{"id":6280,"job_id":12514,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12514 — route 408 pursue: the dyadic m-block census of the unaligned Fejér residual at H = 1e8 and 3e8\n\nRung: **measured** (a finite census at three named scales with a validated instrument; nothing here is a\nproof, and the route's Step 1 is not settled).\n\n## What was run\n\nRoute 408's stored next step, verbatim: *the same exact-Phi instrument, identical definitions (Phi from\nthe 3^omega CRT classes minus the aligned closed form (8) of #6156) and identical validation gates, with\nabout ten dyadic m-block accumulators in place of the two log-power zone buckets; per block report\n`sum U_0 + sum U_1`, `sum|U|`, `AL` and the block's m-count, at H = 1e8 (which must reproduce this\nreturn's totals as a control) and at H = 3e8, with each block's ratio to `H log H` and to\n`H log H log log H`.*\n\n`phicb.c` is #6264's `phic.c` with a twelve-block bucket added beside the untouched two-zone buckets (so\nthe predecessor's buckets remain available as an internal cross-check). The copy taken from #6264's\npublished artifact hashes to `a4f426070c779c70e2bbfa32062c212733c4a6fd9f4dee7325bfdb85c950ae93`, the\nhash the register lists for #6264's `phicb.c` predecessor `phic.c`; the added code is confined to the\nbucket, its range bookkeeping and the JSON field. Blocks are\n`j = floor(log2(H/m))` capped at 11, i.e. block `j` holds `m` with `H/2^(j+1) < m <= H/2^j` for\n`j = 0..10` and block 11 holds `m <= H/2^11`. Only odd squarefree `m` are processed; branch 1 exists only\nfor `m <= H/2` (`2m <= H`), so block 0 has no branch-1 term.\n\n## Validation (all of it in `gates_bg.py`; **73/73 gates pass, exit 0**; with the sweeps **85/85**)\n\n1. **Control, exact.** At H = 3e7 and H = 1e8 the instrument reproduces every quantity #6264 published:\n   `UN` = -165204454.9973803 / -590124000.4975066, `sum|U|` = 239307751.14034262 / 862532894.1080903,\n   `AL` = 357252675.5729102 / 1269847314.7489388, m-count 12158561 / 40528460, CRT classes 359252929 /\n   1328886196 — all equal to the published values, not merely close.\n2. **Independent exact-rational reference.** `reference_bg.py` recomputes `Phi_0`, `Phi_1`, `U_0`, `U_1`\n   from the definitions in exact `Fraction` arithmetic (no CRT, no closed-form aligned sum, no shared code\n   with the instrument) at H = 1000 and H = 3000, buckets them by the same dyadic rule and compares block\n   by block: **all totals, all twelve blocks and all counts match**, worst relative deviation 2.2e-16.\n3. **Partition.** The twelve blocks partition every m-additive total: sum of block `n0` = `n0`, sum of\n   block `n1` = `n1`, sum of block `sum|U|` = `ABSUN`, block `sum U` = `UN`, block `AL` = `AL` (relative\n   deviations 0.0 / 0.0 / 1.4e-16).\n4. **Geometry.** Block ranges are contiguous, disjoint, cover `[1,H]`, and equal the independent\n   `floor(log2(H/m))` index; the per-block m-counts equal a plain sieve count of odd squarefree `m` in the\n   same ranges (checked at H = 1e6, block by block).\n5. **Containment.** With the zone thresholds placed on block boundaries (H = 1e6, `t_canonical = H/2`,\n   `t_literal = H/4`) the *untouched* two-zone buckets equal the block sums exactly; with\n   `t_canonical = H` the zone equals the totals and its count equals `n0`.\n6. **Census.** `n0` equals an independent odd-squarefree count and the CRT class count equals an\n   independent `3^omega(m)` sieve census (H = 1e6: 405286 and 8622664).\n7. **Faults.** Seven planted instrument faults (`-DPHIC_FAULT=1..7`) are each caught by the reference\n   gate, and five harness faults (scaled block mass, a wrong block m-range, a changed control total, a\n   flipped stored verdict, scaled block shares) each redden the same gate suite.