases holds: +(1) `c₃ ≤ −η`: `P ≤ −ηρ³ + |c₂|ρ² + |c₁|ρ + |c₀| < −k`; +(2) `|c₃| < η` and `c₃ρ ≤ (3/4)|c₂|`: `P ≤ (3/4)|c₂|ρ² − |c₂|ρ² + (3k/2)ρ + k/2 ≤ −(μ/4)ρ² + (3k/2)ρ + k/2 < −k`; +(3) `0 < c₃ < η` and `c₃ρ > (3/4)|c₂|`: for every `ρ′ ≥ ρ`, `∂_ρ P(ρ′) = 3c₃ρ′² + 2c₂ρ′ + c₁ ≥ 3c₃ρρ′ − 2|c₂|ρ′ − +3k/2 > (1/4)|c₂|ρ′ − 3k/2 ≥ (μ/4)ρ′ − 3k/2 > 0`; +(4) `c₃ ≥ η`: for every `ρ′ ≥ ρ`, `∂_ρ P(ρ′) ≥ 3ηρ′² − |s|ρ′ − 3k/2 > 0`. +So every point with `|x| ≥ ρ₀` and `P(x) > −k` is of type (3) or (4), and along its ray `P` increases strictly to +`+∞`: the point is connected outward, inside `{P > P(x)}`, to points with `P > 0`. An unbounded component of +`{P > h}`, `h ≥ −k`, contains points with `|x| ≥ ρ₀`, hence points with `P > 0`. ∎ + +**Lemma 2.2 (the death level is a critical value).** For every typed `θ`, `h* := d_P(M)` satisfies +`−k ≤ h* < 0`, and if `h* > −k` then `h*` is the value of an extra strict-window critical point of `P`, so +`n(θ) ≥ 1`. + +*Proof.* `h* ≥ −k` by the axis path `M → S → {u > 1}` (values `≥ −k`, endpoint `A(u) > 0` for `u > 1`); +`h* < 0` because `M` is a strict local maximum, so for `h` close to `0` the `M`-component of `{P > h}` is a +small disk containing no point with `P > 0`. Suppose `h* > −k` is not a critical value. By Lemma 2.1(a) the +critical values are finitely many, so there is `ε > 0` with no critical value in `[h* − ε, h* + ε]`, and +`|∇P| ≥ c > 0` on `P^{−1}[h* − ε, h* + ε]` (a Palais–Smale sequence there would be bounded by Lemma 2.1(a) and +converge to a critical point at a level in the interval). Let `φ_t` be the flow of `∇P/|∇P|²` on the slab +`P^{−1}[h* − ε, h* + ε]`, along which `P` increases at unit rate, and define `η(x) = φ_{h* + ε − P(x)}(x)` for +`x` in the slab and `η(x) = x` for `P(x) ≥ h* + ε`; `η` is continuous, maps `{P ≥ h* − ε}` into `{P ≥ h* + ε}`, +and fixes `{P ≥ h* + ε}`. Shrink `ε` so that `ε < |h*|`. By definition of `h*` there is a path from `M` to a +point `y` with `P(y) > 0` with values `> h* − ε`; `η` of it is a path with values `≥ h* + ε > h*` from `M` +(fixed, since `P(M) = 0 > h* + ε`) to `y` (fixed), contradicting the supremum. So `h*` is a critical value in +`(−k, 0)`. A critical point at a level in `(−k, 0)` is not a pin and is counted by `n` ([CUB] §2: `n` counts +all additional critical points with height strictly between `−k` and `0`), so `n ≥ 1`; by Theorem C it is a +nondegenerate saddle. ∎ + +## 3. Lemma P: every extra window saddle rejects the pair + +**Lemma P.** Let `θ = (s, a, β, c)` be typed with `n(θ) ≥ 1`, and let `(u₁, Z₁)` be an extra strict-window +critical point. Then there is an explicit path `γ` from `M` to a point with `P > 0` whose minimum exceeds +`−k`; consequently `d_P(M) > −k`, and `(M,S)` is **not** the elder pair of `P`. The path is one of: + +* **V (vertical descent at `u₁`)** — when `s + Bu₁ < 0` and `−1 < u₁ < 1/2`: along the axis from `M` to + `(u₁, 0)`, then along the vertical line through `S₁` past `S₁` to `+∞` on the `D` side. Its minimum is + `P(S₁) ∈ (−k, 0)`, attained at `S₁`. +* **U (vertical rise at `u₀`)** — when `B < 0` and `s > B` (equivalently `|B|/2 < |s| < |B|`): along the + axis from `M` to `(u₀, 0)`, `u₀ = −s/B ∈ (−1, −1/2)`, then along the vertical line at `u₀`, on which + `P = A(u₀) + DZ³/3`, to `+∞` on the `D` side. Its minimum is `A(u₀) ∈ (−k, 0)`. + +Let `S_V` be the extra strict-window root with the largest `u`-coordinate. The proof shows that path V +applies at `S_V` whenever `D ≠ 0` (in case (iv) `S_V` is the root on the side `u > u₀` of `u₀ = −s/B`, which +exists both for `n = 1` and for `n = 2`); path U is a second valid path in case (iv), and the `D = 0` subcase +of (iv) has its own path. The margin of Lemma P is `m(θ) := P(S_V) + k` for `D ≠ 0` and `A(u₀) + k` for +`D = 0` in case (iv); it is positive, and it is continuous on the typed `n ≥ 1` domain: `S_V` is a +nondegenerate root ([CUB] Theorem C), so it depends continuously on `θ` by the implicit function theorem; it +persists across `s = B` and across `D² = T` (the root that appears or disappears there is the other one, at +`u_w`); and as `D → 0` in case (iv) its value tends to `A(u₀)`. Theorem LP uses this continuity to make its +jet sets compact. + +*Proof.* Axis parts: by the monotonicity of `A` in §2, the values along the axis from `M` to `(u*, 0)` +lie in `[A(u*), 0]`, and `A(u*) > −k` because `−1 < u* < 1/2` in both cases. Vertical parts: by the +vertical-line fact in §2. It remains to see that every typed jet with `n ≥ 1` falls in V or U. + +(i) `B > 0`. Then `−1/2 < u₁ < 1/2` and `s + Bu₁ < −B/2 + Bu₁ = B(u₁ − 1/2) < 0` by the typed condition: V. + +(ii) `B = 0`. Then `u₁ = −1/2` and `s + Bu₁ = s < 0`: V (the axis part is empty; the vertical line starts at `M`). + +(iii) `B < 0`, `s ≤ B` (`|s| ≥ |B|`). Then `n = 1` by (C6) and `u_w = −1/2 − |B|/(2|s|) ≥ −1`, so +`−1 < u₁ < −1/2`. The linear function `s + Bu` vanishes at `u₀ = −s/B = −|s|/|B| ≤ −1 < u₁` and is +decreasing in `u` (`B < 0`), hence `s + Bu₁ < 0`: V. + +(iv) `B < 0`, `s > B` (`|B|/2 < |s| < |B|`). Then `u₀ = −|s|/|B| ∈ (−1, −1/2)` and `A(u₀) > −k`. If `D ≠ 0`: +in the proof of [CUB] Theorem C the extra roots are the solutions of `g(u) = D` on the `Z > 0` branch and of +`g(u) = −D` on the `Z < 0` branch of the window conic, where `g` is strictly increasing on `(u_w, −1/2)` +([CUB] (C12)) and vanishes at `u₀` (there the line `s + Bu + DZ = 0` reduces to `DZ = 0`). Hence for `n = 2` +the two roots lie on opposite sides of `u₀`, and for `n = 1` the single root solves `g(u) = |D| > 0 = g(u₀)`; +in both cases `S_V` has `u₁ > u₀`, so `s + Bu₁ < 0` (`s + Bu` decreases in `u` for `B < 0`) and +`−1 < u₀ < u₁ < −1/2`: V at `S_V`. Path U — along the axis to `(u₀, 0)`, then the vertical line +`P = A(u₀) + DZ³/3` to `+∞` on the `D` side, minimum `A(u₀)` — is a second valid path. If `D = 0`, +`P(u, Z) = A(u) + (Z²/2) B (u − u₀)` and the vertical line at `u₀` is level; take instead +`M → (u₀, 0) → (u₀, Z) → (u₀ − ε, Z)` with `Z` large: on the last segment `B(u − u₀) ≥ 0`, so +`P ≥ A(u) ≥ A(u₀ − ε) > −k` for small `ε`, and its endpoint value `A(u₀ − ε) + (Z²/2)|B|ε` exceeds `0` for +`Z` large. ([ELDER]'s witness `(−3k/2, 0, −2k, 0)` is this subcase with `u₀ = −3/4`; its polygonal path (E5) +has minimum `−7k/32`, and `pairing_check.py` P2 verifies that both extra saddles sit exactly at `−7k/32`.) + +In V the minimum along the whole path is `P(S₁)`, which lies in `(−k, 0)` by the strict-window hypothesis; +in U it is `A(u₀) > −k`; in the `D = 0` path it is `A(u₀ − ε) > −k`. In all of them the far end has `P > 0`. ∎ + +`pairing_check.py` P1 verifies, exactly, on `6840` rational chart points (three values of `k`), that every +chart point is a typed strict-window critical point with `n ≥ 1`, and the inequalities of the path through +the chart root: V when `s + Bu₁ < 0` and `−1 < u₁ < 1/2` (`4680` points: `A(u₁) > P(S₁) > −k`, +`sign Z₁ = sign D`, the rise past `0`), otherwise U (`2160` points, all in case (iv) with the chart root on the +side `u₁ < u₀`: `u₀ ∈ (−1, −1/2)`, `A(u₀) > −k`, the rise past `0`). For those `2160` points it also computes +the other root exactly (Vieta on [CUB] (C13)), and verifies that it is a strict-window root on the side +`u > u₀`, that V applies to it, and that it is the higher of the two — so the proof's assignment (V at `S_V`) +and the ordering claim of Proposition 3.3 are exercised on every chart point. Smallest margin `1/1200` +(absolute; attained at `k = 1/3`). + +**Lemma P′ (converse).** If `θ` is typed, `n(θ) = 0`, and `S` is the only critical point of `P` at level +`−k` (this excludes the null set `{s ≤ B, D² = T}` of [CUB] (C6)), then `(M,S)` is the elder pair of `P`: +`d_P(M) = −k` and the merge at level `−k` is through `S`. + +*Proof.* By Lemma 2.2 and `n = 0`, `d_P(M) = −k`. Let `C` be the component of `M` in `{P > −k}`. `C` contains +no point with `P > 0` (such a point would give `d_P(M) > −k`), so by Lemma 2.1(b) `C` is bounded. The sheared +axis is `∇P`-invariant with `∇P(u, 0) = (A′(u), 0)`: the segment `(−1/2, 1/2)` is the ascending branch of `S` +into `M` (`A` increases toward `M` on it, and its values lie in `(−k, 0)`, so it lies in `C`), and the ray +`u > 1/2` is the other ascending branch of `S`, on which `A` increases to `+∞` and exceeds `0` for `u > 1`. +So one upper sector of `S` lies in `C` and the other in a component `C′` of `{P > −k}` containing points with +`P > 0`; `C′ ≠ C`. At level `−k` the component of `M` in `{P ≥ −k}` therefore contains `C′` through `S`, and +through no other point, since `S` is the only critical point at that level. `M`, the younger, dies at `S`. ∎ + +**Proposition 3.3 (the death saddle).** If `θ` is typed with `n(θ) ≥ 1`, then `d_P(M) = max{P(S′) : S′ an +extra strict-window saddle}`: `M` dies at the highest extra window saddle. + +*Proof.* By Lemma P, `d_P(M) > −k`; by Lemma 2.2, `d_P(M)` is the value of an extra strict-window saddle. +If `n = 1` there is nothing more to show. `n = 2` occurs only in case (iv) of Lemma P (`B < 0`, `s > B`, +[CUB] (C6)). There, in [CUB]'s notation, the two saddles solve `g(u) = D` on the `Z > 0` branch and +`g(u) = −D` on the `Z < 0` branch of the window conic, with `g` strictly increasing on `(u_w, −1/2)` and +`g(u₀) = 0` at `u₀ = −s/B` (at `u = u₀` the line reduces to `DZ = 0`). So for `D ≠ 0` the two roots lie on +opposite sides of `u₀` (case (iv) of Lemma P); the one with `u₁ > u₀` has `s + Bu₁ < 0` and `−1 < u₀ < u₁`, +hence Lemma P's path V passes through it and `d_P(M) ≥ P(S₁)`. Along the window branch the +height of the root is [CUB] (C9), `P/k = −(u + 1/2)[1 + (2s/B)(u − 1/2)]`, whose derivative is +`−k(B + 4su)/B`; for `B < 0` this is positive by [CUB]'s `B + 4su > 0` on (C11). So the root with the larger +`u` is the higher saddle, and `d_P(M)` equals its value. (`D = 0` puts both saddles at the same height.) ∎ + +Together: **on the limiting local landscape, off the null tie set `{s ≤ B, D² = T}`, `(M,S)` is the elder +pair iff `n(θ) = 0`, and otherwise `M` dies at the highest extra window saddle.** The exploration in `exploration/` (not a proof, not run in CI) +computes the merge tree of `P_θ` numerically from its critical points and gradient branches and agrees: +see `exploration/RESULTS_EXPLORATION.json` for the sampled jets with `n ≥ 1` (death at the highest window +saddle in every case, no self-attachment, including the samples where `P` has a local minimum elsewhere), +the sampled jets with `n = 0` (death at `S` in every case), and the [ELDER] witness (death at `−7/32`). + +## 4. Theorem LP: the identified lower bound for the selection loss + +Let `a₁, a₂` be the local coefficients of [SC] (20): `a_j = ∫ 1{n(θ) = j} dM(θ)`, with `M` the rescaled +typed-jet measure (17) (finite on `{n ≥ 1}` by (19)), so that `ν₁ = a₁ + β_far`, `ν₂ = a₂`. + +**Lemma 4.1 (pathwise normal form on the soft plane).** In the frame (axis, soft eigenvector `e_soft(O)`, +hard eigenvectors) of [SC] (7), with `s` the soft eigenvalue of the midpoint transverse Hessian divided by +`r`, `a = τ_xxz`, `β = τ_xzz`, `c = τ_zzz`, and `K ≥ 1` a bound for `‖f‖_{C⁴}`: for every `R ≥ 1` and +`|(X, Z)| ≤ R`, + + f(rX, rZ, 0) − b = r³ [ P_θ(X, Z) + ρ_r(X, Z) ], sup_{B(R)} |ρ_r| ≤ C_R K r, (4.0) + +with `C_R` depending only on `R` (and `d`); the same holds for the first two derivatives of `ρ_r` in `(X, Z)`. + +*Proof.* Write `g(x, z) = f(x, z, 0)` and expand `g` at the midpoint to third order with the Lagrange +remainder `|R₄| ≤ C K (|x| + |z|)⁴`. The six pin data — values `b`, `b − kr³` and vanishing `x`- and +`z`-derivatives at `(±r/2, 0)` — determine the coefficients: the axial cubic `g(x, 0)` is the Hermite +interpolant of `(b, 0)` and `(b − kr³, 0)` at `±r/2` up to `O(K r⁴)`, which in the scaled variable is +`r³(2kX³ − 3kX/2 − k/2)`; the two conditions `∂_z g(±r/2, 0) = 0` give `g_z(0) = −g_xxz r²/8 + O(K r³)` and +`g_xz(0) = O(K r²)`; and `g_zz(0) = rs`, `g_xxz = a`, `g_xzz = β`, `g_zzz = c` by the definition of the spectral +coordinates. Collecting the `z`-dependent terms in `x = rX`, `z = rZ`: +`g_z z + g_xz xz + g_zz z²/2 + g_xxz x²z/2 + g_xzz xz²/2 + g_zzz z³/6 = r³[(a/2)(X² − 1/4)Z + (s/2)Z² + (β/2)XZ² ++ (c/6)Z³] + O(K r⁴ R⁴)`, which with the axial cubic is `r³ P_θ(X, Z)` up to `O(K r⁴ R⁴)`. Differentiating +the same expansion gives the `C²` statement. ∎ + +This is the quantitative form of [SC] §5's `C²` convergence `[f(rX, rZ, 0) − b]/r³ → P_θ`; in `d = 2` it is +[ELDER] (E7) / [CUB] (G8). It uses no hard implicit function: the path of Lemma P lies in the soft plane. +(4.0) is stated in the unsheared coordinates `(X, Z)` of the frame; the vertical legs of the paths of Lemma P +are the lines `X = u₁ − aZ/(12k)` there, and "lies in `B(R)`" below is read in `(X, Z)`. + +**Lemma 4.2 (compact typed jets).** Let `K ⊂ {typed jets}` be compact in the spectral coordinates +`(s, h, O, τ)` with `h₂ ≥ δ > 0`. Then `r⁻³ Q_r^W(θ_r ∈ K) → M(K)` as `r → 0`. The same proof gives the +statement for bounded Borel `K` on which `M` charges no boundary. + +*Proof.* In the spectral coordinates the rescaled law `r⁻³ Q_r^W(θ_r ∈ dθ)` has density +`(c_m r²/Z_r) h_r(A, T) J_r 1{h₂ > −rs} E[W_r/r⁴ | jet]` ([SC] (7)–(9) and the disintegration below (16); the +finite-`r` type indicator is part of `W_r`). On `K`: `h_r → h₀` pointwise ([SC] §2), `J_r → J_0 = ∏h_j ∏(h_j − +h_i)`, `1{h₂ > −rs} → 1`, `Z_r/r² → z₀` ([SC] §1), and `E[W_r/r⁴ | jet] → ∏h_j² w(s, a, β)` ([SC] §5: the soft +`2×2` blocks of the endpoint Hessians divided by `r` converge to `M_M, M_S`, whose determinants multiply to +`9k²(4s² − B²)` on the typed domain, the hard block to `−diag(h)`, and the mixed soft–hard entries are `O(r)`, +so the determinants and the type indicator converge off the null typed boundary). Domination on `K` by [SC] +(10), (15) and (6) with the bounded prefactor `c_m r²/Z_r`; the exceptional branch of (14) plays no role +because `s` is bounded on `K`. Dominated convergence gives the claim. ∎ + +In `d = 2` this is [CUB] Theorem G summed over the count, which is not consumed here. [SC] displays only the +nonempty-count version (18) and stresses that there is no finite uncentered `n = 0` formula; Lemma 4.2 is +about compact jet sets, on which the total mass is finite whatever the count. + +**Theorem LP.** Conditional on [SC] §§1, 2, 4 (15), 5 and (17)–(20) at their stated fixed-`d, L, b, k`, frame +scope, on [P] §4 (4.1), §5 (5.5), §8 and the maximin characterisation (E1), and on [CUB] Theorem C, + + liminf_{r→0} r⁻³ (1 − p_r) ≥ a₁ + a₂ = ν₁ + ν₂ − β_far. (4.1) + +Combined with [P] Theorem A, `1 − p_r = Θ(r³)` with `a₁ + a₂ ≤ liminf r⁻³(1 − p_r) ≤ limsup r⁻³(1 − p_r) ≤ C`. + +*Proof.* Fix `δ > 0` and `R ≥ 4`. Let `K_{δ,R}` be the set of jets `θ` with `s ≤ −|B|/2 − δ` (typed with +margin), `n(θ) ≥ 1`, `|D| ≥ δ`, `h₂ ≥ δ`, `|s| + |h| + |τ| ≤ 1/δ`, whose path V at `S_V` has margin `m(θ) ≥ δ` +and, truncated at the first point where `P ≥ δ`, lies in `B(R)` in the `(X, Z)` coordinates. It is compact in +the spectral coordinates: it is bounded, and it is closed because along a convergent sequence in it the +roots `S_V` converge (Lemma P: `m` is continuous) to a point with `P ≥ −k + δ`, `Z ≠ 0` (`|D| ≥ δ` and +`s + Bu < 0`) and `u ∈ (−1, 1/2)`, which is again an extra strict-window root of the limit jet, and the +other conditions are closed. It is invariant under the reflection `e_soft → −e_soft` (`n` and `m` are). As +`δ → 0` and `R → ∞`, `K_{δ,R}` increases to `{n ≥ 1}` minus an `M`-null set (`M` is absolutely continuous in +`(s, h, τ)`; the excluded sets are `{D = 0}`, the typed boundary, and the set where the margin vanishes, +which lies on the window boundary), so `M(K_{δ,R}) → a₁ + a₂` by monotone convergence and (19)–(20). + +Transfer. On the event `E_r := {θ_r ∈ K_{δ,R}} ∩ {C_R K r < δ/2}`, Lemma 4.1 shows that the truncated path +`γ_θ`, scaled by `r` and placed in the soft plane of the actual frame, is a path in the torus from `M` along +which `f > b − k r³ + (δ/2) r³ > f(S)` and whose endpoint has `f > b + (δ/2) r³ > b`. By (E1), +`d_f(M) > f(S)`, so `S` is not the elder death partner: `E_r ⊂ {rejected}`. + +Measure. By Lemma 4.2, `r⁻³ Q_r^W(θ_r ∈ K_{δ,R}) → M(K_{δ,R})`. For the complementary event, [SC] §1 (from +[P] (5.3)) gives `W_r/r² ≤ C K^{2d}` and [P] §5 gives `Z_r ≥ z_* r²`, so `E_{Q_r^W}[K^p] ≤ (C/z_*) E_{Q_r}[K^{2d+p}]` +is bounded uniformly in `r` for every `p` (Gaussian moments of the `C⁴` norm under the regression law, [P] +§4); Markov then gives `Q_r^W(C_R K r ≥ δ/2) ≤ E_{Q_r^W}[K^p] (2 C_R r/δ)^p = O_p(r^p)`. Hence + + liminf r⁻³ (1 − p_r) ≥ liminf r⁻³ Q_r^W(E_r) = M(K_{δ,R}), + +and `δ → 0`, `R → ∞` give (4.1). ∎ + +Remarks. (a) The proof uses no window critical point of the field, no gradient trajectory, no near +count: only field values along an explicit path in the soft plane. It is the [ELDER] §2 mechanism run over +the whole `n ≥ 1` domain instead of one witness, with [SC]'s typed-jet measure replacing the one-point +positivity argument of [ELDER] §3. (b) In `d = 2` the soft plane is the whole space. (c) `a₁ + a₂ > 0` by [SC] +(20), so Theorem LP re-derives, conditionally on [SC], the existential lower bounds of [ELDER] / [EDL] / [E_d] +at fixed parameters, with a constant. It does not re-derive their uniformity on compact mark sets, which they +have and this statement does not claim. + +## 5. Conjecture LP= and its one obligation + +**Conjecture LP=.