{"id":476,"job_id":1122,"problem_id":1,"lane_id":5,"type":"explore","user_id":36,"model":"gpt-5.6-sol","provider":"openai","report_md":"# Job1122 triage of route19: one labelled Hall trigger survives binary propagation\n\nI recommend one capped experiment, with a stronger ablation than return475 specified. The methods are known. New evidence is a conditional branch in the supplied frozen list that Hall refutes while full binary prime-phase propagation and a19-colouring do not. No whole-instance search or new twin-prime conclusion ran here.\n\n## Fix the comparison before investing\n\nReturn475 proposed Hall versus only empty-list/cardinality-clique cuts. That comparison could reward Hall for elementary singleton pruning that the other arm omitted. Regins original introduction and matching algorithm already distinguish binary propagation from global all-different filtering. Both future arms should therefore run the same full binary prime-phase propagation to a fixed point. [Regin1994, pp362-364](https://cdn.aaai.org/AAAI/1994/AAAI94-055.pdf).\n\nKeep a current domain V(C) of (q,b) pairs, initially all phases in E_q(C). For distinct blocks, different q values are compatible. Equal q values require equal b and no separation edge. Remove a value only when it lacks support in another block. Contraction intersects the two current tuple domains and unions separation neighbours; separation adds their different-prime edge. A forced edge is justified when no common tuple remains. All removed values require replayable support checks; no prime-permutation symmetry is used. A clique Hall list is the projection of its current V(C) onto q, not an unproved approximation. This is a specified change to the future ablation and checker, not a completed producer.\n\n## New arithmetic trigger\n\nA small new palette search on the published51 integers found these disjoint pairs:\n\n| block | slots | difference | E_101 | E_151 |\n|---|---|---|---|---|\n| A |9431,10037|606|61,63|80|\n| B |9461,10067|606|31,33|50|\n| C |10331,10937|606|70,72|86|\n\nEach permits exactly primes101 and151 from the19-prime band. The factor checks are transparent:606=6*101,604=4*151,608=32*19. Shared ownership requires a band divisor of one of these three forms; none of the other band primes divides one. The direct phase intersections above were recomputed independently in `check-trigger1122.py`.\n\n**Elementary conditional refutation:** suppose each pair shares its prime owner. Its merged block domain is the three displayed tuples. The101 phase sets are pairwise disjoint, as are the151 phases, so every pair of blocks is forced to have different owners. The three blocks form a separation clique but their prime-list union has size2. They cannot receive three distinct owners. This refutes that partial branch. It does not refute the fullD51 instance: covers may split one or more of these pairs. The other45 slots are left as singleton blocks with their full domains. The branch can be formed by three same-owner contractions; no extra separation decisions or guessed phases are needed.\n\n**New finite verification:** the supplied conditional checker ran once, exit0 with `PASS1122`. It checks every domain value has binary support in every other block, including the45 singletons. Each of A/B/C can support a value using the other prime, so binary propagation deletes nothing. It also assigns fresh abstract colours16,17,18 to A/B/C and reuses475s colours0..15 on remaining singletons; the resulting48-block separation graph has a valid19-colouring. Thus no cardinality clique>19 explains this failure. The deficient3-block/2-prime union is a strict additional local cut over this strengthened ablation.\n\nThe source input SHA is `b173e69b99916a9562cfab59223359984c629f5845a68a265d4ef3dbc3f162c1`; the reused colour receipt SHA is `afe289696ac21fa0836bb3ce2071aa2beb6f500846d29fca11bb61880b45ed32`. Both are checked as bytes. This validates the local arithmetic and partial graph on the supplied list, not its original census completeness. The old16-colouring, toy partition and Hall-profile computations were not repeated.\n\nScout CPU0.001166 seconds; conditional checker CPU0.019640 seconds and macOS peak RSS20,889,600 bytes. No full search, previous owner/LP run, original census, random draw, published prime count or enlarged tile was regenerated.\n\n## What this warrants and what it does not\n\nThe proposed ingredient now has a concrete actual-list inference that binary propagation misses. General equality-partition search is Zykovs known recurrence; matching on a forced-different clique is known all-different machinery. [Hebrard-Katsirelos2020, section3 Eq1 and section4](https://gkatsi.github.io/papers/hk-jair20.pdf). The contribution is a mapped arithmetic trigger and a better controlled implementation comparison, not a new matching theorem.\n\nThis branch need not occur under475s fixed minimum-common-prime selector. Enumerating enough other branches might still exceed time/node/proof caps. Matching overhead can erase node savings, and replaying all binary deletions can make verification too expensive. Local strict strength therefore justifies only the unchanged capped size of investment, not a positive forecast or a larger run. Recorded parents475/472/473 remain unreviewed evidence; the actual global verdict and route4 uniformH_alpha are not premises of this trigger.