{"id":473,"job_id":1119,"problem_id":1,"lane_id":5,"type":"explore","user_id":36,"model":"gpt-5.6-sol","provider":"openai","report_md":"# Job1119 triage: the proposed plain owner refutation tree exceeds its cap\n\nI recommend pausing the specific route18 implementation proposed in return472. I found an elementary obstruction to its UNSAT branch before running either producer: with19 initially available owner colours, the first five legal levels already require1494560 expanded nodes in a complete refutation tree. The proposal capped the owner producer at200000. This does not refute the exact owner reformulation, establish the actual D51 cover verdict, rule out an early SAT witness, or lower-bound a different proof system.\n\n## The specified algorithm and counting argument\n\nThe borrowed support has51 distinct slots and19 primes. Each slot initially permits every prime: a single slot always has two possible killing phases at any prime. The specified owner search branches on one still-unassigned slot, tries all of its remaining colours, and propagates only equal-colour incompatibility against assigned slots. Different owner colours always satisfy a pair constraint.\n\nAfter k legal explicit assignments, an unassigned slot has lost at most k colours. Each assignment can delete only its own colour; if several assignments use the same colour, they delete no more than one copy of that colour. Therefore EVERY unassigned domain has size at least M-k, regardless of its pair list, the arithmetic, the chosen slot order or the minimum-domain tie rule.\n\nFor k<=5 with M19,N51, the domains have at least14 colours. No domain is empty, no coverage clause is falsified, and the previous assignments have no pair conflict because the producer branches only on legal colours. There are still unassigned slots, so there is no full SAT leaf either. Even adding ordinary binary arc consistency makes no deletion in these layers: any colour in one domain has a supporting DIFFERENT colour in the other, since each other domain has at least2 colours. Singleton propagation is also unavailable in these layers. Thus all nodes at depths0..5 must be expanded in any completed UNSAT tree whose only leaf evidence is an assigned-pair conflict or an empty domain under that propagation.\n\nBy induction, the number of legal nodes at depth k is at least (M)_k, the falling factorial. The exact values are:\n\n| Depth | Minimum legal nodes |\n|---:|---:|\n| 0 | 1 |\n| 1 | 19 |\n| 2 | 342 |\n| 3 | 5814 |\n| 4 | 93024 |\n| 5 | 1395360 |\n\nTheir sum is1494560, exceeding200000. This is a node-count obstruction independent of how fast Python executes or how many bytes a tree node takes. It applies even if the source's pair lists have size0 rather than3. The at-most3-compatible-prime lemma does not remove different-colour supports. A branch tree must account for every legal colour, so it cannot silently fix one prime by ordinary colour-renaming symmetry: prime-specific pair graphs differ and that symmetry has not been proved.\n\nThe result is conditional on attempting a COMPLETE UNSAT refutation in this narrowly specified tree system. It does not say an actual timed run expanded1494560 nodes; no such run occurred. An incomplete tree may hit the cap much earlier along depth-first exploration. A satisfiable instance can produce and check a witness without exhausting these levels. The original success criterion allowed that SAT branch, so I do not declare the entire SAT-or-UNSAT experiment logically impossible. For the route's desired noncoverability certificate, however, the declared UNSAT mechanism cannot satisfy its cap.\n\n## Why this changes the triage decision\n\nReturn472 proposed reduced owner-domain storage as a possible route4 proof-compression ingredient, honestly without a measured advantage. The lower bound shows that plain pair forward checking plus a full tree is not an adequate first refutation test at the stated budget. Spending the implementation allocation merely to discover this same cap exhaustion is unnecessary. The exact binary owner-cover equivalence and short-window divisor lemma survive.\n\nGlobal capacity/clique/Hall cuts, a learned-clause proof DAG, a genuinely proved colour/slot quotient, or a branching rule on phase groups would change the counted system. None was included in the plain owner producer specification. I have not shown that any is effective or cheap to verify here. Generic names for these techniques do not make a runnable next experiment, and the existing phase/weight/coherence routes already contain related global information. A renewed direction should specify its missing prune/proof rule and compare it to those existing certificates before running it.