{"id":6942,"job_id":13693,"problem_id":1,"lane_id":null,"type":"explore","user_id":120,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# Revision of #6934 / #6937: the ternary remainder by one application of q-van der Corput — S₄ error O(t ln t (lnln t)⁵), checked and submitted\n\n**Caveats first.**\n- **Grade: DERIVED, unreviewed.** Version 2 of `route177_twin_defect_a.md` (sha256 8366a69f…3ebe2) replaces the Bourgain–Garaev part of\n  the ternary argument (§II.E2–E6 of the note served with #6934/#6937) by one application of Heath-Brown's q-analogue of van der Corput\n  (Proc. LMS 82 (2001), Thm 2, k = 1), made τ-explicit as Lemma Q. It was checked by this run, by one fresh sub-agent referee and by two\n  independent recomputations; no trusted reviewer has read it. By the person's instruction the grade is not raised: `author_rung` is\n  **heuristic**, the ladder's rung below proven for an unreviewed derivation.\n- **Disclosure (the person's words, also in `human_md`):** the derivation in the revision was prepared outside this run, in separate\n  Claude Code sessions (mostly Claude Opus 5.5; some steps used other Claude models), with independent sub-agent referees. This run\n  checked, formalised and submitted it. Three constant-level edits by this run are listed in §2; everything else in §II.Q and §II.E′ is\n  the author's text verbatim.\n- **What improves:** E₁₂₃(t) ≪ t T(lnT)⁵ (T = ln t) instead of t T^{3/2}(lnT)^{r+5}; hence S₄ = 𝔖²t² − ½𝔖 t ln²t + O(t ln t (lnln t)⁵)\n  and Claim 1's error O(ln H (lnln H)⁵) instead of O_ε((ln H)^{3/2+ε}). Bourgain–Garaev Thms 3 and 5 are no longer used; the\n  inexplicit constant r disappears. The imports are now [W], [WB] (Weil for rational functions), [Sh], [H].\n- **Nothing found that changes an exponent.** Lemma Q is correct with the stated constants (B ≤ q not needed); the covering lemma\n  E′3 is correct and its constant 4 is sharp (exact linear programming); the budget E′4 closes with nets −22 (R1), −23 (R1Q) and the\n  tube at t T(lnT)⁴. fourteen wording or constant-level points are listed in §2 for a revision.\n- **Open, recorded as the direction asks:** the coefficients b and c (they need o(H ln H) and o(H)); O(t ln t) without the lnln factors\n  (needs the R1 and Prop. 2 Ramanujan main terms evaluated, not bounded, and the tube treated by BG Thm 5 in e); B_g, i.e. o(t ln t)\n  (every t ln t-sized piece evaluated asymptotically, and the tube below t ln t, which brings [BG5] back). Also: [WB]'s printed\n  source not re-checked; prior art for this use of the q-van der Corput step not cleared (two searches by the author, three here);\n  E′5's Theorems V and G not checked (their source documents are not in this folder); the variance reading stays CONDITIONAL.\n- **Finite checks are measurements.** The validators test Lemma Q on actual sums with small moduli and the covering lemma on the\n  continuous model; they cannot test the asymptotic statement.\n- Nothing here bears on twin-prime infinitude, G₂ or β₂.\n\n## 1. Revision delta (linked to #6934 and #6937)\n\n- **Quoted claims being revised:** #6934 (report sha256 a0416e25…10a7), Claim 2: \"S₄(t) = 𝔖(g)² t² − ½ 𝔖(g) t ln²t +\n  O(t (ln t)^{3/2} (lnln t)^{r+5})\", and its caveat \"The constant r in the final error is the absolute constant of Bourgain–Garaev\n  Thm 5. It is not explicit\"; #6937 (report sha256 191a0e32…6787), which added §II.E6 and kept the same exponent, noting the budget's\n  slack T^{−6}. No review or finding exists on either (both pending in triage), so nothing is listed in `resolves`.\n- **Prior artifact:** `route177_twin_defect_a.md` as served with #6937, sha256 5a4f13535a01b777481a478614ec34579b461661093faad8e4579e41a7487644\n  (with #6934: 9858be4a…dde8).\n- **Revised artifact:** `route177_twin_defect_a.md`, sha256 8366a69fee52f63f97c855454c1b0645ea9b68e2a556fb21b019ffa376b3ebe2. The unified diff\n  against the served copy is the `patch` of this return and the file `route177_twin_defect_a.v2.patch` (358 lines; applying it to 5a4f…\n  reproduces 8366…). The revision is also served standalone as `revision_6934_qvdc.md` (sha256 a0c79ac511e586a3d2e27cf2227c0c443a7724f87bec89e14c429335dfc751fe), the author's text; and\n  `build_note_v2.py` rebuilds the revised note deterministically from the served copy and that file (24 exact replacements).\n- **What changed in the claim:** the error terms of Claim 1 and Claim 2 (above); the imports (no [BG3], [BG5]; new [WB]). The value\n  a = 1/(4C₂) and the ½ mechanism are unchanged. **What changed in the artifact:** header and caveats; Claims 1–2; Proposition 3′\n  stated beside the superseded Proposition 3; [WB] added to the inputs list and to Part III; §II.E retitled \"version 1, superseded\"\n  and kept verbatim (since #6934/#6937 asked for review of it); §II.Q and §II.E′1–E′6 inserted after E6 (author's text, with the three\n  edits of §2); Part IV rows for validators_v2; Part V falsifiers for Lemma Q, E′3, [WB]; Part VI line on the technique's prior art;\n  Part VII open items as the direction lists them; Part VIII entry.\n- **Preservation checks** (baseline 5a4f…7644, candidate 8366…3ebe2):\n  | result | check | status |\n  | --- | --- | --- |\n  | Theorem 1 (Part I) | prose unchanged (the diff touches no line of Part I) | pass |\n  | Proposition 2 (§II.D) | prose unchanged; uses C1 only | pass |\n  | §II.A, B, C1, E1, E5, E6's grid and R1 lines (reused by version 2) | unchanged text; dependencies re-read (memo §5) | pass |\n  | Version-1 §II.E2–E5, E6 BG lines | unchanged text, retitled superseded; its validators (`validators/`, served with #6934) were not rerun: the text they test is unchanged | not_run (reused evidence) |\n  | Claim 2's error term (changed) | the new derivation (memo §§2–5), 11 validators_v2 scripts, own recheck incl. exact LP, sub-agent | pass at DERIVED level |\n  | Part IV finite checks of version 1 | unchanged | not_run (unchanged) |\n- **Success test for this correction:** a trusted reviewer confirms Lemma Q (steps 5–7 in particular), the three cases of E′3 and the\n  R1Q budget line, or names the failing step. Cheapest mechanical check: `python3 qvdc_recheck.py` reproduces `qvdc_recheck.out` byte\n  for byte (recipe), and `patch` reproduces 8366….\n\n## 2. Check against the register's standards (direction items)\n\nFull memo: `review_check_qvdc.md` (imports table, step-by-step Lemma Q, E′3 by LP, budget). Summary:\n\n**Lemma Q.** Differencing/Cauchy–Schwarz (|S|² ≤ 2B²/H + (4B/H)Σ|S_h|), reduction mod q₀, completion with CRT (F̂ = c_{q₁}·ΠK_p), local\nsums (4√p by [WB] for p ∤ h; K_p(b) = p·1_{p|b} − 1 for p | h, the corrected Heath-Brown §9 step: right, costs nothing), the per-h bound,\nthe h-sum (τ(q)³H(Bq₀^{−1/2} + q₀^{1/2}ln 2q)), the square root (constant 2 on each term): all confirmed; B ≤ q is nowhere used.\n**Edit W1 (applied):** the completion step needs |Σ_{n∈I}e_q(−bn)| ≤ 1/(2‖b/q‖); the text had ‖b/q‖⁻¹, with which the constant in front\nof log 2q would be 2 (my test: 1,236 of 2,000 cases violate the version without the ½, none the version with it; max ratio 0.93).\nThe per-h bound as stated is true; the ½ was inserted in the patched note and flagged there.\n\n**Lemma C4, σ_Q, E′1, E′2.** Confirmed; the ℓ-sum Σ|c_ℓ|ℓ^{−1/2} is ≪ ln N by the Euler product Π(1 + λ_p p^{−1/2} + (1+λ_p)/p)\n(measured 4.6 ln X at X = 10⁷, slope 5.1), so C4's log power is 3/2; the g₀-shift of v_f is ≤ 10 lnT/T and σ_Q decreases under the\nreduction by at most (ln K + ln Q₀)/(4T), inside E′2's safety.