** `1 − p_r = (a₁ + a₂) r³ + o(r³)`. Equivalently: remote and mixed witnesses never change +the elder partner at leading order, and the entire selection loss is the local nonempty cluster mass. + +By Theorem LP it suffices to show `limsup r⁻³ (1 − p_r) ≤ a₁ + a₂`. Write, for fixed `R`, + + Q_r^W(rejected) ≤ Q_r^W(N_R ≥ 1) + Q_r^W(rejected, N_R = 0), + +where `N_R` counts strict-window critical points of `f` in `B(Rr)` other than the pins. The first term is +`(a₁^R + a₂^R) r³ + o(r³)` by [SC] (18) and `a₁^R + a₂^R → a₁ + a₂` as `R → ∞` by (20). The obligation is + + (O1) limsup_{r→0} r⁻³ Q_r^W(rejected, N_R = 0) → 0 as R → ∞. + +What is known toward (O1), and what is not. On `{N_R = 0}` the analogue of the transfer in Theorem LP would +run Lemma P′ on the coupled field: if the jet `θ_r` has `n = 0` and the cubic landscape has margins — `P` +strictly concave on a ball `B(M, η)` with modulus `μ`, `sup_{C_θ ∖ B(M, η)} P ≤ −μη²/4` for the `M`-component +`C_θ` at level `−k + δ/4`, `C_θ ⊂ B(R − 1)`, the axis ray beyond `S` reaching `P ≥ δ` inside `B(R)`, `θ_r` at +distance `≥ δ` from the typed boundary and from the tie set — and if the `C²` error of Lemma 4.1 on `B(R)` is +below `min(μ/2, μη²/16, δ/8)`, then the rescaled field `F_r = (f(r·) − b)/r³` restricted to the soft plane +has its `M`-component at level `−k + δ/2` inside `{P > −k + δ/4}`, hence inside `B(R)` and away from every +point with `F_r ≥ 0` other than `M` itself; the far sector of its saddle at `S` (exact, since the pins are +exact under `Q_r^W`) reaches `F_r > 0` inside `B(R)`; and the axis gives `d_f(M) ≥ f(S)` (on it +`f − f(S) = r³(X − 1/2)²[2k(X + 1) − O(C_R K r)]`). In `d = 2` this yields `d_f(M) ∈ [f(S), f(S) + (δ/2)r³]` +with `S` the only saddle at such a level inside `B(Rr)` (`N_R = 0`) and the `M`-component confined to the +ball, so the death partner is `S`: **accepted**. In `d ≥ 3` the same conclusion needs the hard-direction +slaving of the superlevel components (fiberwise concavity in the hard coordinates `w`, `f_ww ≤ −h₂/2` on the +cylinder, and the correspondence between components of `{f > h}` and those of the reduced planar potential +`max_w f`), which [SC] §5 carries out for the critical points and this note does not carry out for the +superlevel sets. Granting it, + + {rejected, N_R = 0} ⊂ {n(θ_r) = 0, θ_r outside the margin set} ∪ {C²-error too large} ∪ {n(θ_r) ≥ 1, N_R = 0}, + +where untyped `θ_r` are placed in the first set. The second set is `O(r^p)` (as in Theorem LP). The third +has `r⁻³`-limit `M(n ≥ 1, n_R = 0)` at fixed `R` — its jets have all extra roots outside `B(R)` — by Lemma +4.2 on compact subsets, [SC] §5's joint spectral-jet/count convergence in total variation, and tightness of +`M` on `{n ≥ 1}` ([SC] (19): `|s| ≤ 2|B| + (48kD²)^{1/3}` there, with (10)/(15)/(6) domination); it tends to +`0` as `R → ∞`. So, granting the slaving step, (O1) is equivalent to + + (O1′) limsup_{r→0} r⁻³ Q_r^W(rejected, n(θ_r) = 0, θ_r outside the margin set) → 0 as the margins are relaxed. + +This is **not** a corollary of anything on file. The margin set must be jet-scaled (a fixed `δ` fails: near +`M`, `P(M + Z e_Z) = (Z²/2)(s − B/2) + (D/3)Z³` exceeds any fixed `−δ` for `|Z|` slightly beyond `δ` once +`|s|` is large, so a fixed-margin set would exclude a region of infinite `M`-mass and (O1′) would restate +(O1)); the typed-jet measure is not finite on `{n = 0}` ([SC] after (17)), so its complement cannot be +handled by Lemma 4.2; and the events in the complement — a second soft eigenvalue, a large jet, a jet near +the typed boundary, a landscape whose `M`-component reaches far — are the ones [P] §§6–7 and [SC] §4 control +in unweighted or occurrence-weighted form, not in rejection-weighted form. What is needed is either a direct +argument that a long excursion of `M`'s window component without a window saddle inside `B(Rr)` has +weighted probability `o(r³)`, or the boundary-layer estimates of [P] §7 restated at `o(r³)` on the no-witness +event. That, together with the `d ≥ 3` slaving step, is what separates Theorem LP from LP=. + +Falsifiers. LP= is false if there is a configuration class of weighted probability `≍ r³` in which +`(M,S)` is rejected although no strict-window critical point lies within `O(r)` of the pair — for example +a mechanism where the far sector of `S` reconnects to `M`'s component (self-attachment) through a region +at distance `≫ r`. Lemma P′ shows no such class exists on the accurate planar cubic model with margins; a +counterexample would have to live where the model is inaccurate, unmargined, or where the hard-direction +reduction fails. + +## 6. Consequences for the lifetime law, stated without inflation + +1. **Leading law.** Theorem 2.1 of the manuscript and [P] Theorems B, C need `1 − p_r = O(r³)` only; nothing + here changes the leading term `c ℓ^{−1/3}`. +2. **Compact-window selection loss, `d = 2`.** With [ELDER] (E9)–(E10) (compact `B`, `K`, `r_*`; `r³ = ℓ/k`), + Fatou's lemma and Theorem LP at every fixed `(b, k, u)` give + + liminf_{ℓ↓0} ℓ^{−2/3} [ν_cand(ℓ) − ν_eld(ℓ)] ≥ (1/3) ∫_B ∫_K ∫_{S¹} k^{−5/3} A₀(b,k,u) (a₁ + a₂)(b,k,u) dσ dk db, (6.1) + + using `A_r → A₀` uniformly on the compact parameters ([ELDER] (E9)). Under LP= the limit + `lim ℓ^{−2/3}[ν_cand − ν_eld]` exists and (6.1) is an equality, by dominated convergence with the uniform + constant of [P] Theorem A; (E11)'s existential two-sided bound would then carry an explicit limit. Without + LP=, (6.1) is a lower bound with an identified functional. The every-`d` version follows the same way from + [E_d]'s ledger once its (E9) analogue is cited. +3. **Ranking.** The loss (6.1) is `Θ(ℓ^{2/3})` ([ELDER] (E11), [P] Theorem B): relative order `ℓ` against the + leading `ℓ^{−1/3}`. For the unrestricted density the compact-window difference estimate is explicitly not + asserted ([P] Theorem C), and the rejected density there is bounded with a positive limit (`ρ_rej = O(1)` + by [C7] Theorem K / Corollary T, `ρ_rej → B_{d,L} > 0` by [C7] Theorem Z), so the selection coefficient + bounded here is a compact-window statement; neither [P] §11 nor [C7] claims a second-order expansion of + either density separately, and none is claimed here. What this packet adds is the structure on the local + landscape — which points pair — and a lower bound for its weight; locality of the decision for the field + is Conjecture LP=. +4. **Not claimed.** No numerical value of `a₁ + a₂`; no rate; no uniformity in marks or `d`; no statement + about the joint law of several bars; no change to the C6 / C7 / witness-collision registers. + +## 7. Finite controls + +`pairing_check.py` (standard library, exact rationals): I0–I3 — the shear (2.1) against [CUB] (C1)–(C3), +the conic identity, an exact central-difference check of `∂_Z P`, and `A(u) + k` — on rational points; P1 the +Lemma-P path inequalities on 6840 chart points under the assignment of §3 (path V at the chart root when +`s + Bu₁ < 0`; otherwise — case (iv) with the chart root on the side `u < u₀` — path U at `u₀` is verified +and, in addition, the partner root `S_V` is computed exactly by Vieta on [CUB] (C13), checked to be a +strict-window root with `u > u₀` and the higher of the two, and path V is verified there), with the minimal +margin reported; P2 the [ELDER] witness; N1 a typed `n = 0` grid on which the U hypothesis +`B < 0 < s − B` never holds (consistency of (C6) with case (iv)). +Deterministic JSON, byte-identical in `-O`; mutants `M1` (typed condition broken), `M2` (root pushed below +the window), `M3` (wrong conic identity), `M4` (margin sign) exit 1; unknown label exits 2. +`exploration/elder_cubic.py` (numpy/scipy/sympy; **not** run in CI, **not** a proof) and +`exploration/RESULTS_EXPLORATION.json` record the merge-tree computation of §3 (its known numerical limits +are stated in the script header). None of this certifies the Gaussian steps of §4–§5. + +## 8. Non-claims + +No register, STATUS, GRAPH, PROOF_INDEX or catalog edit; no `lemma_closed`, prize or premise change; no +acceptance of [SC], [CUB], [TSL] or [P] beyond their own records; no numerical constant; no uniformity as +`k ↓ 0`, in `L`, or in `d`; no rate for the `o(r³)`; no claim about the finite-`r` spatial law of the +witnesses; no `d ≥ 3` hard-direction statement beyond what [SC] §5 supplies for the soft plane. + +## 9. Review division + +* **Slice A (deterministic, `d`-free):** §2, Lemma P and Lemma P′, and the exact controls. Any lane. +* **Slice B (Gaussian transfer):** §4 Theorem LP — the coupling identity, the compact-jet convergence read + from [SC] §5, and the `O(r^p)` tail. Best read by a lane not exposed to [SC]'s authorship. +* **Slice C (the obligation):** §5 — whether (O1) follows from [P] §§6–7 and [SC] §4 as stated, or needs + a new estimate. This is the lane where the next theorem is decided. diff --git a/frontiers/local_pairing_20260930/README.md b/frontiers/local_pairing_20260930/README.md new file mode 100644 index 000000000..f76150154 --- /dev/null +++ b/frontiers/local_pairing_20260930/README.md @@ -0,0 +1,28 @@ +# Local pairing (CL-LOCAL-PAIRING-20260930-v1) + +Author-side proof candidate, Anthropic Claude, 30 September 2026. Scientific effect NONE. Nonauthor review required. + +**Question answered.** Near a short-lifetime candidate pair `(M, S)` the account knows how many extra +window critical points there are (`Θ(r³)`, Theorem Q) but not which of them change the elder pairing, nor +whether the decision is local. This packet shows: + +* On the limiting local landscape (the pinned planar cubic of [CUB]), off a null tie set, `(M,S)` is the + elder pair **iff** there is no extra strict-window critical point; otherwise `M` dies at the highest extra + window saddle (Lemmas 2.1–2.2, P, P′, Proposition 3.3 — elementary written proofs, explicit paths; reviewed + twice by clean-context referee agents before landing). +* **Theorem LP** (conditional on [SC] and [P] at their scope): `liminf r⁻³(1 − p_r) ≥ a₁ + a₂ = ν₁ + ν₂ − β_far`, + the local nonempty cluster mass — a lower bound for the `Θ(r³)` selection loss of [P] Theorem A / [ELDER] / + [E_d] by an explicit functional of the cluster measure. +* **Conjecture LP=** (`1 − p_r = (a₁ + a₂) r³ + o(r³)`, i.e. remote witnesses never pair) with its remaining + obligation (O1) stated precisely in §5: a rejection-weighted `o(r³)` estimate on the no-witness event that is + not on file, plus (for `d ≥ 3`) the hard-direction slaving of superlevel components. +* §6 says what this is worth for the lifetime law: a lower bound for the coefficient of the `Θ(ℓ^{2/3})` + compact-window selection loss (relative order `ℓ`); nothing for the leading term; nothing about the + unrestricted density's remainder. + +Files: `LOCAL_PAIRING.md` (the note), `pairing_check.py` (stdlib exact controls; `RESULTS.json` its output, +`-O` identical; mutants `M1`–`M4` exit 1, unknown label exit 2), `SOURCES.json` (exact identities of every +source; [SC] and [TSL] are consumed at frozen unmerged bytes), `exploration/` (numpy/scipy/sympy merge-tree +computation — evidence, not proof, not run in CI). + +Review slices (§9): A deterministic (§§2–3); B the Gaussian transfer (§4); C the obligation (§5). diff --git a/frontiers/local_pairing_20260930/RESULTS.json b/frontiers/local_pairing_20260930/RESULTS.json new file mode 100644 index 000000000..e8e76f78a --- /dev/null +++ b/frontiers/local_pairing_20260930/RESULTS.json @@ -0,0 +1,40 @@ +{ + "checks": { + "I0_I3_identities": { + "info": 144, + "passed": true + }, + "N1_n0_grid": { + "info": { + "n0_points_with_U_hypothesis": 0, + "n0_typed_grid_points": 135 + }, + "passed": true + }, + "P1_path_cases": { + "info": { + "cases": { + "U": 2160, + "V": 4680, + "V_at_other_root": 2160 + }, + "min_margin": "1/1200", + "samples": 6840 + }, + "passed": true + }, + "P2_elder_witness": { + "info": { + "Z1_squared": "15/8", + "n": 2, + "saddle_height": "-7/32", + "u0": "-3/4", + "vertical_line_constant": true + }, + "passed": true + } + }, + "object": "CL-LOCAL-PAIRING-20260930-v1", + "passed": true, + "scientific_effect": "NONE" +} diff --git a/frontiers/local_pairing_20260930/SOURCES.json b/frontiers/local_pairing_20260930/SOURCES.json new file mode 100644 index 000000000..848ec90f9 --- /dev/null +++ b/frontiers/local_pairing_20260930/SOURCES.json @@ -0,0 +1,131 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "object": "CL-LOCAL-PAIRING-20260930-v1", + "kind": "author-side proof candidate (deterministic lemmas complete; Theorem LP conditional; Conjecture LP= with one obligation); nonauthor analytic review required", + "author": "Anthropic Claude", + "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", + "answers": "the 2026-09-30 question relayed by the owner: which window witnesses pair, and whether the pairing is local", + "consumed": [ + { + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "tag": "P", + "use": "§1 p_r and Theorem A; §4 moments; §5 Z_r >= z_* r^2 and (5.3); §8 genericity; §§9-11 ledger" + }, + { + "path": "frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md", + "git_blob": "aaefc8da9668e819591dd9db0b34442394f2362e", + "bytes": 10229, + "sha256": "8ce05a43bde8e0d718f09e2da1ffe75d3ec5893489254afdbf5b58b602cdba0c", + "tag": "ELDER", + "use": "(E1) maximin; §2 path (E5) and C0-stability template; §4 (E9)-(E11)" + }, + { + "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", + "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "tag": "CUB", + "use": "(C1)-(C4), Theorem C, (C6)-(C13); deterministic part only" + } + ], + "consumed_unmerged": [ + { + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "pr": "d6g8k5htny-coder/Math-#162", + "commit": "382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "tag": "SC", + "use": "§1, §2 (7)-(10), §4 (14)-(16), §5 normal form and (17)-(18), §6 (19)-(20), §7 (2), (21)-(24)" + }, + { + "path": "frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md", + "pr": "d6g8k5htny-coder/Math-#166", + "commit": 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"pass_2": "Lemma 2.1(b) contained two false inequalities (replaced by the four-case polar argument); Lemma 2.2 flow wording made an explicit deformation; the S_V assignment in Lemma P made definite and continuous; K_{δ,R} closed by adding s <= -|B|/2 - δ; §5 obligation restated in jet-scaled margins as (O1′) with the d >= 3 slaving named; checker: shear identity (I0), exact central-difference derivative (I2) and Vieta other-root check added, min_margin key; title and README overclaims removed" + }, + "files": [ + { + "path": "LOCAL_PAIRING.md", + "bytes": 37901, + "sha256": "f93effd51bc1dc7f5dc0ac59bed9ebf5d94cfdeb992fd64e1b7235e3ae61420a" + }, + { + "path": "pairing_check.py", + "bytes": 13611, + "sha256": "ff30e7850bd650a9d264ec7537ef955d38a000b085e50834f0ca6c8a933b3cc5" + }, + { + "path": "RESULTS.json", + "bytes": 796, + "sha256": "c012b374ac07c9d51dbab6e2168194b6b27afed6c8f489da5c376230044546b4" + }, + { + "path": "README.md", + "bytes": 2174, + "sha256": 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with P = 0 (maximum), S = (1/2, 0) with P = -k (saddle), typed iff s < -|B|/2. + +Superlevel elder rule: descending the level, components are born at local maxima and merge at +saddles; at a merge the younger component (lower birth level) dies. A component containing a point +of arbitrarily large P (an unbounded component of the cubic) or a local maximum with value > 0 is +OLDER than M. d_P(M) = level of the saddle at which M's component first merges with an older one. +(M,S) is the elder pair iff that saddle is S. + +Merge structure is computed from the Morse data: all critical points (resultant + Newton polish), the two +ascending gradient branches of every saddle (ODE, normalized gradient, escape radius R_far = 60), and +union-find in decreasing saddle order. Exploration only: numpy/scipy/sympy, floating point, not run in CI, +not a proof. Known limits: the escape radius is adequate for the sampled |s| >= 0.15 (the bounded part of +{P > -k} near the zero directions of the cubic form has radius ~ sqrt(2k/(|s| sin^2 phi)) << 60); the +pre-Newton root filter could drop roots for very large coefficients; results are recorded per sample. +""" +import sys, math, random, json +import numpy as np +import sympy as sp +from scipy.integrate import solve_ivp + +X, Z = sp.symbols('X Z', real=True) + + +def cubic_coeffs(k, s, a, beta, c): + k, s, a, beta, c = [sp.Rational(str(round(float(t), 12))) for t in (k, s, a, beta, c)] + P = 2*k*X**3 - sp.Rational(3, 2)*k*X - k/sp.Integer(2) + sp.Rational(1, 2)*s*Z**2 \ + + sp.Rational(1, 2)*a*(X**2 - sp.Rational(1, 4))*Z + sp.Rational(1, 2)*beta*X*Z**2 + c*Z**3/6 + return sp.Poly(P, X, Z) + + +def critical_points(k, s, a, beta, c): + """All real critical points of P via the resultant in Z.""" + P = cubic_coeffs(k, s, a, beta, c).as_expr() + PX = sp.diff(P, X); PZ = sp.diff(P, Z) + res = sp.Poly(sp.resultant(PX, PZ, Z), X) + xs = [complex(r) for r in np.roots([float(cf) for cf in res.all_coeffs()])] + pts = [] + f_PX = sp.lambdify((X, Z), PX, 'numpy'); f_PZ = sp.lambdify((X, Z), PZ, 'numpy') + for xr in xs: + if abs(xr.imag) > 1e-7: + continue + x0 = xr.real + # PX = 0 is quadratic in Z at fixed X + qa = beta/2.0; qb = a*x0; qc = 6*k*x0**2 - 1.5*k + if abs(qa) > 1e-12: + disc = qb*qb - 4*qa*qc + if disc < -1e-9: + continue + disc = max(disc, 0.0) + zs = [(-qb + math.sqrt(disc))/(2*qa), (-qb - math.sqrt(disc))/(2*qa)] + elif abs(qb) > 1e-12: + zs = [-qc/qb] + else: + if abs(qc) > 1e-9: + continue + # a = beta = 0 and X = +-1/2: PZ = s Z + (c/2) Z^2 = 0 + zs = [0.0] + ([-2*s/c] if abs(c) > 1e-12 else []) + for z0 in zs: + if abs(f_PX(x0, z0)) < 1e-6 and abs(f_PZ(x0, z0)) < 1e-6: + # polish with Newton + v = np.array([x0, z0], dtype=float) + H = sp.hessian(P, (X, Z)); f_H = sp.lambdify((X, Z), H, 'numpy') + for _ in range(8): + g = np.array([f_PX(*v), f_PZ(*v)], dtype=float) + Hm = np.array(f_H(*v), dtype=float) + try: + v = v - np.linalg.solve(Hm, g) + except np.linalg.LinAlgError: + break + if not any(np.hypot(*(v - q['pos'])) < 1e-6 for q in pts): + Hm = np.array(f_H(*v), dtype=float) + ev, evec = np.linalg.eigh(Hm) + kind = 'max' if ev[1] < 0 else ('min' if ev[0] > 0 else 'saddle') + val = float(sp.lambdify((X, Z), P, 'numpy')(*v)) + pts.append({'pos': v, 'val': val, 'kind': kind, 'ev': ev, 'evec': evec}) + return pts, P + + +def ascend(P, start, direction, maxima, R_far=60.0): + """Follow the normalized ascending gradient flow from start + eps*direction.""" + PX = sp.lambdify((X, Z), sp.diff(P, X), 'numpy'); PZ = sp.lambdify((X, Z), sp.diff(P, Z), 'numpy') + def rhs(t, y): + g = np.array([PX(y[0], y[1]), PZ(y[0], y[1])], dtype=float) + n = np.hypot(*g) + return g/(n + 1e-12) + y0 = np.array(start, dtype=float) + 1e-3*np.array(direction, dtype=float) + def far(t, y): return R_far - np.hypot(*y) + far.terminal = True + def near_max(t, y): + return min(np.hypot(*(y - m['pos'])) for m in maxima) - 1e-4 if maxima else 1.0 + near_max.terminal = True + sol = solve_ivp(rhs, (0, 400.0), y0, events=[far, near_max], max_step=0.02, rtol=1e-8, atol=1e-10) + yend = sol.y[:, -1] + if np.hypot(*yend) >= R_far - 1e-6: + return ('inf', None) + for i, m in enumerate(maxima): + if np.hypot(*(yend - m['pos'])) < 5e-3: + return ('max', i) + # stalled (near a saddle or slow convergence): decide by gradient ascent continuation + return ('unknown', None) + + +def elder_partner(k, s, a, beta, c): + pts, P = critical_points(k, s, a, beta, c) + maxima = [p for p in pts if p['kind'] == 'max'] + saddles = [p for p in pts if p['kind'] == 'saddle'] + # identify M and S + iM = min(range(len(maxima)), key=lambda i: np.hypot(*(maxima[i]['pos'] - np.array([-0.5, 0.0])))) + assert np.hypot(*(maxima[iM]['pos'] - np.array([-0.5, 0.0]))) < 1e-5, 'M not found' + iS = min(range(len(saddles)), key=lambda i: np.hypot(*(saddles[i]['pos'] - np.array([0.5, 0.0])))) + assert np.hypot(*(saddles[iS]['pos'] - np.array([0.5, 0.0]))) < 1e-5, 'S not found' + # ascending branches of every saddle + branches = [] + for sd in saddles: + e = sd['evec'][:, 1] # eigenvector of the positive eigenvalue (ascent direction) + b1 = ascend(P, sd['pos'], e, maxima); b2 = ascend(P, sd['pos'], -e, maxima) + branches.append((b1, b2)) + # union-find over maxima and the node 'inf' + parent = {('max', i): ('max', i) for i in range(len(maxima))}; parent[('inf', None)] = ('inf', None) + age = {('max', i): maxima[i]['val'] for i in range(len(maxima))}; age[('inf', None)] = math.inf + def find(x): + while parent[x] != x: + parent[x] = parent[parent[x]]; x = parent[x] + return