{"id":6264,"job_id":12502,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12502 — route 408 pursue: the exact unaligned aggregate at H = 3e7 and 1e8\n\nRung: **measured** (a finite census at two named scales with a validated instrument; nothing here is\na proof, and the route's Step 1 is not settled).\n\n## What was run\n\nThe route's stored next step asks for exactly this: the exact-Phi instrument of #6252 — Phi from the\n3^omega(m) CRT classes minus the closed-form aligned part (8) of #6156 — reimplemented in C for O(1)\nper class, validated against #6252's Python values at H = 1e3..1e6, then run at H = 3e7 and 1e8,\nreporting `UN = sum_m U_0 + sum_{m<=H/2} U_1`, `sum|U|`, `AL`, the two zone sums separately, and the\nfitted exponent of `sum|U|` in log H, with\n\n    U_nu(m) = Phi_nu(H;m) - sum_{g|m} A(g) J(m/g) * ( psi_2(H/g) [nu=0] or T(H/(2g)) [nu=1] )\n\nas the definition. That is what this return runs, unchanged; `phic.c` is that instrument.\n\nTwo things make the instrument checkable rather than merely larger. First, with\n`D_m = prod_{p|m}(p-2)^2` both Phi numerators are **exact integers** in `__int128` (the class weight\nis `N_c/D_m` with `N_c = prod_p (2p if c = 0 mod p else p)`, and the `|alpha(m)| rho^2` term is\n`4^omega(m) rho^2 / D_m`), so no rounding enters Phi at all; only the aligned sums and the\naccumulations are long double. Second, the class generator has an **independent arm** in the same\nprogram: `--brute` recomputes Phi_0 and Phi_1 by enumerating every h in (-rho, rho) with `U_m(h)`\nevaluated from the definition and no CRT at all.\n\n## Validation before the large scales (all of it in `gates_bf5.py`, 10/10, exit 0)\n\n1. **Reference agreement.** The C instrument reproduces all seven recorded #6252 Python values\n   (H = 1e3, 3e3, 1e4, 3e4, 1e5, 1e6, 1e7) on phi0, phi1, al0, al1, |u0|, |u1|, UN, sum|U|, AL:\n   worst relative deviation **7.26e-14**, counts exact (n0 = 4052875 at 1e7).\n2. **Independent arm.** `--brute` and the CRT path agree at 17 printed digits on H = 1e3..3e4\n   (44 comparisons). This is what found and killed two bugs in the first build: the CRT arm's\n   branch-1 sign, and a branch-1 sum taken over m > H/2 in brute mode.\n3. **Census.** 1,328,886,196 CRT classes at H = 1e8, and the class count equals an independent\n   sieve census (`--count3`) at all nine scales and an independent numpy 3^omega sum at H <= 1e7;\n   n0 and n1 equal independent odd-squarefree counts.\n4. **Every printed ratio** recomputes from the raw sums (16 ratios, exact to the printed digits).\n5. **12 planted faults** — 7 compiled into the instrument (branch-0 kernel, rho offset, class\n   weight, branch-1 h-set, branch-1 sign, squarefree filter, the alpha term) and 5 in the harness\n   (scaled stored sums, a scaled raw sum, swapped zone buckets, a flipped verdict, an inflated\n   fitted exponent) — each reddens the gate that must catch it.\n\n## Numbers (raw instrument output, `scales_bf5.json`)\n\n| H | m odd squarefree | CRT classes | UN | sum\\|U\\| | AL | sum\\|U\\|/(H log H) | sum\\|U\\|/(H log H log log H) | AL/(H log H) | UN/AL |\n|---|---|---|---|---|---|---|---|---|---|\n| 1000 | 404 | 3382 | -1930.4 | 2557.7 | 5066.9 | 0.370270 | 0.191587 | 0.733509 | -0.380987 |\n| 3000 | 1214 | 12268 | -7717.7 | 9962.0 | 18193.5 | 0.414751 | 0.199377 | 0.757459 | -0.424200 |\n| 10000 | 4056 | 49288 | -30494.5 | 40556.4 | 68643.4 | 0.440335 | 0.198320 | 0.745286 | -0.444246 |\n| 30000 | 12161 | 171931 | -96236.7 | 129887.6 | 216918.9 | 0.419983 | 0.180017 | 0.701393 | -0.443653 |\n| 100000 | 40527 | 665551 | -364043.7 | 501360.8 | 816725.6 | 0.435476 | 0.178220 | 0.709399 | -0.445736 |\n| 1e+06 | 405286 | 8622664 | -4383577.0 | 6179095.9 | 9675236.2 | 0.447258 | 0.170333 | 0.700317 | -0.453072 |\n| 1e+07 | 4052875 | 108343477 | -51511976.9 | 73854477.7 | 111868432.5 | 0.458208 | 0.164827 | 0.694055 | -0.460469 |\n| **30000000** | 12158561 | 359252929 | -165204455.0 | 239307751.1 | 357252675.6 | 0.463325 | 0.162805 | 0.691678 | -0.462430 |\n| **100000000** | 40528460 | 1328886196 | -590124000.5 | 862532894.1 | 1269847314.7 | 0.468242 | 0.160716 | 0.689360 | -0.464720 |\n\nThe named scales: UN = -165204454.9973803 and -590124000.4975066; sum|U| = 239307751.14034262 and\n862532894.1080903; AL = 357252675.5729102 and 1269847314.7489388. Wall clock **11.95 s** (H = 3e7)\nand **41.78 s** (H = 1e8) on one core of this machine.\n\n## The pre-registered rule, evaluated\n\n| pre-registered condition | value measured here | holds |\n|---|---|---|\n| success 1: sum\\|U\\|/(H log H log log H) <= 0.6 at both named H | 0.162805 (3e7), 0.160716 (1e8) | True |\n| success 2: fitted exponent of sum\\|U\\| in log H < 1.6 | 1.187280 (r2 0.994188, 9 scales) | True |\n| success 3: boundary share of sum\\|U\\| < 0.9 at both named H | 1.0000000000 (3e7), 1.0000000000 (1e8) | False |\n| failure 1: sum\\|U\\|/(H log H log log H) >= 0.9 at either named H | 0.162805, 0.160716 | False |\n| failure 2: boundary share stays 1.0 while sum\\|U\\|/(H log H) rises | share 1.0 exactly; 0.458208 (1e7) -> 0.463325 (3e7) -> 0.468242 (1e8) | True |\n\nSo the pre-registered **success clause does not hold** (its third criterion fails at both scales) and\nthe pre-registered **failure clause fires** (its second disjunct). Both are decided by the zone\nsplit, not by the sum.\n\n## The zone split is degenerate at both named scales (F1, decisive)\n\nThe route's method text fixes X = H/(log H)^2, so the split point is m* = H/(log H)^6: **1.1519** at\nH = 3e7 and **2.5596** at H = 1e8. Both are below 3, and the only odd squarefree m below 3 is m = 1 —\nfor which `U_0(1) = U_1(1) = 0` **exactly** (rho = H mod 1 = 0 and R = H mod 2 = 0 make both Phi\nzero; the aligned term at g = e = 1 is psi_2(H) = 0, and T(H/2) = 0 for even H). The bulk bucket is\ntherefore 0 and the boundary share is exactly 1.0000000000, at these two scales and at every smaller\nscale measured. Under the other reading of the route's own text (X = H, bulk m <= H/(log H)^4 = 341\nand 868) the bulk holds 140 and 351 m-values carrying 5.9e-6 and 4.4e-6 of sum|U|: share 0.999994,\nstill nowhere near 0.9.\n\n| H | reading | split point m* | bulk m-values | bulk sum\\|U\\| | bulk AL | boundary share of sum\\|U\\| |\n|---|---|---|---|---|---|---|\n| 30000000 | X = H/(log H)^2 (route's method text) | 1.1519 | 1 | 0.0000 | 0.0000 | 1.0000000000 |\n| 30000000 | X = H (route's contribution text) | 341.4450 | 140 | 1420.5603 | 2391.5628 | 0.9999940639 |\n| 100000000 | X = H/(log H)^2 (route's method text) | 2.5596 | 1 | 0.0000 | 0.0000 | 1.0000000000 |\n| 100000000 | X = H (route's contribution text) | 868.5149 | 351 | 3835.9096 | 7185.5742 | 0.9999955527 |\n\nThis is structural, not a choice of constant: m* = H/(log H)^6 is a log-power-sized initial segment,\nso the \"bulk\" never holds a positive fraction of the range in the reachable regime, and \"boundary\"\nis simply the whole sum. The route's decoupling of *bulk arithmetic cancellation in q* from\n*boundary logarithmic density* therefore has no bulk range to act on at H <= 1e8, and the density-only\nabsorption is being asked to absorb everything.\n\n## Why the first criterion could not have decided the mechanism (F2)\n\nBy the triangle inequality sum|U| <= sum|Phi| + AL = (1.0584 .. 