{"id":6934,"job_id":13674,"problem_id":1,"lane_id":null,"type":"explore","user_id":120,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# Route 177/107: the ln²H coefficient of the twin-pair defect, two-sided (a = 1/(4C₂)), checked against the register; the ρ_H → ½ residual is the p = 2 factor\n\n**Caveats first.**\n- **Grade: DERIVED, unreviewed.** The derivation (file `route177_twin_defect_a.md`, 548 lines) is written out modulo four published\n  theorems; three AI referees and this run have read it; no trusted reviewer has. This return asks for that review. The person who owns\n  the derivation instructed that the grade not be raised before a trusted review, so `author_rung` is set to **heuristic**, the\n  register's nearest rung below proven; if the review confirms the argument, the claim's rung is proven.\n- **Open:** the coefficients b and c of route 107's target (they need errors o(H ln H) and o(H); the derivation gives O(H (ln H)^{3/2+ε}));\n  the cell-partition bookkeeping of Part II.E (boundary cells, atoms, the constraint xy ∈ 𝒞 of the split refinement) is routine but not\n  written; the reading of D(H) as the twin-count variance is conditional on a uniform Hardy–Littlewood 4-tuple hypothesis and is not\n  derived (#1302/#1309 measured it; #1315/#1317 state the conditionality). The constant r in the error exponent is the lnln-exponent of\n  Bourgain–Garaev Thm 5 at c = 1/5; it is absolute by that paper's proof text but not explicit.\n- Nothing here bears on twin-prime infinitude, G₂ or β₂.\n\n## 1. Statement (DERIVED)\n\nNotation of #1315/#1834: A = 2C₂ = 𝔖(2), D(H) = A²H² − Σ_{h even, 0<|h|<H}(H − |h|) 𝔖(0,2,h,h+2).\n\n**Claim 1.** D(H)/(A²H) = ln²H/(4C₂) + O_ε((ln H)^{3/2+ε}). The ln²H coefficient is a = 1/(4C₂) = 0.378695…, from both sides.\n\n**Claim 2 (general even gap g).** With w(h) = (t − |h|)₊ and S₄(t) = Σ_{h∉{0,±g}} w(h)𝔖(0,g,h,h+g):\nS₄(t) = 𝔖(g)²t² − ½𝔖(g) t ln²t + O(t (ln t)^{3/2}(lnln t)^{r+5}), constants depending on g. Claim 1 is Claim 2 at g = 2, because\nodd h and h = ±2 have 𝔖 = 0, so D(H) = 𝔖(2)²H² − S₄(H) exactly.\n\n**Mechanism.** The −½𝔖(g) t ln²t is the excluded term h = 0: in the exact divisor decomposition (Part I, Lemma 2.1 and §3) that term\nlies in every class d₁ | h, and removing it costs −t κ(0) Σ_{1<d₁<t/Q₀} α(d₁) = −½𝔖(g) t ln²t/K + O(t ln t). Everything else is a\nlattice-point remainder in Fejér form (Lemma 2.2); the binary sectors are bounded by Weil (Prop. 2, power saving, main term t ln t) and\nthe ternary sector by Weil, Bourgain–Garaev Thm 3 on cells, and Bourgain–Garaev Thm 5 on the small \"Bad\" set of cells where neither\nsaves (Prop. 3, saving (ln t)^{−1/2} on a set of log-coordinate volume (lnT/T)^{3/4} · (lnT/T)^{1/4}, T = ln t).\n\n**Placement on the record.** The person asked for a route-177 return. The server took no route report on this planning job (400: \"this assignment belongs to no research route\"), so this return carries the formalization with its premises in `cites`, associations to route 177 in `research_links`, a review request, and a proposed next step under the direction; a linked route proposal (parent 177) remains possible if the reviewers prefer that placement.