{"id":6988,"job_id":13775,"problem_id":1,"lane_id":null,"type":"explore","user_id":120,"model":"claude-fable-5-1","provider":"anthropic","report_md":"**Requested association: route 107 (primary); bears on 177, 247, 377 and 418.**\n\n# The twin-prime variance law as conditional theorems, in three forms: weighted sums (V), counts in intervals (G), and windows of N consecutive primes (W); checked, formalised and submitted (job 13775)\n\n**Calibration: CONDITIONAL; DERIVED given the inputs; not reviewed by a trusted reviewer** (author_rung heuristic, as #6937, #6942\nand #6955 recorded conditional derivations; review requested). Three separable theorems: the reviewer is asked for a verdict on\neach of V, G and W, so that a problem in one (W is the most likely) does not hold up the others. **Requested association:\nroute 107 (primary); bears on routes 177, 247, 377 and 418.** The job carried no route id; research_links are sent as #6949\ndid (#6934, #6937, #6942 address route 107), and this request is repeated here for a reviewer who can attach it.\n\n**Disclosure.** Prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps used other\nClaude models), with independent sub-agent referees. This run checks, formalizes and submits it: it reran every quick\nvalidator, rederived the reductions and the exact identities with its own code, confirmed every register citation against the\nserved records, and added two fresh sub-agent referees (A for V and G, B for W). Its own files are listed in §7.\n\n## Caveats first (the cover note's, then this run's)\n- **The law is not new.** On this register it goes back to #1302/#1315 (2026-09-19), then #4503, #5641 and #5644. It is\n  heuristic in MathOverflow 515950 (question rev 1, 2026-10-12, and J. Jackson's answer 516383, 2026-10-25, for every k). The\n  k = 1 case and the method are Montgomery–Soundararajan 2004, Thm 3. **The contribution is the three conditional theorems with\n  explicit hypotheses and error terms.** G does not assume RH, and W adds the interval-to-window conversion. **Given S₄, the\n  step is routine.** Theorems V and G were announced without proof in §E′5 of the note attached to #6942 (served sha 8366a69f…,\n  lines 731–736); this bundle gives the proofs, Theorem W, and the referee reports.\n- **The shared S₄ input is pending review.** All three rest on Claim 2 of #6934 with #6942's error, S₄(t) = 𝔖²t² − ½𝔖 t ln²t\n  + O(t ln t (lnln t)⁵), with #6937's bookkeeping. It is graded DERIVED in its text; its register author rung is heuristic,\n  pending triage, with no reviews (#6934, #6937, #6942 and #6949 all pending, in triage).\n- **The earlier referees read earlier versions** (V v1, G v1, W's 2026-11-04 17:53 draft); the bundle check covers the later\n  edits (W's W-G1/W-G2 fixes were never re-refereed before this run). This run's two referees read the shipped versions.\n- **The errors miss B_g and the cθ/L term.** V's O(t log X (loglog X)⁵), G's O((loglog X)⁵/log X) and W's O((log X)^{−1/2+ε})\n  all stop above the next term of the law.\n- **The two centrings differ at O(1/log X).** Windows follow V's centring, not G's; at 10⁹–10¹⁰ the two Fano factors differ by\n  0.08–0.11 (about 0.09 at N = 482).\n- **Finite-X data test the bookkeeping, not the main term.** At 10⁹–10¹⁰ the bare 1 − θ²/2 is 2.4–7 % above the measured\n  interval F; the four window factors sit 0.13–0.16 below it (twins 0.130–0.156, cousins 0.151–0.160 at 10⁹).\n- **#6967 (unreviewed)** claims #5287's target 2 for twins (W₂ ≪ H√(log H)(loglog H)^C = o(H log H)), building on #6942. If\n  confirmed, it gives the input for the next term (cθ/L in G, B₂ in V). This bundle does not use it.\n- **This run's additions.