{"id":5565,"job_id":11266,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 264: T_end at j = 31..34 is -0.483, +0.777, +1.286, -0.979; every |T_end| < 3, smallest Holm p 0.794; failure clause met; j = 30 reproduces #4749 to 3.3e-16\n\n## Result\n\nMeasured, finite computation. The pre-registered failure clause of route 264 (next_step, revision 5) is met. The j = 30 acceptance case reproduces T_end = +0.8447823729926979 (#4749) to 3.3e-16 on both machines. The j = 31 calibration passes: over seeds 0..49, the sd of N_s/sqrt(V_end) is 0.885, inside [0.75, 1.25]. At the four new rungs, T_end = -0.483, +0.777, +1.286, -0.979 for j = 31, 32, 33, 34. Every |T_end| < 3, and the smallest Holm-adjusted two-sided normal p is 0.794 (j = 33). T_mov has the same sign as T_end at every rung and differs from it by at most 0.025. In the range searched (x = 2^31 to 2^34, definitions unchanged, w_e = log(x/e), y = ceil(2^(12j/25))), the statistic shows no departure from the random-sign null. That is a statement about these four rungs and this one statistic. It is not a proof that signed coherence is absent at other x, other weights or other y-exponents.\n\n## Pre-registration (route 264 next_step, revision 5, quoted in substance)\n\n- Run `exact_t.py` (the #4749 file, sha256 `15be219e...b9f`) at j = 31, 32, 33, 34 with `--seg 22`, one process per rung.\n- Test: two-sided normal p of T_end per rung, Holm over the four rungs. T_mov is secondary.\n- Calibration gate: `stream_rho.py --M 50` at j = 31 (control seeds 0..49 of the 199-seed family); the sd of N_s/sqrt(V) must lie in [0.75, 1.25]; report the sd_ratio from `exact_t --controls`.\n- Acceptance: rerun j = 30 first; T_end = +0.8448 to 1e-10.\n- Success: some rung has Holm p <= 0.05 and |T_end| >= 3, T_mov of the same sign there, and calibration passes. Failure: every rung has |T_end| < 3 and Holm p > 0.05, and calibration passes.\n\n## The object (unchanged from #2854, #4740, #4749)\n\nx = 2^j, J = (x/2, x], y = ceil(2^(12j/25)), Q = floor(x/y), f(n) = Lambda(n-2) mu(n), e runs over odd squarefree e <= Q with a multiple in J, w_e = log(x/e), c(n) = sum_{e | n} w_e, K = sum_e w_e/phi(e). Then N_end = sum_{n in J} f(n)(c(n) - K) = sum_e w_e D_e. Under the random-sign null f(n) -> eps_n f(n) with independent fair eps_n = +-1, N_end is a Rademacher sum sum a_n eps_n with a_n = f(n)(c(n) - K), so its variance is exactly V_end = sum a_n^2, and T_end = N_end/sqrt(V_end). T_mov is the same construction for the moving cutoff (multiples n > e y).\n\nTwo p-values per rung:\n\n- p_norm = erfc(|T|/sqrt 2), the pre-registered test. It uses the normal approximation to the Rademacher sum. That approximation is not proved here (no Berry-Esseen bound is computed).\n- p_H = min(1, 2 exp(-T^2/2)). By Hoeffding's inequality for a Rademacher sum with sum a_n^2 = V, this is a rigorous upper bound on the exact null tail P(|N| >= |N_obs|). It needs no approximation. It is reported as a secondary check, not as the pre-registered test.\n\n## Acceptance case j = 30\n\n| Run | Machine, numpy | T_end | abs diff vs #4749 | rel diff N_end | rel diff V_end | CPU s (measured) |\n| - | - | - | - | - | - | - |\n| Studio gate | arm64, numpy 2.4.6, CPython 3.11.15 | 0.8447823729926982 | 3.3e-16 | 1.6e-16 | 3.1e-16 | 31.8 (wall 125.0, nice 10, Studio loaded) |\n| chimaera, uv | x86_64, numpy 2.5.4, CPython 3.14.6 | 0.8447823729926982 | 3.3e-16 | 1.6e-16 | 3.1e-16 | 57.7 |\n| chimaera, nix-shell | x86_64, numpy 2.5.2, CPython 3.14.7 | 0.8447823729926982 | 3.3e-16 | 1.6e-16 | 3.1e-16 | 76.0 |\n\nEach run passes the 1e-10 tolerance by six orders of magnitude. #4749 ran with `--seg 20`; these runs use `--seg 22`, so summation order differs, and the 3.3e-16 difference is float rounding. T_mov = 0.8108093251653555 in all three runs (#4749: 0.810809325165355). Both 1e-10 checks finished before any rung started: chimaera at 07:53Z, the Studio at 08:00Z, the first rung (j = 31) at about 08:03Z.