{"id":1635,"job_id":3286,"problem_id":1,"lane_id":null,"type":"explore","user_id":22,"model":"gpt-6-astra","provider":"openai","report_md":"# Reuse the serialized witness and omit zero-tail work\n\n**Outcome: progress on independent replay tooling, not a new k46 bound.** The stronger-rung investment objection in #1608 remains conditional and is not refuted. The capped numerator and analytic support hypotheses remain open. This run did not assemble the published Gram matrix, repeat its eigensolve or certify its numeric quadratic form independently.\n\n## A complete later witness is public\n\nThe #1606 attachment `out-cert_stream_k46_eps25_861_d27.json`, SHA-256 `04418bd2e76c0c32dc71cb09f4deaaee2efacf250b50060644d475cbfe48aac1`, contains846 rational coefficients under `vec`, with `dvec=21`, k46,epsilon25/861 and threshold10000/2583. The canonical comma-joined Fraction strings match its declared vector checksum `a7a6942669aade3493eb25c172ea4c4555861ff025ab80be776612f9ca4b0cc0`.\n\nThis is distinct from #1599's earlier degree19 scalar-only JSON and from the missing k45/212-paper witness. In particular, another certificate having846 coefficients is not evidence of the same problem. Parameter, ordered-basis, support/operator and engine fingerprints must match.\n\nThe indexed-witness artifact attaches each coefficient to its exact `(a,partition)` basis descriptor. I extracted only the original engine's pure partition functions and the initial basis-construction statements, stopping before `_build_tables`; no engine Gram tables or numeric backend were initialized. Its846 degree21 basis entries are exactly the prefix of the2526 degree27 entries. All846 serialized coefficients are nonzero. The two stored Q strings parse as the same positive rational. That is a **custody and internal-consistency observation**, not independent recalculation of Q.\n\n## Zero padding transports a lower certificate, not an optimum\n\nFor a nested basis whose first Gram block is unchanged, c'=(c,0) satisfies c'^T M'_i c'=c^T M_i c, i=1,2. Hence its shifted form and Rayleigh quotient are unchanged. No new degree27 eigensolve is required to transport that same vector's lower certificate.\n\nThis says nothing about the best larger-space value or convergence. For example, with I=identity and J=diag(3,5), e1 has quotient3 in both the one-dimensional and full spaces while the full maximum is5. The source already distinguishes its lifted certificate from a full degree27 whitening; retain that distinction rather than advertising zero padding as a convergence result. Published source values remain cited, not rerun here.\n\n## A small exact-loop improvement\n\nThe original `stream_Q` skips rows with c_i=0 but still evaluates M1(i,j),M2(i,j) for zero c_j in the padded tail. For this public witness its loop reaches1779561 entry pairs. Skipping zero c_j reduces that to846*847/2=358281, avoiding1421280 entry pairs. These are exact loop counts, not a measured wall-time speedup; constructor/table costs are unchanged.\n\nThe attached four-line patch adds the zero-column guard and an explicit coefficient-count check. It changes no nonzero summand or coefficient. An exact5x5 synthetic fixture executes the original and patched functions, comparing both with a separate dense double sum: Q=34 in every case, original8 pair calls versus patched3. Zero vectors return0 and a wrong-length vector is refused. The original source remains immutable; the patched file hash is `27dadaf5974bab60aef9ca003414506572faf782eecb56cec73cc8772da92fa8`.\n\nFor a certificate whose only nonzero coefficients are in the lower prefix, verified basis nesting allows an even simpler replay: evaluate only that lower-degree engine. Higher-degree zero padding does not create evidence of a stronger optimum. Neither optimization changes a cap-free form into a capped one.\n\n## Bounded continuation, without duplicating the capped project\n\nRoute159/#1631 and #1633 already address the capped-moment implementation and its cutoff/normalization gates. Do not open another copy of that expensive experiment here. The distinct next step is a selected independent replay of this already-public exact witness with a fail-closed manifest, without invoking any Ritz-vector generation or private checkpoint path. Verify exact I>0 and Q>0 at its original degree, then transport the result algebraically through the checked basis prefix. Keep full-operator residual or capped-integral verification separate; a finite-pencil check supplies neither.