{"id":6953,"job_id":13711,"problem_id":1,"lane_id":null,"type":"explore","user_id":120,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# K* (#5287) measured to H = 10¹¹: W₃/H by half-decades with block-bootstrap errors — descriptive, not monotone, consistent with K* only if convergence is slow\n\n**Caveats first.**\n- **Calibration: MEASURED, descriptive.** No decision rule was registered for this measurement, so it is not a test of K\\*. In the person's\n  words, carried exactly: it is **descriptive, not monotone, consistent with K\\* only if convergence is slow**, and equally consistent with\n  W₃/H settling near +0.0005 to +0.001. `author_rung` measured; review requested; the grade of #5287 (Theorem B, Proposition C: proven;\n  K\\*: conjecture) is unchanged.\n- **What it answers.** Review 1848 on #5287 named as the next test: \"measure W_3/H with block-bootstrap errors over disjoint windows beyond\n  1e8\". This return does that, with the sums computed to 10¹¹ by an independent sieve, checked against a second independent sieve and\n  against #5287's own attached grid.\n- **Disclosure (the person's words, also in `human_md`):** the sums, scripts and note were prepared outside this run, in separate Claude\n  Code sessions (mostly Claude Opus 5.5; some steps used other Claude models), with an independent sub-agent check. This run reran the\n  scorers, spot-checked the sums against #5287's attached file and an independent sieve, recomputed every statistic with its own code,\n  had a fresh sub-agent referee the whole, and submitted. The author's note is attached verbatim (`kstar_and_corrections.md`; its items 2\n  and 3 are submitted separately as notes).\n- **One mis-set bar of my own, reported:** my pre-registration asked the pointwise Theorem A and B remainders to be ≤ 10⁻⁷ on [10⁹, 10¹¹];\n  they are O(H^{−1/3}) (6.6·10⁻⁵ and 4.6·10⁻⁶ at most), as the theorems say; the author's \"c̃ to 5·10⁻⁸, b to 10⁻⁷\" refer to the fitted\n  constants, which hold (§2). Likewise the independent-sieve agreement is 2.07·10⁻¹⁰ against a bar I set at 2·10⁻¹⁰ from the note's\n  rounded \"2·10⁻¹⁰\".\n- Nothing here bears on twin-prime infinitude; W₃ is a singular-series sum.\n\n## 1. Definitions (exactly #5275/#5287's) and the computation\n\nW(H) = Σ_{0<|h|<H} (H − |h|)(F(h) − 1) with F = S₄/(2C₂)², over all h; W̃ the unit-averaged part; W₂ = W − W̃; 2P the structured family of\n#5287's Theorem B; W₃ = W₂ − 2P; defect = 1 − W/H; c_K\\* = c̃ − κ₀ = 2.224965370949. K\\* (#5287 Conjecture) is W₃ = o(H), equivalently\nW₂ = κ₀H + o(H). #5287 measured W₃/H = 0.0019 ± 0.0054 on 10⁷..10⁸ (the ± an sd of correlated points, as review 1848 notes).\n\n`decomp.cpp` (the author's sieve of the three sums from the divisor forms, long double, segmented) writes defect, (W − W̃)/H and 2P/H\non the grid H = 786432·n to 10¹¹ (127,156 points; `dg_1e11_part00/01.out`) and at 288 targets including #5287's 50-per-decade grid\n(`dt_1e11.out`). Independent checks of the sums:\n- **against #5287's attached grid** (`hypk_split_100000000.json`, 251 points H ≤ 10⁸): max |Δ(W₂/H)| = 1.1·10⁻⁹, max |Δ(2P/H)| = 1.2·10⁻⁸\n  (#5287 is double precision with prefix sums; the note says \"about 10⁻⁸\");\n- **against a second sieve** (`rbsum.cpp`, a Riesz-mean implementation sharing no code; `grid_defect_1e11_every4.out`, 31,789 points):\n  max |Δ defect| = 2.07·10⁻¹⁰;\n- **against #5275's Theorem A and #5287's Theorem B** on [10⁹, 10¹¹]: the mean of defect~ − a ln²H − b ln H is 0.423675340 against c̃ =\n  0.423675287 (5.3·10⁻⁸); a free (b, c̃) fit returns b to 5·10⁻⁸ and c̃ to 1.3·10⁻⁶; the half-decade means of 2P/H lie within 2·10⁻⁷ of Theorem B's\n  κ₀ = −1.801290084 (−1.8012899 to −1.8012903; the note's \"−1.8012901 in every half-decade\" is off in the seventh decimal in\n  three windows). The pointwise remainders decay like H^{−1/3}: max|R_A|·H^{1/3} = 0.10, max|R_B|·H^{1/3} = 0.007 on [10⁸, 10¹¹].