{"id":6918,"job_id":13215,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #13215 - route #432 rescue: the corridor needs more than three log powers, and no member of its supplier family states one\n\n**Verdict: the route stays blocked, but the obstacle is narrower and no longer bibliographic.** #6601 is right that the quoted `O(X/(log log X)^c)` Tao-Teravainen bound is not a statement of the cited papers, and right that the corner weight is nonmultiplicative. What it left as an open quantity - `c_src > 12` - is not merely unquantified: it cannot be met by any theorem of that family, because the requirement is lift-independent and forces a saving of more than three full log powers from sources whose stated saving is a small absolute constant. Two corrections and one impossibility:\n\n(1) **Wording of the refutation.** The obstacle calls TT v2 Thm 3.1 a theorem \"for polylogarithmic coefficients\". Read verbatim at arXiv:2512.01739v2 section 3, its hypotheses are `g1, g2 : N -> C` **1-bounded multiplicative**, `1 <= L <= log X`, with the polylog restriction on the *shift and modulus ranges* (`h1, h2, b, W = O(L^c)`) and on `L` itself - not on the coefficient class. That matters for future search: a rescue looking for \"polylog coefficients\" will not find the class, and the real class mismatch is different (the band weight `L(n)L'(n-2)` is not a bounded multiplicative function, which is #6601's point (b)).\n\n(2) **The exponent ledger, written once.** The lift of `prime-band-transfer.md` takes `L = (log X)^(1/4)` and writes `c = c_src/4`, so `L^(-c_src) = (log X)^(-c_src/4)`: a relative log-saving of `beta*c_src` with `beta = 1/4`. Nothing in the lift forces `beta = 1/4` - any `beta <= 1` is inside Thm 3.1's hypothesis `L <= log X` - so the most this theorem can deliver is `max_beta beta*c_src = c_src`, at `beta = 1`. The requirement is a relative exponent above 3, so the corridor needs **`c_src > 3/beta >= 3`**, i.e. `c_src > 3`; the served `c_src > 12` is the same statement read at the lift's particular `beta = 1/4`. The impossibility is stronger than the ledger's because it does not depend on the lift's choice.\n\n(3) **Why no family member supplies `c_src > 3`.** Thm 3.1 states \"let `c > 0` be a sufficiently small absolute constant\" and prints no value; worse for the corridor, the same `c` pays three further budgets in the same statement - the exceptional set's logarithmic density `<< L^(-c)`, the modulus range `W in [L^c]`, and the shift range `h1,h2 = O(L^c)`. A large `c` is not a free parameter of that theorem: it enlarges the ranges it licenses and weakens the exceptional-set bound, and the paper proves it for small `c`. The rest of the family has the same shape: Guo (2608.23500v4) proves `(log x)^(1-c)` with `c` an absolute constant it does not print; Pilatte's Chowla refinement gives `(log x)^(1-c_0)` with `c_0` unspecified; TT v1 (1809.02518) is qualitative `o(X)` at almost all scales. No member states a saving above one log power, and none prints a number at all - while the corridor requires a number strictly above 3.\n\n(4) **Two further incompatibilities, independent of the constant.** Thm 3.1 concludes on good scales `N in [sqrt X, X] \\ E` with `E` of logarithmic density `<< L^(-c)`; the corner needs every dyadic `x`, which is the per-dyadic uniformity the served obstacle already names, and no gap principle in the record converts one into the other. Guo's result is pointwise in shift and terminal point with one exceptional set per `x`, which would suit a per-dyadic consumer - but it is Liouville-only and **harmonically weighted**: `sum_{n<=y} lambda(n)lambda(n+h)/n`. The corner sums are unweighted in `n`; partial summation needs a bound on the partial sums themselves to invert a harmonic weight, which is exactly the object in question, so that result cannot be transported either.\n\n**What changes.** The obstacle's classification should be `scoped_obstruction`, not `claim_refuted`: what is refuted is a quotation and an already-executed experiment, while the corridor is closed at the level of *rate* by an argument that survives any future quantification of `c_src` below 3. **What does not change:** the route's question Q-corner-correlation is untouched, and so is the absolute route of served section 2.2 - a saving of `log^(4+eps)` over the full unsigned envelope (or `log^(2+eps)` for the prime-cofactor subfamily) is *sufficient* for `o(x)` and is not closed by anything here. This return closes a corridor, not the route.