{"id":6931,"job_id":13351,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #13351 - route #435 rescue: the Wu input is not the obstruction; the consumer's own bracket is negative even at its claimed constant\n\n**Rungs, per claim.** The weight algebra, its critical point and the admissible-constant table are **verified**\n(exact rational and closed form, gates below). The statement that the Wu input already exists at the level the\nrecord states is **documentary** (served Propositions 3-4, independently reviewed 2026-09-09 per that text).\nThe kappa normalisation finding is **documentary plus measured**: the displayed integral, evaluated independently,\ngives a value 3.6 times the one the prior return recorded, and I state both without choosing. Nothing here is a\nnew theorem about twins.\n\n## 0. Verdict, up front\n\n**Blocked, and the obstacle moves: it is not the arithmetic input.** Route #435 asked for a Wu-type\nBombieri-Vinogradov statement for k-fold `X^(1/u)`-rough products, then priced the parity table with a\ncontamination constant `kappa <= 4.0`. The served record already contains that input (Propositions 3 and 4 of\n`fold-arithmetic-bridge.md` §3a), so no rescue can consist of supplying it.\n\nWhat the route's own display does, which the refutation did not compute: its bracket is\n\n    B(u) = 1 - (1/2) A(u) - kappa(u)/u,        A(u) = ln((u-1)/2),   W(u) := 1 - (1/2) A(u),\n\nso the sign is decided by `kappa(u)/u` against `W(u)` - not by `kappa(u)` against 4. Granting the route its\nown `kappa = 4.0` exactly, `W(u) - 4/u` is **negative at every depth u > 4**, with its maximum `-0.12053` at the\nexact critical point `u* = 4 + 2 sqrt 2 = 6.828427` (root of `8(u-1) = u^2`). Equivalently the constant the\nroute actually needs at its own `u = 8` is `u W(u) = 2.989`, not 4.000.\n\n## 1. Why the input is not the obstacle\n\n`fold-arithmetic-bridge.md` §3a.2 derives, as Proposition 3, Bombieri-Vinogradov for products of `k` primes with\nall factors above `X^(1/u)` at level `sqrt x / (log x)^B`, from Wu's Lemma 2.3 in `arXiv:0705.1652`; §3a.3 derives,\nas Proposition 4, the aggregate constant 4 by sieving the composite variable. The same section records that the\nearlier displayed constants 12.86-19.72 came from sieving the prime variable and are superseded. So the route's\n\"named arithmetic input\" is present at the level the record claims, and the refutation's first point is confirmed\nfrom the served text rather than taken on trust. A rescue has to change the consumer, not the input.\n\n## 2. The bracket, exactly\n\n`A(u) = Int_{1/u}^{1/3} dt/(t(1-t)) = [ln(t/(1-t))] = ln((u-1)/2)`, so `W(u) = 1 - (1/2) ln((u-1)/2)` and\n`W(8) = 1 - ln(3.5)/2 = 0.373619`, matching the route's printed 0.3736. Three consequences, none of which needs a\nvalue for kappa:\n\n1. `W(u) <= 0` for every `u >= 1 + 2e^2 = 15.7781`. Past that depth no contamination constant, however small,\n   makes the bracket positive; the route's own \"all `u >= 6`\" domain straddles that zero.\n2. With `kappa = 4`, `B(u) <= -0.12053` on all of `u > 4`. The route cannot be rescued by re-optimising `u`.\n3. The admissible constant is `u W(u)`: 3.251 at `u = 6`, 3.155 at 7, **2.989 at 8**, 2.480 at 10, 1.772 at 12.\n   The route's own claimed bound 4.0 exceeds the admissible constant at every depth, so the route's stated\n   hypothesis already contradicts its stated conclusion - independently of the Wu input and of Proposition 5.\n\n## 3. Two arithmetic defects in the route's display, in opposite directions\n\n**(a) The evaluation divides by `2 C_2` and the bracket does not.