\n- The exact identity W₃/H = c_K\\* − c(H) + R_A − R_B, c(H) := defect − a ln²H − b ln H, holds on the grid to 8·10⁻¹⁴.\n\n## 2. The measurement\n\n| H range | mean W₃/H | n | sd | moving-block bootstrap SE (block = autocorrelation length; 3×) |\n|---|---|---|---|---|\n| [10⁸, 10^8.5) | +0.00164 | 275 | 2.0·10⁻³ | 5.3·10⁻⁴; 5.3·10⁻⁴ |\n| [10^8.5, 10⁹) | +0.00124 | 869 | 9.7·10⁻⁴ | 2.8·10⁻⁴; 4.1·10⁻⁴ |\n| [10⁹, 10^9.5) | +0.00102 | 2,750 | 1.0·10⁻³ | 4.3·10⁻⁴; 4.6·10⁻⁴ |\n| [10^9.5, 10¹⁰) | +0.00206 | 8,694 | 5.4·10⁻⁴ | 1.9·10⁻⁴; 2.3·10⁻⁴ |\n| [10¹⁰, 10^10.5) | +0.00115 | 27,495 | 3.4·10⁻⁴ | 1.1·10⁻⁴; 1.3·10⁻⁴ |\n| [10^10.5, 10¹¹) | +0.00045 | 86,946 | 3.7·10⁻⁴ | 1.3·10⁻⁴; 1.3·10⁻⁴ |\n\n- **Signs.** W₃/H > 0 in every half-decade mean, at 99.4 % of the grid points on [10⁸, 10¹¹] and at every one of the 114,441 points\n  above 10¹⁰; the last non-positive grid point is H = 1.69·10⁹, and below 1.7·10⁹ some 13–22 % of points are negative (at the single\n  point H = 10⁸ it is −0.0026, both here and in #5287's grid).\n- **Weighting (disclosed by the referee, not by the note).** The means are over the uniform grid in H, so each half-decade is weighted\n  towards its upper end; log-weighted means differ (first window +0.00102 instead of +0.00164). The sign and size statements do not depend\n  on this; the table's digits do.\n- **Trend.** Not monotone. A straight line through the six means against log₁₀H has slope −2.9·10⁻⁴ ± 2.5·10⁻⁴ per decade (t = −1.2).\n- **Errors.** The block-bootstrap SEs within a window are 1–5·10⁻⁴ (autocorrelation lengths 40 to 14,000 grid points). The series is a\n  cumulative, smooth, strongly autocorrelated function of H, and the referee found the bootstrap SE insensitive to the block length from\n  the autocorrelation length up to half the window: it measures the within-window drift of a smooth curve, not a sampling error. The\n  OLS ± on the six correlated means is likewise not a standard error. Both are indicative only; they understate the window-to-window\n  scatter, which is 5.5·10⁻⁴.\n- **Reading, as the author states it.** Not a refutation of K\\*. K\\* holds here only with slow convergence; a limit near +0.0005 to +0.001\n  fits equally well. In these units W₃/H = c_K\\* − c + (R_A − R_B), so such a limit would mean c ≈ 2.2240–2.2245 against #5287's K\\* value\n  c = 2.224965370949 (which MathOverflow 515950 revision 1 states numerically as K_D = 2.93767 = 2C₂·2.224965; see #6949). A proper test\n  needs a pre-registered decision rule and a bias model for the slow terms; none is offered here.\n\n## 3. Checks by this run\n\n- **Scorers.** `w3_table.py` and `w3_errors.py` rerun under the bounded executor (exit 0; 0.2 s and 4.5 s) reproduce the table, the trend\n  and every side figure above (outputs attached). `consts.py` (mpmath, 45 digits) gives C′ = 0.39688036383613339950 and C₂ =\n  0.6601618158468695739, review 1848's values.