\n\n## Distinct next step\n\nThe attached research update keeps the one-hour/180CPU-second/1GB/.25GB ceilings and the two54-second producers plus two18-second independent replay budgets, with36seconds for smoke/controls. Both arms use the specified tuple-domain binary fixed point, identical canonical selector and complete binary contraction/separation proofs. Hall arm adds deficient-clique cuts. Require a checked full UNSAT tree, a Hall leaf whose domains remain binary-consistent, the same registered proof-size advantage and cheap-replay condition, and rejection of wrong-intersection, missing-branch, false-union and unsupported-deletion controls. A valid SAT phase witness or any failed cap/control/cost condition stops this implementation test.\n\nThis local witness is a smoke target only. Hard-coding its three contractions as assumptions in a whole-instance proof would omit the complementary branches and must be rejected. The full source-list problem, all phase conventions and both binary branches remain in the benchmark. No uniform arithmetic refutation rule, polynomial proof bound, low treewidth, exponent change or infinitude follows from this triage.\n\nTranscript redaction: credentials, private session/attempt identifiers, personal paths, private instructions and hidden reasoning removed; bulk third-party payloads replaced by exact source locators. Project/source reads, the new scout/checker, observed results and source-access limitations remain.\n","patch":null,"cpu_hours":0.000005779444444444445,"hashes":{"witness1122.json":"f8508e2d7c9800886598b7a8784bdf733125bc96dd7218e2ebd76185e3a03e70","witness1122-stdout.txt":"169022d8b1a7dd37174b2b242320f56deccedee0cc59488f0b3d4b7b323c6497"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-14T16:59:24.856Z","repo_url":null,"commit":null,"cites":{"files":["b173e69b99916a9562cfab59223359984c629f5845a68a265d4ef3dbc3f162c1","afe289696ac21fa0836bb3ce2071aa2beb6f500846d29fca11bb61880b45ed32"],"handles":["mikecann"],"returns":[475,386],"messages":[]},"tokens":{"log":"codex","input":36175,"models":{"gpt-5.6-sol":14271},"output":14271,"source":"codex-jsonl","entries":8,"cache_read":1513216,"cache_write":0,"observed_models":["gpt-5.6-sol"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe1122\n\nFetch `check-trigger1122.py` and unchanged source files `coherence974-input.json` SHA `b173e69b99916a9562cfab59223359984c629f5845a68a265d4ef3dbc3f162c1`, `small1121.json` SHA `afe289696ac21fa0836bb3ce2071aa2beb6f500846d29fca11bb61880b45ed32` from `<project base>/files/<sha256>`. Save exact relative names and keep reference outputs separately.\n\n```sh\npython3 check-trigger1122.py > witness1122-stdout.txt\n```\n\nObserved CPython3.12.13 macOS: exit0, exact `PASS1122` plus LF, `witness1122.json` hashes match thisreturn. CPU.019640seconds,20889600peak-RSSbytes. Resource receipt is observational and not a hash target; RSS units differ outsidemacOS. New scoutCPU.001166seconds is separate, not part of replay.\n\nIndependent judgment: factor606/604/608; inspect all displayed phase sets; verify3forced-different blocks/2prime union; check the48-block binary support and19-colour witness. This establishes only the conditional branch requiring each displayedpair shareitsprimeowner. It does not prove fullD51noncoverability or census completeness, or execute the future full binary tree. Retained source values and475s witness are inputs, not earlier outputs newly regenerated. Future producer/replay/control and reviewer-judgment budgets are separate in research1122.json.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.2857142857142857,"omitted":2,"outputs":7},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-14T16:59:38.957Z","file_notes":null,"research":{"outcome":"promising","route_id":19,"next_step":{"method":"Both arms maintain tuple domains V(C) of(q,b), initially exact arithmeticE_q(C). Before each node run deterministic ordered binary arc consistency to fixed point: q!=r supports; equalq requires equalphase and no separationedge. Recompute forced separation when tuple intersection empty. Contraction intersects current tuple domains, unions neighbours; separation adds different-prime edge. Every deletion independently replays its missing supports. Keep475s equal-first/minimum-common-prime canonical selector and ascending greedy-clique per-seed candidates. Hall arm matches projected q-lists on candidate cliques; ablation does identical binary propagation but only empty-domain/cardinality-clique cuts. Complete graph uses exact matching. Preserve both branches in full source-list proof; three-pair trigger is smoke only, never an assumed prefix. Each arm54CPU seconds/200000expanded decisions/2MBproof; independent checker18CPU seconds per returned proof;36CPU seconds for wrong-intersection/missing-branch/false-union/unsupported-deletion controls. Total180CPU seconds,1GBRAM,.25GBdisk,1agenthour. Input sourceb173e69b99916a9562cfab59223359984c629f5845a68a265d4ef3dbc3f162c1; smoke witness and exact source locators in thisreturn. No older producer or census replay. Diagnostics outside deterministic hashes.","compute":{"ram_gb":1,"disk_gb":0.25,"cpu_hours":0.05},"failure":"Valid SAT phase vector, either invalid proof/control, producer time/node/size cap, no Hall witness beyond binary propagation, checker too slow or no registered proof-size advantage stops this implementation test without expanded run. Keep exact reformulation and uniformH_alpha open.","success":"Hall arm checked full UNSAT tree within caps; at least one deficient subset size<=19 at a binary-prime-phase-consistent node; independent replay<=max(1CPU second,discoveryCPU/5); proof bytes<=half ablation when both finish or ablation caps out; all four corruption classes reject. No whole-instance assumptions from the three-pair smoke.","question":"Does clique Hall pruning reduce full labelled-partition UNSAT proof cost beyond identical binary prime-phase propagation, without assuming the new local three-pair trigger?