x + order = sorted(range(len(saddles)), key=lambda i: -saddles[i]['val']) + M = ('max', iM); death = None; self_attach = 0 + for i in order: + b1, b2 = branches[i] + if b1[0] == 'unknown' or b2[0] == 'unknown': + return {'status': 'unresolved', 'saddle': i, 'n_window': None} + r1, r2 = find(b1), find(b2) + rM = find(M) + if r1 == r2: + self_attach += 1 # both branches in one component (a loop); no death + continue + a1, a2 = age[r1], age[r2] + # merge: the older root survives + keep, drop = (r1, r2) if a1 >= a2 else (r2, r1) + if drop == rM and age[keep] > age[rM]: + death = (i, saddles[i]['val']); break + if drop == rM: + # M's component would be dropped only if older exists; equal ages impossible generically + pass + parent[drop] = keep + window = [sd for j, sd in enumerate(saddles) if j != iS and -k < sd['val'] < 0] + return {'status': 'ok', 'self_attachments': self_attach, 'death_saddle': None if death is None else death[0], 'death_val': None if death is None else death[1], + 'is_S': death is not None and death[0] == iS, 'n_window': len(window), + 'window_vals': sorted([sd['val'] for sd in window], reverse=True), + 'n_saddles': len(saddles), 'n_maxima': len(maxima), 'n_minima': len([p for p in pts if p['kind']=='min'])} + + +def cub_n(k, s, a, beta, c): + """[CUB] Theorem C classifier (None if not typed).""" + B = beta - a*a/(12*k); D = (c - a*beta/(4*k) + a**3/(72*k*k))/2 + if not (s < -abs(B)/2): + return None + T = -(s - B)**2*(B + 2*s)/(12*k) + if s > B: + return 2 if D*D < T else 1 + return 1 if D*D > T else 0 + + +def sample_theta_n0(rng, k=1.0): + """Sample typed jets with n = 0 by rejection on a box.""" + while True: + s = -rng.uniform(0.2, 4); a = rng.uniform(-3, 3); beta = rng.uniform(-4, 4); c = rng.uniform(-4, 4) + if cub_n(k, s, a, beta, c) == 0: + return dict(s=s, a=a, beta=beta, c=c) + + +def sample_theta(rng, k=1.0): + """Sample (u,v) in D_shape, Z != 0, a; return (s,a,beta,c) with an extra window root at (u - aZ/(12k), Z).""" + while True: + u = rng.uniform(-1.5, 0.5); b0 = 3 - 12*u*u + lo, hi = abs(b0)/2, 3 - 6*u + if lo < hi: + break + v = rng.uniform(lo, hi) + Zr = rng.choice([-1, 1])*math.exp(rng.uniform(math.log(0.15), math.log(3.0))) + a = rng.uniform(-3, 3) + s = -k*v/Zr**2; B = k*b0/Zr**2; D = k*(v - b0*u)/Zr**3 + beta = B + a*a/(12*k); c = 2*D + a*B/(4*k) + a**3/(144*k*k) + return dict(u=u, v=v, Z=Zr, a=a, s=s, beta=beta, c=c, B=B, D=D) + + +if __name__ == '__main__': + # usage: elder_cubic.py SEED N [n0] (third argument "n0" samples typed jets with no extra window root) + rng = random.Random(int(sys.argv[1]) if len(sys.argv) > 1 else 0) + N = int(sys.argv[2]) if len(sys.argv) > 2 else 40 + mode_n0 = len(sys.argv) > 3 and sys.argv[3] == 'n0' + k = 1.0 + out = [] + for i in range(N): + th = sample_theta_n0(rng, k) if mode_n0 else sample_theta(rng, k) + try: + r = elder_partner(k, th['s'], th['a'], th['beta'], th['c']) + except AssertionError as e: + r = {'status': 'error', 'msg': str(e)} + rec = {**{kk: float(vv) for kk, vv in th.items()}, **{kk: (vv if not isinstance(vv, np.generic) else float(vv)) for kk, vv in r.items()}} + out.append(rec) + print(json.dumps(rec, default=float)) + json.dump(out, open('elder_samples_%s%s.json' % (sys.argv[1] if len(sys.argv) > 1 else '0', '_n0' if mode_n0 else ''), 'w'), default=float) diff --git a/frontiers/local_pairing_20260930/pairing_check.py b/frontiers/local_pairing_20260930/pairing_check.py new file mode 100644 index 000000000..6a3aec4d4 --- /dev/null +++ b/frontiers/local_pairing_20260930/pairing_check.py @@ -0,0 +1,273 @@ +#!/usr/bin/env python3 +"""Exact finite controls for LOCAL_PAIRING.md (standard library only; exact rationals). + +Checked here, exactly: + I0 the shear: [CUB] (C1) evaluated at X = u - aZ/(12k) equals (2.1) with B, D from (C2), at rational points; + I1 the pinned cubic identities: P(u,Z) = A(u) + (Z^2/2)(s+Bu) + (D/3)Z^3 with A(u) = 2k(u-1)(u+1/2)^2, + and A(u) + (B u/6) Z^2 = -k(u+1/2) on the conic 6k(u^2-1/4) + (B/2)Z^2 = 0 (CUB (C8)/(C9) consistency); + I2 at an extra critical point (u1,Z1): P(S1) = A(u1) + Z1^2 (s+Bu1)/6; and the vertical-line derivative + d/dZ P(u1,Z) = Z (s+Bu1+DZ), checked by exact differentiation of the cubic in Z (a central difference + of a cubic equals its derivative plus exactly (D/3) h^2, which is subtracted); + I3 A(u) + k = (k/2)(u+1)(2u-1)^2 and A'(u) = 6ku^2 - 3k/2 at rational points; + P1 the path cases of Lemma P on a rational grid of the chart (u1, v, Z1, a): every sample is a typed jet + with an extra strict-window root (n >= 1 by the [CUB] classifier), at least one of the paths V, U applies + under the V-first assignment (V and U may both apply in case (iv); V is taken first), and the displayed + path has minimum value > -k with the stated margin, rising past 0 at its far end (sampled at 2Z1, 4Z1, + resp. 3, 6, 12 on the D side; the monotonicity itself is the one-line calculus of the text, not a check); + for every chart point whose own root is not V-admissible (case (iv), root on the side u < u0) the OTHER + root is computed exactly by Vieta on [CUB] (C13) and verified to be a strict-window root on the side + u > u0, V-admissible, and higher than the chart root (Lemma P's assignment and Proposition 3.3); + P2 the [ELDER] witness (s,a,beta,c) = (-3k/2,0,-2k,0): both extra saddles at height -7k/32, and the vertical + line at u = -3/4 is constant (D = 0), so the polygonal path of [ELDER] (E5) is needed there (null case); + N1 on a rational grid of typed jets with n = 0, the U hypothesis (B < 0 and s > B) never holds: consistency + of the [CUB] classifier with Lemma P's case (iv), which always has n >= 1; + M semantic mutants (--mutant M1|M2|M3|M4) must fail; an unknown label exits 2. + +The controls certify algebra, case coverage on the grid and the path minima; they do not prove the Gaussian +transfer (Theorem LP) or the weak convergence consumed from [SC]. Scientific effect: NONE. +""" +import argparse +import json +import sys +from fractions import Fraction as F + +K_VALUES = [F(1), F(2), F(1, 3)] + + +def A(k, u): + return 2*k*(u - 1)*(u + F(1, 2))**2 + + +def P(k, s, B, D, u, Z): + return A(k, u) + (Z*Z/2)*(s + B*u) + (D/3)*Z**3 + + +def cub_n(k, s, a, beta, c): + """[CUB] Theorem C classifier on the typed domain; None if not typed.""" + B = beta - a*a/(12*k) + D = (c - a*beta/(4*k) + a**3/(72*k*k))/2 + if not (s < -abs(B)/2): + return None + T = -(s - B)**2*(B + 2*s)/(12*k) + if s > B: + return 2 if D*D < T else 1 + return 1 if D*D > T else 0 + + +def chart(k, u1, v, Z1, a): + b0 = 3 - 12*u1*u1 + s = -k*v/Z1**2 + B = k*b0/Z1**2 + D = k*(v - b0*u1)/Z1**3 + beta = B + a*a/(12*k) + c = 2*D + a*B/(4*k) + a**3/(144*k*k) + return s, B, D, beta, c + + +def chart_grid(): + us = [F(n, 10) for n in range(-14, 5)] # -1.4 .. 0.4 + out = [] + for u1 in us: + b0 = 3 - 12*u1*u1 + lo, hi = abs(b0)/2, 3 - 6*u1 + if not lo < hi: + continue + for t in (F(1, 4), F(1, 2), F(3, 4)): + v = lo + t*(hi - lo) + for Z1 in (F(-2), F(-1), F(-1, 2), F(-1, 4), F(1, 4), F(1, 2), F(1), F(2)): + for a in (F(-2), F(-1, 2), F(0), F(1, 2), F(2)): + out.append((u1, v, Z1, a)) + return out + + +def cub_C1(k, s, a, beta, c, X, Z): + """[CUB] (C1): the unsheared pinned cubic.""" + return 2*k*X**3 - 3*k*X/2 - k/2 + (s/2)*Z*Z + (a/2)*(X*X - F(1, 4))*Z + (beta/2)*X*Z*Z + (c/6)*Z**3 + + +def check_identities(k, mutant): + rows = 0 + # I0: shear (C2) turns (C1) into (2.1) + for (s, a, beta, c) in ((F(-3, 2), F(0), F(-2), F(0)), (F(-2), F(1), F(1, 3), F(-5, 7)), (F(-7, 5), F(-3, 2), F(2), F(4))): + B = beta - a*a/(12*k); D = (c - a*beta/(4*k) + a**3/(72*k*k))/2 + for u in (F(-1), F(-1, 3), F(1, 2), F(5, 4)): + for Z in (F(-2), F(1, 3), F(3)): + X = u - a*Z/(12*k) + if cub_C1(k, s, a, beta, c, X, Z) != P(k, s, B, D, u, Z): + return False, 'I0 shear identity fails' + for u in (F(-3), F(-1), F(-1, 2), F(0), F(1, 2), F(2), F(7, 3)): + lhs = A(k, u) + k + rhs = (k/2)*(u + 1)*(2*u - 1)**2 + dA = 6*k*u*u - 3*k/2 + exact_dA = (A(k, u + F(1, 10**6)) - A(k, u - F(1, 10**6)))/(F(2, 10**6)) # central difference of a cubic: exact up to h^2 term + if lhs != rhs or abs(exact_dA - dA) > F(1, 10**11): + return False, 'I3 axis identity fails at u=%s' % u + for u in (F(-3, 2), F(-1), F(-3, 4), F(-1, 2), F(-1, 5), F(0), F(1, 3), F(1, 2), F(2)): + for Z in (F(-2), F(-1, 3), F(1, 2), F(3)): + for s in (F(-3), F(-1, 2)): + for D in (F(-1), F(2, 5)): + B = -12*k*(u*u - F(1, 4))/(Z*Z) # conic through (u,Z) + lhs = A(k, u) + (B*u/6)*Z*Z + rhs = -k*(u + F(1, 2)) if mutant != 'M3' else -k*(u - F(1, 2)) + if lhs != rhs: + return False, 'I1 conic identity fails at u=%s Z=%s' % (u, Z) + # I2: P at a critical point on the line s + Bu + DZ = 0 -> solve for D given (s,B,u,Z) + Dc = -(s + B*u)/Z + if P(k, s, B, Dc, u, Z) != A(k, u) + Z*Z*(s + B*u)/6: + return False, 'I2 critical value formula fails' + # derivative identity in Z by exact differentiation: for a cubic q(Z) the central + # difference (q(Z+h)-q(Z-h))/(2h) equals q'(Z) + (leading coefficient) h^2 exactly + h = F(1, 7) + central = (P(k, s, B, D, u, Z + h) - P(k, s, B, D, u, Z - h))/(2*h) - (D/3)*h*h + if central != Z*(s + B*u + D*Z): + return False, 'I2 derivative identity fails' + rows += 1 + return True, rows + + +def path_case(k, s, B, D, u1, Z1): + """Return (case, margin, message) for the explicit path of Lemma P at the extra root (u1, Z1). + + V: vertical descent at u1 (needs s + B u1 < 0 and -1 < u1 < 1/2): the path is M -> axis -> (u1,0) + -> straight down the vertical line to S1 -> on to P -> +infinity; minimum P(S1). + U: vertical rise at u0 = -s/B (needs B < 0, s > B, D != 0): M -> axis -> (u0,0) -> vertical line, on + which P = A(u0) + D Z^3/3; minimum A(u0). + Lemma P: every typed jet with a strict-window extra root admits V or U (B = 0 is included in V; D = 0 + with B < 0 < ... is the [ELDER] null case handled by its polygonal path). + """ + P1 = P(k, s, B, D, u1, Z1) + if s + B*u1 < 0 and -1 < u1 < F(1, 2): + if not (A(k, u1) > P1 > -k): + return 'V', None, 'ordering A(u1) > P(S1) > -k fails' + if not ((D > 0) == (Z1 > 0)): + return 'V', None, 'root not on the D side' + far1 = P(k, s, B, D, u1, 2*Z1); far2 = P(k, s, B, D, u1, 4*Z1) + if not (far1 > P1 and far2 > far1 and far2 > 0): + return 'V', None, 'vertical line does not rise past 0' + return 'V', P1 + k, 'ok' + if B < 0 and s > B: + u0 = -s/B + if not (-1 < u0 < -F(1, 2)): + return 'U', None, 'u0 outside (-1,-1/2)' + if D == 0: + return 'U', None, 'D = 0 (null case, [ELDER] path)' + if not (A(k, u0) > -k): + return 'U', None, 'A(u0) <= -k' + Zf = F(3) if D > 0 else F(-3) + if not (P(k, s, B, D, u0, Zf) == A(k, u0) + D*Zf**3/3 and P(k, s, B, D, u0, 2*Zf) > P(k, s, B, D, u0, Zf) > A(k, u0)): + return 'U', None, 'vertical line at u0 not monotone' + if not (P(k, s, B, D, u0, 4*Zf) > 0): + return 'U', None, 'vertical line at u0 does not pass 0 by |Z|=12' + return 'U', A(k, u0) + k, 'ok' + return 'none', None, 'no path case applies (would contradict Lemma P)' + + +def check_paths(mutant): + counts = {'V': 0, 'U': 0, 'V_at_other_root': 0} + total = 0 + min_margin = None + for k in K_VALUES: + for (u1, v, Z1, a) in chart_grid(): + s, B, D, beta, c = chart(k, u1, v, Z1, a) + if mutant == 'M1': + s = -abs(B)/2 + F(1, 100) # break the typed condition + n = cub_n(k, s, a, beta, c) + if n is None or n < 1: + return False, 'chart sample not typed/n>=1 at %s' % ((k, u1, v, Z1, a),) + # (u1,Z1) is a critical point in the strict window + P1 = P(k, s, B, D, u1, Z1) + if mutant == 'M2': + P1 = P1 - k # pretend the root sits below the window + if not (-k < P1 < 0): + return False, 'root not in the strict window' + gu = 6*k*u1*u1 - 3*k/2 + (B/2)*Z1*Z1 # d/du of the sheared cubic + gz = Z1*(s + B*u1 + D*Z1) + if not (gu == 0 and gz == 0): + return False, 'chart point is not a critical point' + case, margin, msg = path_case(k, s, B, D, u1, Z1) + if margin is None: + return False, 'case %s: %s at %s' % (case, msg, (k, u1, v, Z1, a)) + if mutant == 'M4': + margin = -margin + if not margin > 0: + return False, 'nonpositive margin' + counts[case] += 1; total += 1 + min_margin = margin if min_margin is None else min(min_margin, margin) + if case == 'U': + # the chart root is on the side u1 < u0; Lemma P assigns V at the other root S_V (u > u0). + # [CUB] (C13): (12kD^2+B^3) u^2 + 2B^2 s u + B s^2 - 3kD^2 = 0 has the two roots' u-coordinates. + lead = 12*k*D*D + B**3 + if lead == 0: + return False, 'U sample with a single finite root (leading coefficient of (C13) zero)' + u2 = -2*B*B*s/lead - u1 + Z2 = -(s + B*u2)/D + gu2 = 6*k*u2*u2 - 3*k/2 + (B/2)*Z2*Z2 + if gu2 != 0 or not (-k < P(k, s, B, D, u2, Z2) < 0): + return False, 'other root is not a strict-window critical point' + u0 = -s/B + case2, margin2, msg2 = path_case(k, s, B, D, u2, Z2) + if not (u2 > u0 and case2 == 'V' and margin2 is not None): + return False, 'other root is not V-admissible on the side u > u0: %s' % msg2 + if not (P(k, s, B, D, u2, Z2) > P(k, s, B, D, u1, Z1)): + return False, 'other root is not the higher saddle' + counts['V_at_other_root'] += 1 + return True, {'samples': total, 'cases': counts, 'min_margin': str(min_margin)} + + +def check_elder_witness(mutant): + k = F(1) + s, a, beta, c = F(-3, 2), F(0), F(-2), F(0) + B = beta - a*a/(12*k); D = (c - a*beta/(4*k) + a**3/(72*k*k))/2 + n = cub_n(k, s, a, beta, c) + u0 = -s/B # = -3/4 + # the two extra saddles: line s + Bu = 0 (D = 0) and conic + Z2 = -12*k*(u0*u0 - F(1, 4))/B + h = A(k, u0) + Z2*(s + B*u0)/6 + const = all(P(k, s, B, D, u0, Z) == A(k, u0) for Z in (F(0), F(1), F(9, 4), F(-3))) + ok = (n == 2 and u0 == F(-3, 4) and Z2 == F(15, 8) and h == F(-7, 32) and const and D == 0) + if mutant == 'M3': + ok = ok and h == F(-1, 4) + return ok, {'n': n, 'u0': str(u0), 'Z1_squared': str(Z2), 'saddle_height': str(h), 'vertical_line_constant': const} + + +def check_n0(mutant): + """On typed grid jets with n = 0 the U hypothesis (B < 0 and s > B) never holds ([CUB] (C6): s > B gives n >= 1).""" + k = F(1); found = 0; u_hypothesis = 0 + for s in (F(-3), F(-5, 2), F(-2), F(-3, 2), F(-1), F(-1, 2)): + for a in (F(-1), F(0), F(1)): + for beta in (F(-3), F(-1), F(0), F(1), F(2)): + for c in (F(-2), F(0), F(2)): + n = cub_n(k, s, a, beta, c) + if n == 0: + found += 1 + B = beta - a*a/(12*k) + if B < 0 and s > B: + u_hypothesis += 1 + if mutant == 'M4': + u_hypothesis += 1 + return found > 0 and u_hypothesis == 0, {'n0_typed_grid_points': found, 'n0_points_with_U_hypothesis': u_hypothesis} + + +def main(argv=None): + parser = argparse.ArgumentParser(description=__doc__.splitlines()[0]) + parser.add_argument('--mutant', default=None) + args = parser.parse_args(argv) + if args.mutant is not None and args.mutant not in ('M1', 'M2', 'M3', 'M4'): + print('unknown mutant label', file=sys.stderr) + return 2 + report = {'object': 'CL-LOCAL-PAIRING-20260930-v1', 'scientific_effect': 'NONE', 'checks': {}} + ok_all = True + for name, fn in (('I0_I3_identities', lambda: check_identities(F(1), args.mutant)), + ('P1_path_cases', lambda: check_paths(args.mutant)), + ('P2_elder_witness', lambda: check_elder_witness(args.mutant)), + ('N1_n0_grid', lambda: check_n0(args.mutant))): + ok, info = fn() + report['checks'][name] = {'passed': bool(ok), 'info': info} + ok_all = ok_all and bool(ok) + report['passed'] = ok_all + print(json.dumps(report, indent=2, sort_keys=True)) + return 0 if ok_all else 1 + + +if __name__ == '__main__': + raise SystemExit(main()) From 26230f21ed531e6a8ce681e448f73ebcb4513cac Mon Sep 17 00:00:00 2001 From: Claude Date: Wed, 30 Sep 2026 09:08:14 +0000 Subject: [PATCH 2/8] Local pairing: rebase onto main 30106e1; bind [SC] and [TSL] at their merged bytes MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Math-#162 (358f256) and Math-#166 (ab13a08) merged with bytes identical to the reviewed heads the packet froze, so both move from consumed_unmerged to consumed and are now verified by the workflow. Cited-only entries added for the Math-#173 residual-closure record and the Math-#168 numerics note; §0, §1, §6.4 and the README reflect this. §2/§7 wording aligned with the checker (I0-I3; S_V assignment). No mathematical change; scientific effect NONE. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .../local_pairing_20260930/LOCAL_PAIRING.md | 15 ++++--- frontiers/local_pairing_20260930/README.md | 2 +- frontiers/local_pairing_20260930/SOURCES.json | 42 ++++++++++++++----- 3 files changed, 43 insertions(+), 16 deletions(-) diff --git a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md index 8d649da4e..541924b42 100644 --- a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md +++ b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md @@ -22,7 +22,9 @@ scope stated below: the [C6] moments; nonauthor-reviewed at conditional scope). Remote–remote pairs are `O_ρ(r⁵)` ([SC] (2)). The only non-local piece of the *nonempty* event is a **remote singleton** of positive mass `β_far` ([SC] (22); [TSL] Theorem T). So "the rare event is local" is true for pairs and false for singletons. - Nothing in this packet re-proves those statements. + Nothing in this packet re-proves those statements. (The leading-mass localization residual of Math-#160 §5 + is bound to Theorem L as its direct statement by the declarative record [RES], Math-#173; that record proposes + register transitions and executes none.) 2. **Which witnesses pair.