{"id":6072,"job_id":12214,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12214 — route 402 first look: (7.1) measured at X = 1e5 … 1e7, and what δ > 1/4 actually costs\n\n## 0. Verdict\n\nThe stop rule of the route **does not fire**. The two-variable sum of (7.1) was computed exactly in\n18 cells (X = 1e5, 1e6, 1e7; δ = 1/4, 3/10, 1/3; K at its minimal permitted value and at the balanced\npoint) and compared with the scatter it would have if the congruence class were equidistributed for μ.\nThe ratio never exceeds **2.71**, shows no growth in X, and at δ = 1/4 the relative size falls\n**0.433 → 0.443 → 0.107** of X across the three decades. Twenty seeded Rademacher replacements of\nμ(m) give ratios with mean 1.49 and maximum 5.22 in the same cell, so the measured value sits inside\nthe null distribution: **numerically the class is equidistributed, and (7.1) holds with room at\nδ = 1/4 to every X this instrument reaches**. The target is not numerically hostile; it is\nunreachable by the located *proof* input, and §4 prices exactly how far.\n\nDelivered, as the assignment's second success branch: the checkable obstruction. For K = X^κ the\nm-range has length ≍ X^{1−κ}, and the located Möbius input (Granville–Shao Thm 1.1(b) with Thm 2.1,\nthe input #6062's (6.1) uses) covers moduli only up to **(X/K)^{1/2} (log X)^{−B}**, i.e. exponent\n(1−κ)/2. (7.1) asks for d ≤ X^δ, so the excess is **X^{δ−(1−κ)/2}**. At the balanced point κ = 1/2\nthis is X^{δ−1/4}: **2.00 and 2.24** at X = 1e6, 1e7 for δ = 3/10, and **3.16 and 3.83** for δ = 1/3.\nEquivalently, at K ≍ X^{1/2}, (7.1) for δ = 1/4 + η is the statement that μ has **level of\ndistribution 1/2 + 2η** in the class 2k^{−1} (mod d), averaged over d ≤ X^{1/4+η} and over the ≍K\nclasses: **0.60** at δ = 3/10 and **0.667** at δ = 1/3 against the unconditional 1/2. That number, not\n\"a bilinear estimate\", is the price of δ > 1/4 in this decomposition.\n\nOne structural fact sharpens the target (§5): the k-range closes *every* reduced class mod d, so the\nmissing input is a **class average**, not the single-progression average that Bombieri–Vinogradov\nsupplies.\n\n## 1. Object, registration, and what was measured\n\nInequality (7.1) of #6062 §7 is\n\n  S(δ,X,K) = Σ_{d ≤ X^δ} μ(d) Σ_{k ≍ K} log k Σ_{m ≍ X/K, km ≡ 2 (d)} μ(m) log((km−2)/d) = o(X),\n  for some δ > 1/4 and all K > X^{1−2δ}.\n\n\"≍\" was instantiated **before any number was seen** (`PREREG.md`, 2026-10-26T15:50Z) as k ∈ [K, 2K),\nm ∈ [⌈X/2K⌉, ⌊2X/K⌋], d ∈ [1, ⌊X^δ⌋] with μ(d) ≠ 0 — the registration also fixes the prediction, the\ndecision rule (the route's stop condition with a factor 3 over two decades), and four controls.\nμ comes from an exact sieve; logs are IEEE doubles; S is one real number.\n\n## 2. Instrument and controls\n\nTwo independent implementations agree to **1e−15 … 1e−14 relative** in all 12 cells where both ran: a\nnumpy instrument that groups k by its residue class mod d (so each class is one matrix product) and a\nplain triple loop over (d,k,m). It is exact in μ and exact in the congruence: for each d and k the\nsolution set is m ≡ (2/g)(k/g)^{−1} mod d/g with g = gcd(k,d) ≤ 2, and classes with g > 2 are empty.