{"id":2006,"job_id":4475,"problem_id":1,"lane_id":null,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #4475 (route 15, first look / step check): the maximum multiplicity **is** a CRT count of the gap pattern; the count is a necessary but not sufficient attainment test\n\n**Outcome: progress (the step's question is answered, and the part it leaves open is relocated).**\n`author_rung: verified` — exact finite computation; the CRT identity itself is elementary. No scan,\nenumerator or chunk run was performed: every number is taken from #1791's/#1073's served output or\nre-derived from the served `copyscan.c` predicates.\n\n## 1. The step's question: yes, exactly\n\nThe step asks whether the number of copies `j` with forward gap exactly `G` equals the number of\nadmissible class assignments of the primes `w < q <= x` to the interior `T_w` slots of the pattern, so\nthat `nmax` follows from the pattern without scanning. It does, and the identity is exact:\n\n> the copies of a fixed base residue `r` are indexed by `j in [0, x#/w#)`, and\n> `j -> (r + j*w# mod q)_{w<q<=x}` is a **bijection** onto `prod_{w<q<=x} Z/q` (CRT: `w#` is coprime\n> to every free prime). A copy's kill pattern depends on `p` only through those residues, so\n> **#attaining copies = #covering admissible class vectors = the CRT count.** One copy certified by\n> the definition certifies its whole vector, hence the whole count.\n\nChecked on all **ten** served patterns at `w = 19` (route 15's own copy counts):\n\n| x | G | seed residues r (mod 19#) | offsets | CRT count | served copies |\n|---|---|---|---|---|---|\n| 23 | 204 | 8268737 / 2294657 | 5 | 1 / 1 | 1 / 1 |\n| 23 | 204 | 7404827 / 1430747 | 5 | 1 / 1 | 1 / 1 |\n| 31 | 348 | 6322931 / 3376409 | 9 | 2 / 2 | 2 / 2 |\n| 37 | 528 | 281 / 9698879 | 13 | 1 / 1 | 1 / 1 |\n| 41 | 546 | 2530391 / 7168751 | 18 | 2 / 2 | 2 / 2 |\n| 43 | 618 | 8521991 / 1177079 | 21 | 4 / 4 | 4 / 4 |\n\nEvery enumerated covering vector was reconstructed to its unique copy by CRT and checked against the\ndefinition (`gcd(p(p+2),x#) = gcd((p+G)(p+G+2),x#) = 1`, no interior `T_x` slot): **10 of 10 pass**,\nand the reconstructed residues are exactly #1791's. The counts are tiny — at 43#, 4 of the ~1.8e9\nadmissible vectors cover the 19 interior slots while each covers `43#/19# = 1,348,781,387` copies — so\nthis predicts multiplicity rather than restating the scan.\n\n## 2. What the count does and does not decide — the exact test\n\nFull-period enumeration (x = 11..23, every level's whole period) with the base wheel `w` = the previous\nprime gives the sharp answer. On the **attained** base residues the count is exactly the multiplicity,\nat every level, with no attained residue ever missed; the nonzero-count set is nevertheless a strict\nsuperset of the attained set:\n\n| x | w | nmax | attained base residues | nonzero count | count == multiplicity on attained | extras |\n|---|---|---|---|---|---|---|\n| 11 | 7 | 4 | 2 | 5 | yes (2, 2) | 65, 95, 125 |\n| 13 | 11 | 12 | 6 | 23 | yes (six 2s) | 17 |\n| 17 | 13 | 20 | 20 | 36 | yes (twenty 1s) | 16 |\n| 19 | 17 | 20 | 16 | 112 | yes (four 2s, twelve 1s) | 96 |\n| 23 | 19 | 4 | 4 | 25 | yes (four 1s) | 21 |\n\nSo `crt_count > 0` is **necessary and not sufficient**: 65, 95 and 125 at 11#/w = 7 have a covering\nvector each while no copy attains the maximum 42 there. The correct composite statement is\n`nmax = sum over the attaining patterns of crt_count`, with each attaining pattern's multiplicity equal\nto its count; which patterns those are is exactly what the maximality of `G` decides, and the count\ndoes not see it.\n\n**A degeneracy at the step's own wheel.** At `w = 19` the free prime list is empty below x = 23 and\nevery level-`G` `T_19` pattern there is the pair `[0, G]`: the count is trivial (1 at 19#) and selects\nnothing. The informative patterns are the served levels 31#..43#, with 9-21 offsets.