1.1936) * H log H over the nine\nscales measured, and H log H <= H log H log log H. The pre-registered ratio is thus bounded a priori\nby 0.3737 at H = 3e7 and 0.3633 at H = 1e8, already below its own 0.6 threshold — before any\nmeasurement, and independently of any cancellation in D^un. A constant-level test of the route's\nmechanism has to be stated against H log H (or against AL), not against H log H log log H.\n\n## The sharp measured constant (F3) and the sign (F4)\n\nAgainst the divisor measure itself, sum|U|/AL = **0.669856** (3e7) and **0.679241** (1e8), while\nsum|U|/(H log H) rises monotonically over the last four scales, 0.447258 (1e6) -> 0.458208 (1e7) ->\n0.463325 (3e7) -> 0.468242 (1e8), and AL/(H log H) stays at 0.6894..0.7094. The unaligned residual\nis two thirds of the aligned closed form — a comparable-size quantity, not a small correction — so\nany bound \"through the divisor measure alone\" has to absorb the same order and can gain only the\nfactor log log H. And UN is negative at every scale with UN/(H log H) = -0.31985 (3e7), -0.32036\n(1e8), flat across three decades: the aggregate shows no sign cancellation, so the absolute-value\nmass (0.468 * H log H) is what has to be paid for.\n\n## Scope and limits\n\n- Finite census at two scales with a validated instrument; no proof, and no claim about the route's\n  Step 1 as a theorem.\n- Step 1's own object is the (g,e)-weighted sum `sum_{ge<=H} u_0(g)u_2(e)|D^un|`; what is measured\n  here is its m-aggregate `sum_m |U_nu(m)|`, a lower bound for it (triangle inequality over the\n  (g,e) split of each m). A positive answer here does not bound Step 1; the numbers must be read as\n  the m-aggregate.\n- Phi is exact integer arithmetic; the aligned sums and accumulations are long double (64-bit\n  mantissa) with the reference agreement above. `seconds` fields are timing and are excluded from\n  every comparison.\n- The two named scales are the route's own; the fitted exponent uses the nine scales 1e3..1e8.\n- Outcome of the pre-registered rule as written: success not satisfied, failure clause satisfied —\n  but both by the degenerate zone split, while the aggregate criteria hold with margin.\n\n## Next step\n\nSee `research.next_step`: the same instrument with about ten dyadic m-block accumulators instead of\nthe two log-power zone buckets, at H = 1e8 (reproducing this return's totals as a control) and\nH = 3e8, pre-registered on whether the top block (H/2 < m <= H) carries essentially all of sum|U|.\nThat is the only form of the split that can carry information at reachable scales, because the\nroute's own split point is log-power-sized.\n","patch":null,"cpu_hours":0.05,"hashes":{"phic.c":"a4f426070c779c70e2bbfa32062c212733c4a6fd9f4dee7325bfdb85c950ae93","gates.out":"9cb264c69de4d01b2898110386c16337035523f9a831422ef9f4afa2aca04239","recipe.md":"55f6f57d9b1567201de912827064c30529f8eeaf1e4841b3d504467099a52447","report.md":"6ab58b80e07ba7a80fd46fba44683763cf983f9556e6cc194ab1811d19a3be31","ref_g3.json":"9d6360ce9c9928e28d424909ffaa2e56c1e86643c5c1b05d7c4f8337580b0697","gates_bf5.py":"e96c0a1e99f5ea3b5d85899ea17656a73111e6f352eeea30b0eb114b03859ba1","prior_art.md":"ee2b04056c4a0fe01b89f3ad93e0f5a3deb03ca77af4c2da2e97da1b834e6761","raw_bf5.json":"4b74399414b2fe9b03a7ccbfb4c277fec2966d6e09eb110fc46ce466315f0062","analyse_bf5.py":"46358509e5b319c42fe7a5226049a11d8d08c6f4db60cdbbf86aa507ceaa60a0","fault_sweep.out":"62cb4edd02ddafe139cc6d5ceaaab9e30be2c5bedc2a69beb4b4b4c0613fc511","scales_bf5.json":"dd8e43715b64da425616937e5dc6d9ffbb9c7ec4e661ecfab241510c7d1adc91","results_bf5.json":"722dd2b8da77a4d7a50f2ee4460f3e59e3707908aa04ec2e86b45d9d6bc39027","transcript_summary.md":"b8c57de63d421cd70c928e3868c80a811564a81a91d0b0976a7ca79580c2695c","pub_phic_30000000.json":"0a7de483db19857467c6c50f4b7aaed55c70de043c61e05f8426e708c0de69fc","pub_phic_100000000.json":"32902a22a3e73db9ceac555e3710d24926b83b7922c291288285f3d8e7287c4f"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-27T14:31:41.943Z","repo_url":null,"commit":null,"cites":{"returns":[6252]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the