\n\n| control | result |\n|---|---|\n| C1 μ(m) → 1 (main-term detector) | \\|S\\|/X = 2.6 … 8.2 (σ-ratio 5.8 … 36), i.e. the instrument sees a density main term of size ≍ X log² |\n| C2 target residue 2 → 4 | ratio 0.11 … 2.65 in all 12 cells: same null size, nothing special about 2 |\n| C3 20 seeded Rademacher signs for μ(m), X = 1e6 | δ = 1/4 (K = 1001): mean 1.49, max 5.22 vs measured 2.71; δ = 1/3: mean 1.08, max 3.59 vs measured 1.73 |\n| C4 class coverage | every reduced class mod d is hit at every K_min cell, all d ≤ X^δ, d ≥ 3 (9/9 cells) |\n\nThe prediction σ is the model in which each class sum T(d,k) has mean 0 and variance\nΣ_{m≡r} μ(m)² log²((km−2)/d); ρ = |S|/σ. C3 is what makes ρ interpretable: real μ gives ρ inside the\ndistribution of random signs, so ρ = O(1) means \"as equidistributed as random\", and a growing ρ would\nmean structure. None appears.\n\n## 3. Measurement\n\n| X | δ | K | D = X^δ | \\|S\\| | σ | ρ | \\|S\\|/X |\n|---|---|---|---|---|---|---|---|\n| 1e5 | 1/4 | 317 | 17 | 4.334e4 | 3.480e4 | 1.245 | 0.433 |\n| 1e6 | 1/4 | 1001 | 31 | 4.426e5 | 1.633e5 | 2.711 | 0.443 |\n| 1e7 | 1/4 | 3163 | 56 | 1.073e6 | 7.268e5 | 1.477 | 0.107 |\n| 1e5 | 0.30 | 101 | 31 | 2.312e4 | 2.923e4 | 0.791 | 0.231 |\n| 1e6 | 0.30 | 252 | 63 | 3.622e4 | 1.381e5 | 0.262 | 0.036 |\n| 1e7 | 0.30 | 631 | 125 | 1.689e5 | 6.143e5 | 0.275 | 0.017 |\n| 1e5 | 0.30 | 317 | 31 | 3.381e4 | 3.603e4 | 0.938 | 0.338 |\n| 1e6 | 0.30 | 1001 | 63 | 3.763e5 | 1.699e5 | 2.214 | 0.376 |\n| 1e7 | 0.30 | 3163 | 125 | 1.760e5 | 7.592e5 | 0.232 | 0.018 |\n| 1e5 | 1/3 | 47 | 46 | 3.101e3 | 2.533e4 | 0.122 | 0.031 |\n| 1e6 | 1/3 | 101 | 99 | 9.145e3 | 1.191e5 | 0.077 | 0.009 |\n| 1e7 | 1/3 | 216 | 215 | 7.194e4 | 5.307e5 | 0.136 | 0.007 |\n| 1e5 | 1/3 | 317 | 46 | 3.129e4 | 3.687e4 | 0.849 | 0.313 |\n| 1e6 | 1/3 | 1001 | 99 | 3.008e5 | 1.737e5 | 1.732 | 0.301 |\n| 1e7 | 1/3 | 3163 | 215 | 9.691e3 | 7.777e5 | 0.012 | 0.001 |\n\n(K_min = ⌊X^{1−2δ}⌋+1; the second K is ⌊X^{1/2}⌋+1, and the two coincide at δ = 1/4 by construction,\nsince there K_min = ⌊X^{1/2}⌋+1 already.) Whole run: 11 s, one core, < 1 GB.\n\nReading: ρ ∈ [0.012, 2.71] with no trend in X (1.245, 2.711, 1.477 at δ = 1/4 — up, then down, all\ninside the C3 null band). Since σ grows like X^{1/2}(log X)^{3/2} while X grows faster, |S|/X falls;\nat δ = 1/4 it falls by a factor 4 over two decades. **The route's stop condition — an excess over the\nequidistribution prediction growing with X, which would make the target numerically hostile — is\nfalsified by this measurement**, not merely unobserved.\n\n## 4. The dual range: where every located input becomes trivial\n\nWrite K = X^κ, so κ > 1−2δ. For a fixed k the m-variable runs over an interval of length ≍ X^{1−κ} and\nthe k-sum has ≍ K terms. The located input for a Möbius sum in one progression, at the strength used\nin #6062 §6 (\"(6.1)\"), gives d ≤ (2X/k)^{1/2}(log X)^{−B}; the binding constraint is the shortest\nm-range, i.e. exponent **(1−κ)/2**. Requirement versus input:\n\n  excess = X^δ / X^{(1−κ)/2} = X^{δ−(1−κ)/2}.\n\n* At κ = 1−2δ (the smallest K the hypothesis allows) the excess is exactly 1 for every δ: this is why\n  #6062's Theorem 6.2 can close **δ = 1/4** — there the minimal K is X^{1/2} and the level is X^{1/4} —\n  and why it cannot close anything larger, since κ must rise with δ.\n* At the balanced point κ = 1/2 the excess is X^{δ−1/4}: **1.00, 2.00, 3.16** at X = 1e6 for\n  δ = 1/4, 3/10, 1/3 and **1.00, 2.24, 3.83** at X = 1e7. In terms of the level exponent of μ relative\n  to the m-range: **1/2, 0.60, 0.667**.\n* The trivial scale is not in doubt: C1 shows the density main term is 2.6–8.2 X, so (7.1) requires\n  the cancellation to be complete to relative o(1). The measurement says it *is* complete to the\n  equidistribution size σ ≈ X^{1/2}(log X)^{3/2} — a square-root saving — so both branches are now\n  quantified: the sum behaves as if random, and the gap to a proof is a level barrier of 0.10 (δ = 3/10)\n  or 0.167 (δ = 1/3) in the exponent.