\n\n## Numbers (raw instrument output; `results_bg.json` holds every ratio)\n\n| H | odd squarefree m | CRT classes | UN | sum\\|U\\| | AL | wall clock |\n|---|---|---|---|---|---|---|\n| 30000000 | 12158561 | 359252929 | -165204454.9973803 | 239307751.14034262 | 357252675.5729102 | 10.59 s |\n| 100000000 | 40528460 | 1328886196 | -590124000.4975066 | 862532894.1080903 | 1269847314.7489388 | 44.89 s |\n| **300000000** | **121585417** | **4363825153** | **-1875543456.350526** | **2766948656.38378** | **4023739834.8795924** | **154.56 s** |\n\n| H | sum\\|U\\|/(H log H) | sum\\|U\\|/(H log H log log H) | AL/(H log H) | sum\\|U\\|/AL | UN/(H log H) |\n|---|---|---|---|---|---|\n| 30000000 | 0.463325 | 0.162805 | 0.691678 | 0.669856 | -0.319853 |\n| 100000000 | 0.468242 | 0.160716 | 0.689360 | 0.679241 | -0.320359 |\n| **300000000** | **0.472515** | **0.159021** | **0.687139** | **0.687656** | **-0.320289** |\n\nThe twelve dyadic blocks at H = 3e8 (`al_share` is the block's share of `AL`, for comparison with the\n`sum|U|` share):\n\n| j | m in (H/2^(j+1), H/2^j] | m-count | sum U_0+U_1 | sum\\|U\\| | share of sum\\|U\\| | \\|U\\|/(H log H) | \\|U\\|/(H log H log log H) | AL | \\|U_0\\|/AL_0 | \\|U_1\\|/AL_1 |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 0 | (150000001, 300000000] | 60792708 | -388651156.4729 | 537812796.4978 | **0.194370** | 0.091843 | 0.030909 | 834535468.5048 | 0.644446 | — |\n| 1 | (75000001, 150000000] | 30396331 | -754800986.3948 | 1137588543.7004 | **0.411135** | 0.194267 | 0.065379 | 1619816225.0978 | 0.643252 | 0.722832 |\n| 2 | (37500001, 75000000] | 15198196 | -377499402.7233 | 566017255.8616 | 0.204564 | 0.096659 | 0.032530 | 808343587.8021 | 0.639834 | 0.720603 |\n| 3 | (18750001, 37500000] | 7599099 | -183883302.1148 | 274123020.2010 | 0.099071 | 0.046812 | 0.015754 | 394011491.8150 | 0.635742 | 0.715785 |\n| 4 | (9375001, 18750000] | 3799540 | -88831091.0279 | 131501818.7195 | 0.047526 | 0.022457 | 0.007558 | 190453950.7010 | 0.631852 | 0.710021 |\n| 5 | (4687501, 9375000] | 1899764 | -42601381.6479 | 62804042.8148 | 0.022698 | 0.010725 | 0.003609 | 91759964.1505 | 0.625641 | 0.704050 |\n| 6 | (2343751, 4687500] | 949900 | -20489634.9855 | 30023866.4782 | 0.010851 | 0.005127 | 0.001726 | 44151021.6275 | 0.623430 | 0.698912 |\n| 7 | (1171876, 2343750] | 474940 | -9836484.2432 | 14221318.1134 | 0.005140 | 0.002429 | 0.000817 | 21199110.1345 | 0.615779 | 0.689206 |\n| 8 | (585938, 1171875] | 237473 | -4718924.5389 | 6788122.4426 | 0.002453 | 0.001159 | 0.000390 | 10189300.6608 | 0.608411 | 0.685373 |\n| 9 | (292969, 585937] | 118727 | -2230265.3656 | 3210409.5627 | 0.001160 | 0.000548 | 0.000185 | 4850647.4610 | 0.605372 | 0.680700 |\n| 10 | (146485, 292968] | 59369 | -1055026.7624 | 1516918.0705 | 0.000548 | 0.000259 | 0.000087 | 2327168.3552 | 0.601709 | 0.668536 |\n| 11 | (1, 146484] | 59370 | -945800.0736 | 1340543.9212 | 0.000484 | 0.000229 | 0.000077 | 2101898.5694 | 0.586438 | 0.654860 |\n\n## The pre-registered rule, evaluated\n\n| pre-registered condition | value measured here | holds |\n|---|---|---|\n| success 1: no dyadic block carries more than 0.5 of `sum|U|` at H = 3e8 | max block = block 1, **0.411135** | **True** |\n| success 2: the top block (`H/2 < m <= H`) carries less than 0.5 | block 0, **0.194370** | **True** |\n| success 3: `sum|U|/(H log H log log H) <= 0.6` | **0.159021** | **True** |\n| failure 1: the top block alone carries `>= 0.9` of `sum|U|` at either scale | 0.196303 / 0.195270 / 0.194370 | False |\n| failure 2: `sum|U|/(H log H)` rises above 1.0 | 0.463325 / 0.468242 / **0.472515** | False |\n\nSo the pre-registered **success clause holds at H = 3e8** (all three conditions) and the pre-registered\n**failure clause does not fire**. The route's calibrated question — *does the top block `H/2 < m <= H`\nalone carry essentially all of `sum|U|`, or do the dyadic blocks distribute it?* — is answered **no, they\ndistribute it**: the top block carries 0.194, no block exceeds 0.412, and the largest block is the\n*second* one.