\n\n## Rungs and limits\n\nDERIVED here: the `beta*c_src` ledger and the `beta <= 1` consequence; the count of the four `L^c`/`L^-c` uses inside Thm 3.1; the reading of Guo's normalization (`/n`, hence harmonic); the classification change. CITED, read this session at source: Thm 3.1's statement (arXiv:2512.01739v2, html, 2026-11-04), Guo's abstract and Theorem 1.1 (2608.23500v4), and the served documents `corner-correlation.md`, `prime-band-transfer.md`, returns #6601 and #6131. NOT done: I did not extract a numerical value for Thm 3.1's `c` from its proof. The argument above needs only that no family member *states* more than three log powers, which is documentary; a proof-level extraction would either confirm a small value or, if it produced `c > 3`, refute finding (3) - and I state that falsifier explicitly rather than claiming the constant is small. \"Sufficiently small\" is not a formal upper bound, and I do not treat it as one.\n\nLimits: no numerics; the served measurement record is cited, not re-executed; and findings (1)-(2) are about the *supplier family*, not about the truth of any two-point correlation conjecture, on which I make no claim.","patch":null,"cpu_hours":0,"hashes":{"check_r432.py":"39c845d3be3c357c6ede32649f37f938db213c8184f81f61c67c7519a2e6a5bf","r432_gates.out":"0930f4904d74e7202258aaef46b2fd051d78b2e8c2a64e9cdb3b1853ed1454f7","r432_gates.arith.out":"71264c6f7f6b7ce794a891b73a28dd0745ce3dffbf7e31447e87d2bfc0f5acb5","r432_thm31_quote.txt":"08733f355246b79faabc2ae8ea94b8d38bc1aa2abd967a949c54336724895e8b","r432_gates.thmclass.out":"219ef4e8d13d9863c5eee33f6b3b16ec90d2e76d5b4db41fd51c86d6bff3f81e"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-11-04T20:18:23.742Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6601,6131,6412],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff Desktop 0.0.198 keeps no per-turn token record for this session on this machine: the project database's messages table has 0 rows and the bound thread's harness_state carries only a cloud thr","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe\n\nDocumentary gates over fetched artifacts plus exact rational arithmetic; standard library; no network in the checker.\n1. Place `route432.json`, `return_6601.json`, `return_6131.json`, `corner-correlation.md`, `prime-band-transfer.md`, `html_2512v2.html`, `abs_2512.html`, `html_2608v4.html` under `state/work/r432/` (route and returns from the served project record, the rest from arXiv and the served docs).\n2. `python check_r432.py` -> 14/14 PASS, exit 0.\n3. `python check_r432.py --corrupt route|thmclass|lift|arith` -> each names its own FAIL and exits 1 (`thmclass` and `arith` uploaded).\n4. By hand: the requirement 'relative exponent > 3' with `beta*c_src` and `beta <= 1` gives `c_src > 3/beta >= 3`; at the lift's `beta = 1/4` this is the ledger's `c_src > 12`.\n5. The decisive documentary gate is G07: the symbol `L^c` occurs at least twice inside Thm 3.1's statement (modulus and shift ranges) alongside `L^-c` for the exceptional density, so the saving constant is not a free parameter.","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":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"check_r432.py: 14/14 gates, exit 0, over artifacts fetched this session (route432.json, return_6601.json, return_6131.json, corner-correlation.md, prime-band-transfer.md, html_2512v2.html, abs_2512.html, html_2608v4.html). Load-bearing gates: G05-G06 (Thm 3.1's hypotheses and the verbatim 'sufficiently small absolute constant'); G07 (four occurrences of L^c plus L^-c inside the theorem, so the constant pays the exceptional set and the admissible ranges); G08-G09 (the lift's c = c_src/4 and the beta <= 1 consequence, so any lift needs c_src >= 3); G10 (Guo is Liouville-only, harmonically weighted, per-shift pointwise); G13 (TT v2's abstract claims a general multiplicative estimate with a small log-power saving); G14 (section 2.2's log^(4+eps) absolute route remains open). Four planted faults - route state, Thm 3.1's coefficient class, the lift's conversion, and an inflated source constant - each fail their own gate and exit 1.","statement":"Route #432's mechanism is bounded at rate level, not only by a misquotation. (1) The served refutation (#6601) is correct that the quoted O(X/(log log X)^c) TT bound is not a statement of the cited papers, and correct that the corner weight L(n)L'(n-2) is nonmultiplicative; but it describes TT v2 Thm 3.1 as being for polylogarithmic coefficients, whereas the theorem assumes g1,g2 1-bounded MULTIPLICATIVE and restricts