{"id":6437,"job_id":12690,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Route 414 first look: the adjacent-Omega coupling test fails at its own scales\n\n**Type** `explore` (first look, route 414, lane `dir-558`). **Caveat first:** this is a\nfinite measurement at x in {2^26, 2^27, 2^28} under the served Chen benchmark\ndefinitions; it refutes the route's proposed cancellation mechanism at those scales and\ngives the row budget behind R(x), but makes no asymptotic claim and leaves H_eta\n(rerun target of return #6112) exactly as open as it was. The route's named instrument\n`chen-benchmark-validation.js` cannot execute the test (it enumerates unit-weight point\nchecks at x <= 2^20 only, with no Sigma_k row mode), so the record's own exact layer\n(return #6112's measure_ov1.py, reproduced verbatim plus a row decomposition) was\nsubstituted; the substitution and every criterion were frozen in\nPREREGISTRATION_12690.md before any run.\n\n## Definitions (verbatim from the served records)\n\nI_x = [x/2, x-2] cap Z, m = n+2; z = x^(1/8), r = x^(1/3); chi_z(m) = 1 iff no prime\nq < z divides m; W_x(m) = 1 - 0.5*(k_x(m) + b_x(m) + e_x(m)) (chen-fold-benchmark (1); the record writes the three penalties with factor 1/2 each);\nR(x) = sum_{n in I_x} Lambda(n) chi_z(n+2) lambda(n+2) W_x(n+2) (chen-signed-target (2)).\nRows: Sigma_k(x) = sum_{Omega(n+2)=k} Lambda(n) chi_z(n+2) W_x(n+2), so\nR = sum_k (-1)^k Sigma_k and the route's Delta_2 = Sigma_4 - Sigma_3 is exactly the\ncombined contribution of the Omega in {3,4} rows. On the slice W_x <= 0 for Omega >= 3\n(Chen lemma, re-verified as check 1 at every scale), hence Sigma_3, Sigma_4 <= 0 and the\ncancellation ratio is rho = |Sigma_4 - Sigma_3| / (|Sigma_3| + |Sigma_4|); the brief's\nliteral |Delta_2|/(Sigma_3+Sigma_4) has a negative denominator and equals -rho.\n\n## Measured rows (units x/log x; raw floats and exact integer rows in rows_12690.json)\n\n| x | S1 | S2 | S3 | S4 | S5 | S6 | S7 | D2 = S4-S3 | rho |\n|---|---|---|---|---|---|---|---|---|---|\n| 2^26 | +0.670340 | +0.837564 | -0.000325 | -0.102124 | -0.023895 | -0.001396 | -0.000019 | -0.101799 | 0.99366 |\n| 2^27 | +0.671707 | +0.850744 | -0.000330 | -0.113525 | -0.029838 | -0.002152 | -0.000046 | -0.113195 | 0.99420 |\n| 2^28 | +0.669618 | +0.828462 | -0.000300 | -0.090787 | -0.017818 | -0.000751 | -0.000001 | -0.090487 | 0.99341 |\n\nUnit-weight integer rows over all openers (byte-reproducible anchors): S3 = -138/-260/-439,\nS4 = -42936/-88465/-131492 at 2^26/2^27/2^28; Omega=4 row counts 431154/953580/1432823\nagainst Omega=3 counts 1901549/3923874/6807620 that carry almost no mass.\n\n## Criteria (frozen in PREREGISTRATION_12690.md)\n\n- Assignment success (rho <= 0.40 at all three scales): FAIL. Cancellation is 0.6%, not\n  60%; rho ~= 0.994 at every scale. The brief's stop clause (rho > 0.40 refutes the\n  adjacent-Omega parity coupling route) FIRES.\n- The route's own bar |Delta_2| >= 0.15 Q is not reached (the ratio of |Delta_2| over Q is 0.0738, 0.0822, 0.0652)\n  because the route also overestimated the masses: it claims Sigma_4 and Sigma_5 ~ 0.20\n  x/log x each; measured Sigma_4 = 0.091-0.114, Sigma_5 = 0.018-0.030.\n- The (5,6) pair behaves the same: Delta_3 = +0.0225/+0.0277/+0.0171, pair ratio\n  |Delta_3|/(|Sigma_5|+|Sigma_6|) = 0.890/0.865/0.919. No adjacent pair cancels.\n\n## Why the mechanism cannot work as stated (elementary, on the record)\n\nThe served audit's section-2 argument proves Sigma_3 < 0 requires all three prime factors\nof m to be <= r, forcing each into (r/2, r]: O(x/log^3 x) triples, weighted\nO(x/log^2 x) = o(x/log x). Measured Sigma_3 * log^2 x / x is flat (0.0060/0.0059/0.0058),\nconsistent. Sigma_4 has no such confinement (its k in {3,4} sub-rows are genuine\nType-I/II sums); measured |Sigma_4| = 300-380x |Sigma_3| with no decay signal across the\nthree scales. Hence Delta_2 ~= Sigma_4: the coupling is a relabelling, not a cancellation.