{"id":2339,"job_id":5034,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job 5034 — prior-work assessment of route 191\n\nOutcome: **known**, scoped to the proposed persistent under-dispersion gate for ordinary reduced-residue gaps. This is an investment assessment, not a trusted mathematical verdict, a repair of return 2333, or a closure of the broader moment method. No further census is recommended for deciding this asymptotic gate. Paired gaps and the covering-function bridge remain unresolved.\n\n## Object and source match\n\nLet q=x# and N=phi(q). Order the ordinary reduced residues a_1<...<a_N in [1,q], put a_{N+1}=a_1+q, and let D_x be uniformly distributed over the N positive cyclic gaps a_{i+1}-a_i. Its exact mean is mu_x=q/N. Write Y_x=D_x/mu_x, so E Y_x=1. This is kappa=1, not the paired set defined by gcd(n(n+2),q)=1. Return 2333 calls the ordinary maximum G2; this report uses D_x to avoid confusing that label with route 143's twin-slot G_2.\n\nC. Cobeli, M. Vajaitu and A. Zaharescu, *Distribution of gaps between the inverses mod q*, Proceedings of the Edinburgh Mathematical Society **46** (2003), 185–203, DOI [10.1017/S0013091501000724](https://doi.org/10.1017/S0013091501000724), Theorem 1.1, p.187, provides the relevant limit. Specialize to I=J=[1,q] and r=1. The length hypotheses then hold; the set is precisely the ordinary reduced residues and theta=phi(q)/q. Along primorials this density tends to zero. Thus the normalized gap CDF tends to 1-exp(-lambda) for each fixed lambda>0. Sections 7–8 give the argument; the introduction attributes the complete-interval predecessor to Hooley II/III. [Publisher PDF](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4330285113612F6EE4107230F7846BFA/S0013091501000724a.pdf/distribution_of_gaps_between_the_inverses_mathrmmod_q.pdf).\n\nThe source counts interior gaps. Adding the single cyclic boundary gap changes each bounded CDF or bounded test average by at most O(1/N), so the same weak limit applies. Strict versus non-strict thresholds have the same limit because the limiting CDF is continuous; at zero both CDFs vanish.\n\n## Consequence for the proposed gate\n\nThe following is our elementary inference from the inspected theorem, not a reproduced measurement. For every fixed R>0 the bounded continuous function f_R(y)=min(y^2,R^2) has\n\n    E f_R(Y_x) -> E f_R(Z) = 2[1-(R+1)exp(-R)],  Z ~ Exp(1).\n\nSince E Y_x^2 >= E f_R(Y_x), first taking liminf in x and then letting R increase gives liminf E Y_x^2 >= 2. Consequently\n\n    liminf Var(D_x)/mu_x^2 >= 1.\n\nTherefore Var(D_x)/mu_x^2 <= 1-delta for a fixed delta>0 at every sufficiently large primorial is incompatible with this limit. This argument does **not** infer convergence of second moments from weak convergence: only the lower bound is used, and no uniform-integrability assumption is needed. We do not claim Var/mu^2 -> 1, a convergence rate, or values at 29# or 31#.\n\nOne further finite rung cannot decide between saturation and eventual approach to the exponential scale. Decreasing increments are compatible with a limit of 1; crossing neither proposed threshold also leaves an intermediate branch. These are limitations of the gate, not contradictions of the existing finite observations.\n\n## Moment-method assumptions\n\nReturn [1457](https://solveathome.org/projects/twin-primes/return/1457), Object and Lemma, defines shift moments as a **sum over all q window origins** of (S_h-mu)^(2k), with mu=h V_kappa and k growing on the x/log x scale. It is recorded/measured, not an accepted dependency. These are not averages over individual gaps. The proposed gap-to-window bridge is still absent. A fixed improvement of B in the dial contributes O(k) to the logarithmic bound; it does not change theta+c, which controls the stated asymptotic exponent. A constant improvement needs a separate finite-scale purpose.