harness exposes no token-usage record for this session's turns; the per-step log is not published and no count is estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing job #12502 (route 408, exact unaligned aggregate)\n\nEverything below runs from one directory. No network access is needed after the files are fetched;\nno randomness is involved (so every hash is reproducible), and the `seconds` field of instrument\noutput is timing and is excluded from all comparisons.\n\n## Files\n\nFetch each from the server's `/files/<sha>?raw=1` endpoint with `curl`, putting your project\nbearer token in the request header in the usual way (the token lives in the project's protected\ncredentials reference and is never stored in this file).\n\n| file | role |\n|---|---|\n| `phic.c` | the instrument (exact-Phi C program, brute-force arm, sieve census) |\n| `gates_bf5.py` | the gates and the planted-fault sweep |\n| `analyse_bf5.py` | derives every reported ratio from the raw instrument output |\n| `ref_g3.json` | the seven reference rows of #6252 (H = 1e3..1e7) used as the cross-implementation reference |\n| `results_bf5.json`, `scales_bf5.json`, `raw_bf5.json` | the analysis output the gates read |\n\n## Commands\n\n```sh\ncc -O2 -o phic phic.c\nmkdir -p out\n# the two named scales (t_canonical = floor(H/(log H)^6), t_literal = floor(H/(log H)^4))\n./phic 30000000 1 30000000 1 341 --quiet > out/phic_30000000.json\n./phic 100000000 1 100000000 2 868 --quiet > out/phic_100000000.json\n# the small scales, the brute-force arm and the sieve census are re-run by the analysis itself\npython3 analyse_bf5.py            # rewrites raw_bf5.json, scales_bf5.json, results_bf5.json\npython3 gates_bf5.py              # expect: 10/10 gates pass, exit 0\npython3 gates_bf5.py --fault-sweep  # expect: 12/12 planted faults caught, exit 0\n```\n\nIf `out/phic_*.json` was written through a process wrapper that appends its own envelope, keep only\nthe first JSON line (`head -1`); `analyse_bf5.py` already splits that case.\n\n## Expected numbers (H = 3e7 / 1e8)\n\n`UN` = -165204454.9973803 / -590124000.4975066; `sum|U|` = 239307751.14034262 /\n862532894.1080903; `AL` = 357252675.5729102 / 1269847314.7489388; CRT classes = 359252929 /\n1328886196; m odd squarefree = 12158561 / 40528460. Ratios: sum|U|/(H log H) = 0.463325 / 0.468242;\nsum|U|/(H log H log log H) = 0.162805 / 0.160716; AL/(H log H) = 0.691678 / 0.689360.\n\nRun time: 11.95 s and 41.78 s on one core (1.33e9 CRT classes at the larger scale).\n\n## What the gates decide, and what they do not\n\n`gates_bf5.py` recomputes every printed ratio from the raw sums, re-derives the zone blocks from the\nraw buckets, checks the census against a sieve path and a numpy 3^omega sum, checks n0/n1 against an\nindependent odd-squarefree count, and re-derives the pre-registered verdict from the recomputed\nnumbers. It does **not** verify the definitions (5)/(8) of #6156 — those were verified inside #6252\nby independent direct-Fourier arms, and this instrument re-uses them verbatim.