** On the limiting local landscape — the pinned planar cubic `P_θ` of [CUB] — the elder rule is decided exactly: off the null tie set `{s ≤ B, D² = T}`, `(M,S)` is the elder pair **iff** the cubic has no extra strict-window critical point (`n(θ) = 0`) (Lemmas P, P′); if `n(θ) ≥ 1`, `M` dies at @@ -62,10 +64,11 @@ Sources (exact identities in `SOURCES.json`; none is revalidated here): | [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | §1 model and `p_r`; Theorem A upper bound `1 − p_r ≤ C r³`; §8 genericity; §§9–11 radial ledger and (E9)-type intensity. | | [ELDER] | `frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md` | (E1) maximin characterisation; §2 the stable-`C⁰`-connectivity transfer template; §4 (E9)–(E11) density gap in `d = 2`. | | [CUB] | `frontiers/planar_cubic_cluster_20260929/PROOF.md` | (C1)–(C3) pinned cubic and shear; (C4) typed condition; Theorem C classifier `n(θ) ∈ {0,1,2}`; (C7)–(C13). Only the deterministic part. | -| [SC] | `frontiers/spectral_cluster_closure_20260929/PROOF.md` (Math-#162, frozen head `382c0f9`, proof blob `16c56821`) | §2 spectral coordinates and (10); §5 normal form: `[f(rX,rZ,0) − b]/r³ → P_θ` in `C²` on bounded sets, the limit `W_r/r⁴ → ∏h_j² w`, and the rescaled typed-jet measure (17); (18)–(20): `a₁, a₂`; (22): `β_far`; (2), (24) as cited in §0. Unmerged: consumed at its frozen bytes. | -| [TSL] | `frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md` (Math-#166, math head `7c82252`, blob `a32fd5f7`) | Theorem L, Theorem T — cited in §0 only; not consumed by any proof below. | +| [SC] | `frontiers/spectral_cluster_closure_20260929/PROOF.md` (Math-#162, reviewed at head `382c0f9`, merged at `358f256` with identical bytes; proof blob `16c56821`) | §2 spectral coordinates and (10); §5 normal form: `[f(rX,rZ,0) − b]/r³ → P_θ` in `C²` on bounded sets, the limit `W_r/r⁴ → ∏h_j² w`, and the rescaled typed-jet measure (17); (18)–(20): `a₁, a₂`; (22): `β_far`; (2), (24) as cited in §0. | +| [TSL] | `frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md` (Math-#166, reviewed at head `7c82252`, merged at `ab13a08` with identical bytes; blob `a32fd5f7`) | Theorem L, Theorem T — cited in §0 only; not consumed by any proof below. | | [E_d], [EDL] | `frontiers/elder_lower_all_d_20260929/PROOF.md`, `frontiers/elder_dimension_lift_20260928/PROOF.md` | The existential lower bound `1 − p_r ≥ c r³` in every `d` that Theorem LP sharpens; not consumed. | | [C7] | `frontiers/c7_total_bounded_20260929/PROOF.md` (Theorem K / Corollary T), `frontiers/c7_zero_gap_limit_20260929/PROOF.md` (Theorem Z), `frontiers/unrestricted_selection_difference_20260929/PROOF.md` (Theorem U) | Cited in §6.3 only for the order of the unrestricted remainder; not consumed. | +| [RES], [NUM] | `reviews/c6_residual_closure_20260930/RECONCILIATION.md` (Math-#173), `frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md` (Math-#168) | Cited in §0 and §6.4 only (the residual-closure record; uncertified numerical values of the near coefficients); not consumed. | A defect in a consumed interface blocks the corresponding statement here. In particular Theorem LP is conditional on [SC] at exactly the same scope as [TSL]. @@ -408,8 +411,10 @@ reduction fails. either density separately, and none is claimed here. What this packet adds is the structure on the local landscape — which points pair — and a lower bound for its weight; locality of the decision for the field is Conjecture LP=. -4. **Not claimed.** No numerical value of `a₁ + a₂`; no rate; no uniformity in marks or `d`; no statement - about the joint law of several bars; no change to the C6 / C7 / witness-collision registers. +4. **Not claimed.** No numerical value of `a₁ + a₂` (uncertified floating-point values of the planar near + coefficients exist in [NUM], Math-#168, under its own conditions; this packet neither consumes nor confirms + them); no rate; no uniformity in marks or `d`; no statement about the joint law of several bars; no change + to the C6 / C7 / witness-collision registers. ## 7. Finite controls diff --git a/frontiers/local_pairing_20260930/README.md b/frontiers/local_pairing_20260930/README.md index f76150154..d4a6f90b2 100644 --- a/frontiers/local_pairing_20260930/README.md +++ b/frontiers/local_pairing_20260930/README.md @@ -22,7 +22,7 @@ whether the decision is local. This packet shows: Files: `LOCAL_PAIRING.md` (the note), `pairing_check.py` (stdlib exact controls; `RESULTS.json` its output, `-O` identical; mutants `M1`–`M4` exit 1, unknown label exit 2), `SOURCES.json` (exact identities of every -source; [SC] and [TSL] are consumed at frozen unmerged bytes), `exploration/` (numpy/scipy/sympy merge-tree +source; [SC] and [TSL] are consumed at their merged bytes, identical to the reviewed PR heads), `exploration/` (numpy/scipy/sympy merge-tree computation — evidence, not proof, not run in CI). Review slices (§9): A deterministic (§§2–3); B the Gaussian transfer (§4); C the obligation (§5). diff --git a/frontiers/local_pairing_20260930/SOURCES.json b/frontiers/local_pairing_20260930/SOURCES.json index 848ec90f9..ad8c46155 100644 --- a/frontiers/local_pairing_20260930/SOURCES.json +++ b/frontiers/local_pairing_20260930/SOURCES.json @@ -30,9 +30,7 @@ "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", "tag": "CUB", "use": "(C1)-(C4), Theorem C, (C6)-(C13); deterministic part only" - } - ], - "consumed_unmerged": [ + }, { "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", "pr": "d6g8k5htny-coder/Math-#162", @@ -41,7 +39,9 @@ "bytes": 25006, "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", "tag": "SC", - "use": "§1, §2 (7)-(10), §4 (14)-(16), §5 normal form and (17)-(18), §6 (19)-(20), §7 (2), (21)-(24)" + "use": "§1, §2 (7)-(10), §4 (14)-(16), §5 normal form and (17)-(18), §6 (19)-(20), §7 (2), (21)-(24)", + "merged": "358f2562efbccc59f9549e5a79650e2e936395c3", + "note": "reviewed at PR head 382c0f9; merged with identical bytes" }, { "path": "frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md", @@ -51,7 +51,9 @@ "bytes": 17139, "sha256": "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769", "tag": "TSL", - "use": "cited in §0 only (Theorems L, T); not consumed by any proof" + "use": "cited in §0 only (Theorems L, T); not consumed by any proof", + "merged": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "note": "reviewed at PR head 7c82252 (custody repaired in Math-#172, proof bytes unchanged); merged with identical bytes" } ], "cited_only": [ @@ -89,6 +91,22 @@ "bytes": 13028, "sha256": "93a63002d32fd5c35ae994fe28b1205d8098026d6ddde19c67d4977dcf5817a9", "tag": "C7-U" + }, + { + "path": "reviews/c6_residual_closure_20260930/RECONCILIATION.md", + "git_blob": "3cfd17c5ef0356dbd3e8f98a2d08cb404ddd982c", + "bytes": 34758, + "sha256": "b647a186d0cfd1662fc69e9d31dfd447dbb1b2f6ef4364bbd59c4956f18aed25", + "tag": "RES", + "use": "cited in §0 only: the leading-mass localization residual bound to [TSL] Theorem L (Math-#173, merged 30106e1; declarative)" + }, + { + "path": "frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md", + "git_blob": "845881d40520d4acbe15ee196a1ba0d5412eafa1", + "bytes": 11018, + "sha256": "f20c1ac96b00edc071e428bd2e5dc4c91b8546a3ee0693367b783109b26aa518", + "tag": "NUM", + "use": "cited in §6 only: uncertified numerical values of the planar near coefficients (Math-#168, merged 02772ec); not consumed" } ], "referee": { @@ -99,8 +117,8 @@ "files": [ { "path": "LOCAL_PAIRING.md", - "bytes": 37901, - "sha256": "f93effd51bc1dc7f5dc0ac59bed9ebf5d94cfdeb992fd64e1b7235e3ae61420a" + "bytes": 38639, + "sha256": "54a44740198ec77aa5616177676b1e0ef2ffe93115d6bf2cfcca3bba5d180d1c" }, { "path": "pairing_check.py", @@ -114,8 +132,8 @@ }, { "path": "README.md", - "bytes": 2174, - "sha256": "3ce445efcd9d1faadfc7b22451a86cac79fc1be141e6a5f44d471e683ade4cd0" + "bytes": 2207, + "sha256": "7d5585b78aacbd9d018085edf721759120730a965ae704e997839c386806e018" }, { "path": "exploration/elder_cubic.py", @@ -127,5 +145,9 @@ "bytes": 104796, "sha256": "5c80aecdb6db5403a50ab2239361f36b638d1f5e7e5fd0e3d6dfee5dcd661d2b" } - ] + ], + "read_at": { + "branch_base": "30106e1be251e8c18759b6fa4de6176b76380c12", + "note": "Math- main after Math-#173 merged; every consumed source verified at these bytes by the workflow" + } } From e4ecbe20d259c138a79ba7f260efb0254e2d14b5 Mon Sep 17 00:00:00 2001 From: Claude Date: Wed, 30 Sep 2026 09:58:01 +0000 Subject: [PATCH 3/8] Local pairing v1.1: stdlib exploration, D = 0 tie set, relation to Math-#170/#175 recorded MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Applies Codex review 5364265784 on Math-#180: * P1 — exploration/elder_cubic.py (numpy/scipy/sympy) removed; replaced by exploration/elder_grid.py, standard library only: closed-form critical points ([CUB] (C7)/(C13)), discretised elder rule by union-find on a grid, boundary by the ray criterion of Lemma 2.1(b). RESULTS_EXPLORATION.json regenerated (seed 7, spacing 0.02): 120/120 n >= 1 jets die at the highest window saddle (level error <= 1.5e-3), 40/40 n = 0 jets die at level -k (39/40 merge cells at S; one near-typed-boundary jet displaced along the flat valley of S at the correct level), the [ELDER] witness ties at -7/32. * P2 — Proposition 3.3 now determines a unique death saddle only off {D = 0} (for D = 0 in case (iv) both extra saddles sit at A(u0)); summary and §0 reworded accordingly. Also records, in the header, §0, §5, README and SOURCES.json (cited_unmerged), that Math-#170 Theorem E/S and Math-#175 Theorem F prove the deterministic statement and the equality (Conjecture LP=) by the cap route this packet does not use; Theorem LP remains the cap-free lower bound. No mathematical change to Theorem LP; scientific effect NONE. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .../local_pairing_20260930/LOCAL_PAIRING.md | 60 +- frontiers/local_pairing_20260930/README.md | 10 +- frontiers/local_pairing_20260930/SOURCES.json | 45 +- .../exploration/RESULTS_EXPLORATION.json | 9133 ++++++++--------- .../exploration/elder_cubic.py | 205 - .../exploration/elder_grid.py | 277 + 6 files changed, 4763 insertions(+), 4967 deletions(-) delete mode 100644 frontiers/local_pairing_20260930/exploration/elder_cubic.py create mode 100644 frontiers/local_pairing_20260930/exploration/elder_grid.py diff --git a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md index 541924b42..87d889bcd 100644 --- a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md +++ b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md @@ -7,8 +7,17 @@ and Proposition 3.3 are deterministic statements about the pinned cubic with com (reviewed twice by clean-context referee agents before landing; every blocking finding and amendment they raised is applied below). Theorem LP (the lower bound with an identified functional) is conditional: it consumes exact interfaces of [SC], [CUB] and [P] at their own review status. Conjecture LP= (equality) is stated with its -single remaining probabilistic obligation. Scientific effect: NONE — no register, graph, STATUS, prize or -Boolean changes; no numerical constant is claimed. +single remaining probabilistic obligation. **Relation to Math-#170 and Math-#175 (v1.1, added after landing).** +The OpenAI packet Math-#170 (`frontiers/local_elder_geometry_20260930/PROOF.md`, head `79f18f0`, blob +`ef2aa579`, opened before this packet) proves the same deterministic statement as Lemma P′ + Proposition 3.3 +(its Theorem E, by the cubic critical-chord identity and a normalized-gradient collar rather than the explicit +paths used here) and proves the **equality** `r⁻³(1 − p_r) → α₁ + α₂` at fixed planar marks (its Theorem S) by a +route this packet does not use: [P]'s cap implication `F_r ⊂ G_r^c` and a retained-tail integration of [P] §7. +Math-#175 (`frontiers/concave_fibre_elder_20260930/`, head `9c6a734`) lifts it to fixed `d ≥ 3` by a contained +hard-negative tube. So Conjecture LP= below is Math-#170 Theorem S (planar) / Math-#175 Theorem F (`d ≥ 3`), +conditional on their interfaces and review; what this packet adds is an independent same-account derivation +of the deterministic core with exact controls, and the cap-free lower bound Theorem LP. Scientific effect: +NONE — no register, graph, STATUS, prize or Boolean changes; no numerical constant is claimed. ## 0. What this packet answers, and what it does not @@ -32,9 +41,11 @@ scope stated below: Transferred to the field, this gives the lower bound `liminf r⁻³ (1 − p_r) ≥ a₁ + a₂ = ν₁ + ν₂ − β_far` (Theorem LP), where `a₁, a₂` are the local coefficients of [SC] (20) and `β_far` its remote coefficient. Equality — the statement that **remote witnesses never - pair** — is Conjecture LP= with one obligation (§5). This is the bridge from the count law to the - lifetime law: it bounds the existential `Θ(r³)` selection loss of [P] Theorem A / [ELDER] / [E_d] from - below by an explicit functional of the cluster measure, and identifies it under LP=. + pair** — is Conjecture LP= with one obligation (§5); it is proved by Math-#170 Theorem S (planar) and + Math-#175 Theorem F (`d ≥ 3`) by the cap route described in the header, at their conditional scope. This is + the bridge from the count law to the lifetime law: it bounds the existential `Θ(r³)` selection loss of [P] + Theorem A / [ELDER] / [E_d] from below by an explicit functional of the cluster measure, and identifies it + under LP=. 3. **Cluster law.** Nothing new: [SC] (3) identifies `ν = ν₁δ₁ + ν₂δ₂`; this packet uses only its local part. What the bridge is worth for the lifetime density is stated without inflation in §6: it bounds from below @@ -222,7 +233,11 @@ So one upper sector of `S` lies in `C` and the other in a component `C′` of `{ through no other point, since `S` is the only critical point at that level. `M`, the younger, dies at `S`. ∎ **Proposition 3.3 (the death saddle).** If `θ` is typed with `n(θ) ≥ 1`, then `d_P(M) = max{P(S′) : S′ an -extra strict-window saddle}`: `M` dies at the highest extra window saddle. +extra strict-window saddle}`. Off the additional null set `{D = 0}` the maximum is attained at exactly one +extra window saddle, and `M` dies there: the highest extra window saddle. (For `D = 0` in case (iv) the two +extra saddles `(u₀, ±Z₀)` have the same height `A(u₀)` — the [ELDER] witness is such a jet — so only the death +level, not a unique death saddle, is determined; on the Morse distinct-value locus of the field this tie +does not occur.) *Proof.* By Lemma P, `d_P(M) > −k`; by Lemma 2.2, `d_P(M)` is the value of an extra strict-window saddle. If `n = 1` there is nothing more to show. `n = 2` occurs only in case (iv) of Lemma P (`B < 0`, `s > B`, @@ -233,14 +248,21 @@ opposite sides of `u₀` (case (iv) of Lemma P); the one with `u₁ > u₀` has hence Lemma P's path V passes through it and `d_P(M) ≥ P(S₁)`. Along the window branch the height of the root is [CUB] (C9), `P/k = −(u + 1/2)[1 + (2s/B)(u − 1/2)]`, whose derivative is `−k(B + 4su)/B`; for `B < 0` this is positive by [CUB]'s `B + 4su > 0` on (C11). So the root with the larger -`u` is the higher saddle, and `d_P(M)` equals its value. (`D = 0` puts both saddles at the same height.) ∎ +`u` is the higher saddle, and `d_P(M)` equals its value; for `D = 0` both saddles have the height `A(u₀)`, which +is then the death level. ∎ Together: **on the limiting local landscape, off the null tie set `{s ≤ B, D² = T}`, `(M,S)` is the elder -pair iff `n(θ) = 0`, and otherwise `M` dies at the highest extra window saddle.** The exploration in `exploration/` (not a proof, not run in CI) -computes the merge tree of `P_θ` numerically from its critical points and gradient branches and agrees: -see `exploration/RESULTS_EXPLORATION.json` for the sampled jets with `n ≥ 1` (death at the highest window -saddle in every case, no self-attachment, including the samples where `P` has a local minimum elsewhere), -the sampled jets with `n = 0` (death at `S` in every case), and the [ELDER] witness (death at `−7/32`). +pair iff `n(θ) = 0`; otherwise `d_P(M)` is the highest extra window critical value, and off `{D = 0}` `M` dies +at the unique highest extra window saddle.** The exploration in `exploration/` (standard library only; not a +proof, not run in CI) computes the death level of `M` by the discretised elder rule on a grid — cells merged in +decreasing order of `P_θ`, death when `M`'s component first meets a point with `P_θ > 0` — for sampled jets, +and compares it with the closed-form critical points: see `exploration/RESULTS_EXPLORATION.json` (seed 7, +spacing `0.02`, `k ∈ {1, 1/2}`): for all 120 sampled jets with `n ≥ 1` (80 with `n = 1`, 40 with `n = 2`) the +grid death level agrees with the highest window critical value to within `1.5·10⁻³` and the merge cell lies at +that saddle; for all 40 sampled typed jets with `n = 0` the death level is `−k` to within `10⁻⁴`, with the merge +cell at `S` in 39 cases and, in one case with `s + B/2 ≈ −0.024` (a jet close to the typed boundary, where the +saddle at `S` is nearly degenerate in `Z`), displaced `0.18` along the flat valley of `S` at the correct level; +the [ELDER] witness dies at `−7/32`, a tie of its two saddles. ## 4. Theorem LP: the identified lower bound for the selection loss @@ -371,7 +393,12 @@ has `r⁻³`-limit `M(n ≥ 1, n_R = 0)` at fixed `R` — its jets have all extr (O1′) limsup_{r→0} r⁻³ Q_r^W(rejected, n(θ_r) = 0, θ_r outside the margin set) → 0 as the margins are relaxed. -This is **not** a corollary of anything on file. The margin set must be jet-scaled (a fixed `δ` fails: near +This is not a corollary of anything merged. It **is** supplied, at fixed planar marks, by Math-#170 Theorem S +(unmerged; OpenAI; reviewed at head `79f18f0`), whose route avoids the margin set altogether: the failure event +is contained in the bad cap of [P] §8 (`F_r ⊂ G_r^c`), the bad cap's mass restricted to a large residual norm +is `o(r³)` by keeping the indicator inside [P]'s (7.5) scalar integral, and the remaining compact-jet part +converges by dominated convergence with the pointwise limit of the failure indicator (its Theorem E). The +`d ≥ 3` slaving step named below is Math-#175's H1. Read on its own, the margin set must be jet-scaled (a fixed `δ` fails: near `M`, `P(M + Z e_Z) = (Z²/2)(s − B/2) + (D/3)Z³` exceeds any fixed `−δ` for `|Z|` slightly beyond `δ` once `|s|` is large, so a fixed-margin set would exclude a region of infinite `M`-mass and (O1′) would restate (O1)); the typed-jet measure is not finite on `{n = 0}` ([SC] after (17)), so its complement cannot be @@ -428,9 +455,10 @@ margin reported; P2 the [ELDER] witness; N1 a typed `n = 0` grid on which the U `B < 0 < s − B` never holds (consistency of (C6) with case (iv)). Deterministic JSON, byte-identical in `-O`; mutants `M1` (typed condition broken), `M2` (root pushed below the window), `M3` (wrong conic identity), `M4` (margin sign) exit 1; unknown label exits 2. -`exploration/elder_cubic.py` (numpy/scipy/sympy; **not** run in CI, **not** a proof) and -`exploration/RESULTS_EXPLORATION.json` record the merge-tree computation of §3 (its known numerical limits -are stated in the script header). None of this certifies the Gaussian steps of §4–§5. +`exploration/elder_grid.py` (standard library only; **not** run in CI, **not** a proof) and +`exploration/RESULTS_EXPLORATION.json` record the grid elder-rule computation of §3 (closed-form critical +points, discretised maximin by union-find, boundary treated by the ray criterion of Lemma 2.1(b); its +numerical limits are stated in the script header). None of this certifies the Gaussian steps of §4–§5. ## 8. Non-claims diff --git a/frontiers/local_pairing_20260930/README.md b/frontiers/local_pairing_20260930/README.md index d4a6f90b2..47eed8546 100644 --- a/frontiers/local_pairing_20260930/README.md +++ b/frontiers/local_pairing_20260930/README.md @@ -14,15 +14,17 @@ whether the decision is local. This packet shows: the local nonempty cluster mass — a lower bound for the `Θ(r³)` selection loss of [P] Theorem A / [ELDER] / [E_d] by an explicit functional of the cluster measure. * **Conjecture LP=** (`1 − p_r = (a₁ + a₂) r³ + o(r³)`, i.e. remote witnesses never pair) with its remaining - obligation (O1) stated precisely in §5: a rejection-weighted `o(r³)` estimate on the no-witness event that is - not on file, plus (for `d ≥ 3`) the hard-direction slaving of superlevel components. + obligation (O1) stated precisely in §5. That obligation is discharged, at the packets' conditional scope, by + the OpenAI packets Math-#170 (planar, Theorem S: cap implication + retained-tail integration) and Math-#175 + (`d ≥ 3`, hard-negative-fibre