From b5db7dd59842362fb6503bef8bf673d086bbb2da Mon Sep 17 00:00:00 2001 From: Claude Date: Mon, 5 Oct 2026 18:02:20 +0000 Subject: [PATCH] d1 line v1.2: the slice reads' optional notes; Math- #210 recorded as merged (wording only) Scientific effect NONE. v1.1 (2635309, merged with Math- #271 at 4242ab9) passed slices A-D with no required amendment. v1.2 carries their optional notes and the author's staged section 2 wording: - A: s even and the C-infinity version (section 0); the null point x = 1 and the window marks "< r" (section 1); - B: the global-maximum exclusion under Q also for [1D+] Lemma W and the fields f_phi; t1; the Taylor-remainder domain; the inputs and replaced circle parts; narrowed dependency labels (header, section 7); - C: the Ingham rider (all but at most three eigen-directions); - D: the void cap t0 <= L/4 and the source of each term of (5.1). SOURCES.json moves Math- #210 from cited_unmerged (API-bound to head 213dd9e) to cited_only (tree-bound; merged at 0cdc19f, blob b289c7cd), and the workflow's path triggers add that file. The theorem, its hypotheses, constants and rates, line_check.py and RESULTS.json are unchanged. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU --- .github/workflows/d1-line.yml | 1 + frontiers/d1_line_20261004/PROOF.md | 65 ++++++++++++++++++------- frontiers/d1_line_20261004/README.md | 10 ++-- frontiers/d1_line_20261004/SOURCES.json | 23 ++++----- 4 files changed, 65 insertions(+), 34 deletions(-) diff --git a/.github/workflows/d1-line.yml b/.github/workflows/d1-line.yml index 6fcfc541e..615ea2296 100644 --- a/.github/workflows/d1-line.yml +++ b/.github/workflows/d1-line.yml @@ -10,6 +10,7 @@ name: D1 line transfer exact controls - 'frontiers/d1_sharp_remainder_20261001/PROOF.md' - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md' + - 'reviews/literature_classical_neighbors_20261001/ADDENDUM.md' workflow_dispatch: permissions: diff --git a/frontiers/d1_line_20261004/PROOF.md b/frontiers/d1_line_20261004/PROOF.md index 19299a686..aaa591405 100644 --- a/frontiers/d1_line_20261004/PROOF.md +++ b/frontiers/d1_line_20261004/PROOF.md @@ -1,6 +1,6 @@ # The one-dimensional lifetime and crest-to-trough laws on the line: Theorems 1D and 1D⁺ for stationary processes on `ℝ` -Object: CL-D1-LINE-20261004-v1.1. +Object: CL-D1-LINE-20261004-v1.2. Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 4 October 2026. Dylan Roy — delegated AI work. **v1.1 (before any nonauthor pickup; v1 was `f34ffd7`).** The hypothesis loses v1's (R3a), the decay @@ -9,6 +9,15 @@ the cross-covariances between the samples and the pins small. Conditioning on th three eigenvalues of the samples' covariance (interlacing), and the density bound loses three powers of `ε`, which the growing number of samples absorbs. The decay needed elsewhere, `ρ, ρ', ρ'' → 0`, follows from (R1) by the Riemann–Lebesgue lemma (Lemma 4.1_ℝ). Lemma F_ℝ now uses at least `23` cells; its rate, and the theorem, are unchanged. +**v1.2 (wording only; no statement, hypothesis, constant, rate or control changes).** v1.1 (`2635309`) passed all +four nonauthor slice reads with no required amendment: A (Grok Bot agent 8), B (Grok Bot agent 1; OpenAI / GPT-6 +Astra Pro supplemental), C (OpenAI/Codex C137) and D (Grok Bot agent 3). v1.2 carries their optional notes. Slice B's +notes coincide in part with an author-side check of §2 (same provider; not review evidence). §2 now says that the +global-maximum alternative is excluded under `Q` also for [1D⁺] Lemma W and the fields `f_φ` of Proposition M, defines +`t₁`, fixes the domain of a Taylor remainder, and lists the inputs and the replaced circle-only parts in full. §0–§1 +state the evenness of `s`, the smooth version and the window-determined marks. §5 records the void cap `t₀ ≤ L/4` +and names the source of each term of (5.1). The dependency labels are narrowed (header, §7). `SOURCES.json` records +Math- #210 as merged. Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR REVIEW REQUIRED. Scientific effect: NONE — no register, graph, STATUS, PROOF_INDEX, prize or Boolean change. Same GitHub account as every lane; zero organizational independence. Before submission, two clean-context same-family referees (Anthropic Claude subagents) read the draft: slices A–B and @@ -24,7 +33,8 @@ Lemma 4.3 needs a decorrelation hypothesis. This note supplies that bound, and w under the hypothesis (H_ℝ) of §0. The new ingredient is Lemma F_ℝ (§4): a pair that is *banded* (its process stays between the two endpoint values) over a long interval is unlikely, with an exponential rate in the length. The proof uses three standard tools: -- Ingham's inequality, for a uniform lower bound on the conditional covariance of many well-separated derivative samples; +- Ingham's inequality, for a lower bound on all but at most three eigen-directions of the conditional covariance of + many well-separated derivative samples; - Landau's inequality, on cells of fixed length; - Gaussian concentration, for the number of cells where the second derivative is large. @@ -36,9 +46,10 @@ the classical object of Rice (1944/45), Cartwright–Longuet-Higgins (1956) and - Math- #214, `frontiers/d1_third_order_law_20261001/PROOF.md`, blob `1591ecee`, merged at `e4ca2b3` ([1D]). Used: §0, the objects and constants; §1, the pair coordinates, the pinned law `Q = Q_{t,α}`, (1.2)–(1.4), Lemma 1.2, Lemma 1.3 and Remark 1.4; §2, Proposition 2.2; §3, Lemmas 3.1–3.3 and Corollary 3.4; §4, Lemma 4.1, Lemma 4.2 (near form), and the - part of Lemma 4.3 on `[t_*, t₀]`; §5. + part of Lemma 4.3 on `[t_*, t₀]`; in §5, the adjacency estimate only. - Math- #238, `frontiers/d1_sharp_remainder_20261001/PROOF.md`, blob `8dc558a7`, merged at `8404169` ([1D⁺]). Used: §0, and - §§1–4 (Lemma E, Proposition 2.2⁺, Lemmas Φ and W, Proposition M, and the assembly). + §§1–3 (Lemma E, Proposition 2.2⁺, Lemmas Φ and W, Proposition M) and the part of §4 on `(0, t_*]`; the assembly is + redone in §5. **External (standard).** - Ingham's inequality: A. E. Ingham, *Some trigonometrical inequalities with applications to the theory of series*, @@ -55,7 +66,9 @@ the classical object of Rice (1944/45), Cartwright–Longuet-Higgins (1956) and ## 0. Statement **Setting.** `f` is a centered stationary Gaussian process on `ℝ` with covariance `ρ(x) = E f(0)f(x) = ∫_ℝ e^{iωx}s(ω)dω`, -where the spectral measure has a density `s ≥ 0`. Write `λ_{2j} := (−1)^jρ^{(2j)}(0) = ∫ω^{2j}s(ω)dω`. +where the spectral measure has a density `s ≥ 0`. Since `f` is real, `s` is even, so (R2) below also holds on +`[−ω₂, −ω₁]`; and (R1) gives a version with `C^∞` sample paths (Bulinskaya's lemma and the window space of §1 use +`C³`). Write `λ_{2j} := (−1)^jρ^{(2j)}(0) = ∫ω^{2j}s(ω)dω`. > **(H_ℝ)** > - (R1) `∫(1 + |ω|)^n s(ω)dω < ∞` for every `n`. @@ -153,13 +166,14 @@ So the objects of §0, and the marks below, are defined `Q`-almost surely for ev **The Kac–Rice representation on `ℝ`.