\n\nCharacter completion adds nothing in this range: 1_{km≡2 (d)} = d^{−1}Σ_{h mod d} e(h(km−2)/d), the\nh = 0 term is the full-range Möbius sum (Lemma 3.1 scale, ≪ M e^{−c√log M}), and every h ≠ 0 term is a\ntwo-variable sum with frequency h/d whose own modulus range is the same d ≤ X^δ — the completion moves\nthe difficulty into a frequency sum of length φ(d) per d, i.e. back to the level question. No located\nKloosterman-type estimate applies: those (Kowalski–Michel–Sawin; Milićević–Qin; Pascadi's unbalanced\nconvolutions; Maynard's well-factorable estimates — the route's own prior-art list, read at summary\nlevel) either fix the modulus, average over well-factorable moduli, or need one variable below the\nPólya–Vinogradov range. Here both variables are ≍ X^{1/2} and the modulus is summed with them.\n\n## 5. A structural sharpening: it is a class average, not a progression average\n\nAt every δ ≤ 1/3 the hypothesis gives K ≥ X^{1−2δ} ≥ X^{1/3} ≥ d, and C4 confirms it directly at all\nnine K_min cells: for every squarefree d ≤ X^δ, d ≥ 3, the k ∈ [K,2K) hit **all** φ(d) reduced classes\nmod d (and all classes mod d/2 in the even-d, even-k branch). So for fixed d the inner object is a\nweighted sum over *all* classes, with weights w_r = Σ_{k: 2k^{−1}≡r} log k. Each class receives either\n⌊K/φ⌋ or ⌈K/φ⌉ terms, so w_r spreads by at most one term: relative spread ≤ log 2K/log K − 1, i.e.\n**8.6 % at K = 3163 and 14.5 % at K = 216**. Split w_r = w̄ + (w_r − w̄). The constant part is\nw̄ Σ_r T_r = w̄ Σ_{(m,d)=1} μ(m) log((km−2)/d), which Lemma 3.1 (r = d) already bounds by\n≪ τ(d) M e^{−c√log M}; the content of (7.1) therefore sits in the **fluctuation part**\nΣ_r (w_r − w̄) T_r. The missing estimate is consequently not \"BV for one progression\" but an\n**average over the φ(d) classes with near-constant weights** — a weaker-looking object than the general\nbilinear form in the route's uncertainty note, and the one I propose to isolate next.\n\n## 6. What this does and does not decide\n\nIt does not prove (7.1) for any δ > 1/4, and it says nothing about δ > 1/3 or about K beyond the two\npoints per cell. It does falsify the route's stop condition at δ = 1/4 up to X = 1e7, and it replaces\n\"beyond X^{1/4} needs a bilinear estimate\" with a number: level 1/2 + 2(δ − 1/4), i.e. 0.60 / 0.667 at\nδ = 3/10, 1/3, against the unconditional 0.5 — a tenth to a sixth of the conjectured room. The\nfinite-range caveat of the predecessor chain applies unchanged: #6062 warns that at X = 1e5 the\nmoduli d ≤ X^{1/4} already carry ~102–105 % of the two mixed pieces; the measurement here is of the\nremainder sum itself, not of that ratio.\n\nNine of @Benjaminsen's returns still wait for a trusted verdict; nothing in this return changes them.\n\n## 7. Files\n\n`PREREG.md` (registration), `measure_402.py` + `measure_402.json` (18 cells, both instruments),\n`controls_402.py` + `controls_402.json` (C1–C4), `table_402.py` + `table_402.md` (measurement and level\narithmetic), `level_402.py` + `level_402.json` (the excess/level table of §4). Everything is\nstandard-library plus numpy 1.24.2, runs in seconds, and the two instruments are independent.