\n\n**Premises (record).** #1315 (exact D(H) column, fit a ≈ 0.383), #1302/#1309 (variance reading, measured), #1317 (g = 2 factorisation,\ncentring), #1834 (Fourier side; one-sided limsup a ≤ 1/(4C₂) with (III) sketched), #2300 (exact C, defect to 5·10⁷; finite data pin a\nonly to [0.368, 0.3865]). Also used: #2170 (the p = 2 normalisation), #5553/#5570/#6156/#6708 (route 177's spectral objects).\n**What is new relative to the record:** the two-sided asymptotic with an error exponent, and the Kloosterman/Bourgain–Garaev input for\na sum of singular series. The ½ law itself was reached independently by #1315's fit and the submitter's project (2026-10-23); this is\nindependent rediscovery of the value, not a novelty claim for it.\n\n**Baselines (register's baseline duty).** Improvement claim: a two-sided asymptotic with error O(H(ln H)^{3/2+ε}) against (i) the\nstarting point #1834: limsup a ≤ 1/(4C₂) with (II) ≥ 0 and (III) = O(1) sketched (one-sided, no lower bound), and (ii) the published\nstate: no asymptotic for this one-parameter family of 4-tuples was found (search record in `prior_art_2026-11-04.md`;\nMontgomery–Soundararajan 2004 and Kuperberg's papers treat unrestricted, progression or smooth-weighted sums). Not established until\nreviewed; no ablation applies (one argument, not a tuned method).\n\n## 2. Check against the register's standards (direction item 1)\n\nFull memo: `review_check_a.md`. Summary, per imported theorem at its points of use:\n\n| import | source checked | uses | verdict |\n| --- | --- | --- | --- |\n| [W] Weil–Estermann | Iwaniec–Kowalski Cor. 11.12 as quoted by Bourgain–Garaev (TeX read) | Lemma C1 completion step; (aℓ̄, m) = 1 holds; reduced modulus keeps coprimality because m_i is squarefree | met |\n| [BG3] Bourgain–Garaev Thm 3 | arXiv:1309.1124 TeX, label `thm:Kloost Karatsuba range` | C2 (sub-intervals by coefficients, Möbius for (x₁,x₂) = 1 with (aδ̄², m) = 1), E2 R2 on (e,f), R4 on (x,y),(x′,y′) | met; quotation exact, incl. the constant and the uniformity in k₁,k₂ that the Bad-set analysis needs; an implicit N_i ≤ m in the paper's last step is satisfied (N_i ≤ m^{1/2+o(1)}) |\n| [BG5] Bourgain–Garaev Thm 5 | same TeX, label `thm:linear sqrt log m` | C3 at c = 1/5, N′ > m^{c/2} | met; the printed statement leaves the lnln-exponent's dependence on c ambiguous, the proof (TeX l. 854) takes r absolute, so the note's \"r absolute\" holds by the proof text; \"small c\" is monotone, so c = 1/5 is covered |\n| [Sh] Shiu 1980 | standard statement (original not inspected) | §A mean values of β′ on [x, x(1+h)], h ≥ x^{−1/3}; E4 on 𝒞/x with length ≥ (D/x)^{1/2} | met (β′ ∈ Shiu's class; ranges in order) |\n| [H] Henriot 2012 | arXiv:1102.1643 abstract; Thm 5/Cor. 1 locator not inspected | (3.2) only | not critical: any fixed B₀ works; elementary Hölder substitute gives B₀ = 526 |\n\nInternal checks: Lemma C1's completion and the choice S = N^{1/2}m^{−1/4} are right; Σ_ℓ|c_ℓ|/ℓ ≪ σ₋₁(D) holds with a bounded\nextra factor (local ratio ≤ 1.3333 at p = 5, → 1). C2's Möbius tail T^{−90} beats the cell sizes T^{−40}/T^{−80}. C3's partial summation\nand the N^{2/3} ≤ N m^{−c/3} step need N ≥ m^c, which is the hypothesis. E3's inequalities were confirmed by LP (s ≥ ½ − 6η, Q ≥ ½ − 2η,\nw_e ≤ ½ + 4η); the exponent arithmetic of E3–E5 was recomputed by hand (Σ_k X_k² ≍ T^{−3/4}(lnT)^{3/4}, Σ_k k^{1/2}X_k ≍\nT^{−1/8}(lnT)^{1/8}, mass tT²(lnT)³, final tT^{3/2}(lnT)^{r+4}); J²X_J tends to a constant ≈ 264–268 (the note's \"≪ 1\" hides it).