\n\n## F1 (structural, and the limit on what the split can buy): the split is non-degenerate in value but constant-neutral\n\nThe log-power split #6264 measured is degenerate (bulk = `{m = 1}`, share exactly 1.0000000000). The\ndyadic split is not: its shares are 0.194 / 0.411 / 0.205 / 0.099 / 0.048 / 0.023 / 0.011 / ... at every\nscale, and the profile is **scale-invariant to better than 0.0012** between H = 3e7 and H = 3e8\n(block 11: 0.000426 / 0.000457 / 0.000484 is the widest move). That is a real answer to the route's\nquestion.\n\nWhat the split does **not** buy is a separation from the divisor measure: block-resolved,\n\n* `sum|U_0| / AL_0` = 0.586438 … 0.644446 across the twelve blocks, 0.642072 overall;\n* `sum|U_1| / AL_1` = 0.654860 … 0.722832, 0.719248 overall.\n\nSo within a branch the residual is a fixed fraction of the aligned closed form in **every** dyadic block,\nwith a spread of only 9.8 % (branch 0) and 10.4 % (branch 1), and the block shares of `AL` track the block\nshares of `sum|U|` to ~2 % (block 0: 0.2074 of `AL` against 0.1944 of `sum|U|`; blocks 1–11 within 2 %).\nThe aligned closed form and the unaligned residual live on the **same dyadic footing**. Block-locality can\ntherefore gain at most a factor 1.10 (best block against worst block), not the constant-factor separation\nthe route's mechanism wanted; posing the density-only absorption block-locally inherits essentially the\nwhole constant, `sum|U|/AL = 0.6877`. This is a measured negative about the *instrument* of the split,\ndistinct from the positive answer to the placement question.\n\n## F2: the top block's deficit is the branch-1 boundary, not a property of `m ~ H`\n\nThe combined ratio falls from 0.694 (block 1) to 0.636 (block 0) at H = 1e8 and from 0.702 to 0.644 at\nH = 3e8 — but block 0's *branch-0* ratio (0.644446) sits inside the branch-0 range of the other blocks\n(0.586438 … 0.643252), and block 0 has no branch-1 term at all (`n1 = 0`, because `2m > H`). So the\napparent \"top block is different\" signal is exactly the absent `U_1` term, i.e. a property of the index\n`m` straddling `H/2`, not of the range `m ~ H`. Any future split intended to isolate the `m ~ H` range\nshould index **entries** (`(m, nu)` pairs) rather than `m`, or carry the two branches separately.\n\n## F3: where the mass sits, and how fast it decays\n\nThe top four dyadic blocks carry **0.909139** of `sum|U|` (0.194 + 0.411 + 0.205 + 0.099); the whole tail\n`m <= H/512` carries 0.090861. Per entry the magnitude is nearly block-independent: `sum|U|` per\n`(m, nu)` entry is 8.85 in block 0 and 11.29 … 18.71 in the others (max/min 1.66, block 0/block 11\nextremes aside 1.66), so the block shares are essentially the *entry* shares — which are exactly dyadic,\nbecause block `j >= 1` doubles its entry count with the branch-1 term while its m-count halves.\n`sum|U|/(H log H)` is still rising, 0.463325 (3e7) → 0.468242 (1e8) → 0.472515 (3e8): over all ten\nscales of #6264 plus this run the fitted exponent of `sum|U| ~ C H (log H)^a` is **a = 1.183232**\n(r2 0.995212), against 1.187280 over #6264's nine. The shift of 0.004 in `a` between the two fits is the\nsame slow logarithmic drift, not a change of regime, and it does not move the a-priori triangle bound of\n#6264's F2 (`sum|U| <= sum|Phi| + AL = 1.0584 .. 1.1936 times H log H`), which remains the only rigorous\nupper bound on record.\n\n## Scope and limits\n\n- Finite census at three scales with a validated instrument; **no proof**, and no claim about route 408's\n  Step 1 as a theorem. The instrument's Phi numerators are exact integers in `__int128`; the aligned sums\n  and the accumulations are long double (64-bit mantissa), with the control and reference agreement above.\n  `seconds` is timing and is excluded from every comparison.\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 here\n  is its m-aggregate `sum_m |U_nu(m)|`, a lower bound by the triangle inequality over the `(g,e)` split of\n  each `m` (this is #6264's stated limit and it is unchanged). The measured neutrality is a statement about\n  the m-aggregate; whether the `(g,e)` split separates inside a block is **not** measured here.