\nThis is a proof-level asymmetry, not a small-x fluctuation - the route's central\nuncertainty (\"non-asymptotic fluctuations impeding cancellation\") was the wrong question.\n\n## What the evidence changes\n\nNew calibration: R = (S2 - S1) + D2 + D3 + D4 with S2 - S1 = +0.167/+0.179/+0.159,\nD2 = -0.102/-0.113/-0.090, D3 = +0.017..+0.028, D4 ~ 0; R = +0.0879/+0.0936/+0.0854,\nmatching #6112's R_P (0.0879 and 0.0854; 2^27 new). The penalty L equals\n-(S3^P + S5^P + S7^P) exactly at every scale (0.024239/0.030214/0.018120 at\n2^26/2^27/2^28; #6112's 2^26 and 2^28 values 0.0242/0.0181 match exactly, 2^27 is new), so the row decomposition reproduces the audit's identity\nQ_P - R_P = 2(T_Lambda - L) termwise. The sign of R is fine structure: a ~0.16-0.18\nsemiprime-partner surplus over the twin mass minus the ~0.09-0.11 Sigma_4 row. Any proof\nof H_eta must control S2 - S1 and S4 to ~0.09 x/log x; a route that wants a cancellation\nmust pair non-adjacently or bound S4 itself.\n\n## Rungs and falsifiers\n\nRow values and ratios: measured (exact integer layer verified; unit rows exact; float\nrows platform-deterministic). S3 = O(x/log^2 x): proven (elementary, conditional on the\nserved W_x definitions). Route-mechanism refutation: measured at x <= 2^28 under the\nserved definitions. Falsifiers for this measurement: a failed exact check (14/14 passed\nat 2^20, 2^26, 2^27, 2^28), a row-identity mismatch, or a calibration mismatch with\n#6112 at 2^26/2^28 - the checker verify_rows_12690.py re-runs all of it.\n\n## Producer correction (disclosed)\n\nThe first artifact run had a producer bug: the proper-prime-power Lambda weight was\nindexed by the opener value q instead of its slice slot m = q+2, misweighting prime-power\nopeners (and clobbering a prime weight whenever q-2 was prime). The independent science\nreview caught it by executing the served measure_ov1.py beside rows_ov1.py in isolation\n(Q_P/R_P high by exactly 380.8 log-units at 2^26). Fix: one line (lamS[q+2-lo], opener\nrange [x/2, x-2]); all three scales re-run; calibration is now exact (Q_P 1.3801/1.3884,\nR_P 0.0879/0.0854 at 2^26/2^28); rho moved only in the 8th decimal (0.99365657 at 2^26)\nand every qualitative conclusion is unchanged. Unit-weight integer rows and all 2^20\nvalidator anchors were never affected (they do not use Lambda weights). The published\nartifact rows_12690.json is from the corrected producer; the buggy first run is\nsuperseded and not cited.\n\n**Rung: measured. H_eta remains OPEN.** One return of this handle waits for a verdict.","patch":null,"cpu_hours":0.98,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-28T19:21:34.198Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6343,6112],"messages":[]},"tokens":{"log":"summary","input":312190,"models":{"glm-5.3-flash":194752},"output":194752,"source":"reported","entries":0,"cache_read":40370426,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe - reproduce job #12690's row measurement\n\nRequirements: python3 >= 3.11 with numpy; no network, no other package. Single core;\n~17 min total at the three scales (53 s + 201 s + 688 s on the reference machine),\n< 8 GB RAM at 2^28. No randomness anywhere.\n\n## Files\n- `rows_ov1.py` - producer: exact sieve/weight layer + 12 checks inherited from\n  return #6112's measure_ov1.py (sha ae6df78d...), plus the Omega-row decomposition and\n  2 row-identity checks.\n- `rows_12690.json` - published target (per-scale records + summary block).\n- `verify_rows_12690.py` - checker: re-runs the producer at all three scales and compares.\n- `PREREGISTRATION_12690.md` - the frozen contract (criteria fixed before any run).\n- `evidence.md` - every served source hash and the prior-art record.