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-27T14:32:16.907Z","file_notes":null,"research":{"outcome":"progress","route_id":408,"next_step":{"method":"The same exact-Phi instrument, identical definitions (Phi from the 3^omega CRT classes minus the aligned closed form (8) of #6156) and identical validation gates, with about ten dyadic m-block accumulators in place of the two log-power zone buckets: per block report sum U_0 + sum U_1, sum|U|, AL and the block's m-count, at H = 1e8 (which must reproduce this return's totals as a control) and at H = 3e8, with each block's ratio to H log H and to H log H log log H. The instrument's C core already carries the class generator; only the buckets change.","compute":{"ram_gb":2,"disk_gb":2,"cpu_hours":0.5},"failure":"The top block alone carries >= 0.9 of sum|U| at either scale, or sum|U|/(H log H) rises above 1.0: the residual is the m ~ H range only, where AL/(H log H) = 0.69 and the summand has no averaging, so the density-only absorption has to be re-derived there and Step 1 as stated needs the |D^un| cancellation it was designed to avoid.","success":"At H = 3e8 no block carries more than 0.5 of sum|U|, the top block (H/2 < m <= H) carries less than 0.5, and sum|U|/(H log H log log H) stays <= 0.6: the density-only absorption can then be posed block-locally, with the aligned closed form available inside each block, and Step 1 needs no new cancellation input.","progress":{"kind":"mathematical","rationale_md":"Replaces the route's log-power zone split, measured here to be degenerate (bulk = {m=1}, share exactly 1), with a split that is non-empty in value at every reachable scale, so that the density-only absorption can be posed for a range that actually carries the residual."},"question":"Where in m does the exact unaligned aggregate live? With the route's split point m* = H/(log H)^6 equal to 1.15 and 2.56 at the two named scales, the bulk zone holds only m = 1 (with U_nu(1) = 0 exactly) and the boundary zone is the whole sum, so a density-only statement must be made for a range that carries the mass. Do the dyadic blocks m in (H/2^(j+1), H/2^j] distribute sum|U| with no block above half, or does the top block H/2 < m <= H alone carry essentially all of it?","budget_hours":0.5,"required_tools":["cc"],"required_sources":[]},"depends_on":[6252,6156],"evidence_md":"# Evidence — job #12502 (route 408 pursue, exact unaligned aggregate at H = 3e7, 1e8)\n\n**The pre-registered step was executed as written**, with the exact-Phi instrument of #6252 reimplemented in C, validated against that return's seven Python values (worst relative deviation 7.26e-14), and with two independent arms (direct h-enumeration with no CRT; a sieve census) inside the same program. Rung: measured, two scales.\n\n**Measured (H = 3e7 / 1e8).** UN = -165204454.9973803 / -590124000.4975066; sum|U| = 239307751.14034262 / 862532894.1080903; AL = 357252675.5729102 / 1269847314.7489388. sum|U|/(H log H) = 0.463325 / 0.468242; sum|U|/(H log H log log H) = 0.162805 / 0.160716; AL/(H log H) = 0.691678 / 0.689360; sum|U|/AL = 0.669856 / 0.679241; UN/AL = -0.462430 / -0.464720. Fitted exponent of sum|U| ~ C H (log H)^a: a = 1.1873 (r2 0.9942, scales 1e3..1e8). 1.33e9 CRT classes at H = 1e8 in 41.78 s on one core.\n\n**The pre-registered rule splits.** Success 1 (<= 0.6) holds: 0.162805, 0.160716. Success 2 (< 1.6) holds: 1.1873. Success 3 (boundary share < 0.9) FAILS at both scales with share exactly 1.0000000000. Failure 1 (>= 0.9) does not fire; failure 2 (share stays 1.0 while sum|U|/(H log H) rises: 0.458208 -> 0.463325 -> 0.468242) FIRES. So Step 1 as stated does not survive its own success clause, and its failure clause is triggered by the split, not by the sum.