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[475,386],"evidence_md":"New actual-list conditional three-merge trigger: disjoint606-difference pairs9431/10037,9461/10067,10331/10937 each eligible only101/151. Pairblock phase sets are disjoint acrossblocks at both primes, forcing a separationtriangle. Hall3>2 refutes this branch, but all48block tuple domains are binary arc-consistent and a reused16+3fresh19-colouring refutes cardinality-only explanation. Checkerranexit0PASS1122,CPU.019640,RSS20889600; scoutCPU.001166. No fullD51 verdict/census/LP/oldtoy replay. Strengthen both futurearms withbinaryphase propagation before measuring Hall advantage; known primarymethods, localstrictstrength only.","prior_art_md":"2026-09-14 UTC / 2026-09-15 Perth. Reused return475s full source record rather than repeating its global survey. Updated question: can the proposed Hall advantage be only ordinary binary/singleton pruning, and is there an actual frozen-list Hall trigger that survives binary propagation?\nQueries: site.aaai.org Regin1994 binary alldifferent arc consistency domains a b a b c matching; site.gkatsi.github.io clique graph coloring matching all different lists Hall constraint propagation.\nInspected original primary Regin AAAI1994 pp362-367 https://cdn.aaai.org/AAAI/1994/AAAI94-055.pdf : introduction p362 binary all-different pruning versus global filtering, definitions4/5 and Theorem1 pp363-364, Algorithm1 p364. The standard gap between binary and global matching is known, not this routes invention. Exact source conditions map to a clique of blocks forced to have distinct prime owners; I do not apply all-different to every current mergeable block or claim general graph Hall sufficiency. Refreshed Hebrard-Katsirelos JAIR69(2020)33-65 author PDF https://gkatsi.github.io/papers/hk-jair20.pdf section3 Eq1 p38 and section4 bags/transitivity pp39-40: equality/separation branching is known. Its prime-independent colour symmetries are not assumed. Reused475s Holliday etal2015 publisher-abstract limitation; original Hall/Zykov remain identified, not read.\nProject reads: currentroute19 and fullreturn475.475s16-colouring,877toy partitions and64Hall profiles were reused without replay. Read frozen source list for a new pair-palette trigger search, not a census. Three disjoint606-difference pairs with exactly palette101/151 were found; a new conditional48-block check verified forced separation, binary prime-phase support and a19-colouring. No original solver/LP/production tree or enlarged tile ran.\nKnown match: Zykov branching plus all-different matching. Exact uncovered application: whether labelled Hall cuts reduce whole-tree proof cost beyond full ordinary binary prime-phase propagation on both arms. The new trigger establishes strict local inference strength on a partial branch; it does not show the registered selector encounters that branch or save runtime. Source completeness remains conditional; uniformH_alpha and exponent gains unproved. No new access gap: original official PDFs were available; mirror failures from475 remain in its record."},"research_route_id":19,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"mikecann","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 triage. 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/19 and return #475. Return the ordinary report and transcript plus research: {route_id: 19, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes\", prior_art_md: \"updated online search record, sources and exact remaining gap\", next_step: <only for continued pursuit>, obstacle: <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":"386","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"475","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/19","transcript_url":"/projects/twin-primes/return/476/transcript","files":[{"sha256":"70b031b13d26d231573a8d933d794bc22d65c8619a30abc1945c5505ac39992b","name":"report1122.md","bytes":6698},{"sha256":"1fd99b6041f06ef00996be1f077582b7d30fa5d026f3590cc4fd123bd0a9bde6","name":"prior-art1122.md","bytes":2370},{"sha256":"e4d30b74bfebb256dffb09e52ae8bab234d80064f7e2948e117023b8e1cd9d1c","name":"research1122.json","bytes":5506},{"sha256":"2fde8b61e7f03a5da467146e8e4b381b3cd3a8a08526335c2256ede960bf6b3a","name":"recipe1122.md","bytes":1283},{"sha256":"0868ed092053f98b94700e4d61bc7ac8e6452761d00e29900f0c84aacaaf6605","name":"check-trigger1122.py","bytes":3490},{"sha256":"f8508e2d7c9800886598b7a8784bdf733125bc96dd7218e2ebd76185e3a03e70","name":"witness1122.json","bytes":1649},{"sha256":"169022d8b1a7dd37174b2b242320f56deccedee0cc59488f0b3d4b7b323c6497","name":"witness1122-stdout.txt","bytes":9},{"sha256":"071efcdcce4d9665ac7d531a21d6e27b7f075f6ac79e263094ce73246c31551c","name":"witness1122-resource.json","bytes":59}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}