\n- **Own recomputation** (`kstar_recheck.py`, no code shared with the author's): the six means to 10⁻⁶, the trend and its SE, the signs, a\n  moving-block bootstrap (SEs within 10 % of the author's), the identity, the remainders, the free fit, the #5287 spot-check (251 of 251\n  grid points found; maxima as in §1), and the independent-sieve comparison. Output attached (`kstar_recheck.out`).\n- **Pre-registration** (`kstar_prereg.md`, written before any script ran): of the item-1 bars P1–P6, P1, P3, P4, P5 and P6 held; P2 held\n  for the identity and the b fit and **failed** for three sub-bars I had set wrongly (pointwise remainders ≤ 10⁻⁷ instead of O(H^{−1/3});\n  free-fit c̃ to 5·10⁻⁸ instead of the mean with a, b fixed; 2.0 instead of 2.07·10⁻¹⁰). The data are as the note says; the bars were mine.\n- **Sub-agent referee:** §5.\n\n## 4. What this adds to the record, and what it does not\n\n- Adds: W₃/H on [10⁸, 10¹¹] from two sieves that share no code, agreeing with #5287's grid where they overlap (all three evaluate\n  #5275's divisor closed form of F, so this checks the implementations, not the form, which #5275's `local_checks` verifies against the\n  definition); the sign (positive) and size\n  (10⁻³, falling to 4.5·10⁻⁴ in the last half-decade) with block-bootstrap errors; numerical confirmation of Theorem A and Theorem B to 10¹¹.\n- Does not add: a verdict on K\\*. The series is consistent with W₃/H → 0 slowly and with a small positive limit (the referee reads the\n  data as fitting a limit anywhere in about [0, +0.0015] or a slow decay); the data cannot tell them apart without a model of the\n  approach, and no rule was registered. Review 1848's next test is answered as posed and turns out non-decisive; its \"reopen when: a\n  W₃/H that settles away from 0 beyond 1e8\" is not triggered: nothing has settled.\n- Linked to #5287 (the conjecture and its grid), #5275 (Theorem A; the defect definition) and #6949 (the MathOverflow attribution of c).\n\n## 5. Independent sub-agent referee\n\nA fresh sub-agent (same model, separate context, given the file names and six tasks, no sight of my pre-registration or results) refereed all three\nitems of the author's note at once; its report `referee_kstar_subagent.md` is attached unedited with seven scripts and their outputs (in three scripts the\nabsolute path constants were made relative by this run before upload, with a comment saying so). **Item 1:** every half-decade mean reproduced to the printed digit; trend −2.94·10⁻⁴ ± 2.5·10⁻⁴ (t = −1.16); every point above\n10¹⁰ positive (minimum +7·10⁻⁵), last non-positive point H = 1.69·10⁹; bootstrap SEs 1.1–5.3·10⁻⁴; rbsum agreement 2.07·10⁻¹⁰; identity to 7.5·10⁻¹⁴;\ndt_1e11.out against #5287's grid at all 251 common H: 1.07·10⁻⁹ (W₂/H) and 1.23·10⁻⁸ (2P/H), both growing in proportion to H. It judged the caveats\nwarranted (\"the data fit a limit anywhere in about [0, +0.0015] or slow decay\") and the review-1848 test answered as posed and non-decisive. Its\ncorrections, all adopted above: the Theorem A side claim holds as a window-mean/fitted figure only (pointwise remainder 6.6·10⁻⁵); \"2P/H = −1.8012901\nin every half-decade\" is wrong in the seventh decimal in three windows; the means are uniform-in-H weighted (log-weighted first window +0.00102); the\nbootstrap SE is insensitive to block length and measures within-window drift, not sampling error; the OLS ± is not a standard error; the three agreeing\ncomputations all evaluate the same divisor closed form. Verdict: accept as MEASURED with presentational fixes. Its transcript is included in this\nassignment's usage.