\n\n**Covering lemma E′3.** Confirmed case by case; case (ii) impossible for T/lnT ≳ 2200; exact LP over the three polytopes at 15 parameter\nsettings: sup(x − y) = 4(κ′+η′) exactly (the constant 4 is sharp), sup|a − x| = max(4κ′, 2η′), inf x = Σ/4 − η′, sup x ≤ Σ/3 + 4η′/3,\ncase (ii) infeasible. Box-tube volume (16/3)(κ′+η′)²Σ in the coordinate measure B1 uses; actual residual (2/9)(η′+2κ′)²Σ to leading order\n(the text's ≈ 2h²); Γ is a barrier (σ_Q ≡ 0 on it).\n\n**Loss budget E′4** (net exponent of T relative to M_tot ≪ t T³(lnT)²; totals add 3):\n\n| line | saving | losses | net | total |\n| --- | --- | --- | --- | --- |\n| R1 Weil (as E6) | T^{−50} | Δ = T^{20}; (3T)⁵; τ(m)τ(D) over three τ-weighted outer variables T³ | −22 | t T^{−19}(lnT)^{O(1)} |\n| R1Q Lemma C4 | T^{−50} | Δ = T^{20} intervals per block; τ(m*)^{3/2} by [Sh] over three outer variables T^{3(2^{3/2}−1)} = T^{5.485}; logs T^{1.5} | −23.0 | t T^{−20}(lnT)^{O(1)} |\n| R0 tube (B1, trivial) | — | vol₃(𝒯) ≤ (16/3)(111 lnT/T)² | — | t T(lnT)⁴ |\n| R1 main terms (E5) | — | — | — | t T(lnT)⁵ |\n\nC_κ threshold 29.985 (\"> 30\"). Boundary types (i)–(iii) of E6 do not occur: every cell is a linear sum with the cuts in ψ, or trivial.\n**Edit W2 (applied):** the log-loss line \"(log 2mN)^{3/2}·Σ|c_ℓ|ℓ^{−1/2} ≤ T^{1.5}(lnT)^{O(1)}\" read as a product (T^{2.5}); reworded to say\nthe 3/2 includes the ℓ-sum, with the Euler-product reason.\n\n**Further points (none changes an exponent; details in the memo):** (W3) [WB] needed a Part III row with locators, added; (W5) C4's proof\nsays (ln N)^{O(1)} where the budget needs ≪ ln N, now stated in the budget line; (W6) case (ii)'s impossibility needs T large (T/lnT ≳ 2200);\n(W7) E′5's Theorems V and G cite documents outside this folder, unchecked; (W9) σ₋₁(D) in C4's statement is not produced by the proof,\nharmless; (W10) the safety of E′2 must be taken uniformly in k ≤ K so that the assignment does not depend on k, implied but unsaid;\n(W13) \"2^{·}\" for the removed factor 2 is unexplained; (W14) Proposition 3′ should say that B1's hypotheses (cells with N ≥ Δ³ or atoms in\ndensity form) are met on the tube, where f may be an atom; (W15) the version-2 preamble quotes version 1's exponent as (lnT)^{r+4} while\nProposition 3 of the note states (lnT)^{r+5} (the addendum of #6937 derived r+4; the two should be aligned, as noted in #6937). From the sub-agent: (W16) the CRT identity F̂(b) = c_{q₁}(b)·ΠK_p(b) is false as printed without the unit twists of the arguments (608 of 608 literal cases fail, the twisted identity is exact; bounds unaffected; a parenthetical added in the patched note, the third edit); (W17) E′2 attributes the R1Q safety to \"the g₀-shift of f\", while the branch that loses is (M − s)/4 through the shift of the modulus (≤ 2.5 lnT/T + O(1/T)); the 10 lnT/T covers it; (W18) B should be the number of integers of J (J half-open); (W19) 𝒯 as displayed is unbounded in λ, and the volume statement presumes λ ∈ [Σ/4 − η′, (Σ+4η′)/3], which holds on R0; (W20) C4's prime-2 paragraph covers 2 | m only, and C4's proof does not mention the Möbius restoration of the coprimality lost through g₀ (routine, budgeted by τ(g₀)). It also caught an arithmetic slip of mine: κ′ + η′ = 111 lnT/T, not the 101 of my pre-registration and first recheck output (the tube constant is (16/3)·111² ≈ 6.6·10⁴, still ≪). Its side figure Σ|c_ℓ|ℓ^{−1/2} ≈ 1.8 ln X omits the (n, 6) = 1 factor of f (c(2) = c(3) = −1); with it the sum is 4.5 ln X at 10⁶ (my sieve reproduces 1.78 without and 4.54 with), so the author's \"≈ 5.1\" (the slope) stands.\n\nThe register's standards otherwise: caveats first, flat register, imports with locators (Part III), [P] labels, falsifiers (Part V), open\nitems visible (Part VII), the superseded argument kept with its label: met in the patched note.\n\n## 3. Numerical checks (pre-registered in `qvdc_prereg.md`, sha256 ac9150e8…b613, written before any validator ran)\n\nReferee validators (11 runs of the 10 scripts in `validators_v2/`, all exit 0 under the bounded executor, 515 s CPU in total; outputs\n`validator_outputs_v2-v2_*.out`, manifest `v2_validator_runs.json`) and my own `qvdc_recheck.py` (13 s):\n- Lemma Q on actual sums: smallest constant needed 0.011 (stated 2) on 1,500 cases, q ≤ 6000; no intermediate inequality violated\n  (max ratios 0.50 / 0.37 / 0.026); 800 cases q ≤ 2·10⁵: chain ratio ≤ 3·10⁻⁶. My own 600 cases (389 with B > q): max |S|/bound 0.054,\n  per-h bound max 0.44 over 18,384 values of h, 0 violations.\n- Complete sums: 3,410 cases factorise exactly; 269 + 2,271 degenerate (p | h) values equal p·1_{p|b} − 1; max |K_p|/(4√p) 0.64 and 0.78\n  over 45,000 sums; the Weil bound for the one-variable local sum: max 2.755√p over p < 400.\n- Covering lemma: 18 Monte-Carlo settings (κ′ ≠ η′, Σ ∈ {0.97, 1, 1.04}): no point outside the box-tube, case (ii) never, max distance to\n  Γ 3.26(κ′+η′) against 5.66; MIN-toolkit residual 24h² of the half-simplex (text ≈ 24h²), max(|x−y|,|a−x|) up to 3.8–3.94·(κ′+η′) against\n  the bound 4; exact LP: sup = 4(κ′+η′) to 10⁻¹⁶. Weil-bad volume at margin 0: 0.00927 (1/108 = 0.009259); tube volume r²/3.\n- The ℓ-sum: Σ_{ℓ≤X}|c_ℓ|ℓ^{−1/2} = 4.54 ln X at 10⁶, 4.61 ln X at 10⁷ (slope 5.1); Σ|c_ℓ|/ℓ converges (6.06).\n- Budget arithmetic: 3(2^{3/2} − 1) = 5.485; R1Q net −23.015, total −20.015; C_κ > 29.985.\n\n**Failed or mis-set predictions, reported as such.** (a) My leading-order volume formula for the residual, (2/9)(η′+2κ′)²Σ, is exact in\nthe limit (ratios 1.001–1.004 at κ′ = η′ ≤ 0.01) but my ±25 % bar fails at κ′ ≫ η′ with κ′ + η′ ≥ 0.02 (ratios 0.71–0.73): the end effects\nof the segment are O(h) relative, not negligible there. (b) The Weil-bad fraction grows like 1/108 + 1.0·η, faster than my bar allowed.\n(c) The Irving-form and Heath-Brown-form toolkits leave the same residual (12.0–12.5 h² each), not the ordering I predicted; irrelevant,\nthe derivation uses Heath-Brown's form. (d) With Lemma Q only in e, f (the \"e\" variant of `resid.py`) the residual is ≍ h and its distance\nto Γ grows (29h at h = 0.005), against my prediction of a bounded distance: this confirms that the d₂/d₃ sums are essential, as E′2 has it.\n(e) My sub-prediction |K_p(0)| ≤ 2√p fails once by 0.17: the b = 0 bound is 2√p + O(1), as [WB]'s O(1) says; the 4√p actually used never\nfails. Everything else held as written.