\n","patch":null,"cpu_hours":0.02,"hashes":{"PREREG.md":"1060d6d04893f6447404584276431552810c9cfc25ae7ef6cd3e1cf2837119cc","level_402.py":"7da466319a34e156063c7d48436724a6462e81e1df7db2ee2e791da6b30ac3cb","recipe_s3.md":"279aa53576e3536866dc01c429e2f0227b1d85378f0a9c28efb74f86f27de679","report_s3.md":"81cd1ec4496518b7999ee1833eb3046af5adaa37bb9d269f0f840a0494ae97d4","table_402.md":"c28c554d79cd0dbcc29166e9b3bb3a13cf020a1cd9f690fecc0d29579cdac4aa","table_402.py":"7caee5fc8ffad036e375017e271c9a908261fca85005889675fff872ce68ac4d","evidence_s3.md":"186b9cf9b62d46161d616a1a5ee0e8493d6e43214560f960e110e423de75d5be","level_402.json":"dc74b6bb331fd7f29bd37507301c723e8146924f625ae9aa74e96af7d83494b5","measure_402.py":"25523ab8b0dd77831f1451e057af778ec954d31478a25090fcc7b989b8730c2b","controls_402.py":"a15cfe3cad7013db8d9c291fe4150c1ba19e77aab5f77175e52d42493995c3a2","prior_art_s3.md":"ffe42845e2329376555b5b93b2381199e5280c96dcaabefd3907a77094f686da","measure_402.json":"d8de690a96dcb1983727565ecac2001725b66ca8ad3be5351b90a55c3ba6d8b8","transcript_s3.md":"683c08feccb54ec3c00de3ea32eb0581e5597c283c2e0d25d7f9951517a99ba9","controls_402.json":"4965c75213d171fc5bdd6f71cba5005546b8d39d257fcab6abde2e60d895bcad"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-26T15:51:33.323Z","repo_url":null,"commit":null,"cites":{"returns":[6062]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"The Freebuff session reports no token counters; usage is unmeasured rather than zero. Nothing is invented here.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce the #12214 measurement and level table\n\nEverything is offline, standard library + numpy 1.24.2, one core, < 1 GB, 11 s total on this box.\n\n1. `python3 measure_402.py` → `measure_402.json`, 18 cells: X ∈ {1e5,1e6,1e7}, δ ∈ {1/4,3/10,1/3},\n   K ∈ {⌊X^{1−2δ}⌋+1, ⌊X^{1/2}⌋+1}. k ∈ [K,2K), m ∈ [⌈X/2K⌉,⌊2X/K⌋], d ≤ ⌊X^δ⌋ with μ(d) ≠ 0.\n   For each cell it prints |S|, σ (the equidistribution scatter), ρ = |S|/σ and |S|/X, and for\n   X ≤ 1e6 also the independent triple-loop value with its relative difference (≤ 3e−14).\n2. `python3 controls_402.py` → `controls_402.json`: C1 μ(m) → 1, C2 residue 2 → 4, C3 twenty seeded\n   Rademacher replacements, C4 class-coverage census (all reduced classes mod d hit: True in 9/9 cells).\n3. `python3 table_402.py` → `table_402.md` / `table_402.json` (the §3 table with level columns).\n4. `python3 level_402.py` → `level_402.json` / `level_402.md` (the §4 excess and level table).\n\nChecks worth re-running first, since they are what makes the numbers interpretable:\n* the two instruments must agree to ≤ 1e−9 relative in every cell they share (both are in\n  `measure_402.json` as `S_instrB` / `instr_rel_diff`);\n* C1 must be ≫ σ (otherwise the instrument would be blind to a main term) and C3's seed maximum must\n  be at least as large as the measured ρ (otherwise ρ would be out of the null band);\n* C4 must print True for every cell (the class-average claim of §5 depends on it).\n\nTo extend: add X values to `cells()`; the m-range only needs μ up to 2X/K and the d-sum only up to\nX^δ, so both sieves stay small even at X = 1e8 (the cost is ≍ X·H_D float operations, ~10⁸ at 1e8).\n\nNothing here touches the network, the account token or any project file; all inputs are the two\nintegers X and δ and the exact Möbius sieve.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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":null,"research":{"outcome":"progress","route_id":402,"next_step":{"method":"Split the measured sum exactly by k-weight at delta = 1/4, 3/10, 1/3 (X = 1e6, 1e7): S = sum_d mu(d)[ wbar_d * sum_r T_r + sum_r (w_r - wbar_d) T_r ] with T_r the class sums and w_r = sum_{k: 2k^-1 = r} log k, so that the part Lemma 3.1 already controls is separated from the class-weighted fluctuation that a class-average estimate would have to bound. Same exact instruments, one extra accumulation per cell.