** The marked two-point Kac–Rice formula of [1D] §1 holds on `ℝ` for pairs with `x ∈ [0, 1)` and `y = x + t`, `t > 0`. The marks are Borel functions of `f` on a compact window around the pair, or increasing limits of such functions. The argument is this: -- On the domain `{x ∈ [0, 1], t ∈ [ε, 1/ε]}`, the pair law is nondegenerate (Lemma 1.1_ℝ). A mark that depends on `f` +- On the domain `{x ∈ [0, 1], t ∈ [ε, 1/ε]}` (the intensities use `[0, 1)`; the point `x = 1` is a null event), the pair + law is nondegenerate (Lemma 1.1_ℝ). A mark that depends on `f` restricted to `[x − r, x + r]` is a Borel function on the Polish space `C²([−r, 1 + 1/ε + r])`, which also carries the pair's segment `[x, x + t]`. The argument of [1D] §1 ([P] §9 as repaired in [E2]) therefore applies verbatim. - Let `ε ↓ 0`, by monotone convergence. - The crest-to-trough mark `1{f' < 0 on (−τ, τ)}` depends on `f` on the pair's segment. - The elder mark `𝔪 := 1{τ = S(−τ)}` is the increasing limit, as `r → ∞`, of - `𝔪·1{x₋ and x₊ of the maximum −τ lie within distance r of −τ}`. Each term is a Borel function of `f` on + `𝔪·1{x₋ and x₊ of the maximum −τ lie within distance < r of −τ}`. Each term is a Borel function of `f` on `[−τ − r, −τ + r]`. The limit holds `Q`-almost surely by the preceding paragraph, so monotone convergence applies on both sides of the formula. @@ -182,7 +196,8 @@ The following parts of [1D] and [1D⁺] concern a pair at separation `t ≤ t₀ Their proofs use the following, all available on `ℝ`: - the moments `λ_{2j}`; -- the nondegeneracy of finitely many derivatives at one or two points (Lemma 1.1_ℝ, in place of [1D] Lemma 1.1); +- the nondegeneracy of finitely many derivatives at finitely many distinct points: the pins, the midpoint and the + sample points of [1D] Lemma 4.2 (Lemma 1.1_ℝ, in place of [1D] Lemma 1.1); - the parity split of [1D] Remark 1.4, which needs only that `ρ` is even; - Borell–TIS and Sudakov–Fernique for suprema over compact intervals. @@ -191,10 +206,19 @@ So they hold on `ℝ` with the same proofs. The global parts of [1D] §§0–1 a - the normalization `L^{−1}`, the range `t ∈ (0, L)` and the reflection `K(t) = K(L − t)` in (1.2)–(1.3), by (1.2_ℝ)–(1.3_ℝ); - the finiteness of the critical set, by its local finiteness (the Morse property, Lemma 1.1_ℝ); and the exclusion of the - global maximum, by Lemma 1.5_ℝ, under which no global maximum exists. + global maximum, by Lemma 1.5_ℝ and its form under `Q` in §1: `Q`-almost surely, for every `(t, α)`, no global maximum + exists and `x_±` are finite; +- the far form of [1D] Lemma 4.2 and the part of [1D] Lemma 4.3 on `[t₀, L)` with its co-banded bound, by Lemma 3.1_ℝ + and Lemma F_ℝ; and the circle assemblies ([1D] §5 apart from the adjacency estimate, [1D⁺] §4 on `[t_*, L)`), by §5. Three points need a word. -- [1D] Lemma 3.3 holds on `ℝ`, and its global-maximum alternative does not occur. +- [1D] Lemma 3.3 holds on `ℝ`, and its global-maximum alternative does not occur: under `Q` every maximum of `f` has a + finite bar (§1). The same holds for [1D⁺] Lemma W, read with "a function `f_φ` on `ℝ`", and for its use in [1D⁺] + Proposition M on the fields `f_φ = f + (φ − φ_G)m(ψ − ψ(τ))`, for every `φ` given `ω`. There + `ψ(x) = Cov_Q(f_e(x), E₁)/v₁` is built from `ρ′(τ ± x)` and `ρ″(τ ± x)`, so it is bounded by (R1), and each `f_φ` + differs from `f` by a bounded function. So, on the `Q`-almost sure event of §1, each `f_φ` is unbounded above on both + half-lines, and every maximum of `f_φ` has a finite bar. In particular, when both arcs fall below `−1` (the edge `J₋` + with `μ₋(φ) ≤ 0`, or the negative case of [1D] Lemma 3.3 (R)), `𝔖` is