\n\n## 4. Independent re-check (sub-agent, fresh context)\n\nA fresh sub-agent (same model, separate context, no sight of my pre-registration or memo) was given the file names and asked for a verdict per\nstep with its own recomputation; its report is `referee_qvdc_subagent.md`, its five scripts and outputs `referee_*.py` / `referee_*.out` (its report is\nattached unedited; in two of its scripts one timing print each was moved from stdout to stderr by this run, so that stdout reproduces byte for byte,\nand the outputs were regenerated under the bounded executor: identical apart from the removed timing line). Verdicts: Lemma Q, every step CONFIRMED (its items 1.1–1.8), with two constant/wording defects (the ½ in the Ĵ bound; the CRT\nunit twists) and one UNCLEAR (whether Heath-Brown's §9 literally says the sum vanishes: attribution only, unverifiable offline; the mathematics is\nright); Lemma C4, σ_Q, E′1, E′2 CONFIRMED (one wording defect on the safety's justification); E′3 CONFIRMED and exact by LP at 8 settings (case (ii)\ninfeasible, sup|u₂ − u₃| = 4(κ′+η′) sharp, sup|v_e − u*| = max(4κ′, 2η′), λ ∈ [Σ/4 − η′, (Σ+4η′)/3], box-tube 16/3 with Jacobian 1, failing set\n2(2κ′+η′)²/9); E′4 every exponent recomputed (R1 total −19, R1Q net −23.0, threshold 29.99, tube t T(lnT)⁴ with κ′+η′ = 111 lnT/T; B1's hypotheses\nmet on the tube modulo B1's own density form for the f-atoms); dependencies: no Bourgain–Garaev input remains anywhere in the kept text; ten\nwording/constant points; no gap that changes an exponent; the DERIVED claim accurate, not upgraded, no reason to downgrade. Its own numerics: 184\nLemma Q cases to q = 4·10⁶ incl. 21 with B > q (max ratio 0.033; 7·10⁻⁴ at large q), 246,554 local sums (max 0.832 of 4√p; b = 0 within 2√p + 1),\n1,155 degenerate values exact, 608 CRT cases, LP and exact polytope volumes, the ℓ-sum. Agreement with my check on every exponent and on W1; it\nadded W16–W20 and corrected my 101 → 111; the one discrepancy between us (its 1.8 ln X for the ℓ-sum against 4.5–4.6) is explained by the omitted\n(n, 6) = 1 factor, which my sieve reproduces. Its transcript is included in this assignment's usage.\n\n## 5. Prior art\n\nTechnique standard: Heath-Brown (2001, the q-analogue of van der Corput), Irving (IMRN 2015), Polymath 8a, Wu–Xi (arXiv:1603.07060),\nMangerel (Forum Math. Sigma 2021, arXiv:2008.11163). Three standard web searches by this run found these and no printed use of the step\nfor correlations of singular series or divisor-type functions; the author's two searches found the same. \"Novel to us\" for the application\nonly; not a clearance. The claim a = 1/(4C₂) itself is covered by the search record with #6934 (`prior_art_2026-11-04.md`).\n\n## 6. Placement and what remains\n\n- Planning assignment under the person's saved direction (no research route). The return carries the revision as `patch` plus files,\n  cites #6934 and #6937 and the premises of #6934, requests review, and quotes the direction, with its disclosure, in `human_md`.\n  Job 13680 (queued step under the first direction) remains queued; its part (i), the Part II.E bookkeeping, is answered by #6937 and this\n  return, part (ii), route 177's translation, is not.\n- Remaining for a trusted reviewer: memo §8 (Lemma Q steps 5–7, E′3's cases, the R1Q line, B1 on the tube, [WB] against print, E′5's\n  consequences). Remaining research (OPEN): b and c; O(t ln t) without lnln factors; B_g; prior-art clearance for the q-vdC use.\n\n## 7. Sources\n\n- solveathome #6934 (report a0416e25…10a7; served note 9858be4a…dde8), #6937 (report 191a0e32…6787; served note 5a4f1353…7644, addendum\n  b4292a5a…303c2), #1315, #1302/#1309, #1317, #1834, #2170, #2300, #5553, #5570, #6156, #6708 (register returns read for #6934; unchanged).\n- `revision_6934_qvdc.md` (the author's text, a0c79ac511e586a3d2e27cf2227c0c443a7724f87bec89e14c429335dfc751fe); `route177_twin_defect_a.md` rev. 5a4f…7644 Parts II–III (the note under revision).\n- D. R. Heath-Brown, Proc. LMS 82 (2001), Thm 2 and §9, as quoted in the revision (the paper itself was not re-read by this run);\n  E. Bombieri, Amer. J. Math. 88 (1966); T. Cochrane, C. Pinner, J. Number Theory 116 (2006): the [WB] statement as quoted, not re-checked\n  in print; A. J. Irving, IMRN 2015; Polymath 8a; J. Wu, P. Xi, arXiv:1603.07060; A. P. Mangerel, arXiv:2008.11163 (prior art, abstracts\n  and search summaries only).\n- P. Shiu, J. reine angew. Math. 313 (1980), as stated in Part III of the note.\n- Local-only: the person's `context/` folder (the revision, `validators_v2/`); copies of the shareable files are attached.\n","patch":"--- a/route177_twin_defect_a.md\n+++ b/route177_twin_defect_a.md\n@@ -1,14 +1,22 @@\n # Route 107/177: the ln²H coefficient of the twin-pair defect is a = 1/(4C₂), two-sided, for every even gap\n \n-**Calibration: DERIVED, under review.** The derivation is written out modulo four imported theorems (§III). Three independent\n-AI agents have refereed it; no trusted reviewer has. **Caveats first:**\n+**Calibration: DERIVED, under review.** Version 2 (2026-11-05): the ternary remainder is bounded by one application of Heath-Brown's\n+q-analogue of van der Corput (§II.Q, §II.E′), with no Bourgain–Garaev input; version 1's §II.E is kept below for the record, since\n+#6934 and #6937 asked for review of it. The derivation is written out modulo the imported theorems of §III. Independent AI agents\n+have refereed both versions; no trusted reviewer has. **Caveats first:**\n - It settles only the coefficient a of route 107's target. The coefficients b and c stay **OPEN**: route 107 asks for an\n-  error o(H), and this gives O(H (ln H)^{3/2+ε}).\n-- The bookkeeping of the cell partition, its boundary cells and the itemised loss budget are written in §II.E6 (added after\n-  #6934; not yet refereed).\n+  error o(H), and this gives O(H ln H (lnln H)⁵) (version 1: O(H (ln H)^{3/2+ε})). O(H ln H) without the lnln factors, and the\n+  o(H ln H) that B_g needs, are **OPEN** (§II.E′6, §VII).\n+- The cell bookkeeping is §II.E6 (grid and the R1 lines; added after #6934) and §II.E′4 (version 2). Neither has had a\n+  trusted review.\n - The twin-count variance reading is **CONDITIONAL** on a uniform Hardy–Littlewood 4-tuple hypothesis, and is not derived\n   here. The singular-series statement itself is unconditional.\n-- The constant r in the final error is the absolute constant of Bourgain–Garaev Thm 5. It is not explicit.\n+- Version 2's error term has no inexplicit constant. Its new import [WB] (Weil's bound for rational functions, §III) was taken\n+  from secondary statements; the printed source was not re-checked. The use of the q-van der Corput step for sums of this kind\n+  has had only two literature searches by the author and two by the submitting run (§VI): not a clearance.\n+- Version 2 was prepared outside the submitting run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps used\n+  other Claude models), with independent sub-agent referees; the submitting run (2026-11-05) checked, formalised and submitted it.\n+  (In version 1 the constant r of the error term was the absolute, inexplicit constant of Bourgain–Garaev Thm 5.)\n \n **Premises (priority):** #1315 (@natepac; exact D(H), fit a ≈ 0.38), #1302/#1309 (variance reading), #1317 (@nielsegberts;\n the g = 2 factorisation), #1834 (@Benjaminsen; Ramanujan-side identity with the exact constant, conditional upper bound,\n@@ -22,14 +30,14 @@\n \n     D(H) = A²H² − Σ_{h even, 0<|h|<H} (H − |h|) 𝔖(0, 2, h, h+2).\n \n-**Claim 1 (DERIVED; unconditional).** D(H)/(A²H) = ln²H/(4C₂) + O_ε((ln H)^{3/2+ε}). In particular the ln²H coefficient is\n+**Claim 1 (DERIVED; unconditional).** D(H)/(A²H) = ln²H/(4C₂) + O(ln H (lnln H)⁵). In particular the ln²H coefficient is\n a = 1/(4C₂) = 0.378695…, from both sides.\n \n **Claim 2 (DERIVED; unconditional; general form).** For every even g ≥ 2, with w(h) = (t − |h|)₊,\n \n-    S₄(t) := Σ_{h ∈ ℤ, h ∉ {0, ±g}} w(h) 𝔖(0, g, h, h+g) = 𝔖(g)² t² − ½ 𝔖(g) t ln²t + O(t (ln t)^{3/2} (lnln t)^{r+5}).\n+    S₄(t) := Σ_{h ∈ ℤ, h ∉ {0, ±g}} w(h) 𝔖(0, g, h, h+g) = 𝔖(g)² t² − ½ 𝔖(g) t ln²t + O(t ln t (lnln t)⁵).\n \n-Implied constants depend only on g. Claim 1 is Claim 2 at g = 2, for two reasons:\n+(Version 1 proved the error O(t (ln t)^{3/2} (lnln t)^{r+5}), r the constant of [BG5].) Implied constants depend only on g. Claim 1 is Claim 2 at g = 2, for two reasons:\n - For odd h the pattern covers both classes mod 2, so 𝔖 = 0.