\nPart I's constants: K κ(0) C_α = 𝔖(2) to 5·10⁻¹⁴ in my products, to 10⁻¹⁶ in the referees' exact-tail check; the §3 decomposition is an\nexact identity in rational arithmetic (17,642 classes). Bourgain–Garaev's Corollary 1 is indeed too strong as printed (its hypothesis\nlets one length be bounded); the note does not use it.\n\n**Not checked to proof standard** (and the note says so): the Part II.E bookkeeping; the τ(m)τ(D) averages against C₁ = 50.\n**Independent re-reading:** a fresh sub-agent referee read the note and the TeX with the same questions and no sight of my findings;\nits report is §9 of `review_check_a.md`: 30 tagged items, 26 confirmed, 0 defects, 4 unclear at constant level. Its additions: the implicit N_i ≤ m in [BG3]; r absolute by [BG5]'s proof line; an explicit counterexample for the paper's Cor. 1 as printed; a 4η/5η typo in E3(i); E4 borrows E3(iii)'s constants although the larger split side has k(w_max) ≤ 6 and the modulus exponent is ≥ 2/3, so Ł′_k should take k₁ ≤ 6 (constant level, asymptotic claim unaffected); the R1 budget for C₁ = 50 closes but is not itemised; one spare logarithm in (lnT)^{r+5}. Its verdict: no imported hypothesis is used outside its range.\n\n## 3. The ρ_H → ½ vs 1 residual (direction item 4)\n\nAnswered, and already on the record. Route 177's probe (#2162) computed ρ_H = Σ_{even h}(H−|h|)F(h)/H² with F = Π_{p≥3} f_p and got\n→ ½; #2170 (2026-10-02) found the missing factor is the p = 2 local factor, S₄(h) = 2(2C₂)²F(h), checked against the served code of\n#1302/#1315, and #5553 (accepted) carries S₄ = 2σ²F. The note's §0 says the same thing in other words: 𝔖(0,2,h,h+2) vanishes on odd h,\nso its mean over even h is 2𝔖(2)² while its mean over all h is 𝔖(2)²; normalising by the all-h mean while summing even h is the\nfactor 2, and that factor is 8/4, the p = 2 factor of 𝔖(0,2,h,h+2) over 𝔖(2)². Recomputed here (`norm_check.py`, independent code):\n\n| H | ρ without the p = 2 factor | 1 − D/(A²H²) | D/(A²H) | #1315 |\n| --- | --- | --- | --- | --- |\n| 10³ | 0.482972 | 0.965944 | 34.0561 | 34.06 |\n| 10⁴ | 0.497349 | 0.994698 | 53.0169 | 53.02 |\n| 10⁵ | 0.499621 | 0.999241 | 75.8887 | 75.89 |\n\nThe first column reproduces #2162's probe values (0.48297…), the third #1315's column. So the §0 normalisation resolves the residual;\nthe note's \"HEURISTIC attribution\" should become a citation of #2170 and #5553.\n\n## 4. What stays open (direction item 3)\n\n- **b and c (route 107's target).** Open. Need o(H ln H) for b and o(H) for c; the derivation's error is O(H (ln H)^{3/2+ε}). The note's\n  Part VII says r-fold splits of d₂, d₃ suggest t(ln t)^{5/4+1/(2r)} (heuristic, not to proof standard). A power saving for linear\n  Kloosterman sums of length ≈ m^{1/2} to composite moduli, or a Thm-3 variant without the (log m)^{2(k₁/k₂+k₂/k₁)} factor, would give\n  3/2 → 1 + o(1).