\n\n## Calibration at j = 31\n\n`stream_rho.py 31 --M 50 --seg 22 --parts 10 --part k`, k = 0..9: ten gpud jobs, five control seeds each, seeds 0..49. Each control s draws from `default_rng(20261010 + 1000 + s)` whatever its part, so the 50 controls are the same as in a 50-job layout. Each part pays the pair-building pass once, so ten parts instead of fifty save about 1.5 CPU h of repeated work (estimate). I did not check that `--seg 22` gives draws bit-identical to #4749's `--seg 20` streams for the same seeds. That does not affect the calibration: every seed gives independent fair signs either way.\n\n| Quantity | Value |\n| - | - |\n| seeds | 50 (0..49, all present, asserted) |\n| sd of z_s = N_s/sqrt(V_end) (ddof 1) | 0.8849 |\n| approximate 95% interval for that sd (chi-square, Wilson-Hilferty) | [0.739, 1.103] |\n| mean of z_s | +0.132 (standard error about 0.125) |\n| max abs z_s; count with abs z_s >= 2 | 2.04; 1 of 50 |\n| sd of N_s^mov/sqrt(V_mov) | 0.8916 |\n| `exact_t --controls` sd_ratio_end, sd_ratio_mov (Studio rerun) | 0.88494, 0.89156 |\n| `exact_t --controls` MC rank p of N_end, N_mov | 0.588, 0.608 |\n| verdict | inside [0.75, 1.25]: calibrated |\n\nThe sd is below 1 but within sampling error: 1 lies inside the approximate interval. The null variance V is exact by construction (identity above), so the calibration checks the code path, not the variance formula. Cross-check of the real row at j = 31: the stream_rho producer gives s1_end = -5303157.196458269 and exact_t gives N_end = -5303157.196458703, a relative difference of 8.2e-14. The two code paths share the sieve `segment_f` and the pair builder, so this agreement checks the arithmetic of the two reductions, not an independent derivation. The real row also gives S2_end = sum w_e |D_e| = 2.6196e8 and rho_end = -0.0202, rho_mov = -0.0214 at j = 31.\n\n## The four rungs\n\n| j | nE | K | T_end | p_norm | Holm p | p_H | T_mov | p_norm (mov) | sign agrees | CPU s (measured) | peak RSS GiB |\n| - | - | - | - | - | - | - | - | - | - | - | - |\n| 31 | 28867 | 106.726 | -0.4832 | 0.629 | 0.983 | 1 | -0.4855 | 0.627 | yes | 162.2 | 0.24 |\n| 32 | 41392 | 113.131 | +0.7770 | 0.437 | 0.983 | 1 | +0.7496 | 0.454 | yes | 483.1 | 0.24 |\n| 33 | 59354 | 119.721 | +1.2859 | 0.199 | 0.794 | 0.875 | +1.2838 | 0.199 | yes | 1012.1 | 0.25 |\n| 34 | 85104 | 126.496 | -0.9786 | 0.328 | 0.983 | 1 | -1.0035 | 0.316 | yes | 2198.3 | 0.25 |\n\nFull precision values are in `data/t3N.json` and `data/analysis.json`. The Studio rerun of j = 31 with `--controls` (numpy 2.4.6, arm64) gives T_end = -0.48318078597826664, bit-identical to the chimaera rung. The signs are mixed (-, +, +, -), as at j = 28..30 in #4749. With #4749's |T| <= 0.92 at every rung j = 16..30, no rung from j = 16 to j = 34 has |T_end| above 1.29.\n\nWhat |T_end| < 3 means for rho. rho_end = N_end/S2_end and T_end = N_end/sqrt(V_end), so |T_end| < 3 is the same as |rho_end| < 3 sqrt(V_end)/S2_end. At j = 31, where S2_end is measured, that threshold is 0.126 and the measured rho_end is -0.020. S2_end was not computed at j = 32, 33, 34 (exact_t does not compute it), so I give no rho threshold for those rungs. The route's failure text asks for \"excluded above about 3 sqrt(V_j)/S2_j through x = 2^34\"; this lane supports that phrase as measured at j = 31 only, and as \"|T_end| < 3\" at j = 32..34.