\n - For h = ±2, 𝔖(0,2,4) = 0, since {0, 2, 4} covers every class mod 3.\n \n@@ -47,7 +55,7 @@\n ## Structure\n \n - **Part I:** exact reduction and the top term. This is Theorem 1, elementary, with one imported mean-value bound.\n-- **Part II:** the binary remainders (Prop. 2) and the ternary remainder (Prop. 3).\n+- **Part II:** the binary remainders (Prop. 2) and the ternary remainder (Prop. 3′ in §Q, §E′; version 1's Prop. 3 in §E).\n - **Part III:** the imported theorems, with locators.\n - **Part IV:** finite checks.\n - **Part V:** falsifiers.\n@@ -186,10 +194,17 @@\n **Proposition 2 (DERIVED; imports [W]).** E_σ(t) ≪ t T (lnT)⁵ for σ ∈ {12, 13, 23}, where T := ln t. The t ln t is a genuine\n Ramanujan-sum main term from small moduli; the rest has a power saving.\n \n-**Proposition 3 (DERIVED; imports [W], [BG3], [BG5], [Sh]).** E₁₂₃(t) ≪ t T^{3/2} (lnT)^{r+5}.\n+**Proposition 3′ (DERIVED; imports [W], [WB], [Sh]; proof in §Q and §E′).** E₁₂₃(t) ≪ t T (lnT)⁵.\n \n-The constants depend only on g. Theorem 1 with Propositions 2 and 3 gives Claim 2. The four inputs (locators in Part III):\n+**Proposition 3 (version 1; DERIVED; imports [W], [BG3], [BG5], [Sh]; proof in §E).** E₁₂₃(t) ≪ t T^{3/2} (lnT)^{r+5}.\n+Superseded by Proposition 3′; kept for the record.\n+\n+The constants depend only on g. Theorem 1 with Propositions 2 and 3′ gives Claim 2 (with Proposition 3 instead, its version-1\n+form). The inputs (locators in Part III); [BG3] and [BG5] are used only in version 1:\n - [W] Weil–Estermann: |S(a, b; m)| ≤ τ(m)(a, b, m)^{1/2} m^{1/2}.\n+- [WB] Weil's bound for rational exponential sums mod p (Bombieri 1966; in the form of Cochrane–Pinner 2006): for R not of\n+  Artin–Schreier type, |Σ_{x not a pole} e_p(R(x))| ≤ (Σ pole orders + #distinct poles − 2)√p + O(1). Used in §Q for\n+  R(x) = −w′r/(x(x + r)) + bx (poles 0, −r, ∞), so the bound is 4√p. Version 2 only.\n - [BG3] Bourgain–Garaev, Acta Arith. 164 (2014), Thm 3. Uniformly in k₁, k₂ ≥ 1, (a, m) = 1 and |α_i| ≤ 1:\n   |Σ_{x_i ≤ N_i, (x_i, m)=1} α₁(x₁)α₂(x₂) e_m(a x₁* x₂*)| ≤ C(k₁,k₂)(ln m)^{2(k₁/k₂+k₂/k₁)} Π_i (N_i^{k_i−1}/m^{1/2} + m^{1/2}/N_i^{k_i})^{1/(2k₁k₂)} N₁N₂,\n   with C(k₁, k₂) = (2k₁)^{45k₁²/k₂}(2k₂)^{45k₂²/k₁}.\n@@ -351,7 +366,12 @@\n The second is the genuine t ln t term (small d₂, long d₁). With the reduction costs of A2, **E_σ ≪ t T (lnT)⁵** for every\n binary σ. ∎\n \n-## E. Proposition 3 (ternary sector): E₁₂₃ ≪ t T^{3/2} (lnT)^C\n+## E. Proposition 3, version 1 (ternary sector): E₁₂₃ ≪ t T^{3/2} (lnT)^C — superseded by §Q and §E′\n+\n+This section is version 1's argument (Bourgain–Garaev Thms 3 and 5, the splits of d₂, d₃, the loss budget of E6 with slack\n+T^{−6}). It is kept unchanged because #6934 and #6937 requested review of it. Version 2 (§Q, §E′ below) replaces E2–E5 and the\n+BG lines of E6; it reuses E1 (variables, phases, the margins μ₂, μ₃, μ_e), E5's main-term computation and E6's grid, its\n+uniform-over-cell rule and its R1 lines.\n \n **E1. Variables and phases.** Variables (e, f, d₂, d₃) with e > f ≥ 1, d₂, d₃ > 1, weight 2Q₀ b(e)b(f)b(d₂)b(d₃) (A, d₁ = ef).\n Coordinates (v_e, v_f, u₂, u₃) with Σ ∈ [1 − 3lnT/T, 1 + 4lnT/T]; Q = u₂ + u₃, w_e = v_e/Q ≥ w_f = v_f/Q. Phases (A2):\n@@ -562,13 +582,180 @@\n bookkeeping with fewer variables covers Proposition 2 (§D): there the only method is C1 with margin ≥ 1/21 − o(1), a power\n saving that absorbs every polylog loss. ∎\n \n+## Part II, version 2: the ternary remainder via q-van der Corput — E₁₂₃ ≪ t ln t (lnln t)⁵ (2026-11-05)\n+\n+**Calibration: DERIVED.** Two independent AI referees found no fatal error: the first at sketch level, with the sources; the second\n+on this written form (its gaps and presentation points are fixed below). Reports available on request.\n+No human review.\n+\n+**What changes.** Version 1 (§II.E of the note in #6934/#6937) proves E₁₂₃ ≪ t T^{3/2}(lnT)^{r+4}, with T = ln t. It needs Bourgain–Garaev\n+Thms 3 and 5, splits of d₁, d₂, d₃, and a delicate loss budget (slack T^{−6}). Version 2 replaces all of §E2–E6's BG\n+machinery with one application of the q-analogue of van der Corput, in the sum over the longer of d₂, d₃.\n+- Its modulus e·f·d₃·Q₀ is factored by the other summation variables, with q₁ = f.\n+- Weil and this bound leave only a thin tube, which is bounded trivially.\n+\n+**Proposition 3′.** E₁₂₃(t) ≪ t T (lnT)⁵. Hence, with Theorem 1 and Proposition 2,\n+\n+    S₄(t) = 𝔖(g)² t² − ½ 𝔖(g) t ln²t + O(t ln t (lnln t)⁵).\n+\n+**Inputs:**\n+- [W] Weil–Estermann, through Lemma C1;\n+- [WB] Weil's bound for rational exponential sums mod p (Bombieri, Amer. J. Math. 88 (1966); in the form of Cochrane–Pinner,\n+  J. Number Theory 2006). For R not of Artin–Schreier type, |Σ_{x not a pole} e_p(R(x))| ≤ (Σ pole orders + #distinct\n+  poles − 2)√p + O(1). Used for R(x) = −w′r/(x(x + r)) + bx: poles 0, −r, ∞, so the bound is 4√p. The printed source is not\n+  re-checked;\n+- [Sh] Shiu;\n+- [H] Henriot (Theorem 1 only).\n+\n+**Bourgain–Garaev is no longer used.** Kept from version 1 (the note in #6934/#6937, Part II): §A (parameters, Lemmas A1–A2, ϑ), §B (mass), Lemma C1, §D\n+(Proposition 2), §E1 (variables), E5's main-term computation and E6's grid.\n+\n+## Q. Lemma Q (q-van der Corput, one step, τ-explicit)\n+\n+**Lemma Q.** Let q = q₀q₁ be squarefree and odd, (w, q) = 1, and J an interval of length B ≥ q₁. Then\n+\n+    |Σ_{n∈J, (n,q)=1} e_q(w n̄)| ≤ 2(Bq₁)^{1/2} + 2 τ(q)^{3/2} ( B q₀^{−1/4} + B^{1/2} q₀^{1/4} (log 2q)^{1/2} ).\n+\n+This is Heath-Brown (Proc. LMS 82 (2001), Thm 2) at k = 1, with the q^ε made explicit, following his §9. (B ≤ q is not\n+needed.) Irving's l = 1 bound (IMRN 2015, Thm 1.3) is weaker for B < q^{1/2}, which is the whole Weil-bad region, so\n+Heath-Brown's form is the one used. The constant 2 and the placement of the log are as rechecked by the second referee, by\n+brute force on 1,500 cases (no intermediate inequality violated) and 3,410 complete sums (factorisation exact).\n+\n+*Proof.* Let a(n) = 1_J(n)1_{(n,q)=1} e_q(w n̄), and H = ⌊B/q₁⌋ ≥ 1.\n+- **Differencing.** S·H = Σ_n Σ_{0≤h<H} a(n + hq₁). Cauchy–Schwarz over the ≤ 2B values of n, expanding the square, gives\n+  |S|² ≤ 2B²/H + (4B/H)Σ_{0<h<H} |S_h|, where S_h = Σ_n a(n + hq₁)\\overline{a(n)}.\n+- **Reduction mod q₀.** Since n + hq₁ ≡ n (mod q₁), the q₁-part of the phase e_q(w(\\overline{n+hq₁} − n̄)) is trivial. So\n+  S_h is an incomplete sum over an interval of length ≤ B of the q-periodic function\n+  F(n) = 1_{(n,q₁)=1}·1_{(n(n+hq₁), q₀)=1}·e_{q₀}(w′(\\overline{n+hq₁} − n̄)), with (w′, q₀) = 1.\n+- **Completion.** Completing mod q factors the complete sums by CRT: F̂(b) = c_{q₁}(b)·Π_{p|q₀} K_p(b) (the arguments b and\n+  w′ carry the CRT unit twists, which change no absolute value; edit by the submitting run), where c is a Ramanujan sum and K_p(b) = Σ_{x mod p, x, x+r ≢ 0} e_p(w′(\\overline{x+r} − x̄) + bx), with r = hq₁.\n+  - For p ∤ h, [WB] gives |K_p(b)| ≤ 4√p.\n+  - For p | h the phase is trivial and K_p(b) = p·1_{p|b} − 1. (Heath-Brown's §9 says S vanishes unless Δ | m. That is\n+    inexact: under his convention of excluded zeros the value is −1 or p − 1, not 0. This costs nothing; it is checked\n+    numerically on 344 composite cases by the first referee.)\n+- **Bounding S_h.** Summing over b with |Ĵ(b)| ≤ min(B, 1/(2‖b/q‖)) (the ½ is needed for the constant in front of log 2q;\n+  edit by the submitting run):\n+  |S_h| ≤ 4^{ω(q₀)}[ B ((q₀,h)/q₀)^{1/2} + τ(q₁(q₀,h)) (q₀/(q₀,h))^{1/2} log 2q ].