barrier), both opened before this packet and reviewed by this author on + 30 September; this packet's own contribution is the cap-free lower bound and the exact controls. * §6 says what this is worth for the lifetime law: a lower bound for the coefficient of the `Θ(ℓ^{2/3})` compact-window selection loss (relative order `ℓ`); nothing for the leading term; nothing about the unrestricted density's remainder. Files: `LOCAL_PAIRING.md` (the note), `pairing_check.py` (stdlib exact controls; `RESULTS.json` its output, `-O` identical; mutants `M1`–`M4` exit 1, unknown label exit 2), `SOURCES.json` (exact identities of every -source; [SC] and [TSL] are consumed at their merged bytes, identical to the reviewed PR heads), `exploration/` (numpy/scipy/sympy merge-tree -computation — evidence, not proof, not run in CI). +source; [SC] and [TSL] are consumed at their merged bytes, identical to the reviewed PR heads), `exploration/` (standard-library grid +elder-rule computation — evidence, not proof, not run in CI). Review slices (§9): A deterministic (§§2–3); B the Gaussian transfer (§4); C the obligation (§5). diff --git a/frontiers/local_pairing_20260930/SOURCES.json b/frontiers/local_pairing_20260930/SOURCES.json index ad8c46155..d676dc1b4 100644 --- a/frontiers/local_pairing_20260930/SOURCES.json +++ b/frontiers/local_pairing_20260930/SOURCES.json @@ -2,7 +2,7 @@ "schema": 1, "scientific_effect": "NONE", "object": "CL-LOCAL-PAIRING-20260930-v1", - "kind": "author-side proof candidate (deterministic lemmas complete; Theorem LP conditional; Conjecture LP= with one obligation); nonauthor analytic review required", + "kind": "author-side proof candidate (deterministic lemmas complete; Theorem LP conditional; Conjecture LP= = Math-#170 Theorem S / Math-#175 Theorem F at their scope); nonauthor analytic review required", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "answers": "the 2026-09-30 question relayed by the owner: which window witnesses pair, and whether the pairing is local", @@ -117,8 +117,8 @@ "files": [ { "path": "LOCAL_PAIRING.md", - "bytes": 38639, - "sha256": "54a44740198ec77aa5616177676b1e0ef2ffe93115d6bf2cfcca3bba5d180d1c" + "bytes": 41583, + "sha256": "9e7193c16d994ee7f115f79e5e525209ad4faad55331051d3936efcb826c8bb2" }, { "path": "pairing_check.py", @@ -132,22 +132,45 @@ }, { "path": "README.md", - "bytes": 2207, - "sha256": "7d5585b78aacbd9d018085edf721759120730a965ae704e997839c386806e018" + "bytes": 2435, + "sha256": "dcef622428269d6b41730a2e4deb13d24c1d72fce3665d6436cfeef0f917e678" }, { - "path": "exploration/elder_cubic.py", - "bytes": 9959, - "sha256": "67bdede2f46fe74afd23d73c5d12fc3df10d6a82221052e81e8fcc7808a50962" + "path": "exploration/elder_grid.py", + "bytes": 11132, + "sha256": "72d642f4b1ce50ab4133533e8a943a84f1e02229c7c62181d6bce426d731a816" }, { "path": "exploration/RESULTS_EXPLORATION.json", - "bytes": 104796, - "sha256": "5c80aecdb6db5403a50ab2239361f36b638d1f5e7e5fd0e3d6dfee5dcd661d2b" + "bytes": 89311, + "sha256": "a1b8e0ab19dd75f4f595925eeade8bd396b2dc3b82e49cfb542ca18b8e6f85f3" } ], "read_at": { "branch_base": "30106e1be251e8c18759b6fa4de6176b76380c12", "note": "Math- main after Math-#173 merged; every consumed source verified at these bytes by the workflow" - } + }, + "cited_unmerged": [ + { + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "pr": "d6g8k5htny-coder/Math-#170", + "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "bytes": 39722, + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", + "tag": "LEG", + "use": "cited in the header, §0 and §5 only: Theorem E (same deterministic statement as Lemma P′ + Proposition 3.3, different proof) and Theorem S (planar equality by the cap route); not consumed; unmerged, so not verified by the workflow" + }, + { + "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", + "pr": "d6g8k5htny-coder/Math-#175", + "commit": "9c6a73489c5ed2465a3cf65dcc950eeeae074b05", + "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", + "bytes": 23416, + "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e", + "tag": "CFE", + "use": "cited in the header, §0 and §5 only: H1 (the d >= 3 hard-direction slaving named in §5) and Theorem F; not consumed; unmerged, so not verified by the workflow" + } + ], + "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied — exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded" } diff --git a/frontiers/local_pairing_20260930/exploration/RESULTS_EXPLORATION.json b/frontiers/local_pairing_20260930/exploration/RESULTS_EXPLORATION.json index 46b497e7c..bbdc28a6d 100644 --- a/frontiers/local_pairing_20260930/exploration/RESULTS_EXPLORATION.json +++ b/frontiers/local_pairing_20260930/exploration/RESULTS_EXPLORATION.json @@ -1,4735 +1,4406 @@ { - "method": "critical points by resultant + Newton; ascending gradient branches of every saddle by ODE (scipy solve_ivp, normalized gradient, R_far = 60); union-find merge in decreasing saddle order; older = unbounded component or a local maximum with value > 0; a saddle whose two branches lie in one component is a self-attachment (counted, no death)", - "k": 1.0, - "not_a_proof": true, - "n_ge_1_samples": { - "count": 180, - "seeds": [ - 7, - 8, - 9 - ], - "command": "python3 elder_cubic.py 60", - "chart": "u1 in D_shape, |Z1| log-uniform in [0.15, 3], a uniform in [-3, 3]", - "status_ok": 180, - "death_at_S": 0, - "death_at_highest_window_saddle": 180, - "by_n_window": { - "1": 116, - "2": 64 - }, - "with_local_minimum_present": 105, - "self_attachments_observed": 0 - }, - "n_0_samples": { - "count": 30, - "seed": 11, - "command": "python3 elder_cubic.py 11 30 n0", - "status_ok": 30, - "death_at_S": 30, - "self_attachments_observed": 0 - }, - "elder_witness_(-3/2,0,-2,0)": { - 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newline at end of file + } +} diff --git a/frontiers/local_pairing_20260930/exploration/elder_cubic.py b/frontiers/local_pairing_20260930/exploration/elder_cubic.py deleted file mode 100644 index c0f2fe508..000000000 --- a/frontiers/local_pairing_20260930/exploration/elder_cubic.py +++ /dev/null @@ -1,205 +0,0 @@ -"""Exploration: the elder-rule death partner of M in the limiting cubic landscape. - -P(X,Z) = 2kX^3 - 3kX/2 - k/2 + (s/2)Z^2 + (a/2)(X^2-1/4)Z + (beta/2)XZ^2 + (c/6)Z^3 (CUB (C1)) -M = (-1/2, 0) with P = 0 (maximum), S = (1/2, 0) with P = -k (saddle), typed iff s < -|B|/2. - -Superlevel elder rule: descending the level, components are born at local maxima and merge at -saddles; at a merge the younger component (lower birth level) dies. A component containing a point -of arbitrarily large P (an unbounded component of the cubic) or a local maximum with value > 0 is -OLDER than M. d_P(M) = level of the saddle at which M's component first merges with an older one. -(M,S) is the elder pair iff that saddle is S. - -Merge structure is computed from the Morse data: all critical points (resultant + Newton polish), the two -ascending gradient branches of every saddle (ODE, normalized gradient, escape radius R_far = 60), and -union-find in decreasing saddle order. Exploration only: numpy/scipy/sympy, floating point, not run in CI, -not a proof. Known limits: the escape radius is adequate for the sampled |s| >= 0.15 (the bounded part of -{P > -k} near the zero directions of the cubic form has radius ~ sqrt(2k/(|s| sin^2 phi)) << 60); the -pre-Newton root filter could drop roots for very large coefficients; results are recorded per sample. -""" -import sys, math, random, json -import numpy as np -import sympy as sp -from scipy.integrate import solve_ivp - -X, Z = sp.symbols('X Z', real=True) - - -def cubic_coeffs(k, s, a, beta, c): - k, s, a, beta, c = [sp.Rational(str(round(float(t), 12))) for t in (k, s, a, beta, c)] - P = 2*k*X**3 - sp.Rational(3, 2)*k*X - k/sp.Integer(2) + sp.Rational(1, 2)*s*Z**2 \ - + sp.Rational(1, 2)*a*(X**2 - sp.Rational(1, 4))*Z + sp.Rational(1, 2)*beta*X*Z**2 + c*Z**3/6 - return sp.Poly(P, X, Z) - - -def critical_points(k, s, a, beta, c): - """All real critical points of P via the resultant in Z.""" - P = cubic_coeffs(k, s, a, beta, c).as_expr() - PX = sp.diff(P, X); PZ = sp.diff(P, Z) - res = sp.Poly(sp.resultant(PX, PZ, Z), X) - xs = [complex(r) for r in np.roots([float(cf) for cf in res.all_coeffs()])] - pts = [] - f_PX = sp.lambdify((X, Z), PX, 'numpy'); f_PZ = sp.lambdify((X, Z), PZ, 'numpy') - for xr in xs: - if abs(xr.imag) > 1e-7: - continue - x0 = xr.real - # PX = 0 is quadratic in Z at fixed X - qa = beta/2.0; qb = a*x0; qc = 6*k*x0**2 - 1.5*k - if abs(qa) > 1e-12: - disc = qb*qb - 4*qa*qc - if disc < -1e-9: - continue - disc = max(disc, 0.0) - zs = [(-qb + math.sqrt(disc))/(2*qa), (-qb - math.sqrt(disc))/(2*qa)] - elif abs(qb) > 1e-12: - zs = [-qc/qb] - else: - if abs(qc) > 1e-9: - continue - # a = beta = 0 and X = +-1/2: PZ = s Z + (c/2) Z^2 = 0 - zs = [0.0] + ([-2*s/c] if abs(c) > 1e-12 else []) - for z0 in zs: - if abs(f_PX(x0, z0)) < 1e-6 and abs(f_PZ(x0, z0)) < 1e-6: - # polish with Newton - v = np.array([x0, z0], dtype=float) - H = sp.hessian(P, (X, Z)); f_H = sp.lambdify((X, Z), H, 'numpy') - for _ in range(8): - g = np.array([f_PX(*v), f_PZ(*v)], dtype=float) - Hm = np.array(f_H(*v), dtype=float) - try: - v = v - np.linalg.solve(Hm, g) - except np.linalg.LinAlgError: - break - if not any(np.hypot(*(v - q['pos'])) < 1e-6 for q in pts): - Hm = np.array(f_H(*v), dtype=float) - ev, evec = np.linalg.eigh(Hm) - kind = 'max' if ev[1] < 0 else ('min' if ev[0] > 0 else 'saddle') - val = float(sp.lambdify((X, Z), P, 'numpy')(*v)) - pts.append({'pos': v, 'val': val, 'kind': kind, 'ev': ev, 'evec': evec}) - return pts, P - - -def ascend(P, start, direction, maxima, R_far=60.0): - """Follow the normalized ascending gradient flow from start + eps*direction.""" - PX = sp.lambdify((X, Z), sp.diff(P, X), 'numpy'); PZ = sp.lambdify((X, Z), sp.diff(P, Z), 'numpy') - def rhs(t, y): - g = np.array([PX(y[0], y[1]), PZ(y[0], y[1])], dtype=float) - n = np.hypot(*g) - return g/(n + 1e-12) - y0 = np.array(start, dtype=float) + 1e-3*np.array(direction, dtype=float) - def far(t, y): return R_far - np.hypot(*y) - far.terminal = True - def near_max(t, y): - return min(np.hypot(*(y - m['pos'])) for m in maxima) - 1e-4 if maxima else 1.0 - near_max.terminal = True - sol = solve_ivp(rhs, (0, 400.0), y0, events=[far, near_max], max_step=0.02, rtol=1e-8, atol=1e-10) - yend = sol.y[:, -1] - if np.hypot(*yend) >= R_far - 1e-6: - return ('inf', None) - for i, m in enumerate(maxima): - if np.hypot(*(yend - m['pos'])) < 5e-3: - return ('max', i) - # stalled (near a saddle or slow convergence): decide by gradient ascent continuation - return ('unknown', None) - - -def elder_partner(k, s, a, beta, c): - pts, P = critical_points(k, s, a, beta, c) - maxima = [p for p in pts if p['kind'] == 'max'] - saddles = [p for p in pts if p['kind'] == 'saddle'] - # identify M and S - iM = min(range(len(maxima)), key=lambda i: np.hypot(*(maxima[i]['pos'] - np.array([-0.5, 0.0])))) - assert np.hypot(*(maxima[iM]['pos'] - np.array([-0.5, 0.0]))) < 1e-5, 'M not found' - iS = min(range(len(saddles)), key=lambda i: np.hypot(*(saddles[i]['pos'] - np.array([0.5, 0.0])))) - assert np.hypot(*(saddles[iS]['pos'] - np.array([0.5, 0.0]))) < 1e-5, 'S not found' - # ascending branches of every saddle - branches = [] - for sd in saddles: - e = sd['evec'][:, 1] # eigenvector of the positive eigenvalue (ascent direction) - b1 = ascend(P, sd['pos'], e, maxima); b2 = ascend(P, sd['pos'], -e, maxima) - branches.append((b1, b2)) - # union-find over maxima and the node 'inf' - parent = {('max', i): ('max', i) for i in range(len(maxima))}; parent[('inf', None)] = ('inf', None) - age = {('max', i): maxima[i]['val'] for i in range(len(maxima))}; age[('inf', None)] = math.inf - def find(x): - while parent[x] != x: - parent[x] = parent[parent[x]]; x = parent[x] - return x - order = sorted(range(len(saddles)), key=lambda i: -saddles[i]['val']) - M = ('max', iM); death = None; self_attach = 0 - for i in order: - b1, b2 = branches[i] - if b1[0] == 'unknown' or b2[0] == 'unknown': - return {'status': 'unresolved', 'saddle': i, 'n_window': None} - r1, r2 = find(b1), find(b2) - rM = find(M) - if r1 == r2: - self_attach += 1 # both branches in one component (a loop); no death - continue - a1, a2 = age[r1], age[r2] - # merge: the older root survives - keep, drop = (r1, r2) if a1 >= a2 else (r2, r1) - if drop == rM and age[keep] > age[rM]: - death = (i, saddles[i]['val']); break - if drop == rM: - # M's component would be dropped only if older exists; equal ages impossible generically - pass - parent[drop] = keep - window = [sd for j, sd in enumerate(saddles) if j != iS and -k < sd['val'] < 0] - return {'status': 'ok', 'self_attachments': self_attach, 'death_saddle': None if death is None else death[0], 'death_val': None if death is None else death[1], - 'is_S': death is not None and death[0] == iS, 'n_window': len(window), - 'window_vals': sorted([sd['val'] for sd in window], reverse=True), - 'n_saddles': len(saddles), 'n_maxima': len(maxima), 'n_minima': len([p for p in pts if p['kind']=='min'])} - - -def cub_n(k, s, a, beta, c): - """[CUB] Theorem C classifier (None if not typed).""" - B = beta - a*a/(12*k); D = (c - a*beta/(4*k) + a**3/(72*k*k))/2 - if not (s < -abs(B)/2): - return None - T = -(s - B)**2*(B + 2*s)/(12*k) - if s > B: - return 2 if D*D < T else 1 - return 1 if D*D > T else 0 - - -def sample_theta_n0(rng, k=1.0): - """Sample typed jets with n = 0 by rejection on a box.""" - while True: - s = -rng.uniform(0.2, 4); a = rng.uniform(-3, 3); beta = rng.uniform(-4, 4); c = rng.uniform(-4, 4) - if cub_n(k, s, a, beta, c) == 0: - return dict(s=s, a=a, beta=beta, c=c) - - -def sample_theta(rng, k=1.0): - """Sample (u,v) in D_shape, Z != 0, a; return (s,a,beta,c) with an extra window root at (u - aZ/(12k), Z).""" - while True: - u = rng.uniform(-1.5, 0.5); b0 = 3 - 12*u*u - lo, hi = abs(b0)/2, 3 - 6*u - if lo < hi: - break - v = rng.uniform(lo, hi) - Zr = rng.choice([-1, 1])*math.exp(rng.uniform(math.log(0.15), math.log(3.0))) - a = rng.uniform(-3, 3) - s = -k*v/Zr**2; B = k*b0/Zr**2; D = k*(v - b0*u)/Zr**3 - beta = B + a*a/(12*k); c = 2*D + a*B/(4*k) + a**3/(144*k*k) - return dict(u=u, v=v, Z=Zr, a=a, s=s, beta=beta, c=c, B=B, D=D) - - -if __name__ == '__main__': - # usage: elder_cubic.py SEED N [n0] (third argument "n0" samples typed jets with no extra window root) - rng = random.Random(int(sys.argv[1]) if len(sys.argv) > 1 else 0) - N = int(sys.argv[2]) if len(sys.argv) > 2 else 40 - mode_n0 = len(sys.argv) > 3 and sys.argv[3] == 'n0' - k = 1.0 - out = [] - for i in range(N): - th = sample_theta_n0(rng, k) if mode_n0 else sample_theta(rng, k) - try: - r = elder_partner(k, th['s'], th['a'], th['beta'], th['c']) - except AssertionError as e: - r = {'status': 'error', 'msg': str(e)} - rec = {**{kk: float(vv) for kk, vv in th.items()}, **{kk: (vv if not isinstance(vv, np.generic) else float(vv)) for kk, vv in r.items()}} - out.append(rec) - print(json.dumps(rec, default=float)) - json.dump(out, open('elder_samples_%s%s.json' % (sys.argv[1] if len(sys.argv) > 1 else '0', '_n0' if mode_n0 else ''), 'w'), default=float) diff --git a/frontiers/local_pairing_20260930/exploration/elder_grid.py b/frontiers/local_pairing_20260930/exploration/elder_grid.py new file mode 100644 index 000000000..3177e37e4 --- /dev/null +++ b/frontiers/local_pairing_20260930/exploration/elder_grid.py @@ -0,0 +1,277 @@ +#!/usr/bin/env python3 +"""Grid maximin (the elder rule, discretised) on sampled pinned cubics. Standard library only. + +EXPLORATION, NOT PROOF, NOT RUN IN CI. Scientific effect: NONE. + +What it does. For sampled typed jets theta = (s, B, D) of the sheared pinned cubic (LOCAL_PAIRING.md (2.1)) + + P(u, Z) = 2k(u-1)(u+1/2)^2 + (Z^2/2)(s + B u) + (D/3) Z^3, M = (-1/2, 0), S = (1/2, 0), + +it (i) computes every critical point in closed form ([CUB] (C7), (C13): the extra roots lie on the line +s + Bu + DZ = 0 and their u-coordinates solve (12kD^2 + B^3)u^2 + 2B^2 s u + Bs^2 - 3kD^2 = 0; for D = 0 they +sit at u0 = -s/B with Z^2 = (3k - 12k u0^2)/B), counts the strict-window ones (n) and cross-checks n with the +[CUB] classifier; (ii) computes the ordinary superlevel maximin death level of M on a square grid by the +literal elder rule: cells are processed in decreasing order of P and merged with 8-neighbours by union-find; +the death level is the value at which M's component first joins a component containing a point older than +M (P > 0), and the merge cell is recorded; (iii) compares the grid death level with h_* = max window critical +value (or -k when n = 0) and the merge cell with the position of the highest extra window saddle (or S). + +Boundary. The grid covers the box |u|, |Z| <= R with R chosen from the window critical points (R <= 6). A boundary cell +with P > -k is marked "older" when P increases strictly along its outward ray from the origin at 1.5R, 2R, +3R and is positive at 3R (the ray criterion of Lemma 2.1(b): such a cell is connected outward, inside +{P > P(cell)}, to points with P > 0). Other boundary cells are ordinary cells. + +Limits. Grid error near a nondegenerate saddle is of order h^2 (h the spacing) plus one cell of +8-connectivity; ties of the two extra saddle heights closer than the tolerance are reported as ties. Close to +the typed boundary s = -|B|/2 the saddle at S is nearly degenerate in Z (P ~ -k + (Z^2/2)(s + B/2) there), so +the merge cell can sit some distance along that flat valley while the death level is still exact; the level +comparison is the primary check, the merge-cell comparison the secondary one. Runtime about two minutes +for the default sample sizes (pure Python union-find on up to 361k cells per jet). +Nothing here certifies the Gaussian steps of LOCAL_PAIRING.md. + +Usage: python3 elder_grid.py [SEED] [N_SAMPLES] [N0_SAMPLES] [SPACING] +Output: RESULTS_EXPLORATION.json next to this file. +""" +import json +import math +import os +import random +import sys + + +def A(k, u): + return 2*k*(u - 1)*(u + 0.5)**2 + + +def P(k, s, B, D, u, Z): + return A(k, u) + (Z*Z/2)*(s + B*u) + (D/3)*Z**3 + + +def grad(k, s, B, D, u, Z): + return (6*k*u*u - 1.5*k + (B/2)*Z*Z, Z*(s + B*u + D*Z)) + + +def classifier_n(k, s, B, D): + """[CUB] Theorem C in the sheared parameters (typed domain s < -|B|/2), else None.""" + if not (s < -abs(B)/2): + return None + T = -(s - B)**2*(B + 2*s)/(12*k) + if s > B: + return 2 if D*D < T else 1 + return 1 if D*D > T else 0 + + +def quadratic_roots(a2, a1, a0): + if abs(a2) < 1e-14: + return [-a0/a1] if abs(a1) > 1e-14 else [] + disc = a1*a1 - 4*a2*a0 + if disc < 0: + return [] + q = math.sqrt(disc) + return [(-a1 + q)/(2*a2), (-a1 - q)/(2*a2)] + + +def critical_points(k, s, B, D): + """All critical points of P: the pins and the extra roots (closed form).""" + extra = [] + if abs(D) > 1e-12: + for u in quadratic_roots(12*k*D*D + B**3, 2*B*B*s, B*s*s - 3*k*D*D): + Z = -(s + B*u)/D + if abs(Z) > 1e-9: + extra.append((u, Z)) + elif abs(B) > 1e-12: + u = -s/B + Z2 = (3*k - 12*k*u*u)/B + if Z2 > 0: + extra += [(u, math.sqrt(Z2)), (u, -math.sqrt(Z2))] + out = [] + for (u, Z) in extra: + gu, gz = grad(k, s, B, D, u, Z) + if abs(gu) < 1e-7*max(1.0, k) and abs(gz) < 1e-7*max(1.0, k): + out.append((u, Z, P(k, s, B, D, u, Z))) + return out + + +def window_roots(k, cps): + return [(u, Z, val) for (u, Z, val) in cps if -k < val < 0] + + +def grid_death(k, s, B, D, cps, spacing): + """Discrete elder rule on the box |u|,|Z| <= R. Returns (death, merge_u, merge_Z, R, ncells).""" + reach = 1.0 + for (u, Z, val) in cps: + if -k <= val <= 0: # only the window geometry sets the box; Lemma 2.1(b) handles the outside + reach = max(reach, abs(u), abs(Z)) + R = min(6, int(math.ceil(1.6*reach + 1.0))) + nper = int(round(1.0/spacing)) + N = 2*R*nper + 1 # nodes per axis; M and S are exact nodes since 1/2 is a multiple of h + h = 2.0*R/(N - 1) + W = N + 2 # padded width: a never-processed border removes all bounds checks + + def coord(i): + return -R + i*h + + total = W*W + vals = [0.0]*total + older = bytearray(total) + for i in range(N): + u = coord(i) + base = (i + 1)*W + 1 + boundary_i = (i == 0 or i == N - 1) + for j in range(N): + Z = coord(j) + v = P(k, s, B, D, u, Z) + idx = base + j + vals[idx] = v + if v > 0: + older[idx] = 1 + elif v > -k and (boundary_i or j == 0 or j == N - 1): + rho = math.hypot(u, Z) + if rho > 0: + ru, rz = u/rho, Z/rho + seq = [P(k, s, B, D, t*rho*ru, t*rho*rz) for t in (1.5, 2.0, 3.0)] + if v < seq[0] < seq[1] < seq[2] and seq[2] > 0: + older[idx] = 1 + iM = int(round((-0.5 + R)/h)); jM = int(round(R/h)) + idxM = (iM + 1)*W + (jM + 1) + vals[idxM] = 0.0 # exact pin value + older[idxM] = 0 + + parent = list(range(total)) + comp_older = bytearray(older) + processed = bytearray(total) + interior = [(i + 1)*W + (j + 1) for i in range(N) for j in range(N)] + order = sorted(interior, key=vals.