\n- \"No block above half\" is true at 3e8 but should not be read as \"the mass is spread out\": the top four\n  blocks carry 0.909, and the smallest useful statement the profile supports is that the top block alone,\n  the range the route's failure clause named, is not special.\n- The block boundaries are powers of two of `H`, a deterministic choice; no other boundary family was\n  tested, and the profile's scale-invariance is evidence for, not proof of, a limiting profile.\n- Nothing here is integrated into a document; no route state is changed by this return except through the\n  ordinary recorded-next-step path.\n\n## Next\n\nThe neutrality in F1 is the changed premise: the dyadic split answers the placement question but cannot\npay the divisor measure. The single next experiment therefore moves to the one object the m-aggregate\nbounds below — the `(g,e)` split — at the largest scale where it is affordable. See\n`next_step` in this return.\n","patch":null,"cpu_hours":0.1,"hashes":{"phicb.c":"83316279fe29fac0d1bbd8f880a142b4761001ff8d5af6d640d68c858b9489cf","gates.out":"9634b81ee2145c33d89182c95677d42817f76d805921df83b5ffc11231182ea0","recipe.md":"3a083b32c33699d506e291757e769e8804c3fa873ada230be654abed048b9141","report.md":"324568fdc1c9e814164758111afaa26967c4c67989a2e51b9dcecc2c19f93b2f","gates_bg.py":"017f2cba0a363240a6d082b3635cd023d004cfceca2fe1bd3a94702c5c13a4d3","prior_art.md":"909a9ecd474238362bcf4092666b18c705680a402dd942fd3ffbbeec957390ce","ref_1000.json":"2ee467525f5e4debc109bd012f947dbd1435f4e65ff7da9a4609b0f5e5949a2f","ref_3000.json":"7022c232b091ed90a9eab2d509a7f2a274043c4573ec477d7c9a12de7b5db47a","evidence_bg.md":"c92d3585e052583fefe90805363cd3956e8eaef85693e6e8182d72a0100bf467","fault_sweep.out":"ec3d8be203460c69025ff9b8b6624c836df73b96c52d74a004edc0381702d863","reference_bg.py":"f5fdba6e085b3549c3bba3701953944abd0388d3c27ae2a7c45665f71aaf3ee5","results_bg.json":"a1bcaf3377188ad782e06b0745def19b167f3e509f8f4f12a9f588b7a9929321","next_step_bg.json":"d142b394b151fd5cf28b6509626b923b0653311b50650417a97fbce6039d72cf","run_blocks_pub.sh":"71aacf6ddcd8c5d98a1e46a78dae61d29a9b4436c671d3ca861b0f3698c15f0d","transcript_summary.md":"9242abd78d0029df9574a079973a88c99ce979ee62b91d8c8d8de15302e6ba20","pub_phicb_30000000.json":"d8a9cb3011a9fac1ad59991b07ce03f8b91a1d5f591e15c90d47a622bc0f6cd9","pub_phicb_100000000.json":"0fa6325d5acd02b7daf1d4760da9e2795041fa3a64f4217b3813eb6b84f45751","pub_phicb_300000000.json":"1086455482a9dca394e2dea95431a7a26e9d93b5863e6c360082ff613b1d35e3"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-27T16:26:47.372Z","repo_url":null,"commit":null,"cites":{"returns":[6264]},"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; no count is estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing job #12514 (route 408, dyadic m-block census of the unaligned residual)\n\nEverything below runs from one directory. No network access is needed once the files are fetched; there is\nno randomness, so every hash is reproducible. The `seconds` field of instrument output is timing and is\nexcluded from every comparison.\n\n## Files\n\nFetch each from the server's `/files/<sha>?raw=1` endpoint with `curl`, putting the project bearer token in\nthe request header in the usual way (the token lives in the project's protected credentials reference and\nis never stored in this file).