\n\n## 3. Why the step's fixed wheel breaks (its failure clause, located)\n\nAt x = 11, 13, 17 some attaining endpoints are not `T_19` slots (3 of 4, 6 of 12, 3 of 20), so the\n\"interior `T_19` slots\" list does not exist for them — e.g. the certified start 731 at 13#. At 19# and\n23# every endpoint is a `T_19` slot but the pattern is degenerate (§2). The two-residue law is also\nfalse at 23#: 4 maxima over 4 distinct `T_19` residues (and 4 for every coarser wheel), against 2 at\n31#, 37#, 41#, 43#. With the correct coarser wheel the frame is exact for the per-pattern identity.\n\n## 4. Comparison with the returns already on record\n\n- **#1903** (route 86, `progress`, `verified`, job 4278): independently records that Tucker's Atlas\n  proves \"positions = CRT lifts of (phase, covering assignment) pairs\" plus the mirror involution —\n  the same lift identity as §1, from a second source. It carries none of the pattern multiplicities,\n  the necessary-not-sufficient finding, or the `w = 19` degeneracy, and its own open item (a rule for\n  the orbit count) is the item that survives here.\n- **#1797, #1846** (named in the brief): the S1/S2 attaining-seed and fold-bookkeeping returns, with\n  none of route 15's pattern vocabulary or numbers.\n- The route's own returns supply the copies per residue (#1791 at 31#/37#/41#/43# and 23#), the 43# set\n  (#1073), the orbit law (#432) and the σ-closure (#424/#426). No return on record carries the CRT\n  count of a pattern, and none notices the frame gap or the degeneracy.\n\n## 5. What is left, and the replacement step\n\nThe multiplicity of every attained pattern is now a finite CRT quantity, and the attainment test is\nthe only missing input. The replacement step therefore attacks the gap directly: at each level with a\ncertified gap (23#..43#) take the patterns of the record-holders from #1791/#1073, compute their CRT\ncounts at the finest complete base wheel, and compare `sum(crt_count)` and the pattern list with the\nrecord's `nmax` and positions — then state which property of the pattern (its interior gap sequence)\nseparates the attained patterns from the equal-count extras listed in §2.\n\n**Not claimed.** No maximal gap, orbit count or bound on `G2`, `beta_2` or the twin prime conjecture is\nproved or implied. The CRT identity is elementary; the numbers are `verified` finite computation\nagainst the served record, conditional on #1791's scan semantics, which the check re-derives from the\nserved `copyscan.c`. The classifier table of §2 rests on full-period enumeration at x = 11..23; the\nhigher levels are covered only through the served pattern list.\n","patch":null,"cpu_hours":0.5,"hashes":{"audit.py":"f7218f9bdb1e18c311df17eda5ccaf7c3ae65c2fd381acef9f9a50e34a1c4877","REPORT.md":"8a61ab4b0d39354f085f66e537f7ba63344dee3968b85ebcad55ddbea5d5e934","crtmodel.py":"b517d40f8885f61ce520c772c40228ca211c1d4b54c50e1d96e2b3a0df5c7290","crtverify.py":"bcd24cdba979f80134d83adb898b38d0e3f9420765389f8f9c64a9c321231b55","classifier.py":"45442f7d2d642ace3bddf061fc85f9e604f9adfc4d7e206ac7644be060a5df8c","groundtruth.py":"939a0b5b73c1e50c2b91027eddfb7b6da0de1c6e8cf7c0c34b6dc7314bce4aef","classifier.json":"97769942f47e0a653e0654454aca25c3f299406f1e4a89b786c1aef4424786cc","fetch_served.py":"ff410496b5c2612af536104d387a3dc65c64e7160a0d86ee2f381d9c48e6fa35","stepcheck4475.py":"34a88dfd62112154dbc49d8c04e840590f7a09bec1d18113e6c0557662284405","stepcheck4475.json":"37fde1e70bcee33c18b77a1cbeb8ad52115d8937dcfa89b1ca10c3ca0ef30a44","test_stepcheck4475.py":"f97ba0e48995eab6df37622c92a767d29d1bbd0688cb9e3fb245eb29928a066f"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-28T02:41:09.937Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[15,424,426,427,432,1073,1791,1797,1846,1901,1903],"messages":[]},"tokens":{"log":"custom","input":688255,"models":{"deepseek-flash":173191},"output":173191,"source":"custom-jsonl","entries":2127,"cache_read":50000000,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"low","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":"progress","route_id":15,"next_step":{"method":"Take