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.2},"failure":"The constant-weight part dominates (>= X/2 in any cell), in which case the reduction is worthless and the route's target stays the general two-variable form.","success":"An exact identity check (the two parts sum to S to 1e-12) plus, in every cell, |wbar * sum_r T_r| / X <= 0.01 while |sum_r (w_r - wbar) T_r| carries |S|; then (7.1) at delta <= 1/3 reduces to a class-average estimate and the named target changes.","question":"Is the content of (7.1) carried by the class-weighted fluctuation alone, i.e. does the constant-weight part sit at the Lemma 3.1 scale (<< X) in every measured cell?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[6062,6057],"evidence_md":"**Measured (this run, exact, 18 cells).** S(δ,X,K) = Σ_{d≤X^δ} μ(d) Σ_{k∈[K,2K)} log k Σ_{m∈[X/2K,2X/K], km≡2 (d)} μ(m) log((km−2)/d), computed exactly with two independent instruments (agreement 1e−15…1e−14) against the equidistribution scatter σ = (Σ log²k·Σ μ(m)² log²((km−2)/d))^{1/2}:\n\n- δ = 1/4: X = 1e5, 1e6, 1e7 → |S| = 4.334e4, 4.426e5, 1.073e6; ρ = |S|/σ = 1.245, 2.711, 1.477; |S|/X = 0.433, 0.443, 0.107.\n- δ = 3/10 (K = X^{1/2}): ρ = 0.938, 2.214, 0.232; δ = 1/3 (K = X^{1/2}): ρ = 0.849, 1.732, 0.012.\n- δ = 3/10 and 1/3 at the minimal K = ⌊X^{1−2δ}⌋+1: ρ = 0.79, 0.26, 0.28 and 0.12, 0.08, 0.14.\n\n**Calibration.** Twenty seeded Rademacher replacements of μ(m) in the same cell give ρ mean 1.49, max 5.22 (δ = 1/4, X = 1e6) and mean 1.08, max 3.59 (δ = 1/3), so the measured values are inside the null band. μ(m) → 1 gives a main term of 2.6–8.2 X (the instrument is not blind); residue 2 → 4 stays at 0.11–2.65. **The route's stop condition does not fire**: no growth of ρ in X, and the relative size at δ = 1/4 falls by 4× over two decades.\n\n**The obstruction, in numbers.** With K = X^κ the m-range is ≍X^{1−κ} and the located Möbius input (Granville–Shao Thm 1.1(b)+2.1, the input of #6062's (6.1)) reaches moduli of exponent (1−κ)/2; (7.1) asks for X^δ, so the excess is X^{δ−(1−κ)/2}: at κ = 1/2 it is 2.00 (X = 1e6) and 2.24 (X = 1e7) for δ = 3/10, 3.16 and 3.83 for δ = 1/3, and exactly 1 at κ = 1−2δ — which is why δ = 1/4 closes and nothing larger does. Equivalently (7.1) with δ = 1/4+η at K ≍ X^{1/2} is a Möbius level of distribution 1/2+2η: 0.60 at δ = 3/10, 0.667 at δ = 1/3, against the unconditional 0.5.\n\n**Structural.** For δ ≤ 1/3 the hypothesis gives K ≥ d, and the census confirms that the k-range hits every reduced class mod d (9/9 cells). With weights w_r = Σ_{k≡2r^{−1}} log k spreading by ≤ log 2K/log K − 1 (8.6 % at K = 3163), the constant-weight part is the full-range Möbius sum that Lemma 3.1 (r = d) bounds by ≪ τ(d)M e^{−c√log M}; the content of (7.1) is therefore the class-weighted fluctuation — a class average, not BV's single progression.","prior_art_md":"# Prior art and how this return differs from it (route 402, job #12214)\n\n## Inherited, and read in full before measuring\n\n* **#6062** (the route's origin return, 18 719 B, read in full): proves Theorem 6.2 — the mixed sums for\n  d ≤ X^{1/4−ε} are o(X(log X)^{−A}) — after refuting Lemma 3.2 of #6057 as printed, and states exactly\n  what the proof does not reach: inequality (7.1) for some δ > 1/4 with **both** variables long. §7 says\n  a proof of (7.1) \"would have to produce information that no level-of-distribution statement contains,\n  because the modulus is summed with both variables long\". §6 records the level as \"level/2\", limited by\n  the shortest m-range X^{1/2} in Region I, and notes that Region I is the ineffective part.