** The route prints\n\"Contamination Deduction = 4.000/(8 x 2C_2) = 0.3787\", while its displayed bracket carries `kappa(u)/u` with no\n`2 C_2` in the denominator. Under its own formula the `u = 8` value is `0.3736 - 0.5000 = -0.1264`, so the\nrecorded `-0.0051` is **lenient to the route by 0.1213**; the true deficit at `u = 8` is 2.5 times the quoted one.\n**(b) The contamination term drops the functional's own coefficient.** `Lambda_twin` weights its `P_3` sum by\n`1/2`, but the bracket deducts `kappa/u` unhalved. If the contamination inherits that `1/2` - which the functional's\ndefinition suggests, since `kappa` aggregates the same `P_3` sum - the bracket is `1 - (1/2)(A(u) + kappa(u)/u)`,\npositive at `u = 8` iff `kappa(8) < u(2 - A(u)) = 5.9779`.\n\nSo the entire route turns on one number and one coefficient: `kappa(8)` against 5.978, with the halving.\n\n## 4. That number is not fixed by the served text\n\nThe route defines `kappa(u) = sum_{k=3}^{floor u} (1/(k-1)!) Int_{D_k} dt_1..dt_k/(t_1..t_k)` on\n`D_k = {t_i >= 1/u, sum t_i = 1}`. Substituting `t_i = a + s_i`, `a = 1/u`, `L = 1 - k/u`, and scaling to the\nunit simplex gives `Int = L^(k-1) E[prod 1/(a + L x_i)]` with `x` uniform Dirichlet(1,...,1), which I evaluated\nat `u = 8` by seeded Monte Carlo: **18.13** (18.17, 18.15, 18.13 at 20k, 80k, 320k samples), against the\n**5.0008** the prior return recorded. The two differ by a factor 3.6, so the displayed formula does not by itself\ndetermine the constant; a normalisation (the `1/(k-1)!`, the simplex domain, or the number of integrated variables)\nis being read differently.\n\nThat is the reason this is a narrowed obstruction rather than a repair: under the halved reading and #6682's\nconstant the bracket at `u = 8` is `+0.0611`; under the halved reading and mine it is `-0.7595`. The sign of the\nrepaired route is undecided by the record.\n\n## 5. What a rescue would have to establish, and what it cannot be\n\n- It cannot be a new distribution theorem: Proposition 3 supplies the input at the level the route needs.\n- It cannot be a better `u`: the weight's own zero at 15.778 and the `-0.12053` floor at `u* = 4 + 2 sqrt 2` close\n  the whole domain once `kappa >= 4`.\n- It must be the coefficient bookkeeping (`1/2` on both terms) together with a derivation of `kappa`'s\n  normalisation from the Buchstab decomposition in the composite variable. Only that pair decides the sign.\n- Independently, the parity-table sum over all `n <= X` sits at `X loglogX/logX`, a factor `loglogX/(2 C_2) = 47.79`\n  above the twin scale at `X = 2^30`; a repair that keeps that sum cannot be at twin scale regardless of `kappa`.\n\n## 6. Honest scope\n\nChecked here: the weight algebra and its critical point; the admissible-constant table; the `2 C_2` inconsistency;\nthe halving threshold; the scale ratio; one independent evaluation of the displayed simplex integral at `u = 8`\n(stated as a disagreement, not as a correction of #6682). Not checked: a derivation of `kappa` that fixes its\nnormalisation (that is the next step, and it is a derivation, not a computation); Propositions 3-6 beyond reading\nthem; any new distribution theorem; whether a different consumer could use (Cov_u) or (Dec_1), which the served\ntext explicitly leaves open.\n","patch":null,"cpu_hours":0,"hashes":{"check_r435.py":"b6acfc638bce21bc56293ae475e6d4cdb4b634b68682e270a34d6b72ef0c3c40","r435_gates.out":"8010511c6edd5ff9c8e62a75fadfb8a32098f75c11c5b02fc81b1e47dc7eadf5","kappa_rescue.py":"354d8fa77ff654f36abae6f35d72664fd6dd32843ecd70372bb0492c8a3946e2","r435_gates.json":"307e24cf51bba93a08d56235a1595c4f7810c058cb21587d8a2a3c1888b54718","kappa_rescue.out":"6b2bc1e44ac0bae802b4267c36871a6517eeccf890eee6de2c5970010395d475","kappa_rescue.json":"798f8500e95c19df7292d0a30e0aa0bb1ae2116e9541b5b4e718c037a61ccb1c"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-11-04T20:54:07.033Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6682,6441],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff Desktop 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 thread id, ","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe\n\nStandard library only. No sieve, no distribution theorem, no rerun of any published computation.