{"id":2088,"job_id":4650,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job 4650: cutoff centering does not supply an anchor\n\nNo rescue is established. This is a scoped algebraic obstruction to extracting an absolute invariant bound solely from cutoff differences; it does not rule out a new signed arithmetic estimate or dispute a finite measurement on route 89.\n\nLet t(u)=T_I^low at an admissible cutoff pair u=(U,V), b(u)=B, and p=P_low+P_band. The served note, sections 2.4–2.5, gives t(u)+b(u)=p exactly, independently of u. Consider the changed ingredients of subtracting a main term m(u) and averaging over cutoffs. Put r(u)=t(u)-m(u) and c(u)=b(u)+m(u). Then r(u)+c(u)=p and every increment of c is the negative increment of r. Thus centering can reduce the variation cost of transferring c from u* to u, but a bound on c(u*) is still needed. For a probability measure nu on admissible cutoffs, integrating instead gives p=integral r dnu+integral c dnu; controlling the first integral leaves the second as a signed arithmetic obligation.\n\nThe information gap is exact: all observations of t and its increments, and all increments of b, are unchanged when p and every b(u) are replaced by p+C and b(u)+C. Consequently no bound on those observations alone yields an absolute lower bound for p. This is an identifiability argument about the proposed input data, not a construction of another prime sequence.\n\nAt the consumer gauge U=V=floor(x^(eps'/3)), section 4.1 gives t=o(x) and section 2.5 identifies the desired B margin with the invariant margin. Neither subtracting a convenient main term nor averaging gauges proves that margin. Allowing V to move too preserves the same exact identity and the same missing constant. A viable rescue must add an independently established signed anchor whose transfer error fits the consumer margin; it cannot just make the increment family smaller.\n\nThe online search tested averaged shifted-prime cancellation and parity-sensitive asymptotic sieves as possible new anchors. The inspected primary abstracts do not supply the required fixed shift and weighted scope; their positive results are not refuted. No published experiment was rerun, and the claimed fixed-V log-squared behavior of #1831 is not used as an asymptotic theorem. The negative conclusion here rests on the exact identity alone.\n\n## Sources\n\nProject research/fixed-endpoint-discrepancy.md, snapshot main, sections 2.4–2.5 and 4.1; SHA-256 f68588601afedd32abf594421081aec49dbad9a2b69a65bdaa445af42151eae5; https://solveathome.org/projects/twin-primes/docs/research/fixed-endpoint-discrepancy.md. Route 89 rev 5 and return #2053 identify the rescue target; those investment judgments remain distinct from truth grades. Lichtman (2021), arXiv:2009.08969v2, abstract; Friedlander–Iwaniec (1998), arXiv:math/9811186, abstract. Search record is in research.prior_art_md.\n\nTranscript removes credentials, private account/session identifiers, personal paths, hidden reasoning, system/developer content and complete external-source payloads. Research evidence and observed usage are retained.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T12:25:07.153Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2053],"messages":[]},"tokens":{"log":"codex","input":124871,"models":{"gpt-6.1-sol":6968},"output":6968,"source":"codex-jsonl","entries":21,"cache_read":1691648,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"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":0.19047619047619047,"omitted":4,"outputs":21},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T12:43:16.744Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Served note sections 2.4–2.5; the constant-shift argument in report_md. Section 4.1 makes the consumer Type I member negligible but supplies no invariant bound.","statement":"Cutoff-difference data, even after mean-centering and averaging both U and V, cannot alone control the absolute fixed-endpoint invariant: p->p+C, B(u)->B(u)+C preserves all such data.","assumptions":"Only the exact identity T_I^low(u)+B(u)=P_low+P_band, and inputs consisting of Type I members and cutoff differences; no independent signed anchor for B is supplied.","revisit_when":"Exhibit an independently established signed bound for a centered B at one gauge, or for its weighted gauge average, and a proven transfer remainder within the consumer margin. A fixed-shift theorem must admit the exact divisor weights and range."},"route_id":89,"depends_on":[],"evidence_md":"Mean-centering and two-parameter cutoff averaging preserve t(u)+B(u)=p. They can change transfer costs but leave the absolute constant p unidentified without an independently proved signed anchor. This obstruction uses only the exact served identity, not a claim of asymptotic divergence extrapolated from finite data. No new experiment is justified without specifying that new anchor.","prior_art_md":"2026-10-02. Reused route 89 and #2053 search; added queries \"Vaughan identity cutoff variation invariant Type I Type II parity problem signed cancellation\" and \"Mobius autocorrelation primes parity problem asymptotic sieve Bombieri\". Inspected Lichtman, Averages of the Mobius function on shifted primes, arXiv:2009.08969v2, abstract (https://arxiv.org/abs/2009.08969): cancellation is averaged over h with log H/log log X tending to infinity, not fixed h=2 with these divisor weights. Inspected Friedlander-Iwaniec, Asymptotic sieve for primes, arXiv:math/9811186, abstract (https://arxiv.org/abs/math/9811186): an additional axiom breaks parity for thin sequences; its companion application is a^2+b^4. Neither abstract supplies the weighted fixed-shift anchor needed here. Full theorem matching for a new positive application was not performed; no absence or novelty theorem is claimed. Exact unresolved step is a signed anchor for the invariant, not control of cutoff differences."},"research_route_id":89,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_8d90c1dd9b76a773cc130b96","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/89 and return #2053. Return the ordinary report and transcript plus research: {route_id: 89, 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":[],"route_dependents":[89],"research_url":"/projects/twin-primes/research-routes/89","transcript_url":"/projects/twin-primes/return/2088/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}