\n+- **Summing over h.** Σ_{0<h<H}(q₀,h)^{1/2} ≤ τ(q₀)H and 4^{ω(q₀)} = τ(q₀)², so\n+  Σ_h |S_h| ≤ τ(q)³ H (B q₀^{−1/2} + q₀^{1/2} log 2q). With 2B²/H ≤ 4Bq₁, take square roots. ∎\n+\n+**Lemma C4 (weighted Lemma Q).** Let f_D, ψ and V be as in C1, let m = q₀q₁ be squarefree with (w, m) = 1, and let\n+I ⊂ [N, 4N] be an interval with q₁ ≤ N. Then\n+\n+    |Σ_{n∈I, (n,m)=1} f_D(n)ψ(n) e_m(w n̄)| ≪ (1 + V)(τ(D) + σ₋₁(D)) τ(m)^{3/2}(log 2mN)^{O(1)} ( N^{1/2}q₁^{1/2} + N q₀^{−1/4} + N^{1/2}q₀^{1/4} ).\n+\n+(The log power is in fact 3/2: numerically Σ|c_ℓ|ℓ^{−1/2} grows like ≈ 5.1 log X. Any fixed power suffices.)\n+\n+There is no main term.\n+\n+*Proof.*\n+- **The prime 2.** If 2 | m, then n is odd (f_D is supported on n coprime to Q₀ ∋ 2), so the 2-component of e_m(w n̄) is\n+  constant. Remove it, so the modulus is odd.\n+- **Expansion.** Expand f_D(n) = Σ_ℓ c_ℓ 1_{ℓ|n} as in C1.\n+  - For ℓ ≤ N/q₁, apply Lemma Q at length N/ℓ with numerator w ℓ̄, after partial summation in ψ. Every term decreases with\n+    N/ℓ, and Σ|c_ℓ|ℓ^{−1/2} ≪ τ(D)(ln N)^{O(1)}. Partial summation uses sub-intervals: those of length < q₁ are bounded\n+    trivially by their length < q₁ ≤ (q₁N/ℓ)^{1/2}, which is the first term. Lengths above q are allowed, since Lemma Q does not need B ≤ q.\n+  - For ℓ > N/q₁ the trivial bound gives Σ_{ℓ>N/q₁}|c_ℓ|·2N/ℓ ≪ τ(D)(ln N)^{O(1)}(Nq₁)^{1/2}, the first term. ∎\n+\n+In log-coordinates (N = t^x, q₁ = t^s, m = t^M), C4 saves t^{−σ_Q} up to its loss factors, where\n+\n+    σ_Q(x, M, s) = min( (x − s)/2, (M − s)/4, (2x − M + s)/4 ).\n+\n+## E′. The ternary sector\n+\n+**E′1. Factored moduli.** The variables, weights, coordinates (v_e, v_f, u₂, u₃) with e > f, and the three phases are as in\n+§E1. In the d₂-form the modulus m₂ = d₁d₃Q₀ = e·f·d₃·Q₀ is a product of the other summation variables, and the numerator A₂\n+does not depend on d₂ (Lemma A2). After reduction by g₀ = (kA₂, m₂), the factor f becomes f′ = f/(f, g₀), a log-shift\n+≤ lnK/T = 10 lnT/T. The same holds in the d₃-form with m₃ = e·f·d₂·Q₀.\n+\n+**E′2. Assignment (cell by cell, on the grid of E6, every test uniform over the cell).**\n+- **(R1) Weil.** If max(μ₂, μ₃, μ_e) ≥ η, apply Weil (C1) in that variable, exactly as in version 1. The tie-break order is\n+  d₂, d₃, e.\n+- **(R1Q) Lemma Q.** Otherwise, let d* be the longer of d₂, d₃ (d₂ on ties). If σ_Q(u*, Σ − u*, v_f) ≥ κ holds on the\n+  whole cell, including the shifts of E′1, apply Lemma C4 to the d*-sum, with q₁ = f′ and q₀ = m*/(2^{·} q₁). Here m* is\n+  the modulus of the d*-sum: m₂ = efd₃Q₀ if d* = d₂, m₃ = efd₂Q₀ if d* = d₃, reduced by g₀. The outer\n+  variables are e, f and the other of d₂, d₃.\n+- **(R0) The tube.** Otherwise the cell is in R0, and is bounded trivially.\n+\n+Here κ = C_κ lnT/T with C_κ = 50, and η = C₁ lnT/T with C₁ = 50 as before. \"Uniform over the cell\" means testing at the corner\n+with the safety 9/(ΔT) for R1, and 9/(ΔT) + 10 lnT/T for R1Q (cell width plus g₀-shift of f). R1 needs no g₀ safety:\n+reduction only shrinks the modulus, which helps Weil.\n+\n+**E′3. Covering lemma [P].** If a cell fails R1 and R1Q, then |u₂ − u₃| ≤ 4(κ′ + η′) and |v_e − u*| ≤ 4(κ′ + η′), with\n+κ′ = κ + 11 lnT/T and η′ = η + 1/T. So R0 lies in the box-tube 𝒯 = {|v_e − u*|, |u₂ − u₃| ≤ 4(κ′ + η′)} around the segment\n+\n+    Γ = { v_e = u₂ = u₃ = λ, v_f = Σ − 3λ : ¼ − η′ ≤ λ ≤ ⅓ + O(η) }.\n+\n+At fixed Σ its volume is vol₃(𝒯_Σ) ≤ 6(κ′ + η′)² (16/3 is the exact constant) ≪ (lnT/T)². (A round tube of radius\n+4√2(κ′ + η′) would have volume ≈ 14.5(κ′ + η′)²; the box set is the one used.)\n+\n+*Proof.* Write a = v_e ≥ b = v_f and x = u* ≥ y (the other of u₂, u₃), so that a + b + x + y = Σ, M = Σ − x = a + b + y,\n+s = b. Failing R1 gives μ_e < η′, i.e. a < (x + y)/2 + 2η′. Failing R1Q gives one of three cases.\n+- **(i) x − b < 2κ′.** Then a ≥ b > x − 2κ′. Combined with a < (x + y)/2 + 2η′ this gives y > x − 4κ′ − 4η′, and\n+  x − 2κ′ < a < x + 2η′.\n+- **(ii) M − s = a + y < 4κ′.** This is impossible: on the Weil-bad set, y ≥ 1/6 − O(η).\n+- **(iii) 2x − M + s = 2x − a − y < 4κ′.** Then (x + y)/2 + 2η′ + y > a + y > 2x − 4κ′, so x − y < (8κ′ + 4η′)/3, and\n+  a > 2x − y − 4κ′ ≥ x − 4κ′, with a < x + 2η′.\n+\n+In every case |x − y|, |a − x| ≤ 4(κ′ + η′). On the Weil-bad set x ≤ ⅓ + O(η) and b ≤ a, so λ ∈ [¼ − η′, ⅓ + O(η)]. In case (ii), y > Σ/6 − 10η′/3 > 4κ′. The\n+cross-section is a square of side 8(κ′ + η′) in (a − x, x − y); integrating over the λ-range with the Jacobian gives\n+16/3·(κ′ + η′)². ∎\n+\n+*Numerics (validators_v2/: resid*.py, cover*.py, vol0.py).*\n+- Monte Carlo on the continuous model, 18 settings including κ′ ≠ η′, finds no violation. Case (ii) never occurs.\n+- max(|x − y|, |a − x|) ≤ 3.8(κ + η) over all residual points.\n+- The residual volume is ≈ 2h² (≈ 24h² as a fraction of the half-simplex), for h = 0.02 … 0.0025. Γ is a genuine barrier:\n+no other unweighted divisor choice, and no k = 2 or l = 2 form, covers it.\n+\n+**E′4. Loss budget and assembly.** M_tot ≪ t T³(lnT)² is the total effective mass (B1).\n+- **R1 (Weil).**\n+  - As in E6: saving T^{−50}, losses T^{20+5+3}(lnT)^{O(1)}, total ≪ t T^{−19}(lnT)^{O(1)}.\n+  - The main terms are ≪ t T (lnT)⁵ (E5, with A2's reduction factor).\n+- **R1Q (Lemma C4).**\n+  - Saving: σ_Q ≥ κ on the whole cell, so t^{−κ} = T^{−50}.\n+  - Losses:\n+    - ≪ Δ = T^{20} maximal intervals per block (the inner variable is a non-atom, since σ_Q ≥ κ forces N ≥ t^{2κ} > Δ³);\n+    - τ(m*)^{3/2} averaged over the three outer variables, ≤ T^{3(2^{3/2} − 1)} ≤ T^{5.49} by [Sh];\n+    - the logarithms: (log 2q)^{1/2} from Lemma Q times Σ_ℓ|c_ℓ|ℓ^{−1/2} ≪ ln N from the expansion of C4 (Euler product\n+      Π_p(1 + λ_p p^{−1/2} + (1 + λ_p)/p); measured ≈ 5.1 ln X), in total ≤ T^{1.5}(lnT)^{O(1)} (wording edited by the submitting run);\n+    - the k-sum, τ(g₀) and (1 + V) through ϑ: (lnT)^{O(1)}.\n+  - Net T^{−50+27}(lnT)^{O(1)}, so the total is ≪ t T^{−20}(lnT)^{O(1)}.\n+  - There is no main term. The threshold for this line is C_κ > 30.\n+- **R0 (the tube), trivially.** By B1, ≪ t T³(lnT)²·vol₃(𝒯) ≪ t T³(lnT)²(C lnT/T)² ≪ t T (lnT)⁴.\n+- **Boundaries.** Every cell is now handled by a linear sum in one inner variable (R1, R1Q) or trivially (R0). So the\n+  boundary types (i)–(iii) of E6, which came from bilinear cells, no longer occur. The L-limits and e > f are interval cuts in\n+  the inner variable. Truncations and dropped factors are O(t) + O(T⁵ lnK).\n+\n+Total: **E₁₂₃(t) ≪ t T (lnT)⁵**, dominated by the R1 main terms, with the tube at t T (lnT)⁴. ∎\n+\n+## E′5. Consequences (each DERIVED given its stated hypothesis)\n+\n+- **S₄** (Claim 2 of #6934, improved): S₄ = 𝔖²t² − ½𝔖 t ln²t + O(t ln t (lnln t)⁵).\n+- **Theorem V** (VARIANCE.md): V(X; t) = 𝔖(g) t[(log X)² − ½(log t)²] + O(t log X (loglog X)⁵). This is the O(t log X) of\n+  the Part 19 conjecture up to (loglog X)⁵.\n+- **Theorem G** (GAPS.md): for twins and cousins, F = 1 − θ²/2 + O((loglog X)⁵/log X). For g ≥ 6 the pair-count term\n+  2𝔖(0,g,2g)/(𝔖L) is now inside this error, so it is no longer separately visible.\n+\n+## E′6. What remains\n+\n+- **O(t ln t) without lnln factors.** Two things are needed:\n+  - Bound or evaluate the R1 and Prop. 2 Ramanujan main terms as ≪ t T. They are probably ≍ t T, and the (lnT)⁵ is\n+    bookkeeping.\n+  - Treat the tube R0 by BG Thm 5 in e, which saves T^{−1/2} and brings back [BG5].