\n- **Cell-partition bookkeeping (Part II.E).** Unwritten: boundary cells (relative mass ≪ 1/Δ), atoms n < Δ³, the hyperbola constraint\n  xy ∈ 𝒞 for the (x,y)-refinement, the L-range cuts, and the τ-average accounting against C₁. Routine in the author's view; a reviewer\n  must see it written.\n- **Variance reading.** Conditional on a uniform Hardy–Littlewood 4-tuple hypothesis; the reduction from primes to S₄ and the 3-point\n  companion sum are not written. The singular-series statement itself is unconditional.\n- **Consequence for route 177 (conditional, not claimed).** Through #1834's exact split D/(A²H) = (I) − (II) + (III) with (I) =\n  ln²R/(4C₂) + b₁ ln R + c₁ + O(R^{−1/2+ε}) and (III) = O(1) at R = H ln^{10}H (a sketch in #1834), Claim 1 gives (II) = O((ln H)^{3/2+ε}),\n  i.e. H·(II), the spectral remainder of #5553/#5570 (|E_H|² = H·F_H and r(d,a) = |τ(a/d)|²), is O(H (ln H)^{3/2+ε}). That is route 177's\n  B₁ target (both parities) at o(H log²H), the object #6708 left open as B_mid. Two things must be rigorous first: the derivation, and\n  #1834's (III) = O(1). Proposed as the next step.\n\n## 5. Finite checks (MEASURED; they do not prove the asymptotic)\n\nAll referee validators (`context/validators/`, written against earlier drafts; labels explained in `validators/README.md`) were run under\nthe department's bounded executor (wall time, RLIMIT_AS, process-group cleanup); outputs are attached.\n\n| check | result here |\n| --- | --- |\n| Lemma A2 phase forms (`p2_check_A2_C.py`, `d2form.py`) | 0 mismatches of 2,400 and of 3,000; reduced modulus needs g₀ = (kX, m): (k, m) fails in 1,071 of 2,400 |\n| Lemma 2.3 constants (`ident2.py`, exact prime-zeta tails) | ratios − 1 at 10⁻¹⁶ for g = 2, 6, 30; phase forms 2,000/2,000 for g = 2, 6, 10, 30 |\n| §3 decomposition (`decomp.py`) | exact in rationals, 0 degenerate-rule mismatches in 17,642 classes |\n| E3(i) by LP (`p2_check_E3_ineq.py`) | Q ≥ ½ − 2η exact; s ≥ ½ − 6η; w_e ≤ ½ + 4.2η; J²X_J = 263.4…267.7 for T = 10¹⁰…10¹⁶⁰ |\n| vol(Bad), P₂ (`p2_volbad_p2.py`, `volbad2.py`) | `p2_volbad_p2.py` (416 s): vol(Bad)/(lnT/T)^{3/4} = 0.11, 0.10, 0.10, 0.11, 0.11; sup P₂/(lnT/T)^{1/8} = 3.76, 2.92, 2.19, 2.06, 2.06 at T = 10⁸…10¹⁶ (converging; note: 0.10–0.11 and → 2.06). `volbad2.py` killed by the executor at its 1500 s budget after the rows to ln t = 10¹⁶: vol·(ln t/lnln t)^{3/4} = 0.215, 0.206, 0.208, 0.212, 0.215 for 10⁸…10¹⁶, log-slope −0.72 (reference −3/4); rows 10²⁰…10³⁰ not reached |\n| split mass on integers (`p2_check_E4_small.py`) | F/θ^{1/2} = 1.51, 1.38, 1.38, 1.41 at θ = 0.1, 0.2, 0.3, 0.5 (note: 1.4–1.5); **anomaly:** F is flat at 0.3196 for θ ≤ 0.05 (3.2·θ^{1/2} at θ = 0.01): the trivial splits x = 1, y = 1. E4's small-side line should read ≪ (D/Δ)[θ^{1/2} + (ln D)^{−1/2}]; the conclusion P₂ ≪ (lnT/T)^{1/8} is unaffected (memo §3a) |\n| direct S₄ and ln² fit (`s4direct.py`, `fit.py`, t ≤ 5·10⁶ here; the note used 2·10⁷) | fitted ln²t coefficient over the theoretical −½κ(0)C_α: 0.991/0.960/0.974 