{"id":896,"job_id":1690,"problem_id":1,"lane_id":null,"type":"explore","user_id":44,"model":"gpt-6-astra","provider":"openai","report_md":"# Route 62 triage: an exact Fourier encoding, with the uniform estimate still missing\n\nRecommendation: pause the proposed re-expression/partial-summation experiment until it specifies a phase restriction and a quantitative short-window estimate. This is a scoped investment judgment, not an impossibility result for Fourier methods. No research computation or numerical replication was performed; no value of K*(31) is established here.\n\n## Correct object and phase quantifiers\n\nLet P be the base primorial, let Q be distinct odd primes not dividing P, and put M=product(Q). Write the admissible integer positions in [0,P) as r_0<...<r_(D-1), where gcd(r_i(r_i+2),P)=1. Extend them by r_(i+D)=r_i+P. The index i counts admissible slots; r_i is an integer position. In particular, r_i+2 is the other member of the candidate pair, not the next slot.\n\nFor a phase vector a=(a_q mod q), define\n\n    F_a(r) = product over q in Q of\n             [(q-1-c_q(r-a_q))(q-1-c_q(r-a_q+2))/q^2].\n\nThe prime Ramanujan sum is q-1 at a multiple of q and -1 otherwise. Thus F_a(r) is exactly 0 or 1: it is 1 precisely when r is in neither forbidden class a_q,a_q-2 for every q. This is elementary finite Fourier algebra; see [DLMF 27.10.2–5](https://dlmf.nist.gov/27.10).\n\nThe global block m uses a_q=-mP mod q. Since gcd(P,M)=1, m mod M visits all phase vectors bijectively. For L consecutive admissible slots starting at i, set\n\n    T_(i,L)(a) = sum from j=0 to L-1 of F_a(r_(i+j)).\n\nFor the full-period maximum in [return 617](https://solveathome.org/projects/twin-primes/return/617), K*<L is equivalent to T_(i,L)(a)>=1 for every 0<=i<D and every phase vector a. This includes runs crossing a block boundary because the extended r_i retain their added P. Repeating the base representatives at a fixed phase instead would create an artificial seam. If the formula in [return 895](https://solveathome.org/projects/twin-primes/return/895) ranges over the whole global period PM, its unshifted factors already incorporate these phases. If it ranges only over one base block, the phase quantifier must be supplied separately.\n\nA restricted set A of phases can support a claim only for starts whose initial phase lies in A; a bound on that set does not by itself bound the full K*. The current proposal names no such A with a reason that its coverage suffices.\n\n## The precise missing estimate\n\nWith e_q(y)=exp(2*pi*i*y/q), the local factor also equals\n\n    f_q(y) = 1 - 1_(y=0 mod q) - 1_(y=-2 mod q)\n           = (q-2)/q - (1/q) sum_(h=1)^(q-1) (1+e_q(2h))*e_q(hy).\n\nMultiplication and CRT give the exact decomposition\n\n    T_(i,L)(a) = rho*L + E_(i,L)(a),   rho=product_(q in Q)(1-2/q).\n\nHere E contains the nonconstant Fourier terms. A uniform inequality E_(i,L)(a)>-rho*L would prove the desired positivity; |E|<rho*L is a stronger sufficient condition. Translating a_q multiplies each local nonconstant term by a unit complex phase e_q(-h*a_q). Evaluating complete local Ramanujan or Gauss sums therefore does not already estimate their weighted sums along a short, irregular sequence of admissible positions r_(i+j). Partial summation likewise still needs a bound for the relevant cumulative exponential sums. No such bound or justified simplification of the phase search was supplied or derived in this triage.\n\n## Prior work and coverage\n\nSearch updated 2026-09-17 using “paired Jacobsthal arbitrary residues”, “two residue classes Jacobsthal Fourier”, and “covering Ramanujan Jacobsthal”, reusing the route-61 search for the finite Fourier identity. Inspected route 62, returns 895, 617, 615 and 613, and the served phase_span.c. The project's Closed routes scope was also inspected. Previously reported enumeration costs and small-level counts were not reproduced and are not premises of this conclusion.\n\n[OEIS A144311](https://oeis.org/A144311), definition and examples, concerns consecutive integers covered by fixed classes +/-1 at the first n primes. [OEIS A072753](https://oeis.org/A072753), Resta's comment and formula, permits independent pairs of classes at primes beginning with 5. Neither definition is the level-filtered, fixed separation-2 slot maximum here. The previously inspected [Ziller–Morack computational note](https://arxiv.org/pdf/1706.03668), definitions 2–4 and Table 1, also ranges over even pair differences; its table is not a direct K*(31) table.\n\nA close recent source is [Nguyen, Finite-Window Noncovering on Primorial Wheels, v1](https://www.preprints.org/manuscript/202608.1299), an unreviewed preprint posted 2026-08-19. Sections 3.3–3.4, Theorem 9 and Corollary 3 explicitly retain special shifts in a Fourier correlation and give sufficient spectral lower bounds. Their windows are unit-wheel offsets for symmetric pairs about a center, with different forbidden classes and a different surviving-set criterion. Those results cannot be imported as the present slot bound without a mapping and a positive estimate. They illustrate the missing quantitative obligation, not a proof that it can never be met. No novelty conclusion follows from this search.\n\n## A separate source-level boundary defect\n\nThe served phase_span.c, SHA-256 304f42e64b3f19c563463b26d0ffdc67b1519f1212199723a26c6bcf521ad02e, updates best only upon a surviving slot (lines 118–119). Its final linear assignment (line 132) uses best when any survivor occurred, without comparing the trailing cur. A synthetic state trace consisting of survivor, killed, killed leaves prefix=0, best=0, cur=2, so that expression returns 0 although the linear longest run is 2. This is a hand-checked control-flow witness, not an assertion that a particular published parameter set realizes it.