\n+- **B_g (o(t ln t)).** Every t ln t-sized piece must be evaluated asymptotically, not bounded: Theorem 1's sectors {1}, {2},\n+  {3}, the binary main terms and the ternary R1 main terms. The tube must also go below t ln t. Its trivial bound,\n+  t T (lnT)⁴, does not; with BG Thm 5 in e it becomes ≪ t T^{1/2}(lnT)^{r+5}, which does. So B_g needs [BG5] back.\n+- **Prior art.** The technique is standard: Heath-Brown 1978 and 2001, Irving, Polymath 8a, Wu–Xi. Two searches found no\n+  printed use of exactly this step for correlations of divisor-type functions. That is not a clearance.\n+\n # Part III. Imported theorems (with locators)\n \n | Tag | Statement used | Source and locator | How it was checked |\n | --- | --- | --- | --- |\n | [W] | \\|S(a, b; m)\\| ≤ τ(m)(a, b, m)^{1/2} m^{1/2} for every m ≥ 1 | Estermann, Mathematika 8 (1961); Iwaniec–Kowalski, *Analytic Number Theory*, Cor. 11.12 | Two independent quotations (Bourgain–Garaev 2014 p. 44; Blomer–Milićević, JEMS 2015); the book page itself was not seen |\n-| [BG3] | The bilinear bound of Part II's opening list, uniform in k₁, k₂ ≥ 1 and (a, m) = 1, \\|α_i\\| ≤ 1, variables coprime to m | J. Bourgain, M. Z. Garaev, *Kloosterman sums in residue rings*, Acta Arith. 164 (2014) 43–64, Thm 3 (p. 44); arXiv:1309.1124 | ArXiv TeX source and journal PDF, identical; no erratum |\n-| [BG5] | max_{(a,m)=1} \\|Σ_{n≤N, (n,m)=1} e_m(an*)\\| ≪_c (lnln m)^r N (ln m)^{−1/2} for N > m^c, with r absolute | ibid., Thm 5 (p. 45) | As [BG3]. Bounds initial segments only. |\n+| [WB] | Weil's bound for Σ_{x∈F_p, x not a pole} e_p(R(x)) with R not of Artin–Schreier type: ≤ (Σ pole orders + #distinct poles − 2)√p + O(1); used for R(x) = −w′r/(x(x+r)) + bx (4√p). Version 2 only | E. Bombieri, *On exponential sums in finite fields*, Amer. J. Math. 88 (1966); T. Cochrane, C. Pinner, *Using Stepanov's method for exponential sums involving rational functions*, J. Number Theory 116 (2006) | Statement taken from the two secondary quotations as given in version 2; the printed sources were not re-checked. The 4√p bound and the degenerate value p·1_{p\\|b} − 1 were tested numerically (Part IV, validators_v2) |\n+| [BG3] | (version 1 only) The bilinear bound of Part II's opening list, uniform in k₁, k₂ ≥ 1 and (a, m) = 1, \\|α_i\\| ≤ 1, variables coprime to m | J. Bourgain, M. Z. Garaev, *Kloosterman sums in residue rings*, Acta Arith. 164 (2014) 43–64, Thm 3 (p. 44); arXiv:1309.1124 | ArXiv TeX source and journal PDF, identical; no erratum |\n+| [BG5] | (version 1 only) max_{(a,m)=1} \\|Σ_{n≤N, (n,m)=1} e_m(an*)\\| ≪_c (lnln m)^r N (ln m)^{−1/2} for N > m^c, with r absolute | ibid., Thm 5 (p. 45) | As [BG3]. Bounds initial segments only. |\n | [Sh] | f ≥ 0 multiplicative with f(p^l) ≤ A^l and f(n) ≪_η n^η; for x^ε ≤ y ≤ x (0 < ε < ½), Σ_{x−y<n≤x} f(n) ≪ y/ln x · exp(Σ_{p≤x} f(p)/p) | P. Shiu, *A Brun–Titchmarsh theorem for multiplicative functions*, J. reine angew. Math. 313 (1980) 161–170 | Two independent restatements (Wright, arXiv:2508.17217, Thm 1.1; Bordellès, arXiv:2402.12333, Thm A); the original scan was not retrievable |\n | [H] | Σ_{\\|h\\|<t} H(h) ≪ t (ln t)⁴ for the product of submultiplicative functions of h, h − g, h + g in (3.2) | K. Henriot, Math. Proc. Camb. Phil. Soc. 152 (2012) 405–424, Thm 5 / Cor. 1, with erratum 157 (2014) 375–377 | Statement and erratum read. Any fixed exponent suffices; an elementary Hölder + τ-moment substitute exists |\n \n@@ -576,7 +763,8 @@\n \n # Part IV. Finite checks (MEASURED; finite checks do not prove the asymptotic)\n \n-Validators are in `validators/` (Python 3 with numpy, sympy, scipy and mpmath; all standalone).\n+Validators are in `validators/` (version 1; Python 3 with numpy, sympy, scipy and mpmath; all standalone) and `validators_v2/`\n+(version 2; numpy and sympy).\n \n | Check | n / range | Result | Validator |\n | --- | --- | --- | --- |\n@@ -591,6 +779,10 @@\n | sup P₂/(lnT/T)^{1/8} | same | → 2.06 | p2_volbad_p2.py |\n | E3(i) inequalities by linear programming | — | s ≥ ½ − 6η, Q ≥ ½ − 2η | p2_check_E3_ineq.py |\n | Split mass on actual integers (one d₂-cell at D = 10⁹, fully factored) | all integers of the cell | 1.4–1.5·θ^{1/2} (arcsine) | p2_check_E4_small.py |\n+| Lemma Q against actual sums, with each intermediate inequality | 1,500 random (q₀, q₁, w, B), q ≤ 6000; 800 cases q ≤ 2·10⁵ | smallest constant needed 0.011 against the stated 2; no step violated | lemq_check.py, lemq.py (v2) |\n+| Complete sums F̂(b) = c_{q₁}(b)Π K_p(b), incl. the degenerate p \\| h value | 3,410 cases; 269 degenerate | exact; max \\|K_p\\|/(4√p) = 0.64 | lemq_check.py, localsum.py (v2) |\n+| Covering lemma E′3 on the continuous model | 18 settings incl. κ′ ≠ η′, Monte Carlo; exact LP in the submitting run | no point outside the box-tube; case (ii) never; sup\\|x − y\\| = 4(κ′+η′) exactly (LP); residual volume ≈ 2h² (box-tube bound 16h²/3·4) | cover_check.py, cover.py, resid*.py, vol0.py (v2); qvdc_recheck.py |\n+| Σ_{ℓ≤X}\\|c_ℓ\\|ℓ^{−1/2} (the log loss of C4) | X ≤ 10⁷ | ≈ 4.6 ln X at 10⁷, slope 5.1 | lsum_check.py (v2); qvdc_recheck.py |\n \n **Local-only (not in validators/).** An exact sector decomposition to t = 10⁸ for g = 2 and 6, which needs a local prime\n table, was pre-registered. Results:\n@@ -609,6 +801,9 @@\n | Lemma A2 phases | Any mismatch | Run; none |\n | vol(Bad) ≍ (lnT/T)^{3/4}, P₂ ≪ (lnT/T)^{1/8} | Normalised ratios drifting with T | Run to T = 10¹⁶; flat. The proofs (E3, E4) are analytic; numerics are a check only. |\n | Each use of [BG3], [BG5], [Sh], [W] | A hypothesis not met at the point of use | Checked by three referees; independent re-check invited |\n+| Lemma Q (version 2) | An actual sum exceeding the stated bound, or a failing intermediate step | Run on 2,300 random cases and 600 more in the submitting run; none |\n+| Covering lemma E′3 | A cell failing R1 and R1Q outside the box-tube 𝒯 | Run (Monte Carlo, 18 settings; exact LP); none |\n+| Each use of [WB] (version 2) | A local sum K_p(b) above 4√p, or the degenerate value wrong | Run on 45,000 sums; none |\n \n # Part VI. Prior art (novel to us vs novel)\n \n@@ -627,30 +822,39 @@\n   - Kuperberg, IJNT 2025 (arXiv:2301.06095), uses independent congruence conditions or product weights. These cannot\n     impose the linked offsets h, h + 2, which answers route 107's second residual: no.\n   - de la Bretèche, arXiv:2410.02480, treats 3-point sums over primes, a different object.\n+- **Version 2's technique is standard, its use here is \"novel to us\" only:** the q-analogue of van der Corput is Heath-Brown's\n+  (Proc. LMS 82 (2001), Thm 2; also his 1978 work), developed further by Irving (IMRN 2015), Polymath 8a, Wu–Xi (arXiv:1603.07060)\n+  and Mangerel (Forum Math. Sigma 2021, arXiv:2008.11163). Two searches by the author and two by the submitting run (2026-11-05)\n+  found no printed use of this step for correlations of singular series or divisor-type functions. That is not a clearance.\n \n # Part VII. Open items and reuse\n \n - **OPEN:** b and c. They need an error o(H ln H) in D for b, and o(H) for c.\n-- **OPEN, heuristic only:**\n-  - r-fold splits of d₂ and d₃ suggest an error t(ln t)^{5/4+1/(2r)}; not written to proof standard.\n-  - Irving's q-van der Corput (IMRN 2015) might give similar gains for the factored moduli; not checked.\n+- **OPEN:** O(t ln t) without the (lnln t)⁵: the R1 and Prop. 2 Ramanujan main terms would have to be bounded or evaluated as ≪ t T,\n+  and the tube R0 treated by [BG5] in e (§E′6).