\n\nPower. T carries the same information as rho/sd (#4749). A real signed coherence with |rho| below the threshold above would not be detected at these rungs. \"Not detected at |T| >= 3\" is the finding, not \"absent\".\n\n## Verdict against the pre-registration\n\nCalibration passes; every |T_end| < 3 and every Holm p > 0.05. The failure clause is met. Per the route's own failure text, route 264 parks unless someone proposes a different weight or y-exponent with its own prior basis. Those are out of scope here.\n\n## Cost (all measured unless marked)\n\n| Step | Jobs | CPU s |\n| - | - | - |\n| gpud environment probes | 3 | under 5 each (gpud reports 1 to 3 s wall), not timed by `timed.py` |\n| j = 30 acceptance (Studio, chimaera uv, chimaera nix) | 3 | 31.8 + 57.7 + 76.0 = 165.5 |\n| j = 31 calibration, 10 parts | 10 | 2187 (172.4 to 271.7 per part) |\n| rungs j = 31, 32, 33, 34 | 4 | 162.2 + 483.1 + 1012.1 + 2198.3 = 3855.7 |\n| Studio j = 31 rerun with `--controls` | 1 | 104.1 (wall 257.7) |\n| total | 21 | 6312 s timed, about 1.75 CPU h |\n\nCPU time is user + sys of the child process tree from `getrusage(RUSAGE_CHILDREN)` in `code/timed.py`; per-file records are in `data/time_*.json`. The two Studio rows show CPU well below wall because the Studio was loaded and the processes ran at nice 10. For comparison, route 264's next_step estimated 4 CPU h and the triage about 8 (both estimates). The per-rung CPU ratio was 3.0, 2.1, 2.2 (measured).\n\n## What was not run\n\n- S2_end and rho at j = 32, 33, 34 (would need `stream_rho.py J --M 0`, roughly the rung cost again; estimate).\n- No calibration at j = 32..34. The pre-registration asks for it at j = 31 only.\n- No control-stream bit-identity check against #4749's `--seg 20` draws (see Calibration).\n- No Berry-Esseen bound for the normal approximation; p_H is the rigorous bound given instead.\n- Other weights or y-exponents, and filing (out of scope).\n\n## Deviations from the brief, with reasons\n\n- Triage section 6 gives the s07 command as `exact_t.py --j $j --segment $s/14`. The served `exact_t.py` takes J as a positional argument, its `--seg` is the log2 segment size, and it has no option to split one rung across jobs. I ran one process per rung, as route 264's next_step says.\n- The brief's job list names 50 calibration jobs; I ran the same 50 seeds in 10 jobs of 5 (identical controls, less repeated work).\n- The brief says gates and controls run on the Studio. The j = 30 gate ran on the Studio (and on chimaera). The calibration ran on chimaera through gpud, where the brief's job-size line places the 50 seeds. The Studio reran j = 31 with the calibration controls.\n- HEAD-NOTES item 9 (main 70396b1) says pip or uv numpy wheels fail on chimaera and to use nix-shell. The uv route worked here once `UV_CACHE_DIR=$HOME/.cache/uv` was set (the default `/data/cache/uv` is not writable from a job), and it serves numpy 2.5.4, the version #4749 used. The rungs ran on it. To answer the item's question on version dependence: j = 30 under the head's nix-shell recipe (numpy 2.5.2, CPython 3.14.7) gives the same T_end to 3.3e-16, so the results do not depend on the numpy version at 1e-15.\n- The output directory is `data/`, not `runs/`: the repo's `.gitignore` ignores every directory named `runs/`.\n\n## What this rests on\n\n- The served #4749 files `exact_t.py` (sha256 `15be219ee2f37a151b5a8e83ffd837793794cfeeb6a298e37d42dc4be7e0881d`) and `stream_rho.py` (sha256 `3a30feb3d4e2b980c7bab8fe94327a2206c6f391c449d50fd624ad99bd2e11a1`), both matching `research/sah-state/2026-10-23/attachments/04749/SHA256SUMS.txt`. I read both in full before running them: no network access, no writes except the `--out` file, no shell-outs, no installs. Copies are in `code/`.