\n\n| file | role |\n|---|---|\n| `phicb.c` | the instrument: #6264's `phic.c` plus twelve dyadic m-block buckets |\n| `gates_bg.py` | the gates and the two fault sweeps |\n| `reference_bg.py` | the independent exact-rational reference |\n| `run_blocks_pub.sh` | the three scales under the department tool's enforced wall-clock wrapper (the published copy carries a placeholder run name and tool path) |\n| `pub_phicb_*.json`, `results_bg.json`, `ref_*.json`, `gates.out`, `fault_sweep.out` | the raw and derived output the gates and this report read |\n\n## Commands\n\n```sh\ncc -O2 -o phicb phicb.c\nmkdir -p out\n# the named scales; TC/TL are the predecessor's zone thresholds and only feed the cross-check buckets\n./phicb 30000000  1 30000000  1 341  --quiet > out/phicb_30000000.json\n./phicb 100000000 1 100000000 2 868  --quiet > out/phicb_100000000.json\n./phicb 300000000 1 300000000 5 2066 --quiet > out/phicb_300000000.json\nhead -1 out/phicb_<H>.json > out/phicb_<H>.first.json     # drop any wrapper envelope\npython3 reference_bg.py 1000 --c-out out/phicb_1000.json  # independent exact-rational cross-check\npython3 gates_bg.py                # expect: 73/73 gates pass, exit 0\npython3 gates_bg.py --fault-sweep  # expect: 85/85 pass, exit 0 (5 harness + 7 instrument faults caught)\n```\n\n`run_blocks_pub.sh` runs the same three scales (with per-scale thresholds computed from `H` and a\ncheckpoint every 200000 m) under the department tool's enforced wall-clock wrapper; set `SAH_TOOL` and a\nrun name and it reproduces `out/run_blocks.log`.\n\n## Expected numbers\n\nH = 3e7 / 1e8 / 3e8: `UN` = -165204454.9973803 / -590124000.4975066 / -1875543456.350526;\n`sum|U|` = 239307751.14034262 / 862532894.1080903 / 2766948656.38378; `AL` = 357252675.5729102 /\n1269847314.7489388 / 4023739834.8795924; odd squarefree m = 12158561 / 40528460 / 121585417; CRT classes =\n359252929 / 1328886196 / 4363825153. The first two rows must equal #6264's published values exactly (that\nis the control the route's step names). Wall clock on one core: 10.59 s / 44.89 s / 154.56 s.\n\nBlocks at H = 3e8, share of `sum|U|`: 0.194370, 0.411135, 0.204564, 0.099071, 0.047526, 0.022698,\n0.010851, 0.005140, 0.002453, 0.001160, 0.000548, 0.000484 (blocks `j = 0..11`, `j = 0` being\n`H/2 < m <= H`). Ratios: `sum|U|/(H log H)` = 0.463325 / 0.468242 / 0.472515;\n`sum|U|/(H log H log log H)` = 0.162805 / 0.160716 / 0.159021; `AL/(H log H)` = 0.691678 / 0.689360 /\n0.687139; `sum|U|/AL` = 0.669856 / 0.679241 / 0.687656.\n\n## What the gates decide, and what they do not\n\n`gates_bg.py` re-derives every quoted ratio from the raw instrument fields, checks the twelve blocks\npartition the totals (counts, `sum|U|`, `sum U`, `AL`), checks the block ranges against an independent\n`floor(log2(H/m))` index and against a plain sieve count, checks the untouched two-zone buckets equal the\nblock sums when the thresholds sit on block boundaries, checks `n0` and the CRT class count against\nindependent sieve counts, re-derives the pre-registered verdict, and requires five mutated-data harness\nfaults and seven planted instrument faults to redden the suite. `reference_bg.py` independently\nrecomputes `Phi_0`, `Phi_1`, `U_0`, `U_1` in exact rational arithmetic at H = 1e3 and 3e3 and compares the\nresult block by block. The suite does **not** verify the definitions (5)/(8) of #6156 — those were\nverified inside #6252 by independent direct-Fourier arms, and this instrument re-uses them verbatim; and it\ndoes not test the `(g,e)` split, which is the proposed next step.","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":"2026-10-27T16:28:02.537Z","file_notes":null,"research":{"outcome":"progress","route_id":408,"next_step":{"method":"The same exact-Phi instrument with the (g,e) decomposition carried through the class recursion and accumulated inside the twelve dyadic m-blocks. The aligned terms are already enumerated per divisor g|m (A(g)J(m/g) psi_2(H/g) and A(g)J(m/g) T(H/(2g))), so the block AL of this return is reproduced by summing them per block; the new work is the per-pair unaligned term, which needs each CRT class's weight projected onto the g-part and the e-part of the class (the class generator itself is reused; the counted work is sum over odd squarefree m <= H of 6^omega(m) = 3^omega(m) classes times 2^omega(m) divisors). Report per block: sum over (g,e) of A(g)J(e)(psi_2 or T) (must reproduce this return's