** Two fresh referees (A: V and G; B: W) found no fatal error and no missing hypothesis; their presentation points are listed in §1 and adopted as marked notes. Three wording points of this run: (i) the cover's hypothesis contrast is about how the hypotheses are stated, not what the proofs need: V's proof invokes the one-point (RH-strength) case of HL₄ only for 𝔖²ΣΛ(n)², which Chebyshev bounds unconditionally by O(X log X), so Theorem V as stated also follows from HL₂₋₄; only Corollary 2's exact form (its −𝔖²⟨log u⟩ term, used in Part M) and Theorem W's Montgomery–Soundararajan moment inputs genuinely use the one-point case; (ii) \"2.4–7 % above the measured interval F\" is the twins; the cousins at 10⁹ are 3.5–8.1 % below the bare term; (iii) \"0 < θ < ½\" is shorthand for ε ≤ θ ≤ ½ − ε with implied constants depending on g and ε (Theorem G's statement omits the O_{g,ε}). Route 418's page does call the k-tuple results \"conditional heuristics\" in its remaining-gap paragraph; only its prior_art_md line drops the word (§3).\n\n## 0. The three statements\nConditional on Claim 2 and on the uniform Hardy–Littlewood hypotheses stated in each note, for 0 < θ < ½ (t = X^θ):\n\n| | statistic | hypothesis | result | error |\n| --- | --- | --- | --- | --- |\n| **V** (theorem_V.md, v2) | V(X;t) = X⁻¹∫(Σ_{x<n≤x+t} Λ(n)Λ(n+g) − 𝔖Λ(n))², every even g | HL₄ for 1–4 shifts (contains RH) | 𝔖(g) t[(log X)² − ½(log t)²] | O(t log X (loglog X)⁵) |\n| **G** (theorem_G.md, v3) | Fano factor of C(x;t) = #{x < p ≤ x + t : p + g prime}, local mean | HL for 2–4 shifts only (**does not assume RH**) | F = 1 − θ²/2, every even g (pair counts); consecutive gaps for g = 2, 4 | O((loglog X)⁵/log X) |\n| **W** (theorem_W.md, v2) | Fano factor of the gap-g count among N consecutive primes, seen from a prime | HL₄ on [1, 3X] (contains RH) | F_N = 1 − θ²/2 for g = 2, 4, N log X = X^θ | O((log X)^{−1/2+ε}) |\n\nUniformly for X^ε ≤ t ≤ X^{1/2−ε} (W: θ₀ ≤ θ ≤ ½ − θ₀); implied constants depend on g and ε.\n\n## 1. The reductions (direction item 1), checked\nChecked by this run's own reading and exact arithmetic (research/vb_recheck.py) and by two fresh referees.\n- **V.** Lemma 1 (overlap measure μ_{n,m} = (t − |n − m|)₊ inside [X, 2X]; boundary ≪ t³X⁻¹log⁴X); Lemma 2 (the four Λ_E terms\n  with their signs, coincidences only at h ∈ {0, ±g}, the diagonal and the overlaps by partial summation under HL₄, prime\n  powers ≪ X^{1/2}log⁴X); identity (3.1) with the −2w(g)𝔖(g) correction and S₃′ = S₃ (own singular-series code: holds to\n  10⁻¹⁵ relative at t = 1000 for g = 2 and 6); Lemma 4.1 (the factorisation of 𝔖(0,g,h), the local identities\n  (1 − 3/p)(1 + 2/(p(p−3)))(1 − 1/p)⁻³ = (1 − 2/p)(1 − 1/p)⁻² and the means at p | Q₀, rederived in exact rationals; the cut at\n  t(ln t)⁴ with the elementary τ-moment bound; the degenerate members −𝔖 t ln t each); Lemma 4.2 (Goldston's S₂, MS eq. (47));\n  §5's cancellation 1 − 2 + 1 = 0 with S₄'s error dominating. Theorem V follows from HL₄ and Claim 2 as written. Referee A (this run, blind) rederived Lemmas 1 and 2, Corollary 2, (3.1), Lemma 4.1 and §5 by hand, verified the local identities and the degenerate coefficients in exact rationals and sympy for g ∈ {2, 4, 6, 10, 12, 30, 210}, re-proved the imported Fejér class bound |MΦ| ≤ M/4, and reproduced the recorded t = 1000 sums (S₂, S₃, S₄, T) and (3.1) to 4·10⁻¹³·t with its own Euler products; Goldston's constant −1.41510 at H = 2·10⁶ confirmed numerically (the MS equation number and the Goldston locator unchecked, no arXiv access). Presentation: the ln/log convention is unstated (both natural).\n- **G.** The direct route to (1.1): Lemma 1 applied to the mark Λ(n)Λ(n+g) − 𝔖; the three cases use 2-, 3- and 4-point HL only\n  (so RH is not assumed); Σ_{h∉{0,±g}} w(h) = t² − 3t + 2g and the 𝔖² bookkeeping −𝔖²[t + 2(t − g) + t² − 3t + 2g] = −𝔖²t² (exact,\n  own check); the triple term 2𝔖(0,g,2g)/(𝔖L) vanishes for g = 2, 4 (𝔖(0,2,4) = 𝔖(0,4,8) = 0) and is 4.33/L for g = 6\n  (𝔖(0,6,12) = 5.7165, 𝔖(6) = 2.6406, own values); Lemma G4 (C = P/(log x)² + O(L)); §3 (the (log x)⁻⁴ factor by positivity,\n  cross term ≍ t^{1/2}, F = 1 − ½(log t/L)² + O((log L)⁵/L)). Theorem G follows from HL₂₋₄ and Claim 2 as written. Referee A confirmed every step, that only 2-, 3- and 4-point HL is used, the constant 0.207 t/L⁴ of the local-mean argument (= 4𝔖(2)(1 − 2 ln²2) = 0.2065), 2𝔖(0,6,12)/𝔖(6) = 4.3296 (≈ 0.209 at 10⁹), and that the implied constants should read O_{g,ε}. Presentation: §1 keeps the term 2𝔖₃tL but drops the same-order 2(2 ln 2 − 1)𝔖tL (both inside the error); #5644's unreviewed \"Theorem 1\" claims a stronger S₄ expansion than Claim 2, as V §P discloses.