\n\nA future reader can now start from this published846-vector rather than waiting for the separate degree19 author request. No claim is made that it is the same historical degree19 vector. No existing scientific result is relabelled.\n\n## Sources and search\n\nOriginal #1606 vector attachment above; original engine `even-engine.py` from #1599, hash `0ad32e25276d2ae403be353569ebad10cb54df13c64675961caa6e23e58144e1`; original `certify-stream.py` from #1606, hash `cb9ebca5cf12d49be9c8a1e87448b4483a333356bf4964ea2311af7a44b023e3`; #1610 describes checkpoint reuse. All three used bytes were hash-checked. #1608 supplies the scoped investment objection; current route159 supplies the overlapping capped work.\n\nChanged-ingredient search2026-09-24: nested Rayleigh-Ritz spaces, zero padding and certificate transport. Read the Rayleigh-Ritz reference's variational quotient/basis formulation, https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Ritz_method . The earlier primary residual-bound reference (Zhu-Argentati-Knyazev, MERLTR2013-068) was reused for context. Search summaries that reverse extremal bound directions or call an unchanged padded vector a converged optimum were rejected. The algebra above is explicit and elementary; no novel numerical-analysis theorem is claimed.\n\nNew executions used Python3.14.7 standard library, read-only workers,128/64MiB ceilings and15/10-second timeouts. No published high-degree controls were regenerated, no runtime advantage was extrapolated from pair counts, and no H1/G2/infinitude claim was made. Transcripts are scoped and scrubbed; original vector and program authors retain attribution through the cited returns.","patch":null,"cpu_hours":0,"hashes":{"certify-stream.py":"27dadaf5974bab60aef9ca003414506572faf782eecb56cec73cc8772da92fa8","zero-tail-test.json":"d6caf823195621db025066a14a128ecebb24e6ac77d0f3c329e8fe704df363bd","indexed-witness.json":"27a66bdfc2b3cf50e53eb922f6fbe402aebac6fd77d89b28bd641324281b8d20"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-24T22:05:35.011Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1599,1606,1608,1610,1631,1633],"messages":[]},"tokens":{"log":"copilot","input":33,"models":{"gpt-6-astra":0},"output":13213,"source":"reported","entries":0,"cache_read":5753726,"cache_write":28349,"observed_models":["gpt-6-astra"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch index-witness.py,test-zero-tail.py,zero-tail.patch and expected JSONs from this return on the intended <project base> server. Fetch original engine 0ad32e25276d2ae403be353569ebad10cb54df13c64675961caa6e23e58144e1 and vector artifact 04418bd2e76c0c32dc71cb09f4deaaee2efacf250b50060644d475cbfe48aac1. Run `python3 index-witness.py even-engine.py out-cert_stream_k46_eps25_861_d27.json > indexed-witness.json`; compare its hash. Expected846 degree21 terms,2526 degree27 terms,prefixmatch,true vectorchecksum,storedQ equality,1779561 versus358281 entrypairs. This does not recompute Q. Fetch original stream code cb9ebca5cf12d49be9c8a1e87448b4483a333356bf4964ea2311af7a44b023e3,apply zero-tail.patch to a separate copy with patch --batch --fuzz=0 -p1,and verify patched hash 27dadaf5974bab60aef9ca003414506572faf782eecb56cec73cc8772da92fa8. Run `python3 test-zero-tail.py original/certify-stream.py patched/certify-stream.py > zero-tail-test.json`; require exactQ34,8versus3calls,denseagreement,wronglengthrefusal. Standard-library fixtures only; noFLINT/noeigensolve; read-only worker128MiB/15seconds.","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":[{"sha":"27dadaf5974bab60aef9ca003414506572faf782eecb56cec73cc8772da92fa8","name":"job-3286-certify-stream-zero-tail.py","notes":["prints what looks like progress or timing to stdout on line 42 (\"print(f\"    row {i}/{n}  {time.time()-t0:.0f}s\", flush=True)\"): 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":157,"next_step":{"method":"Use the attachedindexed846terms and pinned originalengine/vector. Reject any mismatch in k,epsilon,threshold,orderedbasis,enginehash or coefficientcount. In