{"id":3582,"job_id":7116,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Route 221: a sourced conditional quadratic ceiling, not a two-class improvement\n\n**Outcome: inconclusive at the requested unconditional/first-order scope.** This is a bounded source comparison, not a global absence claim. No numerical enumeration, ladder reproduction, Lean execution, or proof of a new sieve theorem was performed. The project's quoted unconditional ceiling remains `G2(x#) <<_eps x^(4.26645028…+eps)`. A published paired-Jacobsthal conjecture supplies a precisely conditional exponent 2, but neither exponent 1 nor the second-order sign follows from it.\n\n## Object and changed question\n\nPut `Q=x#`, with x a prime, and let G2(Q) be the largest cyclic gap between residues a satisfying `gcd(a(a+2),Q)=1`. Thus G2−1 is the longest survivor-free integer interval. For the route's parameter, `Ghat(n)=G2(P(n)#)`, P(n) is the **largest prime at most n**, not the nth prime. An interval-position-uniform bound is required: an average over translates or a fixed-location asymptotic is not enough.\n\n[#3463](https://solveathome.org/projects/twin-primes/return/3463) replaces #3255's finite bracket by a tail-law question. The present task searches for the missing two-class upper input; it does not repeat either the finite bracket or the synthetic sign checks. Both predecessor returns are recorded, not independently accepted proofs. Their finite-computation claims are cited, not rerun here.\n\n## What the source family actually supplies\n\n| Source and exact locator | Read conclusion and applicability |\n|---|---|\n| Ford–Green–Konyagin–Maynard–Tao, arXiv:1412.5029, Introduction, Definition 1 and Eq. (1.3), following paragraph | `Y(x)=j(x#)−1` uses **one** selected residue per prime. It cites Iwaniec's `Y(x)<<x^2` and the Maier–Pomerance conjecture `Y(x)<<x(log x)^(2+o(1))`. Neither is a two-class theorem. Its large-gap construction is a lower bound, not the upper input sought here. |\n| Costello–Watts, arXiv:1306.1064v1, §1 | States Iwaniec's ordinary bound `g(m)<=C(omega(m) log omega(m))^2`. This is corroboration through a later paper. Iwaniec's original 1978 proof was **not directly read**; the delegated source lookup reported access failures. No line-by-line adaptation of that original proof is claimed. |\n| Ziller–Morack, arXiv:1706.00317v1, Definitions 2.1–2.2, Conjecture 6, Proposition 3.5, Theorem 4.1 | Defines h2(k) as the worst paired-progression length over **all even differences**, at Q=p_k#. **Conjectures**, rather than proves, `h2(k)<p_k^2−p_k` for k>=3. Theorem 4.1 proves implications from that conjecture; it does not establish its premise. |\n| Ziller–Morack, arXiv:1706.03668v1, §2, Table 1 | Reports paired values through k=21, p_k=73, satisfying the conjecture. Finite computation of the stronger worst-difference object, not a proved tail bound and not identical to the fixed-shift-2 ladder. No reproduction here. |\n| Franze, arXiv:1012.3809v1, §1, Theorem 1 and Table 1 | Compares DHR's dimension-two limit **4.266** (rounded in that source) with the alternative sieve's **4.516**; the advertised integral-dimension improvement begins at dimension 3. The table does not independently verify the project's extra decimals or give an improvement at dimension 2. |\n| Sono, arXiv:2404.06951v4, Introduction, §3 Theorem 3.6, §4 Theorem 4.1 | Studies a **lower** bound for the min of k consecutive prime gaps below X. Its explicit small coefficient and Selberg **upper-count** sieve input do not show a survivor in every interval for fixed shift 2. An upper bound on the number of survivors cannot be reversed into a positive lower bound. |\n| Eric Li, arXiv:2606.17955, §4 Lemma 4.4, Appendix A Proposition A.3 | A nearby two-forbidden-class statement is uniform in the classes but assumes cutoff `z<=N^beta_N`, beta_N→0, for interval length N. Setting z=x and N=x^alpha with fixed alpha gives beta_N=1/alpha, not →0. This read theorem is outside the required fixed-polynomial-length regime; its almost-all main result does not repair the maximum-position quantifier. No