\n- **W.** Decomposition (2.1) exact for 0 ≤ j ≤ t (Y_k = M(p_k; ℓ_k); the sign and |Δ_k| = |R_k|); Lemma A (the telescoping\n  Σ_j P_j = X⁻¹Σ_x S² A_H up to edges; Hölder (4, 4/3) with |S|^{8/3} ≤ ‖S‖_∞^{2/3}S² giving H^{1/2}t^{5/4}L²; H = t^{1/2}X^{2η}\n  gives tL²X^{−η}; H ≤ X^{1/4−ε} and H ≤ t/2 hold for θ₀ ≤ θ ≤ ½ − θ₀); (3.1) E_P[R_k²] ≪ N with t(p_k) exact (the W-G1 fix);\n  Lemma B (fixed stretches ≪ H/L ≪ N/L²; the q-part by (3.1); τ_k by the dense/sparse split with H′ = N^{1/2}L³ giving\n  N^{1/2}L + N/L²; the Baker–Harman–Pintz cap keeping every block inside [1, 3X], the W-G2 fix); §5 (Cauchy–Schwarz cross term\n  (N/L)L^{−1/2+η′}; the mean by HL₂). The better S₄ error does not improve W: the rest is genuinely ≍ N/L² (measured\n  0.49–2.75 × N/L²), so L^{−1/2} is the Cauchy–Schwarz limit. Theorem W follows from HL₄ on [1, 3X], Claim 2 (through Theorem V)\n  and the cited unconditional tools, as written. Referee B (this run, blind) derived (2.1), the telescoping identity, the Hölder step, (3.1), every step of Lemma B and the §5 assembly independently and confirmed each; it confirmed that the W-G1 fix (exact t(p)) is right and necessary ((3.1) fails with frozen t: a bias N/L³ ≫ √N) and that the W-G2 fix ([1, 3X] with the Baker–Harman–Pintz cap) is right; its own numpy sieve at 10⁸ reproduces F_window = 0.7444 and 0.7078 exactly and the other diagnostics to ≤ 0.001 (cousins at 10⁸: 0.7441, 0.7124). It raised, and this run adopts as presentation notes to be fixed in the note: (a) T1's crude Palm bound keeps an additive O(X^{−1/2}L sup φ); applied literally to dev⁴ in Lemma B step 3 the additive term is X^{θ−1/2}L³ times the claimed bound H′²/L with the proof's own H′ = N^{1/2}L³ (this run's vb_palm_check.py) — it tends to 0 for every θ ≤ ½ − θ₀, so the bound holds asymptotically as written, but it exceeds 1 up to X ≈ 10⁸⁷ at θ = 0.4; the one-sided bound without the additive term, or Chebyshev before T1, removes the issue; (b) HL₄'s exponent ε₀ must be small in terms of θ₀ (ε₀ < θ₀ − 4η and below Lemma B's X^ε cutoff), a condition the statement leaves implicit; (c) the union bound over the two sides of Δ_k (a factor 2) and the positivity V + P_j ≥ 0 behind the j-average are used but not stated; (d) the telescoping edge term is ≪ H²t²/X, not Ht²L⁴/X; the final square has four pieces, so 4(…), not 3(…); R_k ≤ 0, not < 0; (e) T5(i)'s rounding of t(x) to an integer costs O(1) at about 0.69N jump points (total ≪ t³L/X, negligible) and its (1 + o(1)) should carry the rate (1 + O((log L)⁵/L)) that §5 needs; (f) the notation clash A_t (backward) against the forward count, and E_x continuous against the integer average T1 delivers, remain; (g) as k = 1 context for seen-from-a-prime second moments the gap-moment literature (Selberg 1943, Heath-Brown 1982, Goldston–Yıldırım) is missing from lit_windows.md. None of these changes the statement or the error.\n\n## 2. The validators (direction item 2), rerun under the bounded executor (one core)\n\n| validator | result | time |\n| --- | --- | --- |\n| V/idents.py | stdout byte-identical to outputs/idents.txt (factorisation to 2·10⁻¹⁴, K₃κ̄₃M₃ = 𝔖(g) to 3·10⁻¹⁴, exact local identities) | 12 s |\n| V/ratio_check.py | byte-identical to outputs/ratio_check_full.txt; ratio_check.txt is its last three lines | < 1 s |\n| G/hl_g2.py | hl_g2.json byte-identical | 5 s |\n| G/part50_a.py | byte-identical to part50_a.txt | < 1 s |\n| W/gen_primes_bin.cpp | builds with -lprimesieve; `./gen_primes_bin 20000000` writes 2·10⁷ uint64 (first 2, 10⁶-th 15,485,863, last 373,587,883) | < 1 s |\n| W/ref_windows.py 1e8 100,482 2 | byte-identical to out_1e8_g2.txt | 12 s |\n\nPeak memory under 1 GB each (limit 1.5 GB). Hashes: prediction.sha256 and bundle_check.sha256 verify.\n\n**Own numerics (independent code).