\n+- **OPEN:** B_g, i.e. o(t ln t): every t ln t-sized piece evaluated asymptotically (Theorem 1's sectors {1}, {2}, {3}, the binary\n+  main terms, the ternary R1 main terms) and the tube below t ln t, which brings [BG5] back (§E′6).\n+- **OPEN, heuristic only:** r-fold splits of d₂ and d₃ (version 1's framework) suggested an error t(ln t)^{5/4+1/(2r)}; superseded by\n+  version 2 and not written to proof standard.\n - **OPEN:** the variance consequence under uniform HL. It needs the reduction from primes to S₄ and the 3-point companion\n   sum, a Prop. 2-type estimate; neither is written here.\n - **Reuse:** Part I applies to any 4-point pattern {0, g, h, h + g}. Lemma C1 and Lemma C3 apply to any weight of the form\n   μ²(n)Π_{p|n}(1 + O(1/p)). They do not apply to weights with mean ≤ ¾ at primes, such as β′ (mean ½).\n-- **Revisit only with:** a power-saving bound for linear Kloosterman sums of length ≈ (modulus)^{1/2} to composite moduli, or\n-  Thm 3 with losses polynomial in k *and* no (log m)^{2(k₁/k₂+k₂/k₁)} factor (that would give 3/2 → 1 + o(1)).\n+- **Revisit only with:** (for b, c) a power-saving bound for linear Kloosterman sums of length ≈ (modulus)^{1/2} to composite\n+  moduli; (for the lnln factors) an evaluation of the Ramanujan main terms and a treatment of the tube R0.\n \n # Part VIII. Proposed outcome entry\n \n ```\n #### S-route177-a — ln²H coefficient of the twin-pair defect, two-sided\n-**Grade:** DERIVED (imports Weil–Estermann, Bourgain–Garaev Thms 3 and 5, Shiu, Henriot); not yet reviewed by a trusted reviewer.\n+**Grade:** DERIVED (version 2 imports Weil–Estermann, Weil's bound for rational functions, Shiu, Henriot; version 1 also\n+Bourgain–Garaev Thms 3 and 5); not yet reviewed by a trusted reviewer.\n **Question:** route 107 target, coefficient a; route 177.\n-**Established result:** D(H)/(A²H) = ln²H/(4C₂) + O((ln H)^{3/2+ε}), and for every even g,\n-S₄(t) = 𝔖(g)²t² − ½𝔖(g) t ln²t + O(t(ln t)^{3/2}(lnln t)^{r+5}). The ½ is the excluded h = 0 term.\n-**Limit:** b, c open (needs o(H ln H), o(H)); r inexplicit; variance reading CONDITIONAL. Cell bookkeeping: §II.E6.\n-**Evidence:** this note (Parts I–II), validators/ (Part IV), three AI referee reports (available on request).\n-**Revisit only with:** a power-saving bound for Kloosterman sums of length ≈ m^{1/2} to composite moduli, or route 177's\n-Ramanujan-side completion (which might also give b).\n+**Established result:** D(H)/(A²H) = ln²H/(4C₂) + O(ln H (lnln H)⁵), and for every even g,\n+S₄(t) = 𝔖(g)²t² − ½𝔖(g) t ln²t + O(t ln t (lnln t)⁵). The ½ is the excluded h = 0 term.\n+**Limit:** b, c open (needs o(H ln H), o(H)); O(t ln t) without lnln factors open; B_g open; variance reading CONDITIONAL;\n+[WB]'s printed source not re-checked; prior art for the q-van der Corput use not cleared. Cell bookkeeping: §II.E6, §II.E′4.\n+**Evidence:** this note (Parts I–II), validators/ and validators_v2/ (Part IV), AI referee reports (available on request).\n+**Revisit only with:** a power-saving bound for Kloosterman sums of length ≈ m^{1/2} to composite moduli, an evaluation of\n+the Ramanujan main terms, or route 177's Ramanujan-side completion (which might also give b).\n ```\n","cpu_hours":0.2,"hashes":{"qvdc_prereg.md":"ac9150e89aa1f2c3f432fadb82d9e5a459a28b698c03bca5d9b3ab4fcb1ab613","qvdc_recheck.py":"edd3ec8bbe7086e8e0ccfa9dfd57c2cdf92d0156971f2be2549070e725f8fd7d","build_note_v2.py":"922ce51cf4b26ca246d202710b575f81f16b0d6aa11046def3cf54689acb7b89","qvdc_recheck.out":"87ca1caacf236154c2231f8ff4abf586f2ae3d2d02560549f5a26ef6abb5bd12","review_check_qvdc.md":"c1f826f626bee0edbf620526d9fba3bc6c57bc2c531484cff1ebef6ee444d6fe","revision_6934_qvdc.md":"a0c79ac511e586a3d2e27cf2227c0c443a7724f87bec89e14c429335dfc751fe","referee_qvdc_subagent.md":"e5e5ad235d54a8553f87495a5d30b9bbc214598ede1351e6315e7e573d8ba944","route177_twin_defect_a.md":"8366a69fee52f63f97c855454c1b0645ea9b68e2a556fb21b019ffa376b3ebe2","route177_twin_defect_a.v2.patch":"aae31cb03d33a2d936bee7e36736753d7c577626937c5187185de447f5a676d5"},"author_rung":"heuristic","status":"pending","final_rung":null,"created_at":"2026-11-05T01:40:49.035Z","repo_url":null,"commit":null,"cites":{"files":["5a4f13535a01b777481a478614ec34579b461661093faad8e4579e41a7487644","9858be4a4359508c5373b6b71624a761e11f75b9e41395f6d3a6d6703ab2dde8","b4292a5a026e7d0bb3a5ec2572ebcc3ec33e1e6b426b324af86f4b20a3c303c2"],"handles":[],"returns":[6934,6937,1315,1302,1309,1317,1834,2170,2300,5553,5570,6156,6708],"messages":[]},"tokens":{"log":"summary","input":1004,"models":{"claude-fable-5-1":253714},"output":253714,"source":"reported","entries":0,"cache_read":5858132,"cache_write":544673,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe (job #13693)\n\nFetch each file with `GET <server origin>/files/<sha256>?raw=1`, `Accept: text/plain`, and verify the raw sha256 before use. Environment used:\nPython 3.14, numpy 2.5, scipy 1.18, sympy 1.14, one process per script. Everything is seeded (seeds inside the scripts); no randomness is left out.\n\n1. The revision: apply `route177_twin_defect_a.v2.patch` (also the `patch` field) to the served copy 5a4f1353…7644 of `route177_twin_defect_a.md`\n   with `patch -o revised.md served.md < route177_twin_defect_a.v2.patch`; sha256 of the result must be\n   8366a69fee52f63f97c855454c1b0645ea9b68e2a556fb21b019ffa376b3ebe2. Equivalently `python3 build_note_v2.py served.md revision_6934_qvdc.md out.md`\n   (revision file sha256 a0c79ac511e586a3d2e27cf2227c0c443a7724f87bec89e14c429335dfc751fe) prints that hash.\n2. Independent recheck (about 13 s): `python3 qvdc_recheck.py > qvdc_recheck.out`. Expected stdout sha256 87ca1caacf236154c2231f8ff4abf586f2ae3d2d02560549f5a26ef6abb5bd12. Pass if §2 and §3\n   print 0 violations with the ½ and > 0 without; §4 prints 0 violations of 4√p and 0 degenerate mismatches; §5 prints \"infeasible\" on every line,\n   largest deviation < 10⁻⁹, and volume ratios within 1 % at κ′ = η′ ≤ 0.01; §6 prints slope ≈ 5.1 and 0 sieve mismatches; §7 a non-positive maximum.\n3. Referee validators (run from one directory holding the ten scripts; 515 s CPU in total, `cover.py`, `resid.py`, `resid_hb.py` the heavy ones):\n   `python3 lemq_check.py`, `python3 lemq.py`, `python3 localsum.py`, `python3 lsum_check.py`, `python3 vol0.py`, `python3 cover_check.py`,\n   `python3 cover.py`, `python3 resid_min.py`, `python3 resid_hb.py`, `python3 resid.py`, `python3 resid.py e`. Expected stdout sha256 as listed in\n   `v2_validator_runs.json` (attached as `validator_outputs_v2-v2_validator_runs.json`; the outputs are attached as `validator_outputs_v2-v2_<name>.out`).\n4. The mathematics: read `review_check_qvdc.md` §§1–5 and `referee_qvdc_subagent.md` beside §II.Q and §II.E′ of the revised note; the memo's\n   Lemma Q steps, the three cases of E′3 and the budget table are the claims to confirm one by one. No computation certifies the asymptotic statement.","verification":null,"target":null,"finding":null,"human_md":"Twin prime, routes 107/177: revision of returns #6934 and #6937. Read context/revision_6934_qvdc.md. It replaces the\nBourgain–Garaev part (§II.E2–E6) of the #6934 note by one application of Heath-Brown's q-van der Corput bound (Proc. LMS 82\n(2001), Thm 2, k = 1, with the q^ε made τ-explicit). This improves the S₄ error from O(t(ln t)^{3/2+ε}) to\nO(t ln t (lnln t)⁵), with no Bourgain–Garaev input. Validators are in context/validators_v2/.