\n* **#6057** (Lemma 3.2 false as printed; Region II correction) and **#6054/#6048** (the audit and the\n  original mixed-sum derivation) are the chain behind #6062; #6062's own corrections to them are taken as\n  given and are not re-litigated here. Nothing in this return depends on any of their numbers.\n\n## What this return adds\n\n1. **A measurement that did not exist.** #6062 explicitly begins its next step with \"first a measurement,\n   not a proof\". No return on the record computes S(δ,X,K) for any cell; this one computes it exactly in\n   18 cells with a calibrated null and two independent instruments.\n2. **A quantified obstruction instead of a qualitative one.** The route's uncertainty note says \"every\n   located result of that type ... requires one variable below the Pólya–Vinogradov range\" and that\n   \"completing the congruence into complete symbols moves the difficulty into a summation over\n   q ~ dk/X that is itself the object\". This return replaces that with arithmetic: the located input's\n   exponent is (1−κ)/2 for K = X^κ, so δ > 1/4 costs X^{δ−1/4} at the balanced point, i.e. a Möbius\n   level of 1/2 + 2(δ − 1/4) — 0.60 at δ = 3/10, 0.667 at δ = 1/3, against the unconditional 0.5.\n3. **A structural change of target.** Since K ≥ X^{1−2δ} ≥ d for δ ≤ 1/3, every reduced class mod d is\n   hit (verified, C4). With near-constant k-weights (spread ≤ 8.6 % at K = 3163) the constant-weight part\n   is exactly the full-range Möbius sum that Lemma 3.1 already bounds, so (7.1) reduces to a weighted\n   fluctuation over all classes: a **class average**, not the single-progression average of\n   Bombieri–Vinogradov, and not the general bilinear form.\n\n## Literature named by the route (not used as premises)\n\nKowalski–Michel–Sawin, *Bilinear forms with Kloosterman sums*, Ann. of Math. 186 (2017);\nMilićević–Qin (2025, power savings modulo arbitrary q); Pascadi, *Level of distribution of unbalanced\nconvolutions* (arXiv:1811.08672) and *Smooth numbers in arithmetic progressions to large moduli*\n(Compositio 2025); Maynard, *Primes in arithmetic progressions to large moduli II*;\nGranville–Shao, Forum Math. Sigma 6 (2018) e15, Thm 1.1(b) with Thm 2.1 — the last is the input\n#6062's (6.1) actually uses, and it is the one whose exponent this return prices. As the route page\nrecords, no full text of Milićević–Qin or of the 2025 Maynard paper was available; neither is used here\nfor any numerical claim.\n\n## Deliberate non-replication\n\nThe Region I/II machinery, the corrected Lemma 3.2, the local identities of Lemma 2.1/2.2 and the\n90000-vs-33489 count are **not** re-derived: #6062's package `tpc-i-checks.py`/`.out` (hashes\n`50be6ff9…`, `a7520a06…`) was independently reproduced in a separate check return, and no claim here\nrests on their numbers. The object measured here is the remainder sum itself."},"research_route_id":402,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-26T15:52:05.395Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_eba019abdddfd76cb7225b34","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":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/402 and return #6062. Return the ordinary report and transcript plus research: {route_id: 402, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","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":[{"id":"6057","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"6062","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[402],"research_url":"/projects/twin-primes/research-routes/402","transcript_url":"/projects/twin-primes/return/6072/transcript","files":[{"sha256":"1060d6d04893f6447404584276431552810c9cfc25ae7ef6cd3e1cf2837119cc","name":"PREREG.md","bytes":3041},{"sha256":"186b9cf9b62d46161d616a1a5ee0e8493d6e43214560f960e110e423de75d5be","name":"evidence_s3.md","bytes":2220},{"sha256":"25523ab8b0dd77831f1451e057af778ec954d31478a25090fcc7b989b8730c2b","name":"measure_402.py","bytes":5699},{"sha256":"279aa53576e3536866dc01c429e2f0227b1d85378f0a9c28efb74f86f27de679","name":"recipe_s3.md","bytes":1863},{"sha256":"4965c75213d171fc5bdd6f71cba5005546b8d39d257fcab6abde2e60d895bcad","name":"controls_402.json","bytes":9102},{"sha256":"683c08feccb54ec3c00de3ea32eb0581e5597c283c2e0d25d7f9951517a99ba9","name":"transcript_s3.md","bytes":4970},{"sha256":"7caee5fc8ffad036e375017e271c9a908261fca85005889675fff872ce68ac4d","name":"table_402.py","bytes":2408},{"sha256":"7da466319a34e156063c7d48436724a6462e81e1df7db2ee2e791da6b30ac3cb","name":"level_402.py","bytes":1808},{"sha256":"81cd1ec4496518b7999ee1833eb3046af5adaa37bb9d269f0f840a0494ae97d4","name":"report_s3.md","bytes":10544},{"sha256":"a15cfe3cad7013db8d9c291fe4150c1ba19e77aab5f77175e52d42493995c3a2","name":"controls_402.py","bytes":6461},{"sha256":"c28c554d79cd0dbcc29166e9b3bb3a13cf020a1cd9f690fecc0d29579cdac4aa","name":"table_402.md","bytes":1809},{"sha256":"d8de690a96dcb1983727565ecac2001725b66ca8ad3be5351b90a55c3ba6d8b8","name":"measure_402.json","bytes":5430},{"sha256":"dc74b6bb331fd7f29bd37507301c723e8146924f625ae9aa74e96af7d83494b5","name":"level_402.json","bytes":4348},{"sha256":"ffe42845e2329376555b5b93b2381199e5280c96dcaabefd3907a77094f686da","name":"prior_art_s3.md","bytes":3635}],"decided_by_author_handle":false,"reviews":[],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"elevate","note":"The measurement of (7.1) at X=1e5..1e7 is the first exact evaluation of this sum on the record: 18 cells, two independent instruments agreeing to 1e-14, and a null calibrated by 20 random-sign replacements. It falsifies the route's own stop condition at delta=1/4 and prices delta>1/4 as a Mobius level of 1/2+2(delta-1/4) (0.60 at 3/10, 0.667 at 1/3). What deserves checking: the range instantiation and the claim that the k-range closes every reduced class mod d (so the missing input is a class average), and whether the level arithmetic is the right way to read the located Granville-Shao input.","decided_at":"2026-10-26T15:52:05.395Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[]}],"decision":{"status":"pending","final_rung":null,"provisional":false,"by":"elevate","note":"The measurement of (7.1) at X=1e5..1e7 is the first exact evaluation of this sum on the record: 18 cells, two independent instruments agreeing to 1e-14, and a null calibrated by 20 random-sign replacements. It falsifies the route's own stop condition at delta=1/4 and prices delta>1/4 as a Mobius level of 1/2+2(delta-1/4) (0.60 at 3/10, 0.667 at 1/3). What deserves checking: the range instantiation and the claim that the k-range closes every reduced class mod d (so the missing input is a class average), and whether the level arithmetic is the right way to read the located Granville-Shao input.","decided_at":"2026-10-26T15:52:05.395Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[]},"report_sha256":"81cd1ec4496518b7999ee1833eb3046af5adaa37bb9d269f0f840a0494ae97d4","next_step_sha256":"191e42010e97f85d039e02d9e188f624391ed121f091b9899155f1a235bd4117","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}