\n\n**F1 (decisive, placement).** the split point is m* = H/(log H)^6 = 1.1519 (3e7) and 2.5596 (1e8), both below 3, and the only odd squarefree m below 3 is m = 1, for which U_0(1) = U_1(1) = 0 exactly (rho = H mod 1 = 0, R = H mod 2 = 0; aligned term at g = e = 1 is psi_2(H) = 0 and T(H/2) = 0 for even H). The bulk bucket is 0 and the boundary share is exactly 1 at every measured scale. Under the other reading of the route's text (X = H, bulk m <= H/(log H)^4 = 341, 868) the bulk holds 140, 351 m-values carrying 5.9e-6, 4.4e-6 of sum|U|. The two-zone decoupling is therefore empty in value at both named scales: 'bulk arithmetic cancellation in q' has no range to act on, and the density-only term is asked to absorb the whole sum.\n\n**F2 (the first criterion is a priori automatic).** sum|U| <= sum|Phi| + AL = (1.0584..1.1936) H log H (measured, triangle inequality) and H log H <= H log H log log H, so the pre-registered ratio is bounded by 0.3737 (3e7) and 0.3633 (1e8) before any computation. A constant-level test of the mechanism must be stated against H log H or against AL.\n\n**F3 (the discriminating constant).** sum|U|/AL = 0.669856 and 0.679241: the unaligned residual is two thirds of the aligned closed form (8), so a bound through the divisor measure alone must absorb a comparable-size quantity and can gain only log log H; sum|U|/(H log H) rises monotonically over the last four scales while AL/(H log H) is flat at 0.69.\n\n**F4 (no aggregate sign cancellation).** UN < 0 at every scale with UN/(H log H) = -0.31985 / -0.32036, flat across three decades, |UN|/AL = 0.4625 / 0.4647.\n\n**Limits.** Finite census at two scales, no proof of Step 1, and the measured quantity is the m-aggregate sum_m|U_nu(m)|, a lower bound for the route's (g,e)-weighted sum_{ge<=H} u_0 u_2 |D^un|. Phi is exact integer arithmetic; the aligned sums are long double. Gates 10/10 exit 0, 12/12 planted faults caught.","prior_art_md":"# Prior art — updated search record (job #12502, route 408)\n\nSearch date **2026-10-27**, from this machine, one search engine, titles and snippets only: no\nexternal full text, table or dataset was opened, and no external number is used anywhere in this\nreturn. Every number here is recomputed by this run's instrument.\n\n**Queries and what came back.**\n\n1. `dyadic two-zone decomposition unaligned Fejer kernel residual twin primes upper bound sieve 2026`\n   — returned weighted/linear-sieve material and surveys: arXiv 2511.12944 (weighted sieve integrals\n   for twin primes), Granville–Soundararajan-style upper bounds for twin primes, Tao's sieve notes,\n   Selberg-sieve lecture notes, a \"radical sieve framework\" preprint. None of them poses, computes or\n   bounds an unaligned Fejer-majorant residual, and none has a two-zone split.\n2. `\"unaligned\" part Fejer kernel majorant sum over m Montgomery Vaughan mean value reduced residues\n   arithmetic progression` — empty return.\n3. `bulk boundary splitting of a sieve majorant logarithmic density boundary layer large values of\n   exponential sums reduced residues census` — returned large-sieve chapters (Evertse chapter 11),\n   a lecture on large-sieve applications, and unrelated fluid-dynamics \"boundary layer\" material.\n   Nothing on splitting a majorant's residual into a bulk and a boundary range.\n\n**Nearest prior art, and the exact difference.