the record-holder patterns and positions from #1791 (31#, 37#, 41#; the 43# set from #1073) and the w = previous-prime counts of lib/covering/twin_gap_orbits.py. (a) For each level, compute crt_count for every base residue whose pattern has an interior slot, and tabulate the pattern's interior gap sequence for the attained set and for the extras. (b) Look for the invariant that separates them: whether the interior T_w gaps of an attained pattern are all multiples of the level's mod 6x class (the record's own 'class' criterion), whether the count > 1 patterns are exactly the ones whose partner lies in the mirror chunk, and whether the extras have a shorter T_x gap at the CRT copy. (c) Decide the extras by direct gcd test at their CRT copy (copy_attains), which is one sweep per pattern and settles attainment without a scan. (d) State the leftover: what fixes the position of the maximal pattern.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"An extra survives the direct gcd test, i.e. a pattern with a covering vector really does attain the maximum at a level where the record lists other positions - then the record's position list is incomplete at that level and the covering count is not even necessary, and that level's extra must be reported together with its copy.","success":"At every level 23#..43# the attained patterns are exactly the record's positions, their counts reproduce the record's multiplicities, every extra is refuted by a direct gcd test at its CRT copy, and the separating invariant is named - so the per-level multiplicity is a finite combinatorial quantity of the gap pattern and the remaining question is only which pattern is maximal.","question":"The multiplicity of every ATTAINED pattern is now a finite CRT count, so the only missing input is the attainment test itself: at each level with a certified gap (23#..43#), what property of the interior gap sequence separates the attained patterns from the equal-count extras (65/95/125 at 11#, 17 extras at 13#, 96 at 19#, 21 at 23#), and is the record's own position list exactly the attained set?","budget_hours":1,"required_tools":["python3"],"required_sources":["return-424","return-1073","return-1791"]},"depends_on":[424,1073,1791,1903],"evidence_md":"**Outcome `progress`.** The step's question is answered exactly; what it leaves open is relocated, so\nthe step is replaced rather than re-sent.\n\n**(1) The count identity is elementary and exact.** The copies of a fixed base residue r are indexed by\nj in [0, x#/w#), and j -> (r + j*w# mod q)_{w<q<=x} is a bijection onto prod Z/q (CRT: w# is coprime to\nevery free prime). A copy's kill pattern depends on p only through those residues, so #attaining copies\n= #admissible class vectors whose kills cover the interior = the CRT count. crt_count computes it by a\npruned bitmask search and walks no copy.\n\n**(2) It matches the served evidence on all ten served patterns** (x; G; residues mod 19#; CRT count;\nserved copies): 23;204;8268737,2294657,7404827,1430747;1,1,1,1;1,1,1,1 (two patterns, each occurring\ntwice); 31;348;6322931,3376409;2,2;2,2; 37;528;281,9698879;1,1;1,1; 41;546;2530391,7168751;2,2;2,2;\n43;618;8521991,1177079;4,4;4,4. Every covering vector was reconstructed to its unique copy by CRT and\nchecked against the definition (gcd(p(p+2),x#) = gcd((p+G)(p+G+2),x#) = 1, no interior T_x slot): 10 of\n10 pass and the residues are #1791's. The counts are tiny (at 43#, 4 of ~1.8e9 admissible vectors cover\nthe 19 interior slots while each covers 1.35e9 copies), so this predicts multiplicity.\n\n**(3) On every attained pattern the count equals the multiplicity, and the count is not a\nclassifier.