{"id":2370,"job_id":5077,"problem_id":1,"lane_id":2,"type":"explore","user_id":58,"model":"claude-sonnet-5-5","provider":"anthropic","report_md":"# Job #5077, route 192: a symmetry-respecting screen is still vacuous, but kappa4 already cancels against B_abs for h up to 120 and x up to 47\n\nGrade: **measured** (exact finite arithmetic); the translation-invariance lemma is **proven** (one line, below). Outcome: **progress**.\n\n## What was asked\nRescue route 192, blocked on: the nominated mod-6-mixing partition is sign-separating at 12 rungs (h in 7,10,12). Seek a repair, weaker requirement, new ingredient, or alternate method.\n\n## New ingredient: K is translation invariant\nR_x(d_B) is the probability that all n+d_i, i in B, survive for n uniform mod q. Since n -> n-s permutes Z/q, R_x, hence K_x, depends on d only through differences. Consequences: (i) the earlier candidate (sum of residues mod 6) is not invariant (it shifts by 4s), so it was not a test of the object's own symmetry; (ii) kappa4(h)=sum_(m<h) (h-m) C(m), so the whole h-dependence lives in a 1-D sequence C(m); (iii) enumeration drops from h^4 to O(h^3), which makes h=120 exact and cheap.\n\n## Results\n1. **Screen with invariant partitions (pre-registered):** 5 partitions x 12 rungs, 0 sign-mixing, 12 of 12 sign-separating each; the negative control passed. The label-shuffle criterion is unattainable by any analytic partition (shuffled signs cancel at random, observed classes are coherent); it is stricter than a cancellation bound needs.\n2. **Uniform in h:** |kappa4|/B_abs falls with h (strictly over h>=24 for every x>=11), x in 5..47 (13 rungs, q up to 6.1e17), exact:\n\n| x | h=7 | h=12 | h=24 | h=60 | h=120 |\n|---|---|---|---|---|---|\n| 13 | 5.10e-02 | 2.24e-02 | 4.39e-03 | 6.37e-04 | 9.06e-05 |\n| 29 | 1.15e-02 | 3.12e-02 | 9.98e-03 | 1.00e-03 | 2.21e-04 |\n| 47 | 8.67e-02 | 9.86e-03 | 8.84e-03 | 1.52e-03 | 5.09e-04 |\n\n   For x>=13, |kappa4(120)| stays below 1.1 while B_abs(120) is 1e+03 to 1e+04 (B_abs grows like h^3.4 to h^3.9). The local decay exponent of the ratio between h=60 and 120 is about 3 for x=11..19 and then falls monotonically to 1.6 at x=47: the decay in h slows as x grows.\n3. **Not uniform in x at fixed h:** the h=120 ratio grows with x (increments 9.9e-06 to 3.3e-05 per unit x) (x=17: 7.9e-05; x=47: 5.1e-04); at h=24 it peaks near x=31 to 37. Plotted against mu=h*delta(x) it collapses only partially: for mu>=1 a fit ratio ~ mu^-2.0 with residual factor 1.3 (1 sd), worst point 2.0, band spread up to 4.0x; none for mu<0.5.\n4. **Correction of #2344:** its derived ratio column is wrong in 4 of 5 rows; its exact values are right (reproduced). The sentence \"oracle reaches 0.0085-0.0177 at q=30030\" is unsupported (true: 0.022-0.051).\n\n## Limits\nFinite rungs only; h<=120 (below q in all but the smallest x); 9-12 sample points are not used here, all counts are exhaustive. The mean-gap scaling is observed, not derived. The link from kappa4 to route 190's H(a,A) and to twin primes is NOT tested. Literature: snippets only; Montgomery-Soundararajan likely contains the range reduction, so it is not claimed as new.\n","patch":null,"cpu_hours":0,"hashes":{"exp_result.json":"9e3fe4c8433367800f1b7c187c077b8714735aed1f6c845173736bc064949b17","uniform_h_result.json":"523e6d8c9e1ccf01d450f96b739bae3ad1ea5e07754996515f7b7d16a29c54b1","uniform_h_result_x17_19_23_29.json":"1abc0f8dd040a5b7e942eb5318483a1441d8cb7272b35f803d5c1c90babf9cea","uniform_h_result_x31_37_41_43_47.json":"1534d032e125319d7495b6704c6afac59396f4f25a6efb934c56b91963810032"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-06T02:39:05.353Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2344,2340,2334],"messages":[]},"tokens":{"log":"custom","input":89,"models":{"claude-sonnet-5-5":65155},"output":65155,"source":"custom-jsonl","entries":42,"cache_read":15674501,"cache_write":96630,"observed_models":["claude-sonnet-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Requirements: python3 (stdlib only, no numpy). Single thread.\n1) python3 -I k4.py  -> prints q=30 and q=210 at h=7 with 'OK' against the published controls (-739/15000, 60341/15000) and (-969/9800, 990319/480200).\n2) python3 -I exp.py  (SEED 20261006; ~50 s) -> exp_result.json sha256 9e3fe4c8433367800f1b7c187c077b8714735aed1f6c845173736bc064949b17; H1 violations 0, H2 all True, 60 screen rows, 0 mixing.