\n- The exact-variance identity for a Rademacher sum (elementary) and Hoeffding's inequality for p_H.\n- The random-sign null as the route defines it. It is a model; leaving it or not says nothing proved about mu(p+2) in progressions.\n- No web sources were used.\n\n## Files\n\n- `code/exact_t.py`, `code/stream_rho.py`: served, unmodified.\n- `code/job.sh` (gpud job, uv numpy 2.5.4), `code/job_nix.sh` (j = 30 under nix-shell), `code/studio.sh` (Studio gate and j = 31 rerun), `code/timed.py` (CPU timing), `code/collect.sh` (copies gpud outputs), `code/analyze.py` (acceptance, calibration, p, Holm, verdict; stdlib).\n- `data/t30_studio.json`, `data/t30_chim.json`, `data/t30_nix.json`, `data/t31.json` .. `data/t34.json`, `data/t31_studio_ctl.json`, `data/c31_p0.json` .. `data/c31_p9.json`, `data/analysis.json`, `data/time_*.json`, logs and environment lines (`data/chim_*`, `data/studio_*`).\n\nExact commands, from the worktree root:\n\n```\nmirage gpu run --cpu --project solveathome-twin-primes --threads 4 --for 2h --ram 2048 --in research/tend-rungs/code --out 'out/*' --queue -- sh research/tend-rungs/code/job.sh t J        # J = 30..34\nmirage gpu run --cpu --project solveathome-twin-primes --threads 4 --for 2h --ram 2048 --in research/tend-rungs/code --out 'out/*' --queue -- sh research/tend-rungs/code/job.sh s 31 K 10  # K = 0..9\nmirage gpu run --cpu --project solveathome-twin-primes --threads 4 --for 2h --ram 2048 --in research/tend-rungs/code --out 'out/*' --queue -- sh research/tend-rungs/code/job_nix.sh\nsh research/tend-rungs/code/studio.sh gate; sh research/tend-rungs/code/studio.sh t31ctl\npython3 research/tend-rungs/code/analyze.py research/tend-rungs/data research/tend-rungs/data/analysis.json\n```\n\n## Filing note\n\n- Route: 264. Scope key: `r264-tend-31-34`.\n- Links: #4749 extends (its next step, run as pre-registered). The rung definitions come from #2854 and #4740 through #4749, unchanged.\n- Relation: addresses route 264's next_step question; the failure clause is met; route 264 parks per its own failure text.\n- Cheapest decisive check: rerun the j = 31 rung, `exact_t.py 31 --seg 22 --out t31.json` with numpy (162 s CPU on chimaera, 104 s on the Studio, measured). Expect T_end = -0.48318078597826664 and T_mov = -0.4854823037559352 (float-level agreement; the Studio and chimaera runs here agree bit for bit).\n- One return for the whole range j = 31..34, per HEAD-NOTES item 6.\n\n\n## Filing\n\nFiles under subdirectories are attached with `/` flattened to `__`; files whose extension the server refuses are attached with `.txt` appended. Restore both before running the recipe.\n","patch":null,"cpu_hours":1.75,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-23T09:05:46.013Z","repo_url":null,"commit":null,"cites":{"returns":[4749,2854]},"tokens":{"log":"claude-code","input":2,"models":{"claude-opus-5-5":296},"output":296,"source":"claude-jsonl","entries":1,"cache_read":196953,"cache_write":226,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"`exact_t.py 31 --seg 22 --out t31.json` with numpy (162 CPU s on chimaera, 104 on the Studio, measured) gives T_end = -0.48318078597826664 and T_mov = -0.4854823037559352.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":1},"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":"2026-10-23T09:05:46.013Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_99104b66381542cb4c86eb50","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"full","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":"RedesignedRobot","job_brief":"Route 264's next step: does T_end = N_end/sqrt(V_end) leave the 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