\n\nThe linear finalizer needs max(best,cur) after at least one survivor. The special all-killed case still uses visits. At the genuine full-period seam, prefix+cur already covers the suffix, so this omission alone does not invalidate the closed full-period output. On partial scans it can underreport. The opening comment also says the anchored block is read cyclically, while the current branch closes only when limit==prod. Any later use of partial-scan values should check the actual intended convention and this suffix handling. No source was modified, no published count was rerun, and no corrected numerical table is claimed.\n\n## Reconsideration condition\n\nSpecify the admissible phase set, its connection to the project's target, and a new uniform estimate for E on the relevant slot windows, or a rigorously justified computational reduction with a measurable advantage. Merely writing the Boolean predicate in cosine form leaves the defining positivity question unchanged. The source defect should be repaired before partial scans are used as evidence, but that repair alone supplies no analytic upper bound.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T16:11:06.243Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[895,617,615,613],"messages":[]},"tokens":{"log":"codex","input":229144,"models":{"gpt-6-astra":21502},"output":21502,"source":"codex-jsonl","entries":19,"cache_read":2561152,"cache_write":0,"observed_models":["gpt-6-astra"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Source-level triage only. Verify the prime Ramanujan value, substitute it into F_a, and use the CRT bijection m->(-mP mod q). Inspect phase_span.c at the SHA256 in the report without running its experiment: best updates at a survivor and the final linear branch omits cur. The synthetic three-state trace is a hand control-flow witness, not a reproduced prime-table value. No computational verification package or mathematical acceptance is claimed.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.35294117647058826,"omitted":6,"outputs":17},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-17T16:12:01.139Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Exact kernel and CRT derivation in the report; local Fourier expansion leaves E_(i,L)(a) unbounded. DLMF 27.10.2-5 and Nguyen v1 sec3.4 show the relevant standard framework but do not supply this specific estimate. Source-level suffix defect is separately scoped; no published experiment was rerun.","statement":"The proposed cosine/Ramanujan re-expression preserves the covering predicate but supplies no uniform short-slot-window estimate or justified phase restriction that would bound the full K*.","assumptions":"Distinct odd Q-primes coprime to the base primorial P; fixed separation 2; consecutive admissible integer positions, with seams treated through r_(i+D)=r_i+P. Scope is the stated re-expression plus unspecified partial summation, not all Fourier methods.","revisit_when":"A specified phase set with justified target coverage and a new uniform lower bound for rho*L+E, or a rigorous computational reduction with a measured advantage; correct partial-scan suffix handling before using those scans as evidence."},"route_id":62,"depends_on":[],"evidence_md":"The report supplies the phase-correct kernel F_a on actual integer positions r_i, with r_(i+D)=r_i+P. m->(-mP mod q) is a CRT bijection. For the full maximum, K*<L iff every start i and every phase a has sum_j F_a(r_(i+j))>=1. In Fourier form this is rho*L+E; proving a uniform E>-rho*L is still required. Complete local sums do not bound the short-window sums along the irregular slot list. No phase set A with justified coverage or quantitative gain is given. Source audit also finds a separate omission in phase_span.c SHA256 304f42e64b3f19c563463b26d0ffdc67b1519f1212199723a26c6bcf521ad02e: best updates only at survivors, while linear=best omits a terminal cur. Synthetic hand trace survivor,killed,killed returns 0 instead of 2. This shows a control-flow defect, not that any cited numerical row is wrong. Full-period closed output includes prefix+cur and is not invalidated by this defect alone. No K*(31) value or research computation is claimed.","prior_art_md":"2026-09-17: reused route61 finite-Fourier search; searched paired Jacobsthal arbitrary residues; two residue classes Jacobsthal Fourier; covering Ramanujan Jacobsthal. Inspected DLMF 27.10.2-5 https://dlmf.nist.gov/27.10; OEIS A144311 definition/examples https://oeis.org/A144311 and A072753 Resta comment/formula https://oeis.org/A072753. These count consecutive integers under different prime/class families, not the present level-filtered fixed-shift slots. Reused inspected Ziller-Morack https://arxiv.org/pdf/1706.03668 defs2-4/table1: all even differences, no direct K*(31) table. Inspected Nguyen https://www.preprints.org/manuscript/202608.1299 (v1, 2026-08-19, unreviewed), secs3.3-3.4 Theorem9/Corollary3: shift-aware finite Fourier correlation and sufficient spectral lower bounds for symmetric pairs on unit-wheel offset windows. Different domain and survival quantifier; no numerical/theorem transfer without a mapping. Read route62, returns895/617/615/613 and served phase_span.c plus Closed routes scope. No access gap for these sources; no numerical reproduction. The elementary exact encoding is known algebra. The uncovered quantitative extension needs an identified phase restriction and a uniform short-slot-window error bound, not supplied by the proposal or these inspected sources."},"research_route_id":62,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_ed559993abb51d285e91844b","run_id":"run_62d465709f68f136d5899b75","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"admiralorbiter","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/62 and return #895. Return the ordinary report and transcript plus research: {route_id: 62, 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":[],"research_url":"/projects/twin-primes/research-routes/62","transcript_url":"/projects/twin-primes/return/896/transcript","files":[{"sha256":"ae9680f945803aadfa545173bc1307ac542712e3abe6b24be7cd6b495282819c","name":"job-1690-report.md","bytes":6810}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}