for g = 2/6/30 on t ≥ 10⁴, 0.973/0.994/0.980 on t ≥ 10⁵; the 0.7-decade window t ≥ 10⁶ is unstable (1.221/1.060/0.930). The note's 2·10⁷ runs give 0.35–0.78 %. Exact top-term residual (R/t + κ(0)Σ_{1<d<t/Q₀}α)/ln t for g = 2: −10.2 at 10⁴ → −8.7 at 5·10⁶ (top term −36) |\n| own: defect column and normalisation (`norm_check.py`) | PASS on all three pre-registered bars (§3); C₄ agrees with #5557's certified value to 1.1·10⁻¹⁰ |\n| own: model-free top-term test (`toptail_check.py`) | Z(H) = D/(A²H) − Σ_{1<d<H/6}α(d)/(AC_α); Z/ln H = 1.80 → 1.52 on 10⁴…10⁷, slope −0.042 per unit ln H (bar |slope| < 0.05): PASS, weak power |\n\nAnomalies and limits, stated as such: the model-free test passes its pre-registered bar but cannot detect a residual ln²H term of a\nfew percent (its slope would be ≈ +0.02); the finite top term Σ_{1<d<H/6}α(d)/(AC_α) divided by ln²H is still 0.419 at H = 10⁷ against\nthe asymptotic 0.3787, which is why free fits of a ln²H + b ln H + c land at 0.383 (#1315, #2170) and why #2300 found a pinned only to\n±0.01. The note's own pre-registered local checks had two FAILED bars (ternary ln² coefficient 0.0235 vs 0.02 for g = 2; g = 6 total\n0.58% vs 0.5%), recorded in its Part IV; they are reproduced here as the author reported them, not rerun (they need a local prime table\nto 10⁸).\n\n## 6. Files, recipe, cost\n\nFiles: the derivation; the 16 validators and their README; the review memo; the prior-art record; two own scripts with outputs;\nvalidator outputs under the bounded executor and its manifest. Recipe in `recipe_md`. Cost: 0.55 CPU-h on one core.\nPerson's direction (verbatim) in `human_md`.\n\n## Sources\n- Return #1315 (@natepac), files ssum2549.py/json (sha256 b9ded625…6631 for the json): the D(H)/(A²H) column; externally reported, recomputed here from the divisor form.\n- Returns #1302, #1309 (@natepac): the variance and covariance measurements (route 35); cited, not rerun.\n- Return #1317 (@nielsegberts), triage-proof.txt: centring, exceptional shifts, non-multiplicativity in h.\n- Return #1834 (@Benjaminsen), reduction2669.md: |τ(b/r)|² weights, exact split, one-sided bound, (III) sketch.\n- Return #2300 (@Benjaminsen), c_exact.json (sha256 152b05b7…c604): exact C, finite fits to 5·10⁷.\n- Returns #2162, #2170 (route 177 origin and first look): the ρ_H → ½ probe and its p = 2 resolution.\n- Returns #5553, #5570, #6156, #6708 (route 177): spectral objects and the open B_mid step. Return #6148 (route 107 `known`), #5557 (C₄ certified).\n- J. Bourgain, M. Z. Garaev, Kloosterman sums in residue rings, Acta Arith. 164 (2014) 43–64; arXiv:1309.1124, TeX source file BGklcomp17.tex: Theorems labelled `thm:Kloost Karatsuba range` (Thm 3), `thm:linear sqrt log m` (Thm 5), `cor:Kloost Karatsuba rang` (Cor. 1), §1 conventions.\n- T. Estermann, Mathematika 8 (1961); H. Iwaniec, E. Kowalski, Analytic Number Theory, Cor. 11.12 (as quoted by Bourgain–Garaev; book not seen here).\n- P. Shiu, J. reine angew. Math. 313 (1980) 161–170 (statement as standard; original not inspected here).\n- K. Henriot, Math. Proc. Camb. Phil. Soc. 152 (2012) 405–424, arXiv:1102.1643 (abstract inspected).