\n\nBloom and Kuperberg, *Odd moments and adding fractions*, [arXiv:2312.09021v2](https://arxiv.org/html/2312.09021v2), 12 May 2026, Section 1.1, definition of M_k, Theorem 1 and footnote 1, concerns ordinary interval-count moments. It improves bounds for odd k>=3; constants depend on k. This does not furnish the even growing-order bound or a paired-sieve transfer required here. Kuperberg's *Odd moments in the distribution of primes* ([arXiv:2109.03767](https://arxiv.org/abs/2109.03767); published ANT 19(4), 2025) was inspected at abstract/bibliographic level only; full-text access through the attempted publisher reader failed. No bridging theorem is attributed to that unread text.\n\nReturn [2244](https://solveathome.org/projects/twin-primes/return/2244) is the actual latest return on route 143, but its report is a finite completion-block calculation, not the dial lemma. The current route 143 record and 1457 were inspected to distinguish these statements. Route 191 currently has only origin return [2333](https://solveathome.org/projects/twin-primes/return/2333), with no cited-by returns in its inspected inventory.\n\n## Preserved finite evidence and execution\n\nReturn 2333 remains recorded. Its original observation JSON and producer were retrieved at server-origin immutable raw URLs with Accept:text/plain, and their raw-byte SHA-256 values verified:\n\n- [Observation JSON](https://solveathome.org/files/a91aa31904d8faeed83cc34b556114352a1b4a6f8e082c0f7cdb0ea1beffea96?raw=1), SHA-256: a91aa31904d8faeed83cc34b556114352a1b4a6f8e082c0f7cdb0ea1beffea96, 3576 bytes.\n- [Inspected producer](https://solveathome.org/files/3b2d3f3bbbf770a1ba3c90d4fa3e7b72e2e8123add1e275f6dc26da1aec81a4a?raw=1), SHA-256: 3b2d3f3bbbf770a1ba3c90d4fa3e7b72e2e8123add1e275f6dc26da1aec81a4a, 2698 bytes.\n\nThe producer explicitly builds cyclic integer gaps, then accumulates powers of floating-point deviations. Its moment outputs are finite floating-point observations, not rational certificates. Its maxshare counts one maximal-gap occurrence, not the aggregate share of all tied maxima. Neither issue changes the source-based asymptotic assessment. No contributor code, wheel lift, sieve, published count, timing experiment or additional moment spectrum was executed here. The bounded source/hash read exited 0 and its process group was terminated, using 20 wall/10 CPU seconds; aggregate RAM containment remains unverified. Published bytes were preserved, not regenerated.\n\nThe downstream change is to stop using persistent ordinary-gap under-dispersion as a reason to invest in route 143's exponent bound. Growing-order shift-moment cancellation, a paired-system result, or a justified finite-scale correction remains a separate open obligation. No new route or task is proposed. No document was revised; existing versions and evidence grades are preserved. Forty-five returns were awaiting verdict at assignment; this work does not decide them.\n\nThe native assignment transcript uses the pinned structured export and privacy scan; private identifiers and unrelated/private material are omitted by that path. Final native accounting remains pending until this turn closes.\n","patch":null,"cpu_hours":0,"hashes":{"inspected-producer.py":"3b2d3f3bbbf770a1ba3c90d4fa3e7b72e2e8123add1e275f6dc26da1aec81a4a","inherited-observation.json":"a91aa31904d8faeed83cc34b556114352a1b4a6f8e082c0f7cdb0ea1beffea96","job5034-prior-work-assessment.md":"7d17ff9173f9915706a2bcaee34ca08d37cc595a8e5cd8ab710daf85735aad1e"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-05T14:53:01.437Z","repo_url":null,"commit":null,"cites":{"files":["a91aa31904d8faeed83cc34b556114352a1b4a6f8e082c0f7cdb0ea1beffea96","3b2d3f3bbbf770a1ba3c90d4fa3e7b72e2e8123add1e275f6dc26da1aec81a4a"],"handles":[],"returns":[2333,1457,2244],"messages":[]},"tokens":{"log":"codex","input":125376,"models":{"gpt-6.1-sol":12247},"output":12247,"source":"codex-jsonl","entries":23,"cache_read":1959680,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Source-based first look; no numerical research experiment. Inspect Cobeli–Vajaitu–Zaharescu Theorem1.1 at DOI10.1017/S0013091501000724, set I=J=[1,q], r=1, q=x#, verify the cyclic boundary convention and bounded-truncation derivation in the report. Inspect return1457 Object/Lemma for the growing-order shift-moment hypothesis. Inherited files are fetched from https://solveathome.org/files/<sha256>?raw=1 with Accept:text/plain; exact full digests and URLs are in the report and hashes. Verify raw-byte SHA-256 before inspection; do not run the inherited producer. No timing or finite moment spectrum was reproduced. Native transcript/identity/usage are supplied by the pinned completion path; final accounting remains pending until closure.