__getitem__, reverse=True) + offsets = (-W - 1, -W, -W + 1, -1, 1, W - 1, W, W + 1) + m_seen = False + for idx in order: + processed[idx] = 1 + ra = idx + while parent[ra] != ra: + parent[ra] = parent[parent[ra]] + ra = parent[ra] + for d in offsets: + nb = idx + d + if not processed[nb]: + continue + rb = nb + while parent[rb] != rb: + parent[rb] = parent[parent[rb]] + rb = parent[rb] + if ra != rb: + parent[rb] = ra + if comp_older[rb]: + comp_older[ra] = 1 + if idx == idxM: + m_seen = True + if m_seen: + rM = idxM + while parent[rM] != rM: + parent[rM] = parent[parent[rM]] + rM = parent[rM] + if comp_older[rM]: + i, j = divmod(idx, W) + return vals[idx], coord(i - 1), coord(j - 1), R, N*N + return None, None, None, R, N*N + + +def sample_n1(rng, k): + """Chart sampler: a typed jet with a prescribed extra strict-window root (u1, Z1).""" + while True: + u1 = rng.uniform(-1.45, 0.45) + b0 = 3 - 12*u1*u1 + lo, hi = abs(b0)/2, 3 - 6*u1 + if lo < hi: + break + v = lo + rng.uniform(0.05, 0.95)*(hi - lo) + Z1 = rng.choice((-1.0, 1.0))*math.exp(rng.uniform(math.log(0.3), math.log(1.8))) + s = -k*v/Z1**2 + B = k*b0/Z1**2 + D = k*(v - b0*u1)/Z1**3 + return s, B, D + + +def sample_n0(rng, k): + while True: + B = rng.uniform(-3, 3) + s = -abs(B)/2 - rng.uniform(0.02, 3) + D = rng.uniform(-3, 3) + if classifier_n(k, s, B, D) == 0: + return s, B, D + + +def run_one(k, s, B, D, spacing, tol): + cps = critical_points(k, s, B, D) + win = window_roots(k, cps) + n = len(win) + n_cls = classifier_n(k, s, B, D) + if win: + top = max(win, key=lambda t: t[2]) + h_star = top[2] + target = (top[0], top[1]) + heights = sorted([t[2] for t in win], reverse=True) + tie = len(heights) >= 2 and heights[0] - heights[1] < tol + else: + h_star = -k + target = (0.5, 0.0) + tie = False + death, mu, mz, R, ncells = grid_death(k, s, B, D, cps, spacing) + row = {'s': s, 'B': B, 'D': D, 'n': n, 'n_classifier': n_cls, 'h_star': h_star, + 'death': death, 'R': R, 'cells': ncells, 'tie_within_tol': tie} + if death is None: + row['status'] = 'no_death_found' + return row + row['merge_cell'] = [mu, mz] + row['target'] = list(target) + row['level_error'] = abs(death - h_star) + row['merge_distance'] = math.hypot(mu - target[0], mz - target[1]) + row['level_match'] = row['level_error'] <= tol + row['merge_match'] = row['merge_distance'] <= 3*spacing or tie + row['status'] = 'ok' + return row + + +def main(): + seed = int(sys.argv[1]) if len(sys.argv) > 1 else 7 + n1 = int(sys.argv[2]) if len(sys.argv) > 2 else 40 + n0 = int(sys.argv[3]) if len(sys.argv) > 3 else 15 + spacing = float(sys.argv[4]) if len(sys.argv) > 4 else 0.02 + rng = random.Random(seed) + rows = [] + for k in (1.0, 0.5): + tol = 0.01*max(1.0, k) + for _ in range(n1): + s, B, D = sample_n1(rng, k) + rows.append(dict(k=k, kind='chart_n>=1', **run_one(k, s, B, D, spacing, tol))) + sys.stderr.write('.'); sys.stderr.flush() + for _ in range(n0): + s, B, D = sample_n0(rng, k) + rows.append(dict(k=k, kind='typed_n=0', **run_one(k, s, B, D, spacing, tol))) + sys.stderr.write('o'); sys.stderr.flush() + # the [ELDER] witness (s, B, D) = (-3k/2, -2k, 0), k = 1: both extra saddles at -7/32 (a tie by construction) + rows.append(dict(k=1.0, kind='elder_witness', **run_one(1.0, -1.5, -2.0, 0.0, spacing, 0.01))) + summary = {} + for kind in ('chart_n>=1', 'typed_n=0', 'elder_witness'): + sub = [r for r in rows if r['kind'] == kind] + summary[kind] = { + 'samples': len(sub), + 'classifier_agrees_with_closed_form_count': sum(1 for r in sub if r['n'] == r['n_classifier']), + 'death_found': sum(1 for r in sub if r['status'] == 'ok'), + 'level_match': sum(1 for r in sub if r.get('level_match')), + 'merge_match': sum(1 for r in sub if r.get('merge_match')), + 'ties_within_tol': sum(1 for r in sub if r['tie_within_tol']), + 'by_n': {str(m): sum(1 for r in sub if r['n'] == m) for m in (0, 1, 2)}, + 'max_level_error': max([r.get('level_error', 0.0) for r in sub] or [0.0]), + } + out = {'object': 'CL-LOCAL-PAIRING-20260930-v1', 'kind': 'exploration (stdlib grid maximin); not proof; not CI', + 'scientific_effect': 'NONE', 'seed': seed, 'spacing': spacing, 'n1_per_k': n1, 'n0_per_k': n0, + 'k_values': [1.0, 0.5], 'summary': summary, 'rows': rows} + path = os.path.join(os.path.dirname(os.path.abspath(__file__)), 'RESULTS_EXPLORATION.json') + with open(path, 'w') as fh: + json.dump(out, fh, indent=1, sort_keys=True) + fh.write('\n') + print(json.dumps(summary, indent=1, sort_keys=True)) + + +if __name__ == '__main__': + main() From c8f7e5e83b324991d876ce0df107c9b53cddc460 Mon Sep 17 00:00:00 2001 From: "Claude Fable 5.1" Date: Wed, 30 Sep 2026 14:10:12 +0000 Subject: [PATCH 4/8] =?UTF-8?q?Local=20pairing:=20LEG=20(#170)=20now=20mer?= =?UTF-8?q?ged=20at=20the=20identical=20blob=20=E2=80=94=20moved=20to=20ci?= =?UTF-8?q?ted=5Fonly;=20branch=20merged=20with=20main=203e0a91b=20(no=20p?= =?UTF-8?q?roof=20bytes=20change)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Scientific effect NONE. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- frontiers/local_pairing_20260930/SOURCES.json | 38 +++++++++---------- 1 file changed, 18 insertions(+), 20 deletions(-) diff --git a/frontiers/local_pairing_20260930/SOURCES.json b/frontiers/local_pairing_20260930/SOURCES.json index d676dc1b4..504333932 100644 --- a/frontiers/local_pairing_20260930/SOURCES.json +++ b/frontiers/local_pairing_20260930/SOURCES.json @@ -13,7 +13,7 @@ "bytes": 40261, "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", "tag": "P", - "use": "§1 p_r and Theorem A; §4 moments; §5 Z_r >= z_* r^2 and (5.3); §8 genericity; §§9-11 ledger" + "use": "\u00a71 p_r and Theorem A; \u00a74 moments; \u00a75 Z_r >= z_* r^2 and (5.3); \u00a78 genericity; \u00a7\u00a79-11 ledger" }, { "path": "frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md", @@ -21,7 +21,7 @@ "bytes": 10229, "sha256": "8ce05a43bde8e0d718f09e2da1ffe75d3ec5893489254afdbf5b58b602cdba0c", "tag": "ELDER", - "use": "(E1) maximin; §2 path (E5) and C0-stability template; §4 (E9)-(E11)" + "use": "(E1) maximin; \u00a72 path (E5) and C0-stability template; \u00a74 (E9)-(E11)" }, { "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", @@ -39,7 +39,7 @@ "bytes": 25006, "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", "tag": "SC", - "use": "§1, §2 (7)-(10), §4 (14)-(16), §5 normal form and (17)-(18), §6 (19)-(20), §7 (2), (21)-(24)", + "use": "\u00a71, \u00a72 (7)-(10), \u00a74 (14)-(16), \u00a75 normal form and (17)-(18), \u00a76 (19)-(20), \u00a77 (2), (21)-(24)", "merged": "358f2562efbccc59f9549e5a79650e2e936395c3", "note": "reviewed at PR head 382c0f9; merged with identical bytes" }, @@ -51,7 +51,7 @@ "bytes": 17139, "sha256": "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769", "tag": "TSL", - "use": "cited in §0 only (Theorems L, T); not consumed by any proof", + "use": "cited in \u00a70 only (Theorems L, T); not consumed by any proof", "merged": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", "note": "reviewed at PR head 7c82252 (custody repaired in Math-#172, proof bytes unchanged); merged with identical bytes" } @@ -98,7 +98,7 @@ "bytes": 34758, "sha256": "b647a186d0cfd1662fc69e9d31dfd447dbb1b2f6ef4364bbd59c4956f18aed25", "tag": "RES", - "use": "cited in §0 only: the leading-mass localization residual bound to [TSL] Theorem L (Math-#173, merged 30106e1; declarative)" + "use": "cited in \u00a70 only: the leading-mass localization residual bound to [TSL] Theorem L (Math-#173, merged 30106e1; declarative)" }, { "path": "frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md", @@ -106,13 +106,21 @@ "bytes": 11018, "sha256": "f20c1ac96b00edc071e428bd2e5dc4c91b8546a3ee0693367b783109b26aa518", "tag": "NUM", - "use": "cited in §6 only: uncertified numerical values of the planar near coefficients (Math-#168, merged 02772ec); not consumed" + "use": "cited in \u00a76 only: uncertified numerical values of the planar near coefficients (Math-#168, merged 02772ec); not consumed" + }, + { + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "bytes": 39722, + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "tag": "LEG", + "use": "cited in the header, \u00a70 and \u00a75 only: Theorem E (same deterministic statement as Lemma P\u2032 + Proposition 3.3, different proof) and Theorem S (planar equality by the cap route); not consumed; merged as Math-#170 (fa2e990) at the identical blob, now verified by the workflow" } ], "referee": { "kind": "two clean-context Anthropic Claude subagent passes before landing (same provider, same account; not a nonauthor review)", - "pass_1": "two blocking (mountain-pass step of Lemma P′; the highest-saddle claim unproved) and seventeen amendments; all applied (Lemmas 2.1, 2.2, 4.1, 4.2, Proposition 3.3 added; V-first assignment; D = 0 subcase; tail bound; §5 margins; §6 wording)", - "pass_2": "Lemma 2.1(b) contained two false inequalities (replaced by the four-case polar argument); Lemma 2.2 flow wording made an explicit deformation; the S_V assignment in Lemma P made definite and continuous; K_{δ,R} closed by adding s <= -|B|/2 - δ; §5 obligation restated in jet-scaled margins as (O1′) with the d >= 3 slaving named; checker: shear identity (I0), exact central-difference derivative (I2) and Vieta other-root check added, min_margin key; title and README overclaims removed" + "pass_1": "two blocking (mountain-pass step of Lemma P\u2032; the highest-saddle claim unproved) and seventeen amendments; all applied (Lemmas 2.1, 2.2, 4.1, 4.2, Proposition 3.3 added; V-first assignment; D = 0 subcase; tail bound; \u00a75 margins; \u00a76 wording)", + "pass_2": "Lemma 2.1(b) contained two false inequalities (replaced by the four-case polar argument); Lemma 2.2 flow wording made an explicit deformation; the S_V assignment in Lemma P made definite and continuous; K_{\u03b4,R} closed by adding s <= -|B|/2 - \u03b4; \u00a75 obligation restated in jet-scaled margins as (O1\u2032) with the d >= 3 slaving named; checker: shear identity (I0), exact central-difference derivative (I2) and Vieta other-root check added, min_margin key; title and README overclaims removed" }, "files": [ { @@ -151,16 +159,6 @@ "note": "Math- main after Math-#173 merged; every consumed source verified at these bytes by the workflow" }, "cited_unmerged": [ - { - "path": "frontiers/local_elder_geometry_20260930/PROOF.md", - "pr": "d6g8k5htny-coder/Math-#170", - "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", - "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", - "bytes": 39722, - "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", - "tag": "LEG", - "use": "cited in the header, §0 and §5 only: Theorem E (same deterministic statement as Lemma P′ + Proposition 3.3, different proof) and Theorem S (planar equality by the cap route); not consumed; unmerged, so not verified by the workflow" - }, { "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", "pr": "d6g8k5htny-coder/Math-#175", @@ -169,8 +167,8 @@ "bytes": 23416, "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e", "tag": "CFE", - "use": "cited in the header, §0 and §5 only: H1 (the d >= 3 hard-direction slaving named in §5) and Theorem F; not consumed; unmerged, so not verified by the workflow" + "use": "cited in the header, \u00a70 and \u00a75 only: H1 (the d >= 3 hard-direction slaving named in \u00a75) and Theorem F; not consumed; unmerged, so not verified by the workflow" } ], - "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied — exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded" + "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied \u2014 exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded | v1.1+sources (2026-09-30 14:15 UTC): LEG (#170) moved from cited_unmerged to cited_only after its integration at the identical blob ef2aa579; no other change; branch merged with main 3e0a91b" } From d1b0020e0cd670faef4d9bd9070881a5350a5351 Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 1 Oct 2026 13:28:31 +0000 Subject: [PATCH 5/8] =?UTF-8?q?Local=20pairing:=20status=20repair=20?= =?UTF-8?q?=E2=80=94=20=C2=A75=20attributes=20planar=20LP=3D=20to=20merged?= =?UTF-8?q?=20#170=20Theorem=20S;=20#175=20landed?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Requested in comments 5930621629 and 5930668120 (the one-line repair offered in 5921307897): - §5: the stale "This is not a corollary of anything merged" and "(unmerged; …)" are replaced by the reviewer's sentence: at fixed planar marks the equality is supplied on main by Math-#170 Theorem S (proof blob ef2aa579). - Header and README: #170 (fa2e990) and #175 (eb659bd) recorded as integrated; #175 stays a cited interface, not a premise of Theorem LP. - SOURCES.json: CFE (#175) moved from cited_unmerged to cited_only at the identical blob 923d3236 (now verified from the tree); packet fingerprints refreshed; revision noted. Theorem LP, Proposition 3.3, the checker and RESULTS.json are unchanged. Branch restacked onto main 124c37d. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .../local_pairing_20260930/LOCAL_PAIRING.md | 9 +++--- frontiers/local_pairing_20260930/README.md | 5 +-- frontiers/local_pairing_20260930/SOURCES.json | 31 +++++++++---------- 3 files changed, 22 insertions(+), 23 deletions(-) diff --git a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md index 87d889bcd..a4603c0b4 100644 --- a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md +++ b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md @@ -9,11 +9,12 @@ raised is applied below). Theorem LP (the lower bound with an identified functio interfaces of [SC], [CUB] and [P] at their own review status. Conjecture LP= (equality) is stated with its single remaining probabilistic obligation. **Relation to Math-#170 and Math-#175 (v1.1, added after landing).** The OpenAI packet Math-#170 (`frontiers/local_elder_geometry_20260930/PROOF.md`, head `79f18f0`, blob -`ef2aa579`, opened before this packet) proves the same deterministic statement as Lemma P′ + Proposition 3.3 +`ef2aa579`, opened before this packet; integrated into main at `fa2e990`) proves the same deterministic statement as Lemma P′ + Proposition 3.3 (its Theorem E, by the cubic critical-chord identity and a normalized-gradient collar rather than the explicit paths used here) and proves the **equality** `r⁻³(1 − p_r) → α₁ + α₂` at fixed planar marks (its Theorem S) by a route this packet does not use: [P]'s cap implication `F_r ⊂ G_r^c` and a retained-tail integration of [P] §7. -Math-#175 (`frontiers/concave_fibre_elder_20260930/`, head `9c6a734`) lifts it to fixed `d ≥ 3` by a contained +Math-#175 (`frontiers/concave_fibre_elder_20260930/`, head `9c6a734`, proof blob `923d3236`; integrated into main at +`eb659bd`) lifts it to fixed `d ≥ 3` by a contained hard-negative tube. So Conjecture LP= below is Math-#170 Theorem S (planar) / Math-#175 Theorem F (`d ≥ 3`), conditional on their interfaces and review; what this packet adds is an independent same-account derivation of the deterministic core with exact controls, and the cap-free lower bound Theorem LP. Scientific effect: @@ -393,8 +394,8 @@ has `r⁻³`-limit `M(n ≥ 1, n_R = 0)` at fixed `R` — its jets have all extr (O1′) limsup_{r→0} r⁻³ Q_r^W(rejected, n(θ_r) = 0, θ_r outside the margin set) → 0 as the margins are relaxed. -This is not a corollary of anything merged. It **is** supplied, at fixed planar marks, by Math-#170 Theorem S -(unmerged; OpenAI; reviewed at head `79f18f0`), whose route avoids the margin set altogether: the failure event +At fixed planar marks, this equality is supplied on main by Math-#170 Theorem S (OpenAI; reviewed at head `79f18f0`, +proof blob `ef2aa57959ea9f721bbf2316ce94cf616c1c9113`), whose route avoids the margin set altogether: the failure event is contained in the bad cap of [P] §8 (`F_r ⊂ G_r^c`), the bad cap's mass restricted to a large residual norm is `o(r³)` by keeping the indicator inside [P]'s (7.5) scalar integral, and the remaining compact-jet part converges by dominated convergence with the pointwise limit of the failure indicator (its Theorem E). The diff --git a/frontiers/local_pairing_20260930/README.md b/frontiers/local_pairing_20260930/README.md index 47eed8546..d827edbf4 100644 --- a/frontiers/local_pairing_20260930/README.md +++ b/frontiers/local_pairing_20260930/README.md @@ -16,8 +16,9 @@ whether the decision is local. This packet shows: * **Conjecture LP=** (`1 − p_r = (a₁ + a₂) r³ + o(r³)`, i.e. remote witnesses never pair) with its remaining obligation (O1) stated precisely in §5. That obligation is discharged, at the packets' conditional scope, by the OpenAI packets Math-#170 (planar, Theorem S: cap implication + retained-tail integration) and Math-#175 - (`d ≥ 3`, hard-negative-fibre barrier), both opened before this packet and reviewed by this author on - 30 September; this packet's own contribution is the cap-free lower bound and the exact controls. + (`d ≥ 3`, hard-negative-fibre barrier), both opened before this packet, reviewed by this author on 30 September, + and now on main (#170 at `fa2e990`, #175 at `eb659bd`); this packet's own contribution