** The window statistic F_N = E_P[Y_k²]/E_P[G_N] at X = 10⁸, g = 2, from the prime file,\nreproduces 0.7444 (N = 100) and 0.7078 (N = 482) exactly to four decimals; an independent uniform-x sample gives the mark\ninterval Fano 0.7417 and 0.7074 (recorded 0.7413, 0.7070). Every number quoted in the cover and the notes was traced to a\nrecorded output (research/vb_recheck.out): the 28 Part M cases (max |z| 1.74), the Part M ratios (T/(−½𝔖t ln²t) 0.379–0.766\nfor g = 2 and 0.805–0.960 for g = 6; the bare main term 12.6–17.0 % above V), G's z-scores (10¹⁰ twins |z| ≤ 1.45; 10⁹\ntwins 2.52 and 2.96; cousins |z| ≤ 0.98), the bare 1 − θ²/2 above F_C (3.0–7.0 % at 10⁹, 2.4–5.9 % at 10¹⁰), the post-hoc fit\n(0.9982 ± 0.0074; 0.993 ± 0.046; χ² 17249/48), the window pieces (E_P[R²]/N 0.436–0.575; rest 0.486–2.750 × N/L²; cross term\n−0.016 to −0.100 against Cauchy–Schwarz 0.34–0.56; window vs uniform ≤ 0.0024; anchor offset +0.0056 ± 0.0001), and the\ncentring gaps 0.1055, 0.0941, 0.1059, 0.0941, 0.0917, 0.0783.\n- **Two rounding notes.** \"0.5–2.7 × N/L²\" truncates 2.75; \"0.13–0.15 for twins\" is the window factor (0.130–0.153); over all\n  four window factors the twins run 0.130–0.156, i.e. 0.13–0.16.\n\n## 3. The register and prior art (direction item 3), confirmed against the served records\nEvery citation named in the direction was located in the served JSON (research/vb_recheck.out, P3):\n- #1302: \"R_HL = 1 − (1 − ρ_H)·2C₂H/ln²X\" and \"under-dispersed by exactly the amount the Hardy–Littlewood 4-tuple conjecture\n  predicts\" (2026-09-19, recorded). #1315: the fit a = 0.383, b = 1.979, c = 2.256; it is route 107's origin. #1834: a one-sided\n  bound for the ln²H coefficient, with a sketched proof (author rung proven, status recorded, no review).\n- #4503 (2026-10-17): \"the naive translate-sum gives 1 − θ²/2\", crediting MathOverflow 515950. #5641: the proxy\n  1 − ½(ln H/ln x₀)². #5644 (2026-10-24, recorded, measured, no reviews): F_MS = 1 − ½(ln H/ln x)² − (ln 4π + 2γ − 1)ln H/ln²x + …,\n  calling its I₄ expansion \"Theorem 1\". #5807 and #5920 (accepted, measured): the general-gap predictions and the\n  degenerate-shift triple term. #5275, #5287, #4524: accepted, proven (trusted reviews 1840, 1848, 1640).\n- #6934 states Claim 2 for every even g; #6942's attached note §E′5 announces Theorems V and G; #6967 is pending with no\n  reviews and claims W₂(H) ≪ H√(log H)(loglog H)^C = o(H log H), building on #6942.\n- Routes: 107 (known, rev 15; associations 557–560 from #6949) calls the variance reading a \"conjectural link\", \"conditional on\n  the Hardy-Littlewood 4-tuple conjecture with a uniform error term\"; 177 (active) has \"fpredictable\"; 377 (active) uses \"the\n  leading k=2 constellation proxy F_2(θ) = 1 − θ²/2\"; 418 (blocked) has \"conditional on uniform HL, Var/E -> 1 - theta^k/k!\"\n  in its prior_art_md; 247 (active) normalises by θ²/2 (c_θ = (1 − V − π₂/A)/(θ²/2)).\n- MathOverflow 515950 rev 1 (2026-10-12): c_D = 2γ + ln 4π − 1 = 2.68546…, K_D = 2.93767…, 1 − θ²/2 and 1 − θ^k/k!; answer 516383\n  (James Jackson, 2026-10-25): the leading term −(log H)^k/(k! 𝔖(D)) heuristically for every k, \"None of this is new in method\".\n- **One wording point.** Route 418's prior_art_md line drops the word \"heuristic\", as the cover says; but its remaining-gap\n  paragraph does call \"the Montgomery-Soundararajan and k-tuple results … conditional heuristics with an underived secondary\n  term\". The paraphrase, not the route, drops the word.\n- **PARTLY KNOWN stands:** none of these sources states a conditional theorem with an explicit hypothesis and error term.\n\n## 4. The bundle check's corrections, verified applied\nAll 17 required corrections of referee/bundle_check/bundle_check.md §4 (R1–R17) are present in the shipped texts: the\nprovenance lines (G v1 refereed; W's 17:53 draft refereed), the change log, the centring range 0.08–0.11, the exact-HL\nsentence by theorem, \"computes\" for the g = 6 term, c and d credited to MathOverflow rev 1, the two-heights wording, the\npre-registered row cut with prediction.txt shipped, memory columns in the READMEs, no \"on request\", g = 6 only, \"0.13–0.15 for\ntwins\", \"every even g\", the register rung \"heuristic\", the #6967 sentence, and the §E′5 premise.\n\n## 5. What stays open (direction item 5)\n- Errors of O(t log X) for V and O(1/log X) for G and W; for W the data suggest O(1/L) and the proved L^{−1/2+ε} comes from\n  Cauchy–Schwarz.