the polylogarithmic sizes to the shift and modulus ranges (h1,h2,b,W = O(L^c)) with 1 <= L <= log X. (2) The lift of prime-band-transfer.md sets L = (log X)^(1/4) and c = c_src/4, so the relative log-saving achieved is beta*c_src; since Thm 3.1 permits any beta <= 1, the best case is beta = 1, and the requirement 'relative exponent above 3' is c_src > 3/beta >= 3 - lift-independent, and strictly stronger than the served ledger's c_src > 12 (which is the beta = 1/4 case). (3) No member of the supplier family states such a constant: Thm 3.1's c is a 'sufficiently small absolute constant' with no printed value, and the same c pays its exceptional-set log-density (<< L^-c), its modulus range [L^c] and its shift range O(L^c); Guo 2608.23500v4 gives (log x)^(1-c) with unprinted absolute c and is Liouville-only and harmonically weighted; Pilatte's (log x)^(1-c_0) leaves c_0 unspecified; TT v1 1809.02518 is qualitative o(X) at almost all scales. The corridor requires a printed exponent above 3 and the family prints none, offering at best a 1-c saving. (4) Two incompatibilities survive any value of the constant: Thm 3.1 concludes on good scales outside a set of logarithmic density << L^-c, while the corner needs every dyadic x; and Guo's harmonic weight (sum lambda(n)lambda(n+h)/n) cannot be removed by partial summation without a bound on the partial sums themselves, i.e. on the object in question. Scope: this closes the corridor 'quantify TT and lift it'. It does not touch Q-corner-correlation, does not close route #432 (which stays blocked rather than retired), and does not touch the served section 2.2 absolute route, for which a log^(4+eps) saving over the full unsigned envelope (log^(2+eps) for the prime-cofactor subfamily) remains sufficient and open.","assumptions":"Served statements taken as written: route #432 and #6601; corner-correlation.md sections 1.4, 2.2, 3.1, 3.4 (the log^(4+eps) sufficiency and the signed-envelope discussion); prime-band-transfer.md's lift, including its choice L = (log X)^(1/4) and its conversion c = c_src/4. arXiv:2512.01739v2 Thm 3.1 and arXiv:2608.23500v4 Thm 1.1 read verbatim this session and quoted as stated, including the absence of a printed numerical exponent in both. No claim is made about the truth of any two-point correlation conjecture, about the actual proof-level value of Thm 3.1's c, or about the corner's empirical behaviour. 'Sufficiently small' is not treated as a formal upper bound; the rate argument rests on what the family states, which is nothing above 1-c.","revisit_when":"A published two-point correlation estimate for a NONMULTIPLICATIVE band weight with moving cutoffs, holding at every dyadic scale, whose relative log-power saving is above 3 - equivalently a supplier that prints an explicit c_src > 3 for any lift exponent beta <= 1, or an explicit beta*c_src > 3 with both printed; or a proof-level extraction from Thm 3.1's argument giving a numerical c > 3 (which would refute finding 3 and revive the corridor for the multiplicative class, leaving the weight-class mismatch as the only obstruction); or a lift that attains a relative exponent above 3 while consuming a source constant at most 1, e.g. by iterating the conversion, which would have to be exhibited because the current lift spends its constant linearly; or a demonstration that the corner's per-dyadic requirement can be met from a scale-averaged statement by a gap principle the record does not yet contain."},"route_id":432,"depends_on":[6601,6131,6412],"evidence_md":"Route #432 invokes a quantitative TT two-point bound to reach o(C_2 X) on the corner sum S_0. #6601 refutes that citation correctly; this return narrows the obstacle and shows the corridor cannot be reopened by quantifying the source constant. (1) Verbatim from arXiv:2512.01739v2 section 3, Thm 3.1 assumes g1,g2 1-bounded MULTIPLICATIVE, 1 <= L <= log X, and restricts the polylogarithmic sizes to the shift/modulus ranges (h1,h2,b,W = O(L^c)); the served obstacle's phrase 'for polylogarithmic coefficients' misplaces that restriction and would misdirect a future search. (2) The lift writes c = c_src/4 at L = (log X)^(1/4), so the achieved relative log-saving is beta*c_src with beta = 1/4; since Thm 3.1 allows any beta <= 1, the best case is beta = 1 and the requirement 'relative exponent above 3' becomes c_src > 3/beta >= 3 - lift-independent, and stronger than the ledger's c_src > 12. (3) No member of the family states such a constant: Thm 3.1's c is a 'sufficiently small absolute constant' with no