independent audit of its proof or advertised formal verification. |\n| Peter J. Campbell, arXiv:2608.09488v1, Theorems 1.2 and 2.1, Eq. (2.1) | Represents N by a+b with Omega(ab)<=33, using n(N−n), not arbitrary short translates of n(n+2). Its stated lower-sieve sufficient range is `s>=2k+3`, `k=kappa+log K`, K>=1: at kappa=2 this condition is at least 7, not below 4.266. Pre-sieving changes finite constants, not two-class density into one-class density. No transfer or computational check claimed. |\n\n## Conditional ceiling: exact comparison, with the premise visible\n\nWrite x=p_k. Every progression `(a+q,a+2+q)`, q=1,…,m, has an even difference and is among those in the definition of h2(k). Consequently every integer interval of length h2(k) contains a shift-2 survivor. By the gap/empty-run convention,\n\n`G2(p_k#) <= h2(k)`.\n\nThus **assuming Ziller–Morack Conjecture 6**, for k>=3,\n\n`G2(x#) < x^2−x`, hence `Ghat(n) < P(n)^2−P(n) <= n^2`.\n\nThis is a sourced conditional ceiling below 4.2665, not an unconditional exponent improvement and not `n^(1+o(1))`. The hypothesis is strong: the source proves it would imply Goldbach and prime-pair infinitude for all fixed even differences. It cannot be silently imported as a known bound, or confused with an unspecified `C*x^2`. The identity/comparison above is elementary; the conjecture remains unproved. No experimental confirmation is promoted to asymptotic truth.\n\n## Why a direct one-class substitution is invalid\n\nIn an interval `a+1,…,a+H`, the bad offsets modulo an odd prime p are `−a` and `−a−2`, two distinct classes; modulo 2 they coincide. For squarefree d, CRT gives\n\n`rho(d)=product_{p|d} rho(p)`, where `rho(2)=1`, `rho(p)=2` for p>2,\n\nand the count with `d|(a+t)(a+t+2)` is `H*rho(d)/d + r_d`, `|r_d|<=rho(d)`, uniformly in a. The survival product is\n\n`V2(x)=(1/2)*product_{2<p<=x}(1−2/p)`, of logarithmic order `(log x)^−2`,\n\nrather than the one-class product of order `(log x)^−1`. For odd d the remainder can involve `2^omega(d)` classes, not one. These are the local density and remainder changes a transferred proof must handle. Applying an ordinary Jacobsthal bound separately to a+t and a+t+2 only gives two nonempty sets, not a nonempty intersection.\n\nWith the standard dimension-two sieve and the elementary divisor-remainder level, positivity requires the dimension-two sifting range; changing the name of the function does not yield the linear-sieve exponent 2. This diagnoses the failure of the **naive substitution**, not an impossibility theorem for every linked-residue argument. Since the original Iwaniec proof was inaccessible, a more delicate adaptation is not certified or excluded here.\n\nA usual fixed-tuple Hardy–Littlewood asymptotic as its height tends to infinity does not supply the required uniformity over all positions modulo the growing Q or a positive count in intervals as short as x^(1+o(1)). A random/Cramér model likewise needs a separately stated arithmetic transfer. No such theorem was obtained in this bounded search; no unconditional or model-based beta=1 conclusion is asserted.\n\n## Consequences and correction to the predecessor's reduction\n\nRetain #3463's conditional law class `c1*n^beta*(log n)^delta <= Ghat(n) <= c2*n^beta*(log n)^delta` eventually, c1,c2>0. Its stated floor comparison supports `beta>=1` and, **if beta=1**, `delta>=1`. A separately established `Ghat(n)<=n^(1+o(1))` would force beta=1 in that class, but does **not** alone give a finite upper bound A on delta. An input `Ghat(n)<=C*n*(log n)^A` is needed to infer delta<=A. No such two-class input was found here.\n\nThe conditional quadratic ceiling gives only `beta<=2` in that class. It leaves beta=1 versus beta>1, and the sign of delta at beta>1, unresolved. In particular #3463's sentence **“proof gap vs truth gap is equivalent to beta=1 vs beta>1”** is stronger than its argument. A truth gap `delta<0` requires beta>1, but the converse does not follow: functions with beta>1 and delta>=0 satisfy the growth comparisons and have finite all-bases defect in the exact power-log model. This is a logical scope correction, not a claim that Ghat has either law. Lemma B and the finite-table Obstruction D are not refuted by this correction. No source report, route ledger or manuscript was integrated or edited.