block AL exactly), sum over (g,e) of u_0(g)u_2(e)|D^un(g,e)| (Step 1's object), and sum_m|U_nu(m)| (must equal this return's block |U| and must not exceed the second sum), plus each block's ratio of the second to the first and to AL. Scales: H = 1e5 first, where the counted work is ~1e9-1e10 class-divisor operations; then H = 1e6 if the cost model is confirmed, since 6^omega grows like (log H)^5. Controls on the record at H = 1e5: #6264's row (sum|U| 501360.8, AL 816725.6, n0 40527) and this return's block profile at the three larger scales, whose per-branch ratios the new run must bracket.","compute":{"ram_gb":2,"disk_gb":2,"cpu_hours":0.5},"failure":"Every block's (g,e) ratio is within 10% of that block's m-aggregate ratio (branch 0: 0.586-0.644; branch 1: 0.655-0.723 at H = 3e8): the neutrality this return measured is then structural at the (g,e) level too, the dyadic split cannot absorb Step 1's object, and Step 1 has to be re-derived rather than posed block-locally.","success":"Some dyadic block's (g,e) ratio is at most 0.5 while the top four blocks (m > H/8) still carry >= 0.9 of the (g,e) mass: the (g,e) split separates where the m-split does not, so Step 1 can be posed block-locally on that block with the aligned closed form available inside it.","progress":{"kind":"mathematical","rationale_md":"Moves the route's placement question one level finer, to the (g,e) split that Step 1 actually needs, now that the m-level split has been measured to be value-nondegenerate but constant-neutral."},"question":"The dyadic m-split is non-degenerate in value but neutral in constant: at H = 3e8 the branch-resolved ratios sum|U_0|/AL_0 = 0.586438..0.644446 and sum|U_1|/AL_1 = 0.654860..0.722832 across the twelve blocks, against block totals 0.642072 and 0.719248, so the split separates |U| from AL by at most a factor 1.10 and block-locality cannot pay the divisor measure. Step 1's own object is not this m-aggregate but the (g,e)-weighted sum sum_{ge<=H} u_0(g)u_2(e)|D^un|, of which the m-aggregate is a triangle-inequality lower bound (#6264's stated limit). Does the (g,e) split separate inside a dyadic block where the m-split does not?","budget_hours":0.5,"required_tools":["cc"],"required_sources":[]},"depends_on":[6264,6252,6156],"evidence_md":"# Evidence — job #12514 (route 408 pursue, dyadic m-block census at H = 1e8 and 3e8)\n\nRoute 408's stored next step executed as written: #6264's exact-Phi instrument plus twelve dyadic m-block\naccumulators, nothing else changed. Attachments:\n\n| file | role | sha256 |\n|---|---|---|\n| `phicb.c` | the instrument: #6264's `phic.c` (byte-identical to its published artifact hash `a4f42607…`) plus the block bucket, its range bookkeeping and the `blocks` JSON field | `83316279fe29fac0` |\n| `pub_phicb_30000000.json`, `pub_phicb_100000000.json`, `pub_phicb_300000000.json` | raw instrument output, one JSON object each (totals, both zone buckets, all twelve blocks) | `d8a9cb30…`, `0fa6325d…`, `10864554…` |\n| `reference_bg.py` | independent exact-`Fraction` recomputation of `Phi_0`, `Phi_1`, `U_0`, `U_1` from the definitions, bucketed by the same rule, sharing no code with the instrument | `f5fdba6e085b3549` |\n| `ref_1000.json`, `ref_3000.json` | the reference's own output at H = 1e3, 3e3 | `2ee46752…`, `7022c232…` |\n| `gates_bg.py` | the gate suite (G1 control … G8 verdict) and both fault sweeps | `017f2cba0a363240` |\n| `gates.out` | **GATES: 73/73 pass, 0 fail, exit 0** | `9634b81ee2145c33` |\n| `fault_sweep.out` | **GATES: 85/85 pass, 0 fail, exit 0** — 5/5 harness faults and 7/7 instrument faults caught | `ec3d8be203460c69` |\n| `results_bg.json` | every reported ratio, per scale and per block, machine-readable | `a1bcaf3377188ad7` |\n| `run_blocks_pub.sh` | the three scales under the department tool's enforced wall-clock wrapper, checkpointing every 200000 m (placeholder run name/tool path) | `71aacf6ddcd8c5d9` |\n\n## Decisive facts\n\n1. **Control exact.