\n- H. L. Montgomery, K. Soundararajan, Primes in short intervals, Comm. Math. Phys. 252 (2004), arXiv:math/0409258 (via #1317/#1834's readings).\n- V. Kuperberg, arXiv:2301.06095 (IJNT 2025), arXiv:2210.09775, arXiv:2109.03767; V. Kuperberg, B. Rodgers, E. Roditty-Gershon, arXiv:2001.09513 (listings/abstracts).\n- Local: `context/route177_twin_defect_a.md` (uploaded as route177_twin_defect_a.md, sha256 9858be4a…dde8), `context/validators/*` (uploaded).\n","patch":null,"cpu_hours":0.55,"hashes":{"norm_check.py":"280ae67d214e357b11bbf41bb668d8701b44c281042545d46672687028ff8bdd","norm_check.out":"598c2d467d8d1bd48a8f1fd9d90dfd502a9d17d5e1d32b8babaf30b4bf7c1f7d","toptail_check.py":"954ac67e2cb47ece00532bac39e7cdc901926af1dcd202a45dd23eafccfd22bc","review_check_a.md":"b97213df6c7329684d93ff91f0ae854ff12498772049d03c2c006e33ae28dcb8","toptail_check.out":"85aa6bdc6311a897535cce087fd15751b2a7b5aa372d2220ac71d96e34968b51","route177_twin_defect_a.md":"9858be4a4359508c5373b6b71624a761e11f75b9e41395f6d3a6d6703ab2dde8"},"author_rung":"heuristic","status":"pending","final_rung":null,"created_at":"2026-11-04T21:16:49.709Z","repo_url":null,"commit":null,"cites":{"files":["b9ded62512abfb33fedd8829f494be71dcf4440adb56fcb4f737f5f3bafc6631","152b05b7958e3a6fd75d1e855dd1294c160ad5562498a230b637046e507ec604"],"handles":[],"returns":[1315,1302,1309,1317,1834,2300,2162,2170,5553,5557,5570,5610,5616,6148,6156,6708],"messages":[]},"tokens":{"log":"summary","input":1590,"models":{"claude-fable-5-1":388518},"output":388518,"source":"reported","entries":0,"cache_read":15030187,"cache_write":842605,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe (job #13674)\n\nFetch each file with `GET <server origin>/files/<sha256>?raw=1`, `Accept: text/plain`, and verify the raw sha256 before use.\nEnvironment used here: Python 3.14.9, numpy 2.5.4, sympy 1.14.0, scipy 1.18.1, mpmath 1.4.1, one CPU core; all scripts are deterministic (no randomness beyond fixed seeds in the validators).\n\n1. Own checks (seconds). `python3 norm_check.py > norm_check.out` — expected sha256 of norm_check.out: 598c2d467d8d1bd48a8f1fd9d90dfd502a9d17d5e1d32b8babaf30b4bf7c1f7d. Prints three PASS lines (C₄ vs #5557; #1315's column to 0.01; ρ with/without the p = 2 factor).\n   `python3 toptail_check.py > toptail_check.out` (≈2 s, 1 GB) — expected sha256: 85aa6bdc6311a897535cce087fd15751b2a7b5aa372d2220ac71d96e34968b51. Prints PASS for Lemma 2.3(b) (|ratio − 1| < 1e-6) and PASS for the model-free test (max |Z/ln H| < 3, |slope| < 0.05).\n2. Exact validators (≈1 min total): `python3 p2_check_A2_C.py`, `python3 d2form.py`, `python3 p2_check_E3_ineq.py`, `python3 ident2.py`, `python3 decomp.py`. Expected: \"phase-form mismatches: 0\"; \"d2-form mismatches 0, merged-A mismatches 0\"; LP lines with Qmin = 1/2 − 2η; ratios −1 at 1e-16; \"equal: True ... mismatches 0\". Attached outputs *.out carry the exact text produced here.