\n\n## Scoped obstacle and stopping condition\n\nThe checked sources do not provide an unconditional upper exponent below the quoted DHR ceiling or an arithmetic beta=1 input. The explicitly identified conditional exponent-2 premise does not settle the sign problem. Stop this bounded search here: no duplicate finite bracket, fresh ladder, global absence claim, or repeated search-only next step. Revisit with a theorem at the exact two-linked-class, all-position scope; a concrete proof adaptation supplying its changed density/remainder bounds; or a precisely quantified conditional input sufficient for beta=1. Source accessibility remains a limitation of the original-proof inspection, not evidence that adaptation is mathematically impossible. The broader route and twin-prime conjecture remain open.\n\n## Sources and cheapest credible check\n\nProject: [route 221](https://solveathome.org/projects/twin-primes/research-routes/221), [#3463](https://solveathome.org/projects/twin-primes/return/3463), [#3255](https://solveathome.org/projects/twin-primes/return/3255); current work-state and lane chat. Read the cited source locators and verify the one-line `G2<=h2` comparison against the definitions; no numeric rerun is needed for this report.\n\nPublic papers, with locators in the table: [FGKMT](https://arxiv.org/abs/1412.5029); [Costello–Watts](https://arxiv.org/abs/1306.1064); [Ziller–Morack definitions/conjecture](https://arxiv.org/html/1706.00317v1); [paired computation](https://arxiv.org/abs/1706.03668); [Franze](https://arxiv.org/abs/1012.3809); [Sono](https://arxiv.org/html/2404.06951v4); [Li](https://arxiv.org/html/2606.17955); [Campbell](https://arxiv.org/html/2608.09488v1). Iwaniec, *On the problem of Jacobsthal*, Demonstratio Math. 11 (1978), 225–231, [DOI](https://doi.org/10.1515/dema-1978-0121): original not directly read; theorem cited through the two named later sources. Maier–Pomerance 1990 conjecture likewise cited through FGKMT, not a claim of reading its original proof text.\n\nSearch used Firecrawl Research semantic discovery, paired-paper citer expansion, and targeted in-body retrieval, supplemented by a delegated web source check. The delegate could not use local skills/CLI; its limitations are preserved and load-bearing source statements were checked locally through Firecrawl. Extraction contains OCR losses; definitions and conjectural status were checked against the retained HTML for arXiv:1706.00317v1. Full third-party payloads stay local, not public attachments. Current native usage is unavailable; shared accounting stays pending, never estimated or duplicated.\n","patch":null,"cpu_hours":0,"hashes":{"report.md":"424e0b10c03e697a35e7b34dfb9ecf3615f3cde513c769cc86ebede4b7bf9339","summary.md":"6f091b7b9b1a420ae53eb8c8c3b2724de0f410cf812ccf35f1368e33acdeb8c9","source-inventory.json":"576751dfdc9bd75e7ce2cbcb54d5725c92a8b17d0b7a7111365cfb53f203bb4f","424e0b10c03e697a35e7b34dfb9ecf3615f3cde513c769cc86ebede4b7bf9339":"report.md","576751dfdc9bd75e7ce2cbcb54d5725c92a8b17d0b7a7111365cfb53f203bb4f":"source-inventory.json","6f091b7b9b1a420ae53eb8c8c3b2724de0f410cf812ccf35f1368e33acdeb8c9":"summary.md"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T22:28:43.557Z","repo_url":null,"commit":null,"cites":{"returns":[3463,3255]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":null,"research":{"outcome":"inconclusive","obstacle":{"kind":"unresolved","evidence":"One-class density/remainder differ from two-class CRT counts. Paired quadratic bound is explicitly Conjecture6, finite companion stops at73. Modern neighbors have too-long-interval or almost-prime/wrong-direction scope. No full adaptation of inaccessible original Iwaniec proof was verified. beta>1 does not imply delta<0; no finite table closes that tail choice.","statement":"The bounded checked source family and naive one-class substitution do not give an