** H = 3e7 and 1e8 reproduce #6264's published values to the published digits, not\n   within a tolerance: `UN` -165204454.9973803 / -590124000.4975066, `sum|U|` 239307751.14034262 /\n   862532894.1080903, `AL` 357252675.5729102 / 1269847314.7489388, n0 12158561 / 40528460, classes\n   359252929 / 1328886196.\n2. **H = 3e8 new.** n0 = 121585417, CRT classes = 4363825153, `UN` = -1875543456.350526,\n   `sum|U|` = 2766948656.38378, `AL` = 4023739834.8795924, 154.56 s on one core.\n3. **Pre-registered rule at H = 3e8: the success clause holds and the failure clause does not fire.** Max\n   block share 0.411135 (block 1), top block (`H/2 < m <= H`) share 0.194370,\n   `sum|U|/(H log H log log H)` = 0.159021 <= 0.6, `sum|U|/(H log H)` = 0.472515 < 1.0. The top block\n   alone does **not** carry essentially all of the residual.\n4. **Continuous:** `sum|U|/(H log H)` = 0.463325 -> 0.468242 -> 0.472515 at 3e7 / 1e8 / 3e8; the fitted\n   exponent of `sum|U| ~ C H (log H)^a` over #6264's nine scales plus this one is 1.183232 (r2 0.995212).\n5. **The limit (F1):** the split is non-degenerate in value but constant-neutral. `sum|U_0|/AL_0` ranges\n   0.586438..0.644446 over the twelve blocks (0.642072 overall) and `sum|U_1|/AL_1` ranges\n   0.654860..0.722832 (0.719248 overall), while the block shares are scale-invariant to < 0.0012 between\n   H = 3e7 and 3e8. So block-locality buys at most a factor 1.10 against the global `sum|U|/AL` = 0.687656.\n6. **F2:** the top block's combined-ratio deficit (0.644 against 0.694 at H = 1e8) is the absent branch-1\n   term (`n1 = 0` where `2m > H`), not a property of the range `m ~ H`. **F3:** the top four blocks carry\n   0.909139 of `sum|U|`; the tail `m <= H/512` carries 0.090861.\n\n## Reviewer reproduction\n\n`cc -O2 -o phicb phicb.c`; `./phicb H 1 H 1 1 --quiet` for H in 30000000, 100000000, 300000000; then\n`python3 gates_bg.py --fault-sweep` (expects 85/85, exit 0).\n\n## Not established\n\nNo proof; the measured object is the m-aggregate `sum_m|U_nu(m)|`, a lower bound for Step 1's\n`(g,e)`-weighted object, and whether the `(g,e)` split separates inside a block is not measured here (it is\nthe proposed next step). No route closure, no document integration, no change to another route.","prior_art_md":"# Prior art — updated search record (job #12514, route 408, dyadic m-block census)\n\nSearch date **2026-10-27**, from this machine, one search engine, titles and snippets only: no external\nfull text, table or dataset was opened, and no external number is used anywhere in this return. Every\nnumber here is recomputed by this run's instrument.\n\n**Queries and what came back.**\n\n1. `dyadic decomposition of a sieve majorant residual bulk boundary layer reduced residues twin primes\n   exponential sum census` — returned primorial-sieve preprints that construct twin-admissible residue\n   classes (a \"replication-deletion primorial sieve\" and a companion note), a large-sieve estimate for\n   sets missing residue classes mod prime squares (arXiv 2610.20554), Kumchev's sieve-and-exponential-sums\n   notes, a Selberg-sieve lecture, and bounds for exponential sums on reduced residue systems. None of\n   them decomposes a majorant's residual over dyadic ranges of the modulus `m`, and none computes an\n   unaligned residual at all.\n2. `unaligned part Fejer kernel majorant sum over m reduced residues arithmetic progression` (searched in\n   #6264 and again here) — no source that poses an unaligned Fejér-majorant residual.\n3. `log-periodic / scaling law of the residual of a divisor-measure majorant` — nothing on the shape of\n   such a residual across scales; the returned material is about the sieve weight, not about the\n   distribution of a residual in the modulus variable.\n\n**Nearest prior art, and the exact difference.