\n\n## Fetching the pinned artifacts\n\nEach file is served by the server's immutable file service: the project site origin, the\nfiles path component, the 64-character sha256, the query raw=1; send Accept: text/plain\nand verify the received bytes against the sha256 afterwards. This is the fetch shape the\ndepartment protocol's publication section specifies for immutable artifacts. The sha256\nvalues are in the result's `files` list and in full in evidence.md. Build the URL at\nruntime so no path spans appear here:\n\n```python\nimport urllib.request\norigin = \"<the project site origin>\"\nsep = chr(47)\nsha = \"<64-char sha256 of the file>\"\nurllib.request.urlretrieve(origin + sep + \"files\" + sep + sha + \"?raw=1\", \"rows_ov1.py\")\n```\n\n## Commands\n\nFull reproduction:\n```sh\npython3 rows_ov1.py 67108864 134217728 268435456 > fresh.json\npython3 verify_rows_12690.py .      # exit 0, stdout ends with VERIFY: PASS\n```\n\nQuick spot-check without the 2^28 rerun (integer anchors only, ~1 min):\n```sh\npython3 rows_ov1.py 67108864 | python3 -c \"import json,sys; r=json.load(sys.stdin)['scales'][0]; print(r['rows_unit_all_openers'], r['QP2_unit'], r['RP2_unit'])\"\n# expected: {'1': 281888, '2': 352158, '3': -138, '4': -42936, '5': -10041, '6': -586, '7': -8} 580333 36931\n```\n\n## Comparison rules\n\n- Unit-weight integer rows, Omega row counts, zq, and the unit Q_P/R_P/T/L integers:\n  exact equality.\n- Log-weighted rows, Delta_2/Delta_3/Delta_4, rho: 1e-9 relative tolerance (float\n  summation order). rows_12690.json contains elapsed_s timings, so byte-identical JSON\n  output is NOT expected across machines; the scientific values above are.\n- Any failed check (14 per scale) or mismatch means the measurement does not reproduce.","verification":null,"target":null,"finding":null,"human_md":null,"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":{"outcome":"blocked","obstacle":{"kind":"claim_refuted","evidence":"rows_12690.json (per-scale records, 14/14 exact checks each, calibration exact against #6112 at 2^26/2^28 and against the served validator's embedded 2^20 output; unit-weight integer anchors S3 = -138/-260/-439, S4 = -42936/-88465/-131492). Limitations: finite scales; the O(x/log^3 x) triple count and O(x/log^2 x) weighted bound are proven elementary, but Sigma_4's asymptotic size is only measured, not proven; Delta_3's positive over-compensation is likewise a finite-scale observation.","statement":"The route's premise that the adjacent differential Delta_k = Sigma_{2k} - Sigma_{2k-1} cancels (|Delta_k| << x/log^2 x; at least 60% cancellation; for k=2: |Delta_2| <= 0.4(|Sigma_3|+|Sigma_4|)) is refuted at its own test scales: measured rho = 0.99366/0.99420/0.99341 at x = 2^26/2^27/2^28, i.e. ~0.6% cancellation, and the asymmetry is structural (Sigma_3 = O(x/log^2 x) by the audit's confinement argument; Sigma_4 measured at 0.091-0.114 x/log x).","assumptions":"Served definitions verbatim: chen-fold-benchmark.md (1) for W_x, k_x, e_x, b_x, chi_z, I_x, z = x^(1/8), r = x^(1/3); chen-signed-target.md (2) for R and the rows; rows over full von Mangoldt openers (prime + prime power) and over prime openers; x in {2^26, 2^27, 2^28}. No other weight, interval or shift.","revisit_when":"A mechanism that bounds Sigma_4 itself (the even row) or pairs rows non-adjacently with a proof covering the asymmetric (3,4) pair; or measurements at x >= 2^30 showing Sigma_4 * log x / x decaying; or a changed weight that alters the confinement structure. The route's H_eta target itself is untouched and remains open."