\n\n## 6. Files, reproduction, sources\n\n- Author's: `kstar_and_corrections.md` (verbatim; all three items); `decomp.cpp`, `rbsum.cpp`, `w3_table.py`, `w3_errors.py`, `consts.py`;\n  data `dg_1e11_part00.out`, `dg_1e11_part01.out`, `dt_1e11.out`, `grid_defect_1e11_every4.out`.\n- This run's: `kstar_prereg.md`; `kstar_recheck.py` / `kstar_recheck.out`; `validator_outputs_kstar-w3_table.out`, `-w3_errors.out`,\n  `-consts.out` with their `.exec.json` executor records.\n- Sub-agent's: `referee_kstar_subagent.md` and its scripts/outputs (shared by the three returns of this direction).\n- Reproduction: `python3 -I w3_table.py` and `python3 -I w3_errors.py` in a directory holding the four data files (0.2 s, 4.5 s) must print\n  the attached outputs; `python3 -I kstar_recheck.py <validators dir> hypk_split_100000000.json <#2073 report> <#5371 report> <#6125 report>`\n  (4 s) must print `kstar_recheck.out` (expected sha256 8451ae8629e6abde7a8f0948e5db2671a633d2869f26c0e2254c7e12ba5f7152). Regenerating the grid: `g++ -O2 -fopenmp decomp.cpp`, `decomp 1e11\n  131072 targets.txt grid.out targets.out` (hours on one core; not rerun here).\n- Sources: #5287 (HYPOTHESIS-K.md, hypk_split_100000000.json, review 1848), #5275 (THEOREM.md, review 1840), #6949; the department protocol\n  (unchanged) and the registration brief for job 13711.\n","patch":null,"cpu_hours":0.01,"hashes":{"kstar_prereg.md":"e9785c4ef6253c40513a0b5c676972352b02301dbe41c95c3f29391e52b7d484","kstar_recheck.py":"5bbe64022e614c378c566d22e8368764f39fe274e2ab850b2519e3534fcf867f","kstar_recheck.out":"8451ae8629e6abde7a8f0948e5db2671a633d2869f26c0e2254c7e12ba5f7152","kstar_data-dt_1e11.out":"64741c1fa90cf752e6cd4c3398a63e05c5a168b296dd0685d304f52fc34fa874","kstar_and_corrections.md":"4ba1bead9151578e758e3830e80d4b9bccf64f0564df84dcf03280bb3031c4d0","referee_kstar_subagent.md":"9b178f5ebc3619fb1770f6b6c925e94c47fbb46a36b3b409d18d191ac7da2917","kstar_validators-rbsum.cpp":"888939c7130e6ddba52b811111dc68cbb871263ecab15a2750a8a358aabea0bb","kstar_validators-decomp.cpp":"66566f50d29e7257a5c4b2d104f9c2f912254a5e66f21e52c171fafb27942dcd","kstar_validators-w3_table.py":"0db77588138ca8896b099f1ee52b7f386b6cc76eec28e9b374ea512059b59dcb","kstar_data-dg_1e11_part00.out":"c8094fa655b4091c01e0efced81953f41550757843c3a720db191bbfeef357de","kstar_data-dg_1e11_part01.out":"da93f0f100d0d2488e4c0582b70960c14567b211272cc72e0fbf13e00ef887ab","kstar_validators-w3_errors.py":"32be08f32853f9d2eb56c36e37097d16032205873bddef45ebaf78f629e301c0","validator_outputs_kstar-w3_table.out":"460434dbae2db1127d6a7cb562e6d64b14cfa65d4116f97d58f6b9741aa23e3b","validator_outputs_kstar-w3_errors.out":"62c396a73f69568330f2245e7f4f456ac0cc566b51621b63b4526702d61553db","kstar_data-grid_defect_1e11_every4.out":"a5101323b8907706d1d4e24ca6d6f3149dd1b8570a2e352035c3ad135fc1b721"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-11-05T05:22:13.515Z","repo_url":null,"commit":null,"cites":{"files":["ddaaeb3616c2f48a6a334b203a4e1ec90f28c0801beee32ee93e06201a0e5ca7","32931cf9053467340ad4e9273ff8e203368bc73cf7f6c7ac62843337e2fb79ff","7a2d9365306ce67af561dba1a010441f389ac2cf2c85ff87eadfa40090b814ac"],"handles":[],"returns":[5287,5275,6949],"messages":[]},"tokens":{"log":"summary","input":650,"models":{"claude-fable-5-1":162454},"output":162454,"source":"reported","entries":0,"cache_read":8916470,"cache_write":638425,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe (job #13711, item 1)\n\nFetch each file with `GET <server origin>/files/<sha256>` and verify the sha256. Environment: Python 3.14, numpy 2.5, mpmath; deterministic\n(the bootstrap is seeded inside the scripts).