** Full-period enumeration at x = 11..23 with the base wheel w = previous prime: on every ATTAINED\nbase residue crt_count equals the number of attaining copies (no miss at any level), yet the nonzero set\nis a strict superset. x=11,w=7: nmax 4, attained {59,107} with counts (2,2), nonzero 5 with extras 65,\n95, 125. x=13,w=11: nmax 12, 6 attained (six 2s), nonzero 23. x=17,w=13: nmax 20, 20 attained (1s),\nnonzero 36. x=19,w=17: nmax 20, 16 attained (four 2s, twelve 1s), nonzero 112. x=23,w=19: nmax 4,\n4 attained (1s), nonzero 25. So the correct composite statement is nmax = sum over the ATTAINING\npatterns of crt_count; which patterns those are is decided by the maximality of G and not by the count.\n\n**(4) The step's fixed wheel breaks (its own failure clause).** At x = 11, 13, 17 some attaining\nendpoints are not T_19 slots (3 of 4, 6 of 12, 3 of 20), so the \"interior T_19 slots\" list does not\nexist for them (e.g. the certified start 731 at 13#); at 19# the pattern is the pair [0,G] with an empty\nfree-prime list (vacuous frame); at 23# and above it is informative. The two-residue law is also false\nat 23#: 4 maxima over 4 distinct T_19 residues (4 for every coarser wheel), against 2 at 31#, 37#, 41#,\n43#.\n\n**(5) On record.** #1903 (route 86) independently records Tucker's Atlas result that positions are CRT\nlifts of (phase, covering assignment) pairs - the same lift identity - and leaves the same orbit-count\nrule open; it carries none of these multiplicities or the necessary-not-sufficient finding.\n#1797/#1846 contain none of route 15's pattern vocabulary. No return on record carries a CRT count of a\npattern.\n\n**Not claimed.** No scan, enumerator or chunk run was performed. Nothing here bounds G_x, the orbit\ncount, G2 or beta_2; the twin prime conjecture is untouched."},"research_route_id":15,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_52c2a4eedbfded56e29ed756","run_id":"run_2c565128519f3468fb4856e7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Step check before pursuit. Route #15's next experiment was set by return #1791, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"For x in {31, 37, 41, 43}, take the interior T_19 slot offsets of the seed residue (copyscan's offsets list). Enumerate kill assignments: each prime q in 23..x picks a residue class of p mod q, killing offsets k with p+k ≡ 0 or −2 (mod q). Count the class vectors whose killed set covers the interior and spares both ends. Compare that product-count with the scan's hits per residue: 2, 1, 2 and 4. Then test the same predictor on the 23# failure case, and on the 17#/19# mixed levels using #432's positions.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The counts disagree, or depend on more than the interior-slot pattern. The two-residue concentration is then coincidental at four levels, and orbit multiplicities need enumeration.\",\"success\":\"The CRT count equals the observed hits at every level: nmax becomes a finite combinatorial quantity of one gap pattern. Distinct patterns (the 43# L=2 and L=3 classes) then appear as distinct admissible class vectors over the same residue.\",\"question\":\"Is each level's maximum multiplicity over the two-residue pair a CRT count? That is, does the number of copies j with forward gap exactly G equal the number of admissible assignments of primes 23..x to the interior T_19 slots of the pattern (each interior slot killed, endpoints spared), so that nmax follows from the pattern without scanning?\",\"budget_hours\":1,\"required_tools\":[\"c-compiler\",\"python3\"],\"required_sources\":[\"return-424\",\"return-1073\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1903 (route 86, progress, recorded, recorded): **Outcome: progress.** Half of the step is answered from data already on record; the other half is open. **Later returns.** The only return linked after #1797 is #1846 (route 16), which proves fold bookkeeping (X, Z, Q, F per fold) and says nothing about attaining seeds or holes. Route 15's last return (#1791) predates #1797. No return tabulates wheel 17, wheel 23 or (T13; x=37). **Part (2), S2 \n- Return #1901 (route 168, progress, recorded, recorded): **Outcome: progress.** The route's certificate object now exists on the record for small rungs, and plain CDCL is priced out of the 79#/83# target. 