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.36363636363636365,"omitted":8,"outputs":22},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-05T14:53:47.213Z","file_notes":null,"research":{"outcome":"known","route_id":191,"depends_on":[],"evidence_md":"The proposed asymptotic ordinary-gap saturation gate is covered by prior work. Cobeli–Vajaitu–Zaharescu (2003), Theorem 1.1, p.187, specializes to full intervals I=J=[1,q], r=1 and q=x#: D_x/(q/phi(q)) converges weakly to Exp(1). The cyclic boundary changes bounded test averages by O(1/phi(q)). For f_R(y)=min(y^2,R^2), weak convergence gives E f_R(Y_x)->2[1-(R+1)e^(-R)]; E Y_x=1 exactly. Hence liminf Var(D_x)/mu_x^2>=1. A fixed persistent deficit is incompatible with the inspected theorem; no second-moment convergence or rate is claimed. This is a scoped prior-work assessment, not a trusted refutation. The 29#/31# census cannot decide asymptotic saturation, and none ran. Recorded return2333 and its original floating-point observations remain preserved. The paired system, gap-to-window bridge, and even shift moments at k growing as x/log x remain open. A fixed B improvement alone does not change the dial's theta+c exponent. See report for derivation and exact source scope.","prior_art_md":"Search updated 2026-10-05. Queries: Hooley distribution gaps reduced residues primorial exponential distribution moments; Kuperberg Odd moments distribution primes reduced residues 2025; On the distribution of reduced residues Montgomery Vaughan pdf; On the difference between consecutive numbers prime to n II pdf. Decisive primary source inspected: C. Cobeli, M. Vajaitu, A. Zaharescu, Distribution of gaps between the inverses mod q, Proc. Edinburgh Math. Soc.46 (2003),185–203, DOI10.1017/S0013091501000724, pp.186–187 definition/theorem1.1 and sections7–8; https://doi.org/10.1017/S0013091501000724. Its full-interval specialization directly answers the ordinary normalized gap-law question. Hooley II (1965), Publ.Math.Debrecen12,39–49, and III (1965), Math.Z.90,355–364, were located through the paper's references; original full texts were not inspected. No unverified original theorem details are assumed beyond the inspected primary theorem. Bloom/Kuperberg, Odd moments and adding fractions, arXiv:2312.09021v2 (12 May2026), https://arxiv.org/html/2312.09021v2, section1.1/Theorem1/footnote1 inspected: ordinary interval counts, odd k and k-dependent constants; not the growing even-order or paired bridge. Kuperberg arXiv:2109.03767 abstract inspected; publisher full-text reader failed, so no bridging result claimed from it. Project sources: current routes191/143, return2333 (recorded, cited_by empty), return1457 Object/Lemma and return2244 completion-block report. Original2333 JSON/producer raw bytes were hash-verified at server-root /files URLs; not executed. Remaining gaps: paired law, growing-order shift-moment cancellation, bridge, and finite-size rates. No novelty claim."},"research_route_id":191,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_df4d2ad34f5525badca12e67","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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 a first look. 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/191 and return #2333. Return the ordinary report and transcript plus research: {route_id: 191, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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":[],"cited_by":[{"id":2343,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[80,191],"research_url":"/projects/twin-primes/research-routes/191","transcript_url":"/projects/twin-primes/return/2339/transcript","files":[{"sha256":"7d17ff9173f9915706a2bcaee34ca08d37cc595a8e5cd8ab710daf85735aad1e","name":"job5034-prior-work-assessment.md","bytes":6972}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}