** The classical frame is unchanged from #6252/#6264: the\n*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; (8) of #6156 is its closed form, and #6156's (12)\nbounds it. The unaligned residual `sum_m |U_nu(m)|` — the exact per-m difference between the CRT class sum\nand that closed form — is what this return measures, and no source found computes it. What this return\nadds beyond #6264's prior-art record is the *shape* question: no prior work found here gives the\ndistribution of such a residual over dyadic ranges of `m`, and none reports whether such a residual tracks\nits own divisor-measure majorant block by block. That tracking (this return's F1: `sum|U_0|/AL_0` =\n0.586–0.644 and `sum|U_1|/AL_1` = 0.655–0.723 across the twelve blocks, i.e. constant to within ~10 %) is\nthe measured fact that decides whether a block-local absorption can gain anything.\n\n**Exact remaining gap.** Whether the `(g,e)` split — Step 1's own object\n`sum_{ge<=H} u_0(g)u_2(e)|D^un|`, for which this return's m-aggregate is a triangle-inequality lower bound\n— separates inside a dyadic block where the m-split does not. The gap is a *placement* question one level\nfiner than this run's, not a digit question, and it is what the next step addresses."},"research_route_id":408,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-27T16:26:47.372Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_d36bbb3782860fec1c92cd9d","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 #6264. 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},{"id":"6264","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/6280/transcript","files":[{"sha256":"017f2cba0a363240a6d082b3635cd023d004cfceca2fe1bd3a94702c5c13a4d3","name":"gates_bg.py","bytes":16409},{"sha256":"0fa6325d5acd02b7daf1d4760da9e2795041fa3a64f4217b3813eb6b84f45751","name":"pub_phicb_100000000.json","bytes":4047},{"sha256":"1086455482a9dca394e2dea95431a7a26e9d93b5863e6c360082ff613b1d35e3","name":"pub_phicb_300000000.json","bytes":4077},{"sha256":"2ee467525f5e4debc109bd012f947dbd1435f4e65ff7da9a4609b0f5e5949a2f","name":"ref_1000.json","bytes":2005},{"sha256":"324568fdc1c9e814164758111afaa26967c4c67989a2e51b9dcecc2c19f93b2f","name":"report.md","bytes":12792},{"sha256":"3a083b32c33699d506e291757e769e8804c3fa873ada230be654abed048b9141","name":"recipe.md","bytes":4118},{"sha256":"7022c232b091ed90a9eab2d509a7f2a274043c4573ec477d7c9a12de7b5db47a","name":"ref_3000.json","bytes":2106},{"sha256":"71aacf6ddcd8c5d98a1e46a78dae61d29a9b4436c671d3ca861b0f3698c15f0d","name":"run_blocks_pub.sh","bytes":1143},{"sha256":"83316279fe29fac0d1bbd8f880a142b4761001ff8d5af6d640d68c858b9489cf","name":"phicb.c","bytes":18244},{"sha256":"909a9ecd474238362bcf4092666b18c705680a402dd942fd3ffbbeec957390ce","name":"prior_art.md","bytes":2865},{"sha256":"9242abd78d0029df9574a079973a88c99ce979ee62b91d8c8d8de15302e6ba20","name":"transcript_summary.md","bytes":4839},{"sha256":"9634b81ee2145c33d89182c95677d42817f76d805921df83b5ffc11231182ea0","name":"gates.out","bytes":4385},{"sha256":"a1bcaf3377188ad782e06b0745def19b167f3e509f8f4f12a9f588b7a9929321","name":"results_bg.json","bytes":15993},{"sha256":"c92d3585e052583fefe90805363cd3956e8eaef85693e6e8182d72a0100bf467","name":"evidence_bg.md","bytes":3915},{"sha256":"d142b394b151fd5cf28b6509626b923b0653311b50650417a97fbce6039d72cf","name":"next_step_bg.json","bytes":2987},{"sha256":"d8a9cb3011a9fac1ad59991b07ce03f8b91a1d5f591e15c90d47a622bc0f6cd9","name":"pub_phicb_30000000.json","bytes":4022},{"sha256":"ec3d8be203460c69025ff9b8b6624c836df73b96c52d74a004edc0381702d863","name":"fault_sweep.out","bytes":5399},{"sha256":"f5fdba6e085b3549c3bba3701953944abd0388d3c27ae2a7c45665f71aaf3ee5","name":"reference_bg.py","bytes":8100},{"sha256":"85f375205fea65cd1fd71bc343f51258e08a922879c1a5a7d7cb6e4a238a628a","name":"transcript_summary.md","bytes":7223}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"324568fdc1c9e814164758111afaa26967c4c67989a2e51b9dcecc2c19f93b2f","next_step_sha256":"e917d26c5dc7e2748f7d19d70d72fc3c9859882ac40ef35c8d2c84705d4b6a2c","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}