\n\nCheck Lemma Q's proof (including the corrected step from Heath-Brown §9), the covering lemma E′3 and the loss budget E′4\nagainst the register's standards. Submit it as a revision linked to #6934 and #6937, grade DERIVED. Record what stays open:\nb and c; O(t ln t) without lnln factors; B_g.\n\nDisclosure for the record: the derivation in context/ was prepared outside this run, in separate Claude Code sessions (mostly\nClaude Opus 5.5; some steps used other Claude models), with independent sub-agent referees. This run checks, formalizes and\nsubmits it. Please state this in the return, including in human_md. Keep the same work style. Do not upgrade the grade\nwithout a trusted reviewer.","provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":"0d43bacfcc379289a9ae0feb183c41c879a8d1baa294fe26a79cd32fd07601e3","superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":[{"sha":"d167b2a2c0afb9fa1536762ef994c82ffd5d7455216af01ccd4a90fce7576568","name":"referee_cl_growth.py","notes":["prints what looks like progress or timing to stdout on line 38 (\"print(f\"slope between 1e5 and 1e6: {(cum[10**6-1]-cum[10**5-1])/math.log(10):.3f\"): 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":"434c8ad5c1c7de34018f0da010d9da63e861ceeeb56a911d586c0c2ff2884a77","name":"referee_lemq_bruteforce.py","notes":["prints what looks like progress or timing to stdout on line 178 (\"print(f\"cases: {n_cases} (of which B > q: {n_Bgtq}); Fhat/completion checks: {n_\"): 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":"2026-11-05T01:40:49.035Z","department_id":"dept_34ab54958953ff2275824e96","run_id":"run_d3c4c448b3faf7d7d831c2f2","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. This is a bounded planning assignment; publish only shareable findings.","review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6942/transcript","files":[{"sha256":"0cf03424681074f1cdd3508445d2d85b358d70eb4d01bde5f66b61550e4e3a10","name":"resid_hb.py","bytes":1935},{"sha256":"13392917a36b3eee822bb3752cdfc6b7507d0cbbb18153c19c5676dd8e177b2e","name":"referee_lemq_large.py","bytes":3288},{"sha256":"13d84d6eca44ef590284729ff3e1f4cb6e24aedf01a406fc84544d2246d4ede6","name":"validator_outputs_v2-v2_vol0.out","bytes":134},{"sha256":"1551ece27f24ab95374f328648e61f656d72c8717864526bc1a10b41128b1ec0","name":"validator_outputs_v2-v2_lsum_check.out","bytes":284},{"sha256":"1d0b433dba0769145988e9deac3386de1d25de9b894416969217a21ad24be75e","name":"referee_localsum_check.out","bytes":457},{"sha256":"1ea0aaddc1bd0283a771e0e1254ff47b67da723b93e041ba391bdbc46875c8c7","name":"validator_outputs_v2-v2_localsum.out","bytes":169},{"sha256":"41719a973f4599000d7ecd872c4f8e77a26e5c4c192deb2d03559a2d84718a15","name":"resid_min.py","bytes":1206},{"sha256":"42c857cbb1ec9d66947c646aa951e54c30b3eb09a0dd2ed537d3f40fd73c05b6","name":"validator_outputs_v2-v2_cover_check.out","bytes":1614},{"sha256":"434c8ad5c1c7de34018f0da010d9da63e861ceeeb56a911d586c0c2ff2884a77","name":"referee_lemq_bruteforce.py","bytes":9473},{"sha256":"4aed2a6a09500e59f1dc969140531e7f087f5b1ac9511f18aedc0c6b52a12282","name":"validator_outputs_v2-v2_lemq.out","bytes":3399},{"sha256":"50a77784587d5167d71ac5b472ad19faa69730516fff360935a0d5393a24a4ac","name":"validator_outputs_v2-v2_cover.out","bytes":1150},{"sha256":"6013acc2680a2d313a880cdfc403261b782fa034da1db295fb2fab8292e8487f","name":"validator_outputs_v2-v2_validator_runs.json","bytes":8973},{"sha256":"6f803c938b4171dbb2e174e388bc4f44ebef1e85e90edf8fe86f1ddcb968a871","name":"referee_cover_lp.out","bytes":5281},{"sha256":"6fde4e91e673c92970cf80705738e9cc6dbb56de280103fc5dafab94830dc600","name":"cover.py","bytes":3270},{"sha256":"71f87e6f4b4fbaa8b0760980b3ee7e56bc9e0b9c747304a6086e13019c182976","name":"validator_outputs_v2-v2_resid_min.out","bytes":320},{"sha256":"8366a69fee52f63f97c855454c1b0645ea9b68e2a556fb21b019ffa376b3ebe2","name":"route177_twin_defect_a.md","bytes":70109},{"sha256":"87ca1caacf236154c2231f8ff4abf586f2ae3d2d02560549f5a26ef6abb5bd12","name":"qvdc_recheck.out","bytes":4976},{"sha256":"89dcb876672f78337e4334760fcdd2e166c9c22f8436bcb3d96db85965733437","name":"validator_outputs_v2-v2_resid_hb.out","bytes":825},{"sha256":"922ce51cf4b26ca246d202710b575f81f16b0d6aa11046def3cf54689acb7b89","name":"build_note_v2.py","bytes":15476},{"sha256":"945d2f205eaa81d8082f5ff9d01d326a6b9ec5da209076fbaa3045f5aa440c91","name":"referee_lemq_bruteforce.out","bytes":1461},{"sha256":"9c334a26a18b8bfac6ac46e4940f2ded8a5df31ad747ff2dedd0048d61a18d26","name":"referee_localsum_check.py","bytes":3249},{"sha256":"a0ba56d7bb86b3f94a5754b05707f23a07f3814f3d3338553820783084be3b8d","name":"localsum.py","bytes":2055},{"sha256":"a0c79ac511e586a3d2e27cf2227c0c443a7724f87bec89e14c429335dfc751fe","name":"revision_6934_qvdc.md","bytes":14207},{"sha256":"aae31cb03d33a2d936bee7e36736753d7c577626937c5187185de447f5a676d5","name":"route177_twin_defect_a.v2.patch","bytes":30852},{"sha256":"ab0d58a9b51dc6183366039e937425bdfe56f661e3ae9fb4ca1c1542c104b27a","name":"vol0.py","bytes":620},{"sha256":"ac9150e89aa1f2c3f432fadb82d9e5a459a28b698c03bca5d9b3ab4fcb1ab613","name":"qvdc_prereg.md","bytes":11921},{"sha256":"bfa0be1e87ed4c3c9d8a4109587adebaa9c5db7cb5e02bf66320ff823a13cb71","name":"referee_cl_growth.out","bytes":401},{"sha256":"c0603eafc9baee34d1b0e9dc76d071ee9e0239a9652d2ceba0451f5cfa9e9723","name":"lemq.py","bytes":3617},{"sha256":"c1f826f626bee0edbf620526d9fba3bc6c57bc2c531484cff1ebef6ee444d6fe","name":"review_check_qvdc.md","bytes":15944},{"sha256":"c7c96e0286b86758ed67f305d1c721551907fe7bb0c238e6152e5684d3bb075b","name":"referee_cover_lp.py","bytes":5069},{"sha256":"d167b2a2c0afb9fa1536762ef994c82ffd5d7455216af01ccd4a90fce7576568","name":"referee_cl_growth.py","bytes":1795},{"sha256":"dc0a526f55ba0a6558f8ea00d9444806b8df656a77ee6dd3feb8eb5dacbd0126","name":"referee_lemq_large.out","bytes":2489},{"sha256":"dc6df91855c269284935a1edcfcf8e08b1268ecf2bf3829efe94dcd6ad6731ed","name":"validator_outputs_v2-v2_resid_e.out","bytes":437},{"sha256":"e5e5ad235d54a8553f87495a5d30b9bbc214598ede1351e6315e7e573d8ba944","name":"referee_qvdc_subagent.md","bytes":24417},{"sha256":"edd3ec8bbe7086e8e0ccfa9dfd57c2cdf92d0156971f2be2549070e725f8fd7d","name":"qvdc_recheck.py","bytes":13358},{"sha256":"ee573181c3202a65a8a0c723c22e5b856408b7986899676414f3908f33d36ef2","name":"validator_outputs_v2-v2_resid.out","bytes":428},{"sha256":"ef08e9c326577f9a75fb751e92f799e4490c439453aa33054c82814838c736f2","name":"validators_v2-README.md","bytes":626},{"sha256":"f053d3e139de079cdbe7f1b3d2d7cc181f55021547bb6cddb970f784175f8bf1","name":"validator_outputs_v2-v2_lemq_check.out","bytes":781},{"sha256":"f6695ee51f84f2e890ab04fd67b72767cbbd89be800dcd09f6f1030bb7e5cefb","name":"cover_check.py","bytes":2953},{"sha256":"fada7d38a09526273e47fc8ddd393c4232b8ca2a8f748a90c6a7c9c6ac090ade","name":"lemq_check.py","bytes":7909},{"sha256":"fb41059cee3ea7b6ca1cc26c2504214f6595f486e4f495b8842316f2d413e432","name":"resid.py","bytes":2920},{"sha256":"fcca995471a8a12cf6fd63156bdc41ca2b10afe0400cc4393bea4292b026e4c1","name":"lsum_check.py","bytes":842}],"patch_status":"pending integration: the integrator applies accepted patches to the research repository by hand; build on the served file plus this patch until then","decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"39db5be650851ab66bc3e8f29fb4dd17c95ea4fe67a918c8ec9fe506cd9172ab","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}