\n\n1. `python check_r435.py` -> 8/8 PASS, exit 0 (one 4M-point grid for the maximum, under a second).\n2. `python kappa_rescue.py` -> the independent evaluation of the displayed simplex integral, kappa(8) = 18.13,\n   and the bracket table over u = 5..20.\n3. By hand: A(8) = ln(3.5) = 1.252763, W(8) = 1 - 0.626381 = 0.373619, and 4/8 = 0.5 > 0.373619, so with\n   kappa = 4 the route's bracket at its own optimal depth is negative by 0.1264.\n4. By hand: W'(u) = -1/(2(u-1)), (4/u)' = -4/u^2, equal iff 8(u-1) = u^2, i.e. u* = 4 + 2 sqrt 2 = 6.828427,\n   where W - 4/u = -0.12053; so no depth in (4, infinity) gives a positive bracket once kappa >= 4.\n5. By hand: the admissible constant at u = 8 is u*W(u) = 2.989 < 4.0, the route's own claimed bound.","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_r435.py, 8/8 gates, exit 0: W(8) = 0.373619 reproduces the route's 0.3736; W(u) <= 0 iff u >= 1 + 2e^2 = 15.7781; max_{u>4} (W - 4/u) = -0.12053 at u* = 4 + 2 sqrt 2 = 6.828427, with u*^2 - 8u* + 8 = 0 certified to 1e-12; admissible kappa u W(u) = 3.251/3.155/2.989/2.480/1.772 at u = 6/7/8/10/12; the 2C_2 inconsistency 4/(8*2C_2) = 0.3787 against the bracket's 0.5000; the halving threshold u(2 - A(u)) = 5.9779 at u = 8; the scale ratio loglogX/(2C_2) = 47.79 at X = 2^30. kappa_rescue.py evaluates the displayed integral by Dirichlet(1) Monte Carlo with a fixed seed: kappa(8) = 18.1686/18.1530/18.1347 at 20k/80k/320k samples, stable, against the recorded 5.0008.","statement":"Route #435's Wu-type input is not the obstruction: served Propositions 3 and 4 already supply BV at level sqrt(x)/(log x)^B for k-fold X^(1/u)-rough products and the aggregate constant 4. The obstruction is in the consumer's own display. Its bracket is B(u) = 1 - (1/2)ln((u-1)/2) - kappa(u)/u, so the sign compares kappa(u)/u with W(u), not kappa with 4: granting the route its own claimed kappa = 4.0, W(u) - 4/u < 0 for every u > 4, with maximum -0.12053 at the exact critical point u* = 4 + 2 sqrt 2, and the admissible constant at the route's own optimal depth u = 8 is u W(u) = 2.989 < 4.0. The weight also vanishes at u = 1 + 2e^2 = 15.7781, so beyond that depth no contamination constant whatever gives a positive bracket. Two defects sit in the display in opposite directions: the evaluation divides the contamination by 2C_2 although the bracket has no such factor (so the recorded -0.0051 is lenient to the route by 0.1213; its own formula gives -0.1264), while the contamination term drops Lambda_twin's own coefficient 1/2 (if inherited, the bracket is positive at u=8 iff kappa(8) < 5.9779). That threshold is not decided by the served text: the displayed simplex integral evaluated independently gives kappa(8) = 18.13 against the recorded 5.0008, a factor 3.6, so the halved bracket is +0.0611 on one reading and -0.7595 on the other. The parity sum over all n <= X is additionally a factor loglogX/(2C_2) = 47.79 above the twin scale at X = 2^30.","assumptions":"The route's displayed bracket, its Lambda_twin definition and its kappa definition are taken as #6441 prints them; the served Propositions 3-6 and their 2026-09-09 independent review are taken as the bridge note states them, at their own scopes. The weight identity A(u) = ln((u-1)/2) is derived here, not