not the death point of `𝔐`. - In the step of [1D] Lemma 4.3 on `[t₁, t₀]`, `sup|f''|` is the supremum over the band `[−τ, τ]`; Landau's inequality on the band needs nothing more. (On `ℝ`, `sup_ℝ|f''| = ∞` almost surely.) With this reading, the near part of [1D] Lemma 4.3 gives @@ -202,14 +226,18 @@ Three points need a word. ∫_{t_*}^{t₀} K^{band}(t, h) dt = O(h^{3/4}). (2.1) Here `K^{band}(t, h) := (12/t⁴)p_t(α)E_Q[|f''(−τ)f''(τ)|1{banded}]`, and "banded" means `f(τ) ≤ f ≤ f(−τ)` on `[−τ, τ]`. - Also `t_* := (12h)^{1/(5−δ)}` with `δ ∈ (0, 1/10]`, as in [1D] Proposition 2.2; §5 takes `δ = 1/10`. + Also `t_* := (12h)^{1/(5−δ)}` with `δ ∈ (0, 1/10]`, as in [1D] Proposition 2.2; §5 takes `δ = 1/10`. Here + `t₁ := h^{1/10}` is [1D]'s notation (its Lemma 4.3), unrelated to `T₁` of Lemma 4.1_ℝ. The same band reading applies + to the Taylor remainders `τ^K sup|f^{(K+1)}|` in the proof of [1D] Lemma 1.3: the supremum is over `[−τ, τ]`. - Beyond the list above, [1D⁺] uses the following: - `E_Q(K^e)² ≤ C` with `K^e = sup_{[−2t,2t]}|f_e^{(6)}|`, in (Φ.1) and, through `K^⊥`, in (Φ.3) and Proposition M; - the invertibility of `Cov(U₁, U₃)` for `t > 0` in (Φ.2), now by Lemma 1.1_ℝ; - uniform Gaussian tails of `K̃^o` and `K^⊥` in (Φ.3); - `E_Q[O(3/2)⁴] ≤ C` in Proposition M. - Each is a supremum or a moment over `[−2t, 2t]`, bounded by Sudakov–Fernique and Borell–TIS on a compact interval. + The first, third and fourth are suprema or moments over `[−2t, 2t]`, bounded by Sudakov–Fernique and Borell–TIS on + a compact interval. The second also gives `sup_{t≤t₀}sup_{[−2t,2t]}|μ_t^{(5)}| < ∞` in (Φ.2), through the continuity + of `(t, x) ↦ μ_t^{(5)}(x)` (by (R1)) and the nondegenerate limit of [1D] Lemma 1.3. Two facts from [1D] §4 are used in the following form. - An adjacent typed pair is banded: `f' < 0` on `(−τ, τ)`, so `f` is monotone there. @@ -380,8 +408,9 @@ sample values.) ∎ ## 5. Proof of Theorem 1D_ℝ -Fix `δ = 1/10` and `t₀` as in [1D⁺] §4 (small enough for [1D] Lemma 4.3), and `T₀ ≥ t₀` as in Lemma F_ℝ. By (2.1), -Lemma 3.1_ℝ and Lemma F_ℝ, +Fix `δ = 1/10` and `t₀` as in [1D⁺] §4 (small enough for [1D] Lemma 4.3), and `T₀ ≥ t₀` as in Lemma F_ℝ. On `ℝ` the cap +`t₀ ≤ L/4`, which [1D⁺] §4 inherits from the far form of [1D] Lemma 4.2, is void: that form is not used. The three +terms of (5.1) come from (2.1) on `[t_*, t₀]`, Lemma 3.1_ℝ on `[t₀, T₀]` and Lemma F_ℝ on `[T₀, ∞)`: ∫_{t_*}^∞ K^{band}(t, h) dt = O(h^{3/4}) + O(h²log(1/h)) + O(h²) = O(h^{3/4}). (5.1) @@ -397,7 +426,7 @@ Lemma 3.1_ℝ and Lemma F_ℝ, - On `[t_*, ∞)`, an elder pair is banded (§2), so `K ≤ K^{band}`, and (5.1) applies. Doubling gives (1D_ℝ.2). The constants are those of [1D] §0. They depend only on `λ₂, λ₄, λ₆, λ₈`, through `p₁₂`, `σ₃`, `σ₄`, -`D` and `𝒬`. ∎ +`D` and `𝒬`. For `e^{−x²/2}` their values, and the per-crest constants, are given in §0. ∎ ## 6. Remarks @@ -435,10 +464,10 @@ Doubling gives (1D_ℝ.2). The constants are those of [1D] §0. They depend only | Tag | Path | Role | |---|---|---| -| [1D] | `frontiers/d1_third_order_law_20261001/PROOF.md` (blob `1591ecee`, merged `e4ca2b3`) | consumed: §§0–5 as listed above | -| [1D⁺] | `frontiers/d1_sharp_remainder_20261001/PROOF.md` (blob `8dc558a7`, merged `8404169`) | consumed: §§0–4 | +| [1D] | `frontiers/d1_third_order_law_20261001/PROOF.md` (blob `1591ecee`, merged `e4ca2b3`) | consumed: the parts listed in §2 (§§0–4, and the adjacency estimate of §5) | +| [1D⁺] | `frontiers/d1_sharp_remainder_20261001/PROOF.md` (blob `8dc558a7`, merged `8404169`) | consumed: §§0–3, and §4 on `(0, t_*]` | | [P], [E2] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d`), `reviews/d1_section9_borel_repair_20260925/REPAIR.md` (blob `fe9b9ce4`) | the marked Kac–Rice convention, through [1D] §1 — cited | -| #210 | `reviews/literature_classical_neighbors_20261001/ADDENDUM.md` (Math- #210, open; head `213dd9e`, blob `b289c7cd`) | comparison only (Remark 3) | +| #210 | `reviews/literature_classical_neighbors_20261001/ADDENDUM.md` (Math- #210, merged at `0cdc19f`; blob `b289c7cd`, as read at head `213dd9e`) | comparison only (Remark 3) | ## 8. Exact controls (`line_check.py`; standard library; exact rationals except C3; byte-identical under `-O`) diff --git a/frontiers/d1_line_20261004/README.md b/frontiers/d1_line_20261004/README.md index 5f68c466e..8b750eb13 100644 --- a/frontiers/d1_line_20261004/README.md +++ b/frontiers/d1_line_20261004/README.md @@ -2,7 +2,11 @@ **Author-side proof candidate (Anthropic Claude). Scientific effect: NONE. Nonauthor review required.** -Object `CL-D1-LINE-20261004-v1.1`. Full text: [`PROOF.md`](PROOF.md). Dylan Roy — delegated AI work. +Object `CL-D1-LINE-20261004-v1.2`. Full text: [`PROOF.md`](PROOF.md). Dylan Roy — delegated AI work. + +**v1.2** is wording only. v1.1 (`2635309`) passed all four nonauthor slice reads with no required amendment (A Grok Bot +agent 8; B Grok Bot agent 1, with OpenAI / GPT-6 Astra Pro; C OpenAI/Codex C137; D Grok Bot agent 3). v1.2 carries +their optional notes and records Math- #210 as merged. The theorem, its hypotheses and its rates are unchanged. **v1.1** was made before any nonauthor pickup; v1 was `f34ffd7`. It drops v1's decay hypothesis on `ρ, ρ', ρ''`, so (H_ℝ) is now purely spectral. Lemma 4.3_ℝ now uses interlacing: conditioning on the three pin observations lowers at @@ -57,8 +61,8 @@ the number of sample points has to grow with the length of the band. - Math- #238, `frontiers/d1_sharp_remainder_20261001/PROOF.md`, blob `8dc558a7`, merged at `8404169` ([1D⁺]). **Cited.** `[P]` (`UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, blob `dfed3b8d`) and `[E2]` (`REPAIR.md`, blob `fe9b9ce4`), the -marked Kac–Rice convention, through [1D] §1. Math- #210 (open; head `213dd9e`, addendum blob `b289c7cd`) is cited for -comparison only and is bound through the repository API. +marked Kac–Rice convention, through [1D] §1. Math- #210 (merged at `0cdc19f`; addendum blob `b289c7cd`, the blob read +at head `213dd9e`) is cited for comparison only. `SOURCES.json` now binds it in the tree, with the other cited sources. ## Controls diff --git a/frontiers/d1_line_20261004/SOURCES.json b/frontiers/d1_line_20261004/SOURCES.json index 999b00621..214ba1251 100644 --- a/frontiers/d1_line_20261004/SOURCES.json +++ b/frontiers/d1_line_20261004/SOURCES.json @@ -1,13 +1,13 @@ { "schema": 1, "scientific_effect": "NONE", - "object": "CL-D1-LINE-20261004-v1.1", + "object": "CL-D1-LINE-20261004-v1.2", "kind": "author-side proof candidate: Theorems 1D and 1D+ for stationary Gaussian processes on the line under (H_R) = (R1) all spectral moments finite, (R2) s >= s0 > 0 on [omega1, omega2] with omega1 > 0, (R3) ess sup (1 + omega^6) s < infinity (no decay rate of rho is assumed): nu_+ = (C0/2) h^(-1/3) + (I/2) h^(1/4) + B2 h^(1/3) + O(h^(3/4)) and nu = C0 l^(-1/3) + C1 l^(1/4) + 2 B2 l^(1/3) + O(l^(3/4)), with the constants of Math- #214. New: Lemma 1.5_R and the process under Q; the Kac-Rice representation on R; Lemma 3.1_R (the compact middle range); Lemma 4.1_R (the pin