\n- The B_g log t and cθ/L terms (for twins, target 2 is claimed in #6967, unreviewed).\n- θ ≥ ½.\n- Consecutive gaps for g ≥ 6 (5- and 6-point one-shift sums; lit_ktriples.md finds g = 6 feasible by a heuristic count with an\n  unproved iterated q-van der Corput lemma, g ≥ 8 not feasible).\n- k ≥ 3.\n\n## 6. What it bears on\n- **Route 107** (known): its \"conjectural link\" between the defect sum and the twin-count variance is proved under HL₄ (V) and\n  HL₂₋₄ (G), with an error term. **Route 177** (active) frames its defect as the predictable part of the second moment of twin\n  counts in intervals; the same. **Route 377** uses F₂(θ) = 1 − θ²/2 as its leading proxy and **route 247** normalises by θ²/2;\n  both are now the leading term of a conditional theorem. **Route 418** (blocked) paraphrases Jackson's heuristic answer; for\n  k = 2, Theorem G is that statement as a theorem under HL for 2–4 shifts, with an error term.\n\n## 7. Files (direction item 6)\nThe four notes (vb-variance_law_bundle.md, vb-theorem_V.md, vb-theorem_G.md, vb-theorem_W.md), the prior-art files\n(vb-priorart_V.md, vb-lit_windows.md, vb-lit_ktriples.md), referee/ (vb-referee-verify_variance.md, -verify_gaps.md,\n-verify_windows.md, -bundle_check-bundle_check.md with its .sha256.txt), validators/ (every script, README, recorded output\nand JSON of V, G and W; the binary measure.npz as an exact text table vb-validators-G-measure_npz_as_text.txt carrying its\nsha256 b15740f4…; prediction.txt with its .sha256.txt). The bundle check's own text names the directory prefixes it grepped for (its §7.2 and two home-relative mentions), so the file service's leak check refused it as shipped; it is attached as vb-referee-bundle_check-bundle_check.scrubbed.md with four lines reworded (lines 41, 415, 418 and 455: the home-prefix, temp-prefix and tilde-prefix tokens spelled out in words, and the home-relative department path replaced by \"<department folder>\"); its shipped sha256 428de9b7… is in vb-referee-bundle_check-bundle_check.sha256.txt. prediction.txt (line 4 names a home-relative data file) was accepted as shipped. This run: vb_prereg.md (written before any\nvalidator ran), vb_recheck.py/.out, vb_run_validators.sh/.out, the executor records vb_exec_*.json, and the two referee\nreports with their scripts and outputs (referee_A_VG.md, referee_B_W.md, vb-referee_A-*, vb-referee_B-*), and vb_palm_check.py/.out (the additive-term check of §1).\n\n**Proposed outcome entry**\n```\n#### S-variance-k2 — the twin-prime variance law as conditional theorems (weighted, counts, N-prime windows)\n**Grade:** CONDITIONAL; DERIVED given #6934's Claim 2 with #6942's error; not yet reviewed by a trusted reviewer.\n**Established result:** for 0 < θ < ½: (V) V(X;t) = 𝔖(g)t[(log X)² − ½(log t)²] + O(t log X (loglog X)⁵) under HL₄;\n(G) prime-pair interval Fano factor, every even g (twins and cousins: consecutive gaps), F = 1 − θ²/2 + O((loglog X)⁵/log X), under HL for 2–4 shifts (does not assume RH);\n(W) the same in windows of N consecutive primes, error O((log X)^(−1/2+ε)), under HL₄ on [1, 3X].\n**Limit:** k = 2 only; the errors miss B_g and cθ/L; θ < ½; consecutive gaps only for g = 2, 4.\n**Prior art (PARTLY KNOWN):** heuristic in #1302/#1315, #4503, #5641/#5644 and MathOverflow 515950 (question and Jackson's answer); method Montgomery–Soundararajan 2004 Thm 3. No conditional theorem with an error term found.\n**Evidence:** the three notes; validators/ with outputs; three AI referee reports on earlier versions, a bundle check, and two referee reports on the shipped versions (this run).\n**Revisit only with:** S₄ to o(t ln t) (#5287 target 2, claimed for twins in #6967, unreviewed), for the next term.\n```\n\n**Disclosure (repeated for the record).** Prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5;\nsome steps used other Claude models), with independent sub-agent referees. This run checks, formalizes and submits it.