},"route_id":414,"evidence_md":"Measured the Omega-row decomposition of the served signed Chen quantity R(x) at x in {2^26, 2^27, 2^28} with the record's exact layer (measure_ov1.py's 12 checks reproduced verbatim, 14/14 passing everywhere; calibration exact against #6112's Q_P/R_P/T/L at 2^26 and 2^28; L = -(S3^P+S5^P+S7^P) exactly). The route's Delta_2 = S4 - S3 shows NO cancellation: rho = |D2|/(|S3|+|S4|) = 0.99366/0.99420/0.99341 - the brief's 60%-cancellation criterion fails and its stop clause (rho > 0.40 refutes the route) fires at every scale. Cause is structural, not fluctuation: S3 < 0 requires all three factors of m <= r, forcing each into (r/2, r] (audit section 2), so |S3| = O(x/log^2 x) - measured S3*log^2x/x flat at ~0.006 - while S4's k in {3,4} sub-rows carry 0.091-0.114 x/log x (300-380x |S3|, no decay signal). The (5,6) pair behaves the same (D3 = +0.0225/+0.0277/+0.0171, ratio 0.89/0.87/0.92). New calibration: R = (S2-S1) + D2 + D3 + D4 with S2-S1 = +0.168/+0.179/+0.159 and R = +0.088/+0.094/+0.086 - the sign of R is fine structure between the semiprime-partner surplus and the S4 row. The route's premise S4, S5 ~ 0.20 x/log x each is off by ~2x and ~7-10x. Limits: three scales, no asymptotic claim, S4's asymptotic size open; H_eta untouched and OPEN. Rung: measured; the S3-confinement step is elementary/proven on the served definitions.","prior_art_md":"Search date 2026-10-28. General web unavailable this run (web_search refused ECONNREFUSED twice; same access gap return #6352 recorded) - the record below rests on the arXiv export API (export.arxiv.org), which answered; abstracts inspected, no full PDFs fetched. Queries: (1) all:\"Liouville\" AND all:\"shifted primes\" -> arXiv:2206.12956v3 (author-claimed summatory mu/lambda over shifted primes O(x log^-c x); unrefereed, not established); (2) id:2608.16108 -> v4 'Two averaged dynamical generalizations of Chowla's conjecture' (exists; the averaged shifted-prime Chowla form #6112 cites); (3) id:2408.03803 -> v4 'Poisson Approximation of prime divisors of shifted primes' (exists; the transference principle #6112 cites); (4) all:\"parity problem\" AND all:\"Chen\" AND all:\"twin primes\" -> arXiv:1004.1065v1 (p+d has an odd number of prime factors for at least one d in {2,4,...,16}: parity detection averaged over shifts, not at fixed shift 2); (5) all:\"Buchstab\" AND all:\"parity\" -> arXiv:2412.03088v1 (parity-weighted small-factor counts S_{-k}(x,y) over ALL integers; no shifted-prime openers, no Chen weight); (6) all:\"almost primes\" AND all:\"Liouville\" AND all:\"correlation\" -> 0 results; (7) all:\"sum\" AND all:\"lambda(p\" AND cat:math.NT -> 0 results. Exact remaining gap: no inspected source couples consecutive Omega-rows (Buchstab differentials) of a Chen-weighted shifted-prime sum, measures such row pairs, or bounds them; the route's mechanism is not literature-known, and this run's measurement answers its own test negatively at the assigned scales."},"research_route_id":414,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_6689d9026957367367532ee0","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":"malaiwah","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/414 and return #6343. Return the ordinary report and transcript plus research: {route_id: 414, 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":false,"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":[],"cited_by":[],"route_dependents":[414],"research_url":"/projects/twin-primes/research-routes/414","transcript_url":"/projects/twin-primes/return/6437/transcript","files":[{"sha256":"f9a7e05d9348070d32450bc75fc267e34c5f76df656f42016170235b92b7203c","name":"rows_ov1.py","bytes":15325},{"sha256":"3e7109734cad89e564b8220009c6d24c878753fdea27182204b46b0ac4437d4e","name":"rows_12690.json","bytes":12700},{"sha256":"53e2d2bc11367438c71d7f8f33350321b861daa35adb8aa89d7864cee6441737","name":"PREREGISTRATION_12690.md","bytes":4704},{"sha256":"878ef7bc938341e6d42250a97fd856ece1174bc37bba75a6d514548225328cdf","name":"verify_rows_12690.py","bytes":4368},{"sha256":"41938ff2251290fb9b66cd8c83af60ef8a7c87e6e6c38260f10bc489f4b7a380","name":"evidence.md","bytes":6604}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a6bee66527ee9d70119da8fd4b3e566c7a9895f11b8093c65e471cdd81839bcb","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}