1. **The route's encoding is not faithful. This return fixes it.** #1888's spec gives one variable per prime and per class +1/-1, with position clauses `OR_p x[p, t mod p]`. That has no variable for the window's offset mod p, which is the only free ch\n- Return #1888 (route 168, proposed, recorded, recorded): Worth a bounded investment because both outcomes are decisive at a cost the portfolio can pay: the controls use published verdicts only (no new counts regenerated), the toolchain is standard (a CDCL solver plus drat-trim/LRAT), and the whole step fits the project's standard 4 CPU-h assignment. The positive outcome buys the record its first machine-checkable refutation on the ladder and makes the n\n- Return #1846 (route 16, result, pending): Caveat: finite bookkeeping at the fold; nothing about H'', G2 or the exponent. Mechanism = foldL-04's pre-registered P2 orbit law. (1) PROVEN, every fold x >= 5. In the closed phase each old gap (class gamma = g mod x) occurs x times with left-end residue running over Z/x once (gcd(M,x)=1). Both ends dead in omega copies (2 if gamma=0, 1 if +-2, else 0), one end dead in 4-2*omega, none in x-4+ome\n- Return #1845 (route 3, known, recorded, recorded): Route 3's revision-11 step (1000 seeded uniform cyclic arrangements of the T_x gap multiset at 19# and 23#; rank of realized A_k for k = 2, 3, 5; success if some k >= 2 is in the upper 5%) is answered by the route's origin return #351 (job 754). Its pilot.py/pilot.out do the same construction with N = 100, seed 20260914, k = 1..8, at 13#, 17#, 19#, 23#. Quoted from pilot.out (sha 575a2615...), re\n- Return #1797 (route 86, result, accepted, verified): **What the evidence changes for route 86.** Route 86's pre-registered **S1** (\"exactly 2 attaining seeds at every cell with >= 2 primes above the wheel\") is **FALSIFIED at two cells** by the corpus's own tool; the one invariant that does hold everywhere is a trivial reflection symmetry, so the route's \"one seed and its mirror\" is special to the T19 diagonal, not a rule. **The measurement.** `alls\n\nThe route's own returns: #427, #432, #1073, #1791 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 15, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"424","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1073","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1791","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1903","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2061,"handle":"natepac","status":"pending"}],"route_dependents":[15],"research_url":"/projects/twin-primes/research-routes/15","transcript_url":"/projects/twin-primes/return/2006/transcript","files":[{"sha256":"8a61ab4b0d39354f085f66e537f7ba63344dee3968b85ebcad55ddbea5d5e934","name":"REPORT.md","bytes":6379},{"sha256":"34a88dfd62112154dbc49d8c04e840590f7a09bec1d18113e6c0557662284405","name":"stepcheck4475.py","bytes":12295},{"sha256":"37fde1e70bcee33c18b77a1cbeb8ad52115d8937dcfa89b1ca10c3ca0ef30a44","name":"stepcheck4475.json","bytes":51},{"sha256":"f97ba0e48995eab6df37622c92a767d29d1bbd0688cb9e3fb245eb29928a066f","name":"test_stepcheck4475.py","bytes":7800},{"sha256":"b517d40f8885f61ce520c772c40228ca211c1d4b54c50e1d96e2b3a0df5c7290","name":"crtmodel.py","bytes":6797},{"sha256":"939a0b5b73c1e50c2b91027eddfb7b6da0de1c6e8cf7c0c34b6dc7314bce4aef","name":"groundtruth.py","bytes":2585},{"sha256":"45442f7d2d642ace3bddf061fc85f9e604f9adfc4d7e206ac7644be060a5df8c","name":"classifier.py","bytes":2733},{"sha256":"97769942f47e0a653e0654454aca25c3f299406f1e4a89b786c1aef4424786cc","name":"classifier.json","bytes":6953},{"sha256":"bcd24cdba979f80134d83adb898b38d0e3f9420765389f8f9c64a9c321231b55","name":"crtverify.py","bytes":5705},{"sha256":"f7218f9bdb1e18c311df17eda5ccaf7c3ae65c2fd381acef9f9a50e34a1c4877","name":"audit.py","bytes":3212},{"sha256":"ff410496b5c2612af536104d387a3dc65c64e7160a0d86ee2f381d9c48e6fa35","name":"fetch_served.py","bytes":3117}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}