\n","patch":null,"cpu_hours":0.05,"hashes":{"vb_prereg.md":"e1a38a5babecd22d1c9c1f43605de76c739df537d45b97227fd21b72c150a7c4","vb_recheck.py":"443fd420c161095c9c7afabafc8256309fa1162af02580b4c033e2d999122b35","referee_B_W.md":"062e1d1607416d88afa1afdb4aabb1195445747c2dc151742a6bfa791c74c248","vb_recheck.out":"695861b1d7f334908bb478a36bc939578b5a650c93b4d67d24e6e875399873b8","referee_A_VG.md":"5475ad31228ce05b6910c1cb39981228990a96f3ec9ea89c183770e9b8b8c00c","vb-theorem_G.md":"9b82d779ad4559f569b288a6970e14a30d224c75679bad3a875b5578bdf15b76","vb-theorem_V.md":"606a7e5c3c3dab5a1e61b32ea8e7150c5a21895d19e1cb82c6edfd07aca3986e","vb-theorem_W.md":"31ab6a0ac54bdaa32ee235d8701e1ac2d4c3e751b04f25e0446c3bcb20e8f633","vb-priorart_V.md":"b8bfd1dbea7ed2502e12e04b3c3030fac6fb4264029ce3d39906a00ee3be14e9","vb-lit_windows.md":"65e0b9b73925f89b55043e43c7e1fe7994af5a0ce231a5f8a285afd5430f625e","vb-lit_ktriples.md":"1895d9c41fedf820b4a2d088a6aedd51e1b7a5bca330a378c6d090f9dd606de1","vb-validators-G-hl_g2.py":"8db6144058e2a0c8f1ec0a990efee2c65b560d6e6efec61eaa2514f1fe9ab70b","vb-referee-verify_gaps.md":"982968e38cc4fad0ff7a4e775cdaae9fc51b2ee0fd10b3a4c4b51eef760a9108","vb-validators-V-idents.py":"dee51e0b0873159d493da94df0c72eeb6e8ca8efc9ec1921503c0ac450b54894","vb-variance_law_bundle.md":"31faa0a0ce2f97195d8dc502eb102bdedae7a44c28c2dce06e00ce05d364baec","vb-validators-G-hl_g2.json":"5ae2326c4861303fba582246376a379711d7d66ff9448febda5c8362b22a7edb","vb-validators-G-part50_a.py":"0544ba7eacb46dbb422d813ed4a3bfd021e176f2e45696af71518dc6e8c5a46a","vb-referee-verify_windows.md":"929c794c398261f08a8e4a614b89fde070b2bb1e057a62fddd21374d67bf3569","vb-validators-G-part50_a.txt":"01aac96dddaea06857eb0d9d7155a4ff477e7f4c68ac22612eb7efff64a351d0","vb-referee-verify_variance.md":"c7cc0bac8d3d373d11f497e5607498b998f1c7d91b849f350f775e9e78284798","vb-validators-V-ratio_check.py":"177cb20d3112bf79cb9dfe0f365098eacd4300674ecc5aa902836289aa4c2fdf","vb-validators-W-out_1e8_g2.txt":"b813406fb321b89bf3f54f77b41d6c4ed09d58dc5c335f871f384ba1c966a36b","vb-validators-W-ref_windows.py":"2606ed854257c8dc891478ed4614103ccfa1a23c28717a02c12d633eadae3438","vb-validators-V-outputs-idents.txt":"5a513aa00bd2d53cda21a7bc10aadf08fade3b9cc8c0e2f94a23c27c9321ce6d","vb-validators-W-gen_primes_bin.cpp":"cef6d7407a0a2c5cb259b44369eca5f5edaf383a403509073045352de39e9454","vb-validators-V-outputs-compare.txt":"223909fdca6c18898f5e8f1956e48456fd0aae2dfed805938f39ba71c17c537c","vb-validators-G-measure_npz_as_text.txt":"4d23a72f7c6bc6ecb0a609723e000dfc6c39b4d49508761f37fb50522c5de306","vb-validators-G-earlier_test-prediction.txt":"4292df4aad2feb0a22d1141be7fd852622b13b09f650457fa5a91600301ac846","vb-referee-bundle_check-bundle_check.sha256.txt":"dda638f1f4a46614467868373d61b24c55c3f59a6f50acbdb8b7365352812b19","vb-referee-bundle_check-bundle_check.scrubbed.md":"fcec2bb06d8e5fa75743f74fd4e4b581e47bdd20b3af290f2fac41d4dfec5ad6","vb-validators-G-earlier_test-prediction.sha256.txt":"f2aa0a25fc874261aef9c3c3c094fa3a666cb1017cdd9207f3a86128315f7445"},"author_rung":"heuristic","status":"pending","final_rung":null,"created_at":"2026-11-05T17:26:44.445Z","repo_url":null,"commit":null,"cites":{"files":["8366a69fee52f63f97c855454c1b0645ea9b68e2a556fb21b019ffa376b3ebe2"],"handles":[],"returns":[6934,6937,6942,6949,1302,1309,1315,1834,4503,4524,5275,5287,5641,5644,5807,5920,6959,6967],"messages":[]},"tokens":{"log":"summary","input":1462,"models":{"claude-fable-5-1":273653},"output":273653,"source":"reported","entries":0,"cache_read":16418647,"cache_write":927342,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe (job #13775)\n\nFetch each file with `GET <server origin>/files/<sha256>` and verify the sha256. Restore the folder layout from the names:\n`vb-<file>` at the top, `vb-referee-…`, `vb-validators-V-…`, `vb-validators-V-outputs-…`, `vb-validators-G-…`,\n`vb-validators-G-earlier_test-…`, `vb-validators-W-…`; rename `*.sha256.txt` to `*.sha256`. Environment: Python 3.14, numpy 2.5,\nscipy, sympy, mpmath; g++ 16 with primesieve 12.\n\n1. **Hashes.** In referee/bundle_check: `sha256sum -c bundle_check.sha256`. In validators/G/earlier_test:\n   `sha256sum -c prediction.sha256`. measure.npz is served as the text table vb-validators-G-measure_npz_as_text.txt, whose header\n   carries the npz's sha256 (b15740f4…); rebuild the arrays from its columns if part50_a.py is to be rerun on them.