covariance, Riemann-Lebesgue); Lemma 4.2_R (Landau on an interval); Lemma 4.3_R (Ingham + interlacing: conditioning on the pins lowers at most three eigenvalues); Lemma 4.4_R (Gaussian concentration for the number of rough cells); Lemma F_R (the far band is O(h^2))", "author": "Anthropic Claude", "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", "attribution": "Dylan Roy - delegated AI work; actual performer Anthropic Claude", "read_at": { - "branch_base": "f34ffd748ac00507fb3ef2abd01cdeb6eb7bb43e" + "branch_base": "d47b90e7d5baaf599b6fcd2e3f1bc94ca4dc9648" }, "consumed": [ { @@ -48,20 +48,17 @@ "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", "tag": "E2", "use": "the Borel repair of [P] section 9, through [1D] section 1" - } - ], - "cited_unmerged": [ + }, { "path": "reviews/literature_classical_neighbors_20261001/ADDENDUM.md", "bytes": 16207, "sha256": "5e3903ad05b0566a1e999b0e69b309a4165578705d687f4a3021d22ebd1e1e42", "git_blob": "b289c7cd4be29fad6913aa4d753899cb0ef851aa", "tag": "210", - "pr": "Math- #210", - "head": "213dd9eb8ba83de242516b10c999fbe56d17fd3f", - "use": "Math- #210 v1.3 (open): the classical crest-to-trough literature and the d = 1 remark whose leading constant 0.19971814 this note proves on the line (Remark 3); comparison only" + "use": "Math- #210 v1.3 (merged at 0cdc19f; the blob read at head 213dd9e): the classical crest-to-trough literature and the d = 1 remark whose leading constant 0.19971814 this note proves on the line (Remark 3); comparison only" } ], + "cited_unmerged": [], "cited_external": [ { "tag": "Ingham", @@ -98,7 +95,7 @@ "M6": "C6", "M7": "C6" }, - "label_history": "v1 (f34ffd7, opened 2026-10-04 06:46Z) assumed |rho^(j)(x)| <= C (1 + |x|)^-1 for j <= 2, and Lemma 4.3_R bounded the conditional covariance by a Schur complement with small cross-covariances. v1.1, before any nonauthor pickup, drops that hypothesis: Lemma 4.3_R uses interlacing (at most three eigenvalues drop) and loses three powers of eps; Lemma 4.1_R gets the pin covariance from Riemann-Lebesgue; Lemma F_R uses at least 23 cells. The theorem, its rates and its constants are unchanged; checker C5 and C6 updated, mutant M7 added.", + "label_history": "v1 (f34ffd7, opened 2026-10-04 06:46Z) assumed |rho^(j)(x)| <= C (1 + |x|)^-1 for j <= 2, and Lemma 4.3_R bounded the conditional covariance by a Schur complement with small cross-covariances. v1.1, before any nonauthor pickup, drops that hypothesis: Lemma 4.3_R uses interlacing (at most three eigenvalues drop) and loses three powers of eps; Lemma 4.1_R gets the pin covariance from Riemann-Lebesgue; Lemma F_R uses at least 23 cells. The theorem, its rates and its constants are unchanged; checker C5 and C6 updated, mutant M7 added. v1.2 is wording only: after the four nonauthor slice reads of v1.1 (2635309, merged with Math- #271 at 4242ab9) it carries their optional notes and records Math- #210 as merged. The theorem, its hypotheses, rates and constants, and line_check.py, are unchanged.", "replayed": [ "line_check.py" ], @@ -106,13 +103,13 @@ "files": [ { "path": "PROOF.md", - "bytes": 38349, - "sha256": "e8e401bd82cabd3f876812c1089f5c0bf3a6240c48b836776aed8f535489c7a9" + "bytes": 41713, + "sha256": "6866f364450ecc393b652e74728abcf5b6b498b99279fdcfa0c9a2deb038a10e" }, { "path": "README.md", - "bytes": 7259, - "sha256": "1ace3aa3cbaa0bb2b34ab954c9d9a742ffd9e1ef33845dcbc1ee9a1317a51f5b" + "bytes": 7670, + "sha256": "c47b3d46d64198c4efe780e3337d0253e85be8d4c62a6e2a28698f2c27f6e0f6" }, { "path": "line_check.py",