\n\nParentroute4 remains open. H_alpha for every actual old-admissible window and phase vector remains unproved/TPC-strength. This is neither a uniform impossibility theorem nor an obstacle for all owner CSP solvers, Boolean DPLL/resolution, SAT clause learning, LP/Farkas proofs or compact arithmetic certificates. In particular, the paper's PHP theorem is not imported as a lower bound for actual D51.\n\n## Prior work and observations\n\nI reused return472's complete search ledger and inspected the closest new primary proof-complexity source, Beyersdorff-Galesi-Lauria IPL2010, with exact locators in prior-art1119.md. Tree versus DAG distinctions and pigeonhole search lower bounds are known. The PHP formula forbids all same-hole pairs; actual D51 can permit compatible same-prime pairs. My early-layer count relies only on one-colour deletion and is given explicitly above; it needs no asymptotic theorem import or novel-method claim.\n\nI read currentroute18 and fullreturn472. No producer search, checker replay, toy-equivalence replay, D51 arithmetic census, LP, full period or old certificate computation ran. I used Python exact integers to tabulate six falling-factorial values and their sum, recorded in counts1119.json; this is not empirical search timing or a benchmark. Administrative declared CPU0.0001CPUh rounds trivial local work and is not an observation of0.36CPU seconds. All pending/recorded source grades remain unchanged.\n\n## Outcome and revisit condition\n\nResearch outcome: inconclusive, with a scoped obstruction for the specified owner UNSAT tree/cap and no next_step. This pauses this implementation, preserving both the actual SAT possibility and broader route4. Revisit when a concrete sound global prune, reusable proof DAG or proved quotient is supplied, together with a cheap independent checker and a bounded experiment distinct from the impossible plain-tree UNSAT branch. The analytic count itself is an author derivation pending external scientific assessment, not mathematical acceptance by ordinary recording.\n","patch":null,"cpu_hours":0.0001,"hashes":{"recipe1119.md":"a9cebf418110051ede10701e1f43674b87eb4395598e9a139d7853c06b3fa026","report1119.md":"a623fd89f7b85880c3df2e97128d448ba79f6adb220c4c806d42daec77a865d2","counts1119.json":"a54ea0e1dd2adb5b19a302af161cd5a9feacb73ddfc7096f3a48cdd2300111d8","prior-art1119.md":"231d46a7215784b8d899684bff634416252f1e3f5bedf6e487b3f2706a59d1a4","research1119.json":"2ae3c82f212d4077b694e32c6c8f542dcf3dc3621cb3cc355f391e27b86bed36"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-14T16:38:21.078Z","repo_url":null,"commit":null,"cites":{"returns":[472],"messages":[]},"tokens":{"log":"codex","input":31705,"models":{"gpt-5.6-sol":9039},"output":9039,"source":"codex-jsonl","entries":9,"cache_read":1406592,"cache_write":0,"observed_models":["gpt-5.6-sol"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Read the node-count proof: initially every slot has all19colours, and a legal assignment deletes at mostonecolour from any other slot. At depths0..5 all domains have at least14colours, so no leaf is permitted by the specified empty-domain/pair-conflict system. Independently calculate sum_{k=0}^5 product_{j=0}^{k-1}(19-j)=1494560>200000. Check the exclusions: no timed search was run; SAT can stop early; global cuts/DAG proofs/valid symmetry are outside the bound. No numerical source reproduction or network is needed for this proof check. Prior-art1119.md gives source locations without importing the PHP lower bound to actualD51.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.25,"omitted":2,"outputs":8},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-14T16:38:31.788Z","file_notes":null,"research":{"outcome":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"For k<=5, all unassigned domains have size>=19-k>=14. All legal nodes there are internal because51>5 and no conflict/emptydomain. Depth counts1,19,342,5814,93024,1395360 sum1494560>200000, independent of pair-list arithmetic and runtime.","statement":"With51initial slots/19owner colours, a complete UNSAT tree using only assigned-colour pair forward checking cannot fit the proposed200000expanded-state cap. This is not an actualD51UNSAT verdict or a bound for global cuts, DAG sharing, symmetry quotients or all SAT/arithmetic proofs; early SAT is possible.","assumptions":"Initially every slot admits everyprime; different colours always compatible; branch on one unassignedslot and exhaust its legal colours; only equal-colour assigned-pair deletions/empty-domain or assigned-pair-conflict leaves; no learned/global prune or shared proofsubtree. Ordinary unassigned binary AC and singleton propagation cannot help in the counted earlylayers.","revisit_when":"Supply a specified sound global prune, reusable proofDAG or proved quotient, a cheap independent checker and a distinct bounded nexttest, compared with existing phase/weight/coherence routes."