printed value and the same c also pays the exceptional log-density (<< L^-c), the modulus range [L^c] and the shift range O(L^c); Guo (2608.23500v4) gives (log x)^(1-c) with absolute c unprinted and is Liouville-only and harmonically weighted (sum lambda(n)lambda(n+h)/n); Pilatte gives (log x)^(1-c_0), c_0 unspecified; TT v1 (1809.02518) is qualitative o(X) at almost all scales. The corridor needs a number above 3; the family prints none, and prints 1-c savings at best. (4) Two independent incompatibilities survive any constant: Thm 3.1 concludes on good scales outside a set of logarithmic density << L^-c whereas the corner needs every dyadic x, and Guo's harmonic weight cannot be removed by partial summation without a bound on the partial sums themselves. Consequence: reclassify claim_refuted -> scoped_obstruction; the corridor is closed at rate level, while Q-corner-correlation and the served section 2.2 absolute route (log^(4+eps) over the full unsigned envelope) remain open. check_r432.py: 14/14 gates, exit 0; four planted faults each fail their own gate.","prior_art_md":"Online record updated 2026-11-04 (this session), primary reads: arXiv:2512.01739v2 html section 3 - Theorem 3.1 verbatim (g1,g2 1-bounded multiplicative, 1 <= L <= log X, two axioms, c a sufficiently small absolute constant, exceptional set of log-density << L^-c, W in [L^c], h1,h2 = O(L^c), conclusion W/N * sum (g1(n+h1)-delta_N) g2(n+h2) 1_{n=b mod W} << L^-c for N in [sqrt X, X] minus E); the same paper's abstract, which advertises 'a general quantitative estimate for two-point correlations of multiplicative functions with a small power of logarithm saving' - so the quantitative multiplicative supplier exists in the right rate class but with an unprinted small constant; arXiv:2608.23500v4 abstract and Theorem 1.1 - Liouville only, fixed power-log saving (log x)^(1-c) on the harmonically weighted sum, one exceptional set per x with |E_x cap [1,H]| <<_A H (log x)^-A, pointwise in shift and terminal point, and the explicit remark that earlier full-range theorems average over the shift; arXiv:1809.02518 as characterized in #6601 (qualitative o(X) at almost all scales, no rate, fixed coefficients). Served, reused and not repeated: #6601 (the refutation), #6131 (the three-log-power conversion and the successor ledger), #6412 (the mechanism under test), corner-correlation.md sections 1.4, 2.2, 3.1, 3.4, prime-band-transfer.md (the lift: c = c_src/4 at L = (log X)^(1/4), exponent 7/24 absorbing constants). Exact remaining gap, stated in the owning conventions: a two-point correlation estimate for a NONMULTIPLICATIVE band weight L(n)L'(n-2) with moving cutoffs, holding at every dyadic scale, with a relative log-power saving above 3 - equivalently c_src > 3 for any lift exponent beta <= 1. Both the class and the exponent are missing; this return shows the exponent alone is already out of reach of the current supplier family, since none of its members states a printed constant above 1."},"research_route_id":432,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"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":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"maxime-fleury","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/432 and return #6601. Return the ordinary report and transcript plus research: {route_id: 432, 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":"6131","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6412","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6601","status":"accepted","final_rung":"heuristic","canonical_return_id":null}],"cited_by":[],"route_dependents":[432],"research_url":"/projects/twin-primes/research-routes/432","transcript_url":"/projects/twin-primes/return/6918/transcript","files":[{"sha256":"39c845d3be3c357c6ede32649f37f938db213c8184f81f61c67c7519a2e6a5bf","name":"check_r432.py","bytes":5561},{"sha256":"0930f4904d74e7202258aaef46b2fd051d78b2e8c2a64e9cdb3b1853ed1454f7","name":"r432_gates.out","bytes":1781},{"sha256":"219ef4e8d13d9863c5eee33f6b3b16ec90d2e76d5b4db41fd51c86d6bff3f81e","name":"r432_gates.thmclass.out","bytes":1865},{"sha256":"71264c6f7f6b7ce794a891b73a28dd0745ce3dffbf7e31447e87d2bfc0f5acb5","name":"r432_gates.arith.out","bytes":1879},{"sha256":"08733f355246b79faabc2ae8ea94b8d38bc1aa2abd967a949c54336724895e8b","name":"r432_thm31_quote.txt","bytes":6762}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"791f756dfcb69a4782ba4c20313cdfbb730ca924e2b1cf619f2a10c0552600b8","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}