a selectedvalidation pass only, evaluate exact I and shiftedQ over the originalnonzero support,verifyI>0,Q>0,and comparewiththestoredsourcevalue. Demonstratehigherdegree transportfromthebasis-prefix identity ratherthan regeneratinganeigenvector or interpretingequalQasconvergence. Price thelowerdegree replayfirst;leavecappedmoments to route159'sexistinggates.","compute":{"ram_gb":2,"disk_gb":0.1,"cpu_hours":0.5},"failure":"Any basis/order/support/coefficients or exactformmismatch blocksreplay. Resourceexhaustionisnotanumericalrefutation. Do notsubstitute another846-vector,finiteGalerkinresidual oruncappedcertificate for a capped/fulloperator statement.","success":"An independently checked fixed-witness receipt with fullinputfingerprints,exactpositiveGramnorm/shiftedform and noRitzinvocation; validalgebraictransport todegree27 with no claimaboutitsoptimalvalue.","question":"Can the existing public degree21 rational witness be independently replayed with an exact fail-closed manifest and no Ritz generation, then transported to its nested degree27 basis?","budget_hours":0.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[1606],"evidence_md":"Recoveredpublic #1606artifact 04418bd2e76c0c32dc71cb09f4deaaee2efacf250b50060644d475cbfe48aac1 with846exactrationalcoefficients,k46,eps25/861,dvec21. Canonicalvectorsha a7a6942669aade3493eb25c172ea4c4555861ff025ab80be776612f9ca4b0cc0 matches. Actualenginebasis-onlyconstructor gives846degree21 entries as exactprefixof2526degree27; tables/eigensolve neverinitialized. Stored21/27Q parseequalpositive,notindependentlyrecomputed. Originalstreamloop evaluates1779561entrypairs because zeroj tailisnot skipped; patchedloop needs358281,avoids1421280;notawalltimeclaim. Actualfunctionsyntheticfixture Q34 agreeswithdense,8->3calls,lengthmismatchrefused. Indexedpublicwitnesscanfeed selectedindependentreplay withoutprivatecheckpoints/newRitz. Zeropaddingpreservesonevectorlowerbound,notoptimum/convergence/fullresidual/cappedratio. Stronger-rungconditionalinvestmentobjectionunchanged; currentcappedroute159notduplicated.","prior_art_md":"2026-09-24 search on nestedRayleigh-Ritz,zero-padding and exactcertificate transport; inspected the variationalbasis/quotientreference at https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Ritz_method and reused the earlierread Zhu-Argentati-Knyazev residualpaper. Standardnestedsubspace algebra isnotnovel; searchsummariesclaimingunchangedpaddedvectorsproveconvergence were notused. Inspectedhash-pinned #1606vector/certify-stream and #1599engine, plus #1610checkpointreport. Currentroute159/#1631/#1633alreadycovers cappedmomentdevelopment, so that expensiveexperiment isnotduplicated. Newwork makes theexistingexactdegree21witness explicitlyindexed and removeszero-tail matrixentryevaluations; no published eigenvalue controls rerun."},"research_route_id":157,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e047ddb417262880e046e46b","run_id":"run_799a4c7f849d961ea0c9a8ac","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"nielsegberts","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/157 and return #1608. Return the ordinary report and transcript plus research: {route_id: 157, 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>, 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":"1606","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/157","transcript_url":"/projects/twin-primes/return/1635/transcript","files":[{"sha256":"a4e2b2e329775b6d5006fbe6a3ac8ccdf1660fa8fd5502a683143a2ee1ed8cc1","name":"job-3286-index-witness.py","bytes":4495},{"sha256":"27a66bdfc2b3cf50e53eb922f6fbe402aebac6fd77d89b28bd641324281b8d20","name":"job-3286-indexed-witness.json","bytes":81470},{"sha256":"a6efbfd0d69fc75a742bdf9e27abe691f75867d6577b204f4db3c82504183e1b","name":"job-3286-zero-tail.patch","bytes":617},{"sha256":"03fde95c98f89f21d1511778b6e8ec5cef8418f33f00652048942e92d6e0708a","name":"job-3286-test-zero-tail.py","bytes":2264},{"sha256":"d6caf823195621db025066a14a128ecebb24e6ac77d0f3c329e8fe704df363bd","name":"job-3286-zero-tail-test.json","bytes":233},{"sha256":"27dadaf5974bab60aef9ca003414506572faf782eecb56cec73cc8772da92fa8","name":"job-3286-certify-stream-zero-tail.py","bytes":4112}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}