\n3) python3 -I uniform_h.py 120 5 7 11 13  (~45 s) -> uniform_h_result_x5_7_11_13.json (served as uniform_h_result.json, same content) sha256 523e6d8c9e1ccf01d450f96b739bae3ad1ea5e07754996515f7b7d16a29c54b1; then 17 19 23 29 (~70 s) sha256 1abc0f8dd040a5b7e942eb5318483a1441d8cb7272b35f803d5c1c90babf9cea; then 31 37 41 43 47 (~2 min) sha256 1534d032e125319d7495b6704c6afac59396f4f25a6efb934c56b91963810032. Each run prints 'validação(h<=12 exata)=True'.\nExpected anchors: q=30030 h=120 |kappa4|/B_abs = 0.000091; q=6.1e17 (x=47) h=120 = 0.000509.\nDisclosure: k4.py was edited AFTER the pre-registration only to extend the PRIMES list from 29 to 47 (behavior for x<=29 identical; hash at pre-registration d2f136a7...). Pre-registration hash: bc5c3074fd03cc6a7d8e4e68ebdf2eb8d234a582c8b68073eb17947fc6c9220a.\nRun time total ~5 min wall. cpu_hours not measured separately.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"medium","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-06T02:39:25.921Z","file_notes":[{"sha":"128e88012c87f4de8a8af1fb3db7337a96199a234faafe2926d925143450af41","name":"exp.py","notes":["prints what looks like progress or timing to stdout on line 70 (\"print(\"x=%d q=%d pronto (%.0fs) H1 violações=%d\" % (x, q, time.time() - t0, res[\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"progress","route_id":192,"next_step":{"method":"Extend the exact translation-reduced enumeration (sorted forms (0,a,b,m)) to x in 53..97 with h up to ceil(5/delta(x)); validate against the h^4 census at h<=8; fit log(ratio) against x at fixed mu and tabulate sign and periodicity of C(m).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"ratio at fixed mu increases with x at 3 or more consecutive x with a positive fitted slope; then no uniform-in-x statement holds in the mean-gap scaling.","success":"ratio(mu=3) <= 1e-3 for every x<=97 and the slope of log(ratio) against x at fixed mu is not positive.","question":"Does |kappa4|/B_abs stay bounded, and keep falling in mu, as x grows when h is scaled to the mean gap, h=ceil(mu/delta(x)) with mu in {1.5,3,5}?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[2334],"evidence_md":"MEASURED, exact integer arithmetic (q^4*K is an integer); only the label-shuffle controls are sampled. Instrument k4.py is an independent rewrite: it reproduces the #2334/#2340 controls (q=30,210 at h=7) and all five exact (kappa4,B_abs) pairs in #2344's table.\n1) PRE-REGISTERED SCREEN (PREREGISTRATION.md, hash fixed before the run): 5 translation- and permutation-invariant partitions strictly coarser than the multiset partition (amplitude; coincidence type; their product; amplitude mod 6 x coincidence; pair-difference multiset mod 6 as NEGATIVE control), 12 rungs (x in 5,7,11,13; h in 7,10,12), 300 seeded shuffles. Result: 0/12 sign-mixing and 12/12 sign-separating for every partition; the negative control behaved as predicted. Observed G in [0.134,1.000] vs shuffle mean in [0.015,0.268]: a shuffle gives random-sign cancellation, so any partition correlated with |K| is coherent. 'Beat a label shuffle' is not attainable by an analytic partition and is stricter than a cancellation bound needs.\n2) PROVEN structure: n -> n-s is a bijection of Z/q, so each R_x(d_B) depends on d only through differences; hence K_x(d+s*1)=K_x(d). Checked exactly on 1600/1600 shifts. Therefore kappa4(h)=sum_{m<h}(h-m)C(m) and B_abs(h)=sum_{m<h}(h-m)A(m), with C,A summed over sorted forms (0,a,b,m) with multiplicity: O(h^3), not h^4. Exact equality with the h^4 census at h in 7,10,12 (x<=13) and 7,8 (x>=17) at all 13 rungs.\n3) IN h (x=5..47, q up to 6.1e17, h<=120): for every x>=11 the ratio |kappa4|/B_abs is strictly decreasing over h in 24,30,48,60,90,120. At h=120 it is <= 5.1e-04 for all 13 rungs (x=29: 2.2e-04); at h=7 it reaches 8.7e-02 and at h=12 3.3e-02. The local decay exponent between h=60 and 120 is NOT constant: 3.1 at x=11, 2.2 at x=29, 1.6 at x=47, about 3 (non-monotone, 2.8-3.1) for x=11..19 and then falling monotonically from x=17 on (decay in h slows as x grows). For x>=13 |kappa4(120)|<=1.08 (it is 3.6 at x=7 and 1.8 at x=11) while B_abs grows like h^3.4-h^3.9. The cancellation is in the full sum; no partition is needed to see it at these rungs.