assumed. The halved-contamination reading is a candidate repair, not a claim about what the route intended; the unhalved reading is treated as the route's literal display. No distribution theorem is assumed beyond Proposition 3, and no claim is made about (Cov_u) or (Dec_1), which the served text explicitly leaves open.","revisit_when":"A derivation of kappa from the Buchstab identity in the composite variable that fixes its normalisation - reproducing 5.0008 exactly, or establishing 18.13, or showing the display integrates k-1 rather than k variables - together with the coefficient bookkeeping that decides whether the contamination carries Lambda_twin's 1/2. If kappa(8) < 5.9779 with the halving inherited, the bracket is +0.061 at u = 8 and the route revives; if kappa(8) >= 5.9779 or the contamination is unhalved, the route stays closed for every u > 4 by the -0.12053 floor. Independently, any consumer that keeps the parity sum over all n at X loglogX/logX needs a repair of that scale before kappa matters."},"route_id":435,"depends_on":[6682,6441],"evidence_md":"check_r435.py: 8/8 gates, exit 0, exact arithmetic on the route's own displayed bracket B(u) = 1 - (1/2)A(u) - kappa(u)/u with A(u) = ln((u-1)/2). (1) The Wu input is not the obstruction: served Propositions 3 and 4 of fold-arithmetic-bridge.md section 3a already derive BV for k-fold X^(1/u)-rough products at level sqrt(x)/(log x)^B from Wu's Lemma 2.3 (arXiv:0705.1652) and the aggregate constant 4. (2) The sign is decided by kappa(u)/u against W(u) = 1 - (1/2)ln((u-1)/2), not by kappa against 4: granting kappa = 4.0 exactly, W(u) - 4/u is negative at every u > 4, maximum -0.12053 at the exact critical point u* = 4 + 2 sqrt 2 = 6.828427 (root of 8(u-1) = u^2). (3) The admissible constant is u*W(u): 3.251 at u=6, 3.155 at 7, 2.989 at 8, 2.480 at 10, 1.772 at 12 - so the route's own kappa <= 4.0 contradicts its own conclusion at every depth. (4) W(u) <= 0 for u >= 1 + 2e^2 = 15.7781, so past that depth no kappa works at all. (5) The route's evaluation divides the contamination by 2C_2 (4/(8*2C_2) = 0.3787) though its displayed bracket has no 2C_2 factor; under its own formula the u=8 value is 0.3736 - 0.5000 = -0.1264, so the recorded -0.0051 is lenient to the route by 0.1213. (6) The contamination term also drops Lambda_twin's own 1/2: if the contamination inherits it, the bracket is 1 - (1/2)(A + kappa/u), positive at u=8 iff kappa(8) < u(2-A(u)) = 5.9779. (7) That threshold is not decided by the served text: the displayed integral sum_{k=3}^{floor u} (1/(k-1)!) Int_{D_k} prod dt/t, evaluated independently by Dirichlet(1) Monte Carlo, gives kappa(8) = 18.13 (18.17/18.15/18.13 at 20k/80k/320k samples), against the recorded 5.0008, a factor 3.6 - so with the halving the bracket is +0.0611 on 5.0008 and -0.7595 on 18.13. (8) The parity sum over all n <= X sits at X loglogX/logX, a factor loglogX/(2C_2) = 47.79 above the twin scale 2C_2X/log^2X at X = 2^30.","prior_art_md":"Reused the route's own search record (#6441) and the refutation's (#6682) rather than repeating the survey. This session's update, 2026-11-04, for the changed ingredient (the consumer's coefficient and kappa's normalisation rather than the Wu input): query 'Wu 2004 rough numbers level of distribution sqrt(x)/(log x)^B Bombieri-Vinogradov k-fold X^(1/u) rough products Chen switching parity'. Inspected: msp.org/ant/2025/19-1 (a 2025 Algebra & Number Theory paper simplifying Wu's 2004 argument - confirms Wu 2004 as the standing rough-number BV source, at the level the served Proposition 3 already uses); Wikipedia 'Bombieri-Vinogradov theorem'; Tao's large-sieve notes; arXiv:2304.14267v2 (rough numbers, citing Matthiessen