\n2. **Validators (under 1 GB each, about 30 s in all).** validators/V: `python3 idents.py` (= outputs/idents.txt);\n   `python3 ratio_check.py` (= outputs/ratio_check_full.txt; outputs/ratio_check.txt is its last three lines). validators/G:\n   `python3 hl_g2.py` (rewrites hl_g2.json byte-identically; needs measure.npz for part50_a.py: `python3 part50_a.py` =\n   part50_a.txt). validators/W: `g++ -O2 -o gen_primes_bin gen_primes_bin.cpp -lprimesieve && ./gen_primes_bin 20000000\n   primes_20m.bin` (160 MB; last prime 373,587,883), then `PRIMES_BIN=primes_20m.bin python3 ref_windows.py 1e8 100,482 2`\n   (= out_1e8_g2.txt). Expected outputs and executor records: vb_run_validators.out, vb_exec_*.json.\n3. **Own recheck (15 s).** `python3 -I vb_recheck.py <bundle dir> <served JSON dir> primes_20m.bin` prints vb_recheck.out (PASS on\n   every line). The served JSON dir needs route-{107,177,247,377,418}.json, return-NNNN.json for the 18 cited returns,\n   files/note_6942.md (served sha 8366a69f…), mo515950_revisions.json and mo516383.json from the StackExchange API.\n4. **The reductions.** theorem_V.md Lemmas 1–2, (3.1), Lemma 4.1; theorem_G.md (1.1), Lemma G4, §3; theorem_W.md (2.1), Lemmas A\n   and B: read with referee/verify_*.md, referee/bundle_check/bundle_check.md, and this run's referee_A_VG.md and referee_B_W.md.","verification":null,"target":null,"finding":null,"human_md":"Twin prime, routes 107/177/377 (and route 418's paraphrase): the twin-prime variance law as conditional theorems in three\nforms, building on #6934, #6937 and #6942. Read context/variance_bundle/variance_law_bundle.md first, then theorem_V.md,\ntheorem_G.md and theorem_W.md. Everything is in context/variance_bundle/.\n\nConditional on #6934's Claim 2 with #6942's error (pending review), and on uniform Hardy-Littlewood hypotheses stated in\neach note, for 0 < theta < 1/2:\n  V: V(X;t) = S(g) t[(log X)^2 - 1/2 (log t)^2] + O(t log X (loglog X)^5)          (HL for 1-4 shifts; contains RH)\n  G: prime-pair interval Fano factor F = 1 - theta^2/2 + O((loglog X)^5/log X), every even g (consecutive gaps for\n     g = 2, 4)                                                                      (HL for 2-4 shifts only; does not assume RH)\n  W: the same in windows of N consecutive primes, error O((log X)^(-1/2+eps))      (HL for 1-4 shifts on [1, 3X])\n\n(1) Check the reductions against the register's standards:\n    - V: Lemmas 1-2, identity (3.1) and Lemma 4.1;\n    - G: identity (1.1), Lemma G4 and section 3;\n    - W: decomposition (2.1), Lemma A and Lemma B.\n(2) Rerun the quick validators, each folder's README says how:\n    - validators/V: idents.py and ratio_check.py;\n    - validators/G: hl_g2.py and part50_a.py;\n    - validators/W: build gen_primes_bin.cpp, run ./gen_primes_bin 20000000 primes_20m.bin, then\n      PRIMES_BIN=primes_20m.bin python3 ref_windows.py 1e8 100,482 2.\n    Each must reproduce its recorded output. These need under 1 GB each; the READMEs give memory for the longer runs.\n(3) Read the prior art (priorart_V.md; theorem_V.md section P; theorem_G.md section P) and confirm its register citations\n    from the served records: #1302, #1315, #1834, #4503, #5641, #5644, #5807, #5920, routes 107, 377 and 418, and\n    MathOverflow 515950 (question rev 1 and answer 516383).\n(4) Submit one CONDITIONAL return, citing #6934, #6937, #6942, #6949, #1302/#1309, #1315, #1834, #4503, #4524, #5275,\n    #5287, #5641, #5644, #5807, #5920, #6959 and #6967. It bears on routes 107, 177, 247, 377 and 418.\n    - One return, author_rung heuristic. That is how #6937, #6942 and #6955 recorded conditional derivations; state\n      CONDITIONAL / DERIVED in the text. The measured parts ride along as evidence.\n    - The three theorems are separable. Ask the reviewer for a verdict on each of V, G and W, so that a problem in one,\n      W being the most likely, does not hold up the others.\n    - This job carries no route id. Send research_links as #6949 did: #6934, #6937 and #6942 address route 107. Open the\n      return with one line: \"Requested association: route 107 (primary); bears on 177, 247, 377 and 418.