is the cap-free lower bound + and the exact controls. * §6 says what this is worth for the lifetime law: a lower bound for the coefficient of the `Θ(ℓ^{2/3})` compact-window selection loss (relative order `ℓ`); nothing for the leading term; nothing about the unrestricted density's remainder. diff --git a/frontiers/local_pairing_20260930/SOURCES.json b/frontiers/local_pairing_20260930/SOURCES.json index 504333932..5e3564135 100644 --- a/frontiers/local_pairing_20260930/SOURCES.json +++ b/frontiers/local_pairing_20260930/SOURCES.json @@ -115,6 +115,14 @@ "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", "tag": "LEG", "use": "cited in the header, \u00a70 and \u00a75 only: Theorem E (same deterministic statement as Lemma P\u2032 + Proposition 3.3, different proof) and Theorem S (planar equality by the cap route); not consumed; merged as Math-#170 (fa2e990) at the identical blob, now verified by the workflow" + }, + { + "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", + "bytes": 23416, + "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e", + "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", + "tag": "CFE", + "use": "cited in the header, \u00a70 and \u00a75 only: H1 (the d >= 3 hard-direction slaving named in \u00a75) and Theorem F; not consumed and not a premise of Theorem LP; merged as Math-#175 (eb659bd) at the identical blob, now verified by the workflow" } ], "referee": { @@ -125,8 +133,8 @@ "files": [ { "path": "LOCAL_PAIRING.md", - "bytes": 41583, - "sha256": "9e7193c16d994ee7f115f79e5e525209ad4faad55331051d3936efcb826c8bb2" + "bytes": 41691, + "sha256": "5234896586fed780a084e82fa44e0ffc890dc9d20f0a5afadeec8901a397378a" }, { "path": "pairing_check.py", @@ -140,8 +148,8 @@ }, { "path": "README.md", - "bytes": 2435, - "sha256": "dcef622428269d6b41730a2e4deb13d24c1d72fce3665d6436cfeef0f917e678" + "bytes": 2490, + "sha256": "b1930a8b248e88558790f4466e233336d5b4fbefccb1d43ec3b00db2c59562b5" }, { "path": "exploration/elder_grid.py", @@ -158,17 +166,6 @@ "branch_base": "30106e1be251e8c18759b6fa4de6176b76380c12", "note": "Math- main after Math-#173 merged; every consumed source verified at these bytes by the workflow" }, - "cited_unmerged": [ - { - "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", - "pr": "d6g8k5htny-coder/Math-#175", - "commit": "9c6a73489c5ed2465a3cf65dcc950eeeae074b05", - "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", - "bytes": 23416, - "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e", - "tag": "CFE", - "use": "cited in the header, \u00a70 and \u00a75 only: H1 (the d >= 3 hard-direction slaving named in \u00a75) and Theorem F; not consumed; unmerged, so not verified by the workflow" - } - ], - "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied \u2014 exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded | v1.1+sources (2026-09-30 14:15 UTC): LEG (#170) moved from cited_unmerged to cited_only after its integration at the identical blob ef2aa579; no other change; branch merged with main 3e0a91b" + "cited_unmerged": [], + "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied \u2014 exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded | v1.1+sources (2026-09-30 14:15 UTC): LEG (#170) moved from cited_unmerged to cited_only after its integration at the identical blob ef2aa579; no other change; branch merged with main 3e0a91b | status repair (2026-10-01): \u00a75's stale 'not a corollary of anything merged' / 'unmerged' sentence replaced by the attribution to Math-#170 Theorem S on main (requested in comments 5930621629 and 5930668120); #175 (CFE) moved from cited_unmerged to cited_only after its integration at eb659bd with the identical blob 923d3236; header and README status updated; mathematics, checker and RESULTS unchanged; branch merged with main 124c37d" } From 9a114b9acb2b1d5d12d0e3534d52c2c4fbb7cace Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 21:14:37 +0000 Subject: [PATCH 6/8] =?UTF-8?q?Local=20pairing=20v1.2:=20apply=20Slice=20C?= =?UTF-8?q?=20finding=20C-01=20(=C2=A75=20acceptance=20through=20the=20exa?= =?UTF-8?q?ct=20saddle=20level)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Codex's Slice C review (C85, 5396856086) showed that v1.1's optional §5 margin sketch, which confined M's component at one level above f(S), does not imply acceptance; its exact falsifier is recorded. The sketch is withdrawn as a proof. §5 now states Lemma 5.1 (common-radius acceptance), consumed from QS-E (main#229 comment 5961415030; author-side candidate), with exhausting jet-scaled margin sets M(eta, R), the corrected decomposition, the review's normalization and Gaussian-conditioning notes, and marks (O1') => (O1) as conditional on Lemma 5.1. Lemmas 2.1-2.2, P, P', Proposition 3.3, Theorem LP, pairing_check.py, RESULTS.json, the exploration and the workflow are unchanged. SOURCES.json updates the two packet identities and the revision record. Scientific effect: NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .../local_pairing_20260930/LOCAL_PAIRING.md | 147 ++++++++++++++---- frontiers/local_pairing_20260930/README.md | 7 + frontiers/local_pairing_20260930/SOURCES.json | 10 +- 3 files changed, 128 insertions(+), 36 deletions(-) diff --git a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md index a4603c0b4..05afeb2bd 100644 --- a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md +++ b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md @@ -19,6 +19,14 @@ hard-negative tube. So Conjecture LP= below is Math-#170 Theorem S (planar) / Ma conditional on their interfaces and review; what this packet adds is an independent same-account derivation of the deterministic core with exact controls, and the cap-free lower bound Theorem LP. Scientific effect: NONE — no register, graph, STATUS, prize or Boolean changes; no numerical constant is claimed. +**v1.2 (2 October 2026), answering the Slice C review.** Codex's Slice C review (C85, 5396856086) found +C-01: v1.1's optional §5 margin sketch confined `M`'s component at one level above `f(S)` only, and it gave an +exact falsifier. v1.2 withdraws that sketch as a proof. In its place §5 states Lemma 5.1, a common-radius +acceptance lemma that controls the component through the exact saddle level. It is consumed from QS-E (main#229 +5961415030; author-side candidate, nonauthor review requested), with exhausting jet-scaled margin sets +`𝓜(η, R)`. §5 also records the review's notes on the weighted normalization and on Gaussian conditioning, and +now says that the reverse reduction (O1′) ⇒ (O1) is conditional on Lemma 5.1. Lemmas 2.1–2.2, P, P′, +Proposition 3.3, Theorem LP, the controls and the cap bypass are unchanged. ## 0. What this packet answers, and what it does not @@ -367,45 +375,118 @@ where `N_R` counts strict-window critical points of `f` in `B(Rr)` other than th (O1) limsup_{r→0} r⁻³ Q_r^W(rejected, N_R = 0) → 0 as R → ∞. -What is known toward (O1), and what is not. On `{N_R = 0}` the analogue of the transfer in Theorem LP would -run Lemma P′ on the coupled field: if the jet `θ_r` has `n = 0` and the cubic landscape has margins — `P` -strictly concave on a ball `B(M, η)` with modulus `μ`, `sup_{C_θ ∖ B(M, η)} P ≤ −μη²/4` for the `M`-component -`C_θ` at level `−k + δ/4`, `C_θ ⊂ B(R − 1)`, the axis ray beyond `S` reaching `P ≥ δ` inside `B(R)`, `θ_r` at -distance `≥ δ` from the typed boundary and from the tie set — and if the `C²` error of Lemma 4.1 on `B(R)` is -below `min(μ/2, μη²/16, δ/8)`, then the rescaled field `F_r = (f(r·) − b)/r³` restricted to the soft plane -has its `M`-component at level `−k + δ/2` inside `{P > −k + δ/4}`, hence inside `B(R)` and away from every -point with `F_r ≥ 0` other than `M` itself; the far sector of its saddle at `S` (exact, since the pins are -exact under `Q_r^W`) reaches `F_r > 0` inside `B(R)`; and the axis gives `d_f(M) ≥ f(S)` (on it -`f − f(S) = r³(X − 1/2)²[2k(X + 1) − O(C_R K r)]`). In `d = 2` this yields `d_f(M) ∈ [f(S), f(S) + (δ/2)r³]` -with `S` the only saddle at such a level inside `B(Rr)` (`N_R = 0`) and the `M`-component confined to the -ball, so the death partner is `S`: **accepted**. In `d ≥ 3` the same conclusion needs the hard-direction -slaving of the superlevel components (fiberwise concavity in the hard coordinates `w`, `f_ww ≤ −h₂/2` on the -cylinder, and the correspondence between components of `{f > h}` and those of the reduced planar potential -`max_w f`), which [SC] §5 carries out for the critical points and this note does not carry out for the -superlevel sets. Granting it, - - {rejected, N_R = 0} ⊂ {n(θ_r) = 0, θ_r outside the margin set} ∪ {C²-error too large} ∪ {n(θ_r) ≥ 1, N_R = 0}, - -where untyped `θ_r` are placed in the first set. The second set is `O(r^p)` (as in Theorem LP). The third -has `r⁻³`-limit `M(n ≥ 1, n_R = 0)` at fixed `R` — its jets have all extra roots outside `B(R)` — by Lemma -4.2 on compact subsets, [SC] §5's joint spectral-jet/count convergence in total variation, and tightness of -`M` on `{n ≥ 1}` ([SC] (19): `|s| ≤ 2|B| + (48kD²)^{1/3}` there, with (10)/(15)/(6) domination); it tends to -`0` as `R → ∞`. So, granting the slaving step, (O1) is equivalent to - - (O1′) limsup_{r→0} r⁻³ Q_r^W(rejected, n(θ_r) = 0, θ_r outside the margin set) → 0 as the margins are relaxed. +What is known toward (O1), and what is not (amended in v1.2). The v1.1 sketch here confined the +`M`-component of the rescaled field at the single level `−k + δ/2`. It did not control the levels between +`f(S)` and `f(S) + (δ/2)r³`, where a remote saddle can still kill `M` on `{N_R = 0}`. Codex's Slice C review +(C85, 5396856086, finding C-01) gives an exact falsifier of that finite-radius implication. +- *The construction.* Take `k = δ = 1` and `P = A(X + HZ) − σZ²/2`, with `H = 10⁹`, `σ = 10¹⁴` and `R = 125`. +- *Why the v1.1 margins hold.* The cubic is typed with `n = 0`, its component at `−3/4` lies in `B(124)`, and + `N_R = 0`. A bump placed outside `B(125)` leaves the actual `C²` error on `B(R)` equal to zero. +- *Why acceptance fails.* The sheared vertical path from `M` stays above `−7/8` and leaves `B(125)`. A bump + there, outside `B(125)`, makes the pair rejected. + +The v1.1 sketch is therefore withdrawn as a proof. What (O1) needs is a *common-radius acceptance lemma*: +one that controls the component through the exact saddle level and slightly below it. For the planar cubic, +QS-E supplies it. + +Notation for the lemma. +- *Coordinates.* Work in [CUB]'s sheared coordinates `(u, Z)`, normalized by `k`, so that `P_θ/k` has the + pins `0` and `−1`. These are the coordinates of main#229 5961415030, where `u = X + aZ/(12k)`. +- *The saddle margin.* `μ(θ) := 1 + max{P_θ(Y)/k : Y an extra nondegenerate saddle}`, and `μ = −∞` if there is + none. Every extra critical point lies below `0`, so `n(θ) = 0` iff `μ ≤ 0`. Off the tie set `Σ`, `n = 0` + forces `μ < 0`. +- *The QS sets* (QS §§1–4): + - the tolerances `m_S` and `ε_M`; + - the ellipses `E_S` and `E_M`; + - the cut trap `T_ℓ`: the `M`-component of `{P > ℓ}` cut along the stable curve of `S`; + - the axis segment `A = [M, (3/2, 0)]`. + +**Lemma 5.1 (common-radius acceptance; QS-E).** Let `θ` be typed, off `Δ ∪ Σ`, with `n(θ) = 0`. Let +`0 < η < min(|μ|, m_S, ε_M)`, and let `V_η(θ) := T̄_{−1−η} ∪ A ∪ E_S ∪ E_M`, taken in raw coordinates. +Suppose `V_η(θ) ⊂ B(R)`. Let `F` be a planar function with the exact pins of `P_θ/k` such that: +- `|F − P_θ/k| < η` on `T̄_{−1−η} ∪ A`; +- QS's rescaled `C²` bounds (E2) and (E3) hold on `E_S` and `E_M`. + +Then `d_F(M) = F(S)` (elder), whatever `F` is outside `B(R)`. + +Both of QS-E's bounds use only paths inside `V_η ⊂ B(R)`: +- the exit from the trap, for the upper bound; +- the axis `A`, for the lower bound. + +So the lemma applies to the torus field through the chart, exactly as in #170 §5. + +This is Theorem QS-E of main#229 5961415030, an author-side candidate whose nonauthor review is requested; +this packet consumes it at that status. QS Remark 4 shows why the axis beyond `S` must be in `V_η`. The +lemma needs no `N_R = 0`. In the C-01 cubic, along the sheared vertical path, the level-`(−1 − η)` component +reaches `|X| ≈ H(2/σ)^{1/2} ≈ 142`, beyond `B(125)`. So the containment hypothesis fails there, as it must. +The exact C-01 cubic, with `B = D = 0`, also lies on `Δ`. + +The margin sets. Put + + 𝓜(η, R) := {θ typed : n(θ) = 0, θ ∉ Δ ∪ Σ, η < min(|μ|, m_S, ε_M), V_η(θ) ⊂ B(R − 1)}. + +These sets are jet-scaled, and they increase to `{n = 0} \ (Δ ∪ Σ)` as `η ↓ 0` and `R ↑ ∞`: QS Lemma 4 +makes the trap bounded at each such `θ`, although its size is not explicit. Two bounds hold on +`𝓜(η, R)`: +- `E_S ⊂ B(R − 1)` gives `κ_S ≥ r̃²/(R − 1)²`, and `η < m_S = (2/5)r̃²` then gives + `1/κ_S < 2(R − 1)²/(5η)`. +- In the same way, `1/κ_M < (R − 1)²/(2η)`. + +So the raw-coordinate thresholds of (E2) and (E3) are at least `c(η, R)/(1 + |a|/(12k))²`. + +The decomposition in `d = 2`. Lemma 4.1's planar `C²` error on the fixed ball `B(R)` is `O(C_R K r)`, so + + {rejected, N_R = 0} ⊂ {rejected, n(θ_r) = 0, θ_r ∉ 𝓜(η, R)} ∪ {θ_r ∈ 𝓜(η, R), error on B(R) above θ_r's thresholds} + ∪ {n(θ_r) ≥ 1, N_R = 0} ∪ {θ_r ∈ Δ ∪ Σ}, + +where untyped `θ_r` are placed in the first set. +- *The second set* is `O(r^p)`, as in Theorem LP, by the Gaussian moments of `a` and of `K`. +- *The third set* has `r⁻³`-limit `M(n ≥ 1, n_R = 0)` at fixed `R`, since its jets have all their extra roots + outside `B(R)`. This follows from three facts: + - Lemma 4.2, on compact subsets; + - [SC] §5's joint spectral-jet/count convergence in total variation; + - the tightness of `M` on `{n ≥ 1}` ([SC] (19): `|s| ≤ 2|B| + (48kD²)^{1/3}` there, with (10)/(15)/(6) + domination). + + This limit tends to `0` as `R → ∞`. +- *The last set* is null. + +Three conditions apply to these statements. +- Do not condition the Gaussian regression on `{N_R = 0}`; that would destroy its Gaussian structure. Keep + every intersection indicator under the original law `Q_r^W`, with positive majorants. +- In `d ≥ 3` the same conclusion needs the hard-direction slaving of the superlevel components: + - fiberwise concavity in the hard coordinates `w`; + - `f_ww ≤ −h₂/2` on the cylinder; + - the correspondence between the components of `{f > h}` and those of the reduced planar potential + `max_w f`. + + [SC] §5 carries this out for the critical points; this note does not carry it out for the superlevel sets. +- (O1) ⇒ (O1′) holds already. At fixed regular `R`, [SC] §5's nonempty joint convergence gives + `r⁻³Q_r^W(n(θ_r) = 0, N_R ≥ 1) → 0`. The converse direction is the decomposition above. So, granting + Lemma 5.1 (and, in `d ≥ 3`, the slaving step), (O1) is equivalent to + + (O1′) limsup_{r→0} r⁻³ Q_r^W(rejected, n(θ_r) = 0, θ_r ∉ 𝓜(η, R)) → 0 as η ↓ 0 and R ↑ ∞. At fixed planar marks, this equality is supplied on main by Math-#170 Theorem S (OpenAI; reviewed at head `79f18f0`, proof blob `ef2aa57959ea9f721bbf2316ce94cf616c1c9113`), whose route avoids the margin set altogether: the failure event is contained in the bad cap of [P] §8 (`F_r ⊂ G_r^c`), the bad cap's mass restricted to a large residual norm is `o(r³)` by keeping the indicator inside [P]'s (7.5) scalar integral, and the remaining compact-jet part converges by dominated convergence with the pointwise limit of the failure indicator (its Theorem E). The -`d ≥ 3` slaving step named below is Math-#175's H1. Read on its own, the margin set must be jet-scaled (a fixed `δ` fails: near +`d ≥ 3` slaving step named above is Math-#175's H1. Read on its own, the margin set must be jet-scaled (a fixed `δ` fails: near `M`, `P(M + Z e_Z) = (Z²/2)(s − B/2) + (D/3)Z³` exceeds any fixed `−δ` for `|Z|` slightly beyond `δ` once `|s|` is large, so a fixed-margin set would exclude a region of infinite `M`-mass and (O1′) would restate (O1)); the typed-jet measure is not finite on `{n = 0}` ([SC] after (17)), so its complement cannot be handled by Lemma 4.2; and the events in the complement — a second soft eigenvalue, a large jet, a jet near the typed boundary, a landscape whose `M`-component reaches far — are the ones [P] §§6–7 and [SC] §4 control -in unweighted or occurrence-weighted form, not in rejection-weighted form. What is needed is either a direct +in unweighted or occurrence-weighted form, not in rejection-weighted form. C85's review records the +normalization: +- [P] §§6–7 keep the double soft factor `W_r ≲ r²λ₁(λ₁ + CrU)`. So the soft interval gives an `O(r⁵)` weighted + numerator, and one division by `Z_r ≥ z_*r²` gives `O(r³)`. +- Exhausting the margins needs more than that displayed bound: a written rejection-tail deduction that keeps + the residual and hard-spectrum tail indicators inside the same integral. +- [SC] §4's domination covers only the nonempty near-occurrence measure. + +What is needed is either a direct argument that a long excursion of `M`'s window component without a window saddle inside `B(Rr)` has weighted probability `o(r³)`, or the boundary-layer estimates of [P] §7 restated at `o(r³)` on the no-witness event. That, together with the `d ≥ 3` slaving step, is what separates Theorem LP from LP=. @@ -413,9 +494,10 @@ event. That, together with the `d ≥ 3` slaving step, is what separates Theorem Falsifiers. LP= is false if there is a configuration class of weighted probability `≍ r³` in which `(M,S)` is rejected although no strict-window critical point lies within `O(r)` of the pair — for example a mechanism where the far sector of `S` reconnects to `M`'s component (self-attachment) through a region -at distance `≫ r`. Lemma P′ shows no such class exists on the accurate planar cubic model with margins; a -counterexample would have to live where the model is inaccurate, unmargined, or where the hard-direction -reduction fails. +at distance `≫ r`. Lemma P′ and Lemma 5.1 show that no such class exists on the accurate planar cubic model with margins. +The margins must control the component down to the exact saddle level; C-01 shows that confinement at +one level above `f(S)` is not enough. A counterexample would have to live where the model is inaccurate +or unmargined, or where the hard-direction reduction fails. ## 6. Consequences for the lifetime law, stated without inflation @@ -475,3 +557,6 @@ witnesses; no `d ≥ 3` hard-direction statement beyond what [SC] §5 supplies f from [SC] §5, and the `O(r^p)` tail. Best read by a lane not exposed to [SC]'s authorship. * **Slice C (the obligation):** §5 — whether (O1) follows from [P] §§6–7 and [SC] §4 as stated, or needs a new estimate. This is the lane where the next theorem is decided. + Reviewed by Codex (C85, 5396856086): the remaining weighted obligation and the cap bypass are sound at + their conditional scope. Finding C-01 is applied in v1.2: the v1.1 sketch is withdrawn, Lemma 5.1 is consumed + from QS-E, and the reverse reduction is marked conditional. diff --git a/frontiers/local_pairing_20260930/README.md b/frontiers/local_pairing_20260930/README.md index d827edbf4..ccadb61e2 100644 --- a/frontiers/local_pairing_20260930/README.md +++ b/frontiers/local_pairing_20260930/README.md @@ -29,3 +29,10 @@ source; [SC] and [TSL] are consumed at their merged bytes, identical to the revi elder-rule computation — evidence, not proof, not run in CI). Review slices (§9): A deterministic (§§2–3); B the Gaussian transfer (§4); C the obligation (§5). + +**v1.2 (2 October 2026).