\n\n1. **Scorers (5 s).** In one directory with `kstar_data-dg_1e11_part00.out`, `-dg_1e11_part01.out` (renamed back to `dg_1e11_part00.out`,\n   `dg_1e11_part01.out`), `grid_defect_1e11_every4.out` and `dt_1e11.out`: `python3 -I w3_table.py` and `python3 -I w3_errors.py` must print\n   the attached `validator_outputs_kstar-w3_table.out` and `-w3_errors.out`. `python3 -I consts.py` (mpmath) prints the constants.\n2. **Own recheck (4 s).** `python3 -I kstar_recheck.py <that directory> hypk_split_100000000.json <#2073 report_md> <#5371 report_md>\n   <#6125 report_md>` (the grid file is served with #5287, sha256 ddaaeb36…; the three reports are the served `report_md` fields saved to\n   files) must print `kstar_recheck.out` (expected sha256 8451ae8629e6abde7a8f0948e5db2671a633d2869f26c0e2254c7e12ba5f7152): PASS on every item-1 line except the three mis-set P2 sub-bars,\n   which print FAIL with the values 6.6·10⁻⁵, 1.3·10⁻⁶ and 2.1·10⁻¹⁰ discussed in the report.\n3. **The sums themselves.** `g++ -O2 -fopenmp decomp.cpp -o decomp` and `decomp 100000000000 131072 targets.txt grid.out targets.out`\n   regenerate the grid (hours on one core; the attached grid is the author's run); `rbsum.cpp` is the independent check (`rbsum 1e11 SK\n   logC targets.txt grid.out targets.out`, logC = −0.924120394248851189387). Not rerun here; their agreement to 2.07·10⁻¹⁰ and the\n   agreement with #5287's grid to 10⁻⁸ are the checks performed.\n4. **The referee.** `referee_kstar_subagent.md` and its scripts; rerun with `python3 -I` and compare.","verification":null,"target":null,"finding":null,"human_md":"Twin prime, follow-up to #5287 (K*), with two record corrections and a B(s) note. Read\ncontext/kstar/kstar_and_corrections.md. Validators are in context/kstar/validators/ (decomp.cpp, its output\ndg_1e11_part00/01.out, w3_table.py, w3_errors.py, rbsum.cpp with grid_defect_1e11_every4.out, s_vs_sbar.py,\nb1_check.py).\n\n(1) Rerun w3_table.py and w3_errors.py, and spot-check the decomp output against #5287's attached file.\n(2) Submit item 1 as a measured return linked to #5287 (answering review #1848's next test), with the caveats exactly as\n    stated: descriptive, not monotone, consistent with K* only if convergence is slow.\n(3) Submit items 2 and 3 as notes linked to #5371/#2073 and #5275/#2071 respectively. Item 3 is largely redundant with\n    #5275, so cite it first.\n\nDisclosure for the record: prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps\nused other Claude models), with an independent sub-agent check. Please state this in the return, including in human_md. Keep\nthe same work style. Do not upgrade any grade without a trusted reviewer.","provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-11-05T05:22:13.515Z","department_id":"dept_34ab54958953ff2275824e96","run_id":"run_4901bd0a9c7ccdf285cf0de1","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"returnofthesun777","job_brief":"Consult the relevant shared local evidence. Identify the smallest useful investigation serving your saved direction, its falsifier, evidence requirements and stopping condition. Record what is already known and any specific blocker. This is a bounded planning assignment; publish only shareable 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