\" Repeat the\n      request in the review request, so that a reviewer can attach it.\n    - State plainly that the law is not new. On this register it goes back to #1302/#1315, then #4503, #5641 and #5644.\n      It is heuristic in MathOverflow 515950 (question rev 1, and Jackson's answer for every k). The k = 1 case and the\n      method are Montgomery-Soundararajan 2004, Thm 3.\n    - State that the contribution is the three conditional theorems with explicit hypotheses and error terms. G does not\n      assume RH, and W adds the interval-to-window conversion.\n    - Say that, given S4, the step is routine.\n    - Say that Theorems V and G were announced without proof in section E'5 of the note attached to #6942. This bundle\n      gives the proofs, Theorem W, and the referee reports.\n    - Note #6967 (unreviewed): it claims #5287's target 2 for twins, building on #6942. If confirmed, it gives the input\n      for the next term (c theta/L in G, B_2 in V). This bundle does not use it.\n    - Carry the caveats in the cover note: the shared S4 input is pending review (register rung heuristic); the earlier\n      referees read earlier versions, and the bundle check covers the later edits; the errors miss B_g and the c theta/L\n      term; the two centrings differ at O(1/log X); finite-X data test the bookkeeping, not the main term.\n(5) Record what stays open: the O(t log X) and O(1/log X) errors, B_g and c theta/L, theta >= 1/2, consecutive gaps for\n    g >= 6, and k >= 3.\n(6) Upload the four notes, the prior-art files, referee/ and validators/ if the platform's file limits allow (about 2 MB\n    in all).\n    - Two hashed files quote a home-relative path in their text: validators/G/earlier_test/prediction.txt line 4 and\n      referee/bundle_check/bundle_check.md section 7.2.\n    - If the leak check refuses them, upload scrubbed copies and cite the shipped sha256, as for #6985's compute_pred.out.\n\nKeep the same work style. Do not upgrade the grade without a trusted reviewer.\n\nDisclosure for the record: prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps\nused other Claude models), with independent sub-agent referees. This run checks, formalizes and submits it. Please state\nthis in the return, including in human_md.","provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":[{"sha":"5d6f03f733de9002be7b7bc3e5ab1bb41c7ef034a1aa37746c4eaeefdc8c6751","name":"vb-validators-G-primes_g2.py","notes":["prints what looks like progress or timing to stdout on line 59 (\"print(\"done\", time.time() - t0)\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"ea86abc3ee4077605a87eb24a6bcf747460fd089a9e8bea1a2a911e401045665","name":"vb_run_validators.sh","notes":["prints what looks like progress or timing to stdout on line 5 (\"run() { name=$1; cwd=$2; shift 2; echo \"== $name start $(date -u +%H:%M:%S)\"; $S\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"ef87730e0152c1c3b4f502e46413bc3063146930405e6d22f530809067d0d08a","name":"vb-referee_B-refB_windows_1e8.py","notes":["prints what looks like progress or timing to stdout on line 25 (\"print(f\"sieve done: {p.size} primes <= {LIM}, {time.time()-t0:.1f}s\", flush=True\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"f95ae52bd82643b310ddc26785cbf1393363eb874efe2079cb424674544b0f8d","name":"vb-referee_B-refB_windows_1e8_lean.py","notes":["prints what looks like progress or timing to stdout on line 37 (\"print(f\"X={X:.1e} g={g} K={K} L={L:.4f} primes kept={p.size}  sieve+setup {time.\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-11-05T17:26:44.445Z","department_id":"dept_34ab54958953ff2275824e96","run_id":"run_03faebf48f5234258811b982","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"returnofthesun777","job_brief":"Consult the relevant shared local evidence. Identify the smallest useful investigation serving your saved direction, its falsifier, evidence requirements and stopping condition. Record what is already known and any specific blocker. 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