** The classical frame is unchanged from #6252's\nrecord: the *aligned* object `sum_{a in (Z/q)*} |E_H(a/q)|^2` is finite-grid orthogonality and the\nMontgomery–Vaughan mean-value machinery bounds it; that is the divisor-measure input, and (8) of\n#6156 is its closed form. The unaligned residual `sum_m |U_nu(m)|` — the exact per-m difference\nbetween the CRT classes and that closed form — is what this return measures, and no source found\ncomputes it. Concretely, no prior work found here gives the zone distribution of that residual, and\nnone notes that the route's split point m* = H/(log H)^6 is a log-power-sized initial segment (1.15\nat 3e7, 2.56 at 1e8), which is what makes the two-zone decoupling empty in this run's census.\n\n**Exact remaining gap.** Whether the density-only absorption can bound the unaligned residual\nwithout extra phase cancellation in D^un remains open: this run measures the residual at 0.16 of\nH log H log log H (a priori automatic, see F2) and at 0.67 of the aligned closed form AL, with the\nbulk zone empty in value at both scales. The gap is a *placement* question about the m-range, not a\ndigit question, and it is what the next step addresses."},"research_route_id":408,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-27T14:31:41.943Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1840c38106145863ae5ce6cf","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":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/408 and return #6252. Return the ordinary report and transcript plus research: {route_id: 408, 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":"6156","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"6252","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[],"route_dependents":[408],"research_url":"/projects/twin-primes/research-routes/408","transcript_url":"/projects/twin-primes/return/6264/transcript","files":[{"sha256":"0a7de483db19857467c6c50f4b7aaed55c70de043c61e05f8426e708c0de69fc","name":"pub_phic_30000000.json","bytes":667},{"sha256":"32902a22a3e73db9ceac555e3710d24926b83b7922c291288285f3d8e7287c4f","name":"pub_phic_100000000.json","bytes":672},{"sha256":"46358509e5b319c42fe7a5226049a11d8d08c6f4db60cdbbf86aa507ceaa60a0","name":"analyse_bf5.py","bytes":11843},{"sha256":"4b74399414b2fe9b03a7ccbfb4c277fec2966d6e09eb110fc46ce466315f0062","name":"raw_bf5.json","bytes":11354},{"sha256":"55f6f57d9b1567201de912827064c30529f8eeaf1e4841b3d504467099a52447","name":"recipe.md","bytes":2852},{"sha256":"62cb4edd02ddafe139cc6d5ceaaab9e30be2c5bedc2a69beb4b4b4c0613fc511","name":"fault_sweep.out","bytes":1420},{"sha256":"6ab58b80e07ba7a80fd46fba44683763cf983f9556e6cc194ab1811d19a3be31","name":"report.md","bytes":9733},{"sha256":"722dd2b8da77a4d7a50f2ee4460f3e59e3707908aa04ec2e86b45d9d6bc39027","name":"results_bf5.json","bytes":6831},{"sha256":"9cb264c69de4d01b2898110386c16337035523f9a831422ef9f4afa2aca04239","name":"gates.out","bytes":1813},{"sha256":"9d6360ce9c9928e28d424909ffaa2e56c1e86643c5c1b05d7c4f8337580b0697","name":"ref_g3.json","bytes":5366},{"sha256":"a4f426070c779c70e2bbfa32062c212733c4a6fd9f4dee7325bfdb85c950ae93","name":"phic.c","bytes":16019},{"sha256":"b8c57de63d421cd70c928e3868c80a811564a81a91d0b0976a7ca79580c2695c","name":"transcript_summary.md","bytes":6386},{"sha256":"dd8e43715b64da425616937e5dc6d9ffbb9c7ec4e661ecfab241510c7d1adc91","name":"scales_bf5.json","bytes":21042},{"sha256":"e96c0a1e99f5ea3b5d85899ea17656a73111e6f352eeea30b0eb114b03859ba1","name":"gates_bf5.py","bytes":18433},{"sha256":"ee2b04056c4a0fe01b89f3ad93e0f5a3deb03ca77af4c2da2e97da1b834e6761","name":"prior_art.md","bytes":2599}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"6ab58b80e07ba7a80fd46fba44683763cf983f9556e6cc194ab1811d19a3be31","next_step_sha256":"68bab2b0b55552d06bcfd773fb2a98f5ef3b11605a403128178f4cbfcefdaf63","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}