** Codex's Slice C review (C85, 5396856086) found C-01: v1.1's optional §5 margin sketch +confined `M`'s component at one level above `f(S)` only. v1.2 withdraws that sketch as a proof and states +Lemma 5.1 instead. It is a common-radius acceptance lemma, consumed from QS-E (main#229 5961415030; author-side +candidate), with exhausting jet-scaled margin sets. §5 also records the review's notes on normalization and +conditioning, and marks (O1′) ⇒ (O1) as conditional on Lemma 5.1. Nothing else changes: not the deterministic +lemmas, not Theorem LP, and not the controls. diff --git a/frontiers/local_pairing_20260930/SOURCES.json b/frontiers/local_pairing_20260930/SOURCES.json index 5e3564135..2688d7c91 100644 --- a/frontiers/local_pairing_20260930/SOURCES.json +++ b/frontiers/local_pairing_20260930/SOURCES.json @@ -133,8 +133,8 @@ "files": [ { "path": "LOCAL_PAIRING.md", - "bytes": 41691, - "sha256": "5234896586fed780a084e82fa44e0ffc890dc9d20f0a5afadeec8901a397378a" + "bytes": 46711, + "sha256": "649c8015d88c230af84660455a02a7f8acb1397b89d308ff462a973bd1f22fe7" }, { "path": "pairing_check.py", @@ -148,8 +148,8 @@ }, { "path": "README.md", - "bytes": 2490, - "sha256": "b1930a8b248e88558790f4466e233336d5b4fbefccb1d43ec3b00db2c59562b5" + "bytes": 3092, + "sha256": "5b3c3662171ffe53618b0346d7550bb79ab7050b43fe8d1e5dc87b3809cc3dc0" }, { "path": "exploration/elder_grid.py", @@ -167,5 +167,5 @@ "note": "Math- main after Math-#173 merged; every consumed source verified at these bytes by the workflow" }, "cited_unmerged": [], - "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied \u2014 exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded | v1.1+sources (2026-09-30 14:15 UTC): LEG (#170) moved from cited_unmerged to cited_only after its integration at the identical blob ef2aa579; no other change; branch merged with main 3e0a91b | status repair (2026-10-01): \u00a75's stale 'not a corollary of anything merged' / 'unmerged' sentence replaced by the attribution to Math-#170 Theorem S on main (requested in comments 5930621629 and 5930668120); #175 (CFE) moved from cited_unmerged to cited_only after its integration at eb659bd with the identical blob 923d3236; header and README status updated; mathematics, checker and RESULTS unchanged; branch merged with main 124c37d" + "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied \u2014 exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded | v1.1+sources (2026-09-30 14:15 UTC): LEG (#170) moved from cited_unmerged to cited_only after its integration at the identical blob ef2aa579; no other change; branch merged with main 3e0a91b | status repair (2026-10-01): \u00a75's stale 'not a corollary of anything merged' / 'unmerged' sentence replaced by the attribution to Math-#170 Theorem S on main (requested in comments 5930621629 and 5930668120); #175 (CFE) moved from cited_unmerged to cited_only after its integration at eb659bd with the identical blob 923d3236; header and README status updated; mathematics, checker and RESULTS unchanged; branch merged with main 124c37d | v1.2 (2026-10-02): Slice C review C85 5396856086 finding C-01 applied \u2014 the v1.1 \u00a75 single-level margin sketch is withdrawn as a proof; Lemma 5.1 (common-radius acceptance, consumed from QS-E, main#229 comment 5961415030, author-side candidate) with exhausting jet-scaled margin sets M(eta, R) replaces it; the review's normalization and Gaussian-conditioning notes are recorded; (O1') => (O1) marked conditional on Lemma 5.1; LOCAL_PAIRING.md and README.md only; checker, RESULTS, exploration and workflow unchanged" } From abe38bef71355c7523017a28ebde8e9815f8e5bc Mon Sep 17 00:00:00 2001 From: Claude Date: Fri, 2 Oct 2026 22:22:44 +0000 Subject: [PATCH 7/8] =?UTF-8?q?Local=20pairing:=20status=20refresh=20?= =?UTF-8?q?=E2=80=94=20QS-E=20consumed=20at=20its=20delivered=20nonauthor?= =?UTF-8?q?=20review?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The v1.2 delta review 5397276533 (item 8) found one nonblocking reporting drift: LOCAL_PAIRING.md and README.md still called QS-E (main#229 5961415030) an author-side candidate awaiting review. Its nonauthor review 5961620598 returned ACCEPT / PASS_TECHNICAL_SCOPED. Three status phrases are updated to cite it, and SOURCES.json records the new file identities and the revision note. Lemma 5.1, all mathematics, pairing_check.py, RESULTS.json, exploration and the workflow are unchanged. No merge of main: integration restacks. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- frontiers/local_pairing_20260930/LOCAL_PAIRING.md | 6 +++--- frontiers/local_pairing_20260930/README.md | 4 ++-- frontiers/local_pairing_20260930/SOURCES.json | 10 +++++----- 3 files changed, 10 insertions(+), 10 deletions(-) diff --git a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md index 05afeb2bd..1f8c9455b 100644 --- a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md +++ b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md @@ -23,7 +23,7 @@ NONE — no register, graph, STATUS, prize or Boolean changes; no numerical cons C-01: v1.1's optional §5 margin sketch confined `M`'s component at one level above `f(S)` only, and it gave an exact falsifier. v1.2 withdraws that sketch as a proof. In its place §5 states Lemma 5.1, a common-radius acceptance lemma that controls the component through the exact saddle level. It is consumed from QS-E (main#229 -5961415030; author-side candidate, nonauthor review requested), with exhausting jet-scaled margin sets +5961415030; nonauthor review ACCEPT / PASS_TECHNICAL_SCOPED, 5961620598), with exhausting jet-scaled margin sets `𝓜(η, R)`. §5 also records the review's notes on the weighted normalization and on Gaussian conditioning, and now says that the reverse reduction (O1′) ⇒ (O1) is conditional on Lemma 5.1. Lemmas 2.1–2.2, P, P′, Proposition 3.3, Theorem LP, the controls and the cap bypass are unchanged. @@ -415,8 +415,8 @@ Both of QS-E's bounds use only paths inside `V_η ⊂ B(R)`: So the lemma applies to the torus field through the chart, exactly as in #170 §5. -This is Theorem QS-E of main#229 5961415030, an author-side candidate whose nonauthor review is requested; -this packet consumes it at that status. QS Remark 4 shows why the axis beyond `S` must be in `V_η`. The +This is Theorem QS-E of main#229 5961415030, whose nonauthor review 5961620598 returned ACCEPT / PASS_TECHNICAL_SCOPED; +this packet consumes it at that reviewed scope. QS Remark 4 shows why the axis beyond `S` must be in `V_η`. The lemma needs no `N_R = 0`. In the C-01 cubic, along the sheared vertical path, the level-`(−1 − η)` component reaches `|X| ≈ H(2/σ)^{1/2} ≈ 142`, beyond `B(125)`. So the containment hypothesis fails there, as it must. The exact C-01 cubic, with `B = D = 0`, also lies on `Δ`. diff --git a/frontiers/local_pairing_20260930/README.md b/frontiers/local_pairing_20260930/README.md index ccadb61e2..958dea0ce 100644 --- a/frontiers/local_pairing_20260930/README.md +++ b/frontiers/local_pairing_20260930/README.md @@ -32,7 +32,7 @@ Review slices (§9): A deterministic (§§2–3); B the Gaussian transfer (§4); **v1.2 (2 October 2026).** Codex's Slice C review (C85, 5396856086) found C-01: v1.1's optional §5 margin sketch confined `M`'s component at one level above `f(S)` only. v1.2 withdraws that sketch as a proof and states -Lemma 5.1 instead. It is a common-radius acceptance lemma, consumed from QS-E (main#229 5961415030; author-side -candidate), with exhausting jet-scaled margin sets. §5 also records the review's notes on normalization and +Lemma 5.1 instead. It is a common-radius acceptance lemma, consumed from QS-E (main#229 5961415030; nonauthor +review ACCEPT / PASS_TECHNICAL_SCOPED, 5961620598), with exhausting jet-scaled margin sets. §5 also records the review's notes on normalization and conditioning, and marks (O1′) ⇒ (O1) as conditional on Lemma 5.1. Nothing else changes: not the deterministic lemmas, not Theorem LP, and not the controls. diff --git a/frontiers/local_pairing_20260930/SOURCES.json b/frontiers/local_pairing_20260930/SOURCES.json index 2688d7c91..031c192ce 100644 --- a/frontiers/local_pairing_20260930/SOURCES.json +++ b/frontiers/local_pairing_20260930/SOURCES.json @@ -133,8 +133,8 @@ "files": [ { "path": "LOCAL_PAIRING.md", - "bytes": 46711, - "sha256": "649c8015d88c230af84660455a02a7f8acb1397b89d308ff462a973bd1f22fe7" + "bytes": 46742, + "sha256": "e01e1d097fb087fee013559f508ad58849da06f9a764eed1464286946d78331a" }, { "path": "pairing_check.py", @@ -148,8 +148,8 @@ }, { "path": "README.md", - "bytes": 3092, - "sha256": "5b3c3662171ffe53618b0346d7550bb79ab7050b43fe8d1e5dc87b3809cc3dc0" + "bytes": 3130, + "sha256": "745ef6547c7e93993649f5cb7a978af87b175a908f202d2e398e4eeca2e460ae" }, { "path": "exploration/elder_grid.py", @@ -167,5 +167,5 @@ "note": "Math- main after Math-#173 merged; every consumed source verified at these bytes by the workflow" }, "cited_unmerged": [], - "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied \u2014 exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded | v1.1+sources (2026-09-30 14:15 UTC): LEG (#170) moved from cited_unmerged to cited_only after its integration at the identical blob ef2aa579; no other change; branch merged with main 3e0a91b | status repair (2026-10-01): \u00a75's stale 'not a corollary of anything merged' / 'unmerged' sentence replaced by the attribution to Math-#170 Theorem S on main (requested in comments 5930621629 and 5930668120); #175 (CFE) moved from cited_unmerged to cited_only after its integration at eb659bd with the identical blob 923d3236; header and README status updated; mathematics, checker and RESULTS unchanged; branch merged with main 124c37d | v1.2 (2026-10-02): Slice C review C85 5396856086 finding C-01 applied \u2014 the v1.1 \u00a75 single-level margin sketch is withdrawn as a proof; Lemma 5.1 (common-radius acceptance, consumed from QS-E, main#229 comment 5961415030, author-side candidate) with exhausting jet-scaled margin sets M(eta, R) replaces it; the review's normalization and Gaussian-conditioning notes are recorded; (O1') => (O1) marked conditional on Lemma 5.1; LOCAL_PAIRING.md and README.md only; checker, RESULTS, exploration and workflow unchanged" + "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied \u2014 exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded | v1.1+sources (2026-09-30 14:15 UTC): LEG (#170) moved from cited_unmerged to cited_only after its integration at the identical blob ef2aa579; no other change; branch merged with main 3e0a91b | status repair (2026-10-01): \u00a75's stale 'not a corollary of anything merged' / 'unmerged' sentence replaced by the attribution to Math-#170 Theorem S on main (requested in comments 5930621629 and 5930668120); #175 (CFE) moved from cited_unmerged to cited_only after its integration at eb659bd with the identical blob 923d3236; header and README status updated; mathematics, checker and RESULTS unchanged; branch merged with main 124c37d | v1.2 (2026-10-02): Slice C review C85 5396856086 finding C-01 applied \u2014 the v1.1 \u00a75 single-level margin sketch is withdrawn as a proof; Lemma 5.1 (common-radius acceptance, consumed from QS-E, main#229 comment 5961415030, author-side candidate) with exhausting jet-scaled margin sets M(eta, R) replaces it; the review's normalization and Gaussian-conditioning notes are recorded; (O1') => (O1) marked conditional on Lemma 5.1; LOCAL_PAIRING.md and README.md only; checker, RESULTS, exploration and workflow unchanged | status refresh (2026-10-02): QS-E's consumed status updated from 'author-side candidate, nonauthor review requested' to its delivered nonauthor review 5961620598 (ACCEPT / PASS_TECHNICAL_SCOPED), as requested by the v1.2 delta review 5397276533 (item 8); three status phrases in LOCAL_PAIRING.md and README.md only; Lemma 5.1, all mathematics, checker, RESULTS, exploration and workflow unchanged" } From 49e24985518cb3c2ba7d6653627e547c4620e21a Mon Sep 17 00:00:00 2001 From: Dylan Roy <249851642+d6g8k5htny-coder@users.noreply.github.com> Date: Sat, 3 Oct 2026 08:54:31 -0700 Subject: [PATCH 8/8] Local pairing: re-pin cited-only [RES] after Math-#183 landed (source metadata only) Integration lane (xAI/Grok, Cursor) on behalf of Dylan Roy. After the source-preserving merge of main 7858329 into the branch (8ec18cc), exact-controls failed: merged source drifted for reviews/c6_residual_closure_20260930/RECONCILIATION.md. Math-#183 (merge a4c3199) changed that record from blob 3cfd17c5 (34758 B) to 9668d06a (36574 B): v1.6 history entry, installed-state LIVE/mutant rules and a fifteenth mutant for the record's own checker. The Theorem L direct-statement binding cited in section 0 and the declarative-only, executed=false status are unchanged, so the cited-only identity is re-pinned. No other source changed. LOCAL_PAIRING.md, README.md, pairing_check.py, RESULTS.json, exploration and workflow are byte-identical. Scientific effect NONE; LP= not proved here; no GRAPH/STATUS change; #183 not executed. --- frontiers/local_pairing_20260930/SOURCES.json | 11 ++++++----- 1 file changed, 6 insertions(+), 5 deletions(-) diff --git a/frontiers/local_pairing_20260930/SOURCES.json b/frontiers/local_pairing_20260930/SOURCES.json index 031c192ce..c6abe1c6f 100644 --- a/frontiers/local_pairing_20260930/SOURCES.json +++ b/frontiers/local_pairing_20260930/SOURCES.json @@ -94,11 +94,12 @@ }, { "path": "reviews/c6_residual_closure_20260930/RECONCILIATION.md", - "git_blob": "3cfd17c5ef0356dbd3e8f98a2d08cb404ddd982c", - "bytes": 34758, - "sha256": "b647a186d0cfd1662fc69e9d31dfd447dbb1b2f6ef4364bbd59c4956f18aed25", + "git_blob": "9668d06a41e14057fe6e03ac1bf469aa1ed18580", + "bytes": 36574, + "sha256": "60ed9ca2df666bc31cf50dc1fc7f3f8e8527d5efe4b16c265c205e1e9d1e6b34", "tag": "RES", - "use": "cited in \u00a70 only: the leading-mass localization residual bound to [TSL] Theorem L (Math-#173, merged 30106e1; declarative)" + "use": "cited in \u00a70 only: the leading-mass localization residual bound to [TSL] Theorem L (Math-#173, merged 30106e1; declarative)", + "note": "re-pinned 2026-10-03 from blob 3cfd17c5ef0356dbd3e8f98a2d08cb404ddd982c (34758 bytes) to the bytes landed on main by Math-#183 (merge a4c3199, record v1.6): the change adds the v1.6 history entry, the installed-state LIVE/mutant rules and a fifteenth mutant (partial-install) for the record's own checker; the Theorem L direct-statement binding cited in \u00a70 and the declarative-only, executed=false status are unchanged" }, { "path": "frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md", @@ -167,5 +168,5 @@ "note": "Math- main after Math-#173 merged; every consumed source verified at these bytes by the workflow" }, "cited_unmerged": [], - "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied \u2014 exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded | v1.1+sources (2026-09-30 14:15 UTC): LEG (#170) moved from cited_unmerged to cited_only after its integration at the identical blob ef2aa579; no other change; branch merged with main 3e0a91b | status repair (2026-10-01): \u00a75's stale 'not a corollary of anything merged' / 'unmerged' sentence replaced by the attribution to Math-#170 Theorem S on main (requested in comments 5930621629 and 5930668120); #175 (CFE) moved from cited_unmerged to cited_only after its integration at eb659bd with the identical blob 923d3236; header and README status updated; mathematics, checker and RESULTS unchanged; branch merged with main 124c37d | v1.2 (2026-10-02): Slice C review C85 5396856086 finding C-01 applied \u2014 the v1.1 \u00a75 single-level margin sketch is withdrawn as a proof; Lemma 5.1 (common-radius acceptance, consumed from QS-E, main#229 comment 5961415030, author-side candidate) with exhausting jet-scaled margin sets M(eta, R) replaces it; the review's normalization and Gaussian-conditioning notes are recorded; (O1') => (O1) marked conditional on Lemma 5.1; LOCAL_PAIRING.md and README.md only; checker, RESULTS, exploration and workflow unchanged | status refresh (2026-10-02): QS-E's consumed status updated from 'author-side candidate, nonauthor review requested' to its delivered nonauthor review 5961620598 (ACCEPT / PASS_TECHNICAL_SCOPED), as requested by the v1.2 delta review 5397276533 (item 8); three status phrases in LOCAL_PAIRING.md and README.md only; Lemma 5.1, all mathematics, checker, RESULTS, exploration and workflow unchanged" + "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied \u2014 exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded | v1.1+sources (2026-09-30 14:15 UTC): LEG (#170) moved from cited_unmerged to cited_only after its integration at the identical blob ef2aa579; no other change; branch merged with main 3e0a91b | status repair (2026-10-01): \u00a75's stale 'not a corollary of anything merged' / 'unmerged' sentence replaced by the attribution to Math-#170 Theorem S on main (requested in comments 5930621629 and 5930668120); #175 (CFE) moved from cited_unmerged to cited_only after its integration at eb659bd with the identical blob 923d3236; header and README status updated; mathematics, checker and RESULTS unchanged; branch merged with main 124c37d | v1.2 (2026-10-02): Slice C review C85 5396856086 finding C-01 applied \u2014 the v1.1 \u00a75 single-level margin sketch is withdrawn as a proof; Lemma 5.1 (common-radius acceptance, consumed from QS-E, main#229 comment 5961415030, author-side candidate) with exhausting jet-scaled margin sets M(eta, R) replaces it; the review's normalization and Gaussian-conditioning notes are recorded; (O1') => (O1) marked conditional on Lemma 5.1; LOCAL_PAIRING.md and README.md only; checker, RESULTS, exploration and workflow unchanged | status refresh (2026-10-02): QS-E's consumed status updated from 'author-side candidate, nonauthor review requested' to its delivered nonauthor review 5961620598 (ACCEPT / PASS_TECHNICAL_SCOPED), as requested by the v1.2 delta review 5397276533 (item 8); three status phrases in LOCAL_PAIRING.md and README.md only; Lemma 5.1, all mathematics, checker, RESULTS, exploration and workflow unchanged | source re-pin (2026-10-03, integration lane): after the source-preserving merge of main 7858329 (head 8ec18cc), the cited-only [RES] record changed under Math-#183 (a4c3199) from blob 3cfd17c5 to 9668d06a; non-substantive for this packet (record v1.6 checker procedure only; Theorem L binding and declarative status unchanged), so its identity is re-pinned; no other source changed; LOCAL_PAIRING.md, README.md, checker, RESULTS, exploration and workflow unchanged" }