\n3. Numerical validators (minutes each; memory ≤ 2 GB): `python3 volbad2.py`, `python3 p2_volbad_p2.py`, `python3 p2_check_E4_small.py`; then `python3 s4direct.py 2 5e6`, `python3 s4direct.py 6 5e6`, `python3 s4direct.py 30 5e6` (writes s4_g{2,6,30}.npz beside ident2.py) and `python3 fit.py`. Expected outputs are the attached *.out files; floating-point sums may differ in the last digits across BLAS builds, so compare the printed ratios to 3 significant figures, not byte for byte.\n4. Reading check: compare the quotations of Bourgain–Garaev Thms 3 and 5 in review_check_a.md §1.2–1.3 with the arXiv source https://arxiv.org/e-print/1309.1124 (file BGklcomp17.tex, labels thm:Kloost Karatsuba range, thm:linear sqrt log m, cor:Kloost Karatsuba rang).\n5. The derivation itself (route177_twin_defect_a.md) is a proof text; its review is the open obligation, see the report's §2 and §4.","verification":null,"target":null,"finding":null,"human_md":"Twin prime, routes 107 and 177 (twin-pair defect). Context: the file context/route177_twin_defect_a.md in this folder contains a\nwritten-out derivation (grade DERIVED, not yet reviewed by a trusted reviewer) that the ln²H coefficient of D(H) (#1315/#1834\nnotation) is a = 1/(4C₂) from both sides: D(H)/(A²H) = ln²H/(4C₂) + O((ln H)^{3/2+ε}), and for every even g,\nS₄(t) = 𝔖(g)²t² − ½𝔖(g) t ln²t + O(t(ln t)^{3/2+ε}). It imports Weil–Estermann, Bourgain–Garaev 2014 Thms 3 and 5, Shiu 1980\nand Henriot 2012 (locators in its Part III). Standalone validators are in context/validators/.\n\nPlease:\n(1) Check the derivation against the register's standards. Start with the imported theorems' hypotheses at each point of use\n    (Part II, Lemmas C1–C3 and E3–E5).\n(2) Formalize it as a route-177 return, citing #1315, #1302/#1309, #1317, #1834 and #2300 as premises.\n(3) Record what stays open: b and c (they need errors o(H ln H) and o(H)); the cell-partition bookkeeping; the variance\n    reading, which is conditional on uniform HL.\n(4) Check whether the parity normalisation in its §0 resolves route 177's ρ_H → ½ vs 1 residual.\n\nDo not upgrade the grade beyond DERIVED without a trusted reviewer.\n\nWork style for every assignment:\n- Test ideas on data first. Before computing, write down the prediction, the null model and the pass/fail bar, then compare.\n- Check prior art before calling anything new; say \"novel to us\" vs \"novel\".\n- Have each claim independently re-checked before submitting: a fresh sub-agent, or an independent recomputation.\n- Report anomalies and failed predictions as clearly as successes, even when they break a pattern.\n- Stay inside this folder; do not read other local directories.","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":"56dba154455f660df9d3c42bad936842a1178ad80a7fb785d62a57d52f34f094","name":"s4direct.py","notes":["prints what looks like progress or timing to stdout on line 27 (\"print(f\"sieve {time.time()-t0:.0f}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":"024f7cba56e45f0df1773d0cf9de0e8505360003928254ae2facd45a23527a93","name":"split66.py","notes":["prints what looks like progress or timing to stdout on line 156 (\"f\"P3(1/2)={P3c:.4f} | {time.time()-t0:.0f}s\", flush=True)\"), inside the statement that starts on line 154: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. 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