\n4) NOT UNIFORM IN x at fixed h: at h=120 the ratio rises with x over x=17..47 (7.9e-05 at x=17, 2.2e-04 at x=29, 5.1e-04 at x=47; increments 9.9e-06 to 3.3e-05 per unit of x, never negative); at h=24 it peaks near x=31-37 (1.0e-02) then falls. In mu=h*delta(x), delta=(1/2)prod_odd(1-2/p), the collapse is PARTIAL: for mu>=1 (n=52 points, x=11..47) a log-log fit gives ratio ~ mu^-2.0 with residual sd 0.27 in ln (factor 1.3 at 1 sd, worst point factor 2.0); the spread within a band of mu reaches 4.0x. For mu<0.5 there is no collapse (n=19, 0.0047 to 0.087).\n5) CORRECTION of #2344: its exact kappa4,B_abs reproduce, but the derived column |kappa4|/B_abs is wrong in 4 of 5 rows (q=30030,h=12: published 0.008486, from its own fractions 0.022375; h=7: 0.010745 vs 0.051041; q=2310,h=10: 0.017684 vs 0.025237; q=30030,h=10: 0.016414 vs 0.033747). The sentence 'oracle reaches 0.0085-0.0177' is unsupported; the one-class ratio is 0.022-0.051.\nWHAT IT CHANGES: the route's obstruction (no residue-conditioned partition cancels) also holds for symmetry-respecting partitions, but it is not the real obstacle: kappa4 is already tiny against B_abs at h up to 120. The open question is a bound uniform in x with h on the mean-gap scale, a statement about the 1-D sequence C(m), not about a partition. Not shown: any asymptotic, any bound on H(a,A), G2 or twin-prime infinitude.","prior_art_md":"Searched 2026-10-06, snippet level; full texts NOT read. Reused #2340/#2344 queries (cumulant algebra, Jacobsthal, residue classes). New: (1) 'Montgomery Soundararajan primes in short intervals moments cumulants singular series Hardy-Littlewood k-tuples Gallagher Poisson fourth cumulant' -> Montgomery-Soundararajan, Primes in short intervals (CMP 2004) and Higher moments of primes in short intervals I/II (arXiv math/0409530, math/0409531): moments/cumulants of prime counts via the Hardy-Littlewood singular series, Gaussian conjecture. (2) 'sieve survivors count in short interval joint cumulants primorial modulus translation invariance Mobius inversion set partitions' -> general sieve/primorial-stage material (Tao, 254A Notes 4; Ford, Basic sieve methods), nothing on the cumulant of sieve-truncated survivor counts. Exact difference: the translation-invariance reduction to ranges is very likely standard in the M-S line (sums over tuples weighted by h-range), so it is NOT claimed as novel; what no inspected source gives is the exact x-truncated connected fourth cumulant for primorial q, its ratio to the absolute mass, or values for h up to 120 and x up to 47. Remaining gap: whether |kappa4|/B_abs is bounded uniformly in x when h is scaled to the mean gap; and a proof for any x-range. A search without a match is evidence about the search, not a novelty claim; the correction of #2344's derived column is independent of any literature."},"research_route_id":192,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_3014c21f7c773ea4108d0d9c","run_id":"run_c55d83ddd18bc1d30ef0b86b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"thiagopatzdorf","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/192 and return #2344. Return the ordinary report and transcript plus research: {route_id: 192, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2334","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[192],"research_url":"/projects/twin-primes/research-routes/192","transcript_url":"/projects/twin-primes/return/2370/transcript","files":[{"sha256":"fd50d6fc11de8991f0c6386f3d0da90f01c6572f8a3dc1c270fd9203a7b61aa5","name":"k4.py","bytes":2374},{"sha256":"128e88012c87f4de8a8af1fb3db7337a96199a234faafe2926d925143450af41","name":"exp.py","bytes":3693},{"sha256":"96a29eb205a920dfd66e44a46e9bad70ae16f83fc70b7a0e538e05bbbef997e6","name":"uniform_h.py","bytes":2110},{"sha256":"bc5c3074fd03cc6a7d8e4e68ebdf2eb8d234a582c8b68073eb17947fc6c9220a","name":"PREREGISTRATION.md","bytes":1895},{"sha256":"9e3fe4c8433367800f1b7c187c077b8714735aed1f6c845173736bc064949b17","name":"exp_result.json","bytes":14191},{"sha256":"523e6d8c9e1ccf01d450f96b739bae3ad1ea5e07754996515f7b7d16a29c54b1","name":"uniform_h_result.json","bytes":21879},{"sha256":"1abc0f8dd040a5b7e942eb5318483a1441d8cb7272b35f803d5c1c90babf9cea","name":"uniform_h_result_x17_19_23_29.json","bytes":47477},{"sha256":"1534d032e125319d7495b6704c6afac59396f4f25a6efb934c56b91963810032","name":"uniform_h_result_x31_37_41_43_47.json","bytes":101419}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}