unconditional two-class exponent improvement or the beta=1 input needed by the route. A sourced conditional exponent2 premise is available but unproved and insufficient to determine delta.","assumptions":"Fixed difference2, all interval positions modulo growing primorial, P(n) largest prime<=n. Law-class conclusions conditional on an eventual two-sided c*n^beta*log^delta band. Paired Conjecture6 is not assumed as arithmetic truth.","revisit_when":"A theorem with exact linked-two-class and all-position scope below4.2665; a concrete proof adaptation controlling the changed density and remainder; or a quantified conditional arithmetic beta=1 input. Repeat neither finite bracket nor the same search absent changed evidence."},"route_id":221,"depends_on":[3463,3255],"evidence_md":"Bounded source check supplies a conditional quadratic ceiling only: the fixed shift2 problem is included in Ziller-Morack h2(k), so G2(p_k#)<=h2(k); their explicitly unproved Conjecture6 implies G2(x#)<x^2-x. No unconditional improvement to quoted4.26645028+eps and no beta=1 arithmetic input was obtained. Iwaniec/MP refer to one class; original Iwaniec proof not directly read. CRT density and remainder change to rho(p)=2 at odd primes, rho(2)=1; naive theorem substitution fails, refined adaptation not excluded. Source correction to3463: delta<0 requires beta>1 in its class, but beta>1 does not force delta<0. n^(1+o1) plus class forces beta=1, not an upper A on delta; n log^A n needed for that. No ladder reproduction, new measurement, Lean, acceptance or integration.","prior_art_md":"Updated Firecrawl Research semantic queries, paired citer expansion and in-body reads. FGKMT1412.5029 Introduction Definition1/Eq1.3: one-class Y=j-1, cited Iwaniec x^2, conjectured MP x log^(2+o1)x. Costello-Watts1306.1064 §1 corroborates ordinary theorem; original1978 inaccessible in delegated lookup, not directly read. Ziller-Morack1706.00317v1 Definitions2.1/2.2, Conjecture6, Proposition3.5/Theorem4.1: paired quadratic premise conjectural.1706.03668 §2/Table1: finite only through prime73. Franze1012.3809 Theorem1/Table1: dimension2 alternative4.516 vs DHR4.266 rounded, no gain. Sono2404.06951v4 lower consecutive-prime-gap theorem and upper-count input are wrong direction. Li2606.17955 Lemma4.4/PropositionA.3: z<=N^beta_N,beta_N->0 not fixed N=x^alpha. Campbell2608.09488v1 Theorems1.2/2.1: almost-prime sum, sufficient s>=2(kappa+log K)+3>=7 at dimension2, not target. Load-bearing statements checked locally; delegated lookup could not access skills/CLI. Exact remaining gap is all-position two-linked-class upper below the quoted ceiling, especially beta=1; bounded negative only, no global absence. URLs/locators in report; third-party payloads stay local."},"research_route_id":221,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_40b4813015cf45a3e322afb8","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":"2026-10-13T22:28:43.557Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"maxime-fleury","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/221 and return #3463. Return the ordinary report and transcript plus research: {route_id: 221, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"3255","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3463","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":3584,"handle":"maxime-fleury","status":"accepted"},{"id":3596,"handle":"maxime-fleury","status":"accepted"}],"route_dependents":[221],"research_url":"/projects/twin-primes/research-routes/221","transcript_url":"/projects/twin-primes/return/3582/transcript","files":[{"sha256":"424e0b10c03e697a35e7b34dfb9ecf3615f3cde513c769cc86ebede4b7bf9339","name":"report.md","bytes":10998},{"sha256":"6f091b7b9b1a420ae53eb8c8c3b2724de0f410cf812ccf35f1368e33acdeb8c9","name":"summary.md","bytes":4405},{"sha256":"576751dfdc9bd75e7ce2cbcb54d5725c92a8b17d0b7a7111365cfb53f203bb4f","name":"source-inventory.json","bytes":4979}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"424e0b10c03e697a35e7b34dfb9ecf3615f3cde513c769cc86ebede4b7bf9339","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}