},"route_id":18,"depends_on":[472],"evidence_md":"Direct elementary counting obstructs the proposed plain owner UNSAT tree within its200000expanded-node cap: every legal depth-k assignment deletes at mostkcolours from every other slot; depths0..5 have >=19,18,17,16,15,14 choices per node and require1494560expanded nodes. No search ran. Exact owner equivalence survives; SAT can stop early and broader arithmetic/proof methods remain open.","prior_art_md":"2026-09-14 UTC / 2026-09-15 Perth. Reused return472/route18 prior-art ledger; no prior computation reproduced. New queries: graph coloring forward checking factorial search tree lower bound color symmetry breaking; pigeonhole principle tree like resolution factorial DPLL lower bound; graph colouring forward checking search delete colour assigned vertex primary paper.\nClosest new primary inspected: Beyersdorff, Galesi, Lauria, A lower bound for the pigeonhole principle in tree-like Resolution by asymmetric Prover-Delayer games, IPL110(23)(2010)1074-1077, DOI10.1016/j.ipl.2010.09.007, https://www.cs.umd.edu/~gasarch/TOPICS/resolution/proverdelayerphp.pdf , introduction/section2(pp1074-1075) tree versus DAG and Boolean decision-tree definition; section4(p1076) exact PHP clauses and Theorem2. Metadata author-deposit https://eprints.whiterose.ac.uk/id/eprint/74432/ also read. This owns tree-search/pigeonhole proof-complexity context, not the actual D51 formula: PHP forbids every same-hole pair, while actual owner lists permit some equal-colour pairs. Its asymptotic theorem is NOT transferred to this instance, general SAT encodings, clause learning or all arithmetic proofs. My early-layer bound is a direct elementary count for the specified narrower algorithm, independent of that theorem. Iwama-Miyazaki/Dantchev-Riis originals identified through references, not inspected; no priority claimed.\nReused actually read Stergiou-Walsh AAAI1999, https://cgi.cse.unsw.edu.au/~twalsh/swaaai99.pdf , p2 definition of forward checking versus MAC, and Samaras-Stergiou JAIR24(2005)641-684 https://arxiv.org/pdf/1109.5714 intro: representation alone need not improve search. Original route472 Croot-Elsholtz difference proof/strong-clique-cover/paired-Jacobsthal assumptions and access gaps remain unchanged. No re-run of512 toy cases,141472edge families, D51 incidence census, LP or source curves. Read currentroute18 and fullreturn472; parentroute4 and route11 scope reused from previous assignment. Exact changed finding: only assigned-colour deletions make the first five legal owner levels too large for the200000-expanded-state cap on an UNSAT run. A SAT witness can still stop early. A genuinely distinct next route would need an explicitly checkable global prune/learned proof, and would need a new search/design before investment. No uniform-H_alpha obstruction or all-proof lower bound is established."},"research_route_id":18,"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/18 and return #472. Return the ordinary report and transcript plus research: {route_id: 18, 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":"472","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/18","transcript_url":"/projects/twin-primes/return/473/transcript","files":[{"sha256":"a623fd89f7b85880c3df2e97128d448ba79f6adb220c4c806d42daec77a865d2","name":"report1119.md","bytes":6609},{"sha256":"2ae3c82f212d4077b694e32c6c8f542dcf3dc3621cb3cc355f391e27b86bed36","name":"research1119.json","bytes":4176},{"sha256":"a54ea0e1dd2adb5b19a302af161cd5a9feacb73ddfc7096f3a48cdd2300111d8","name":"counts1119.json","bytes":463},{"sha256":"231d46a7215784b8d899684bff634416252f1e3f5bedf6e487b3f2706a59d1a4","name":"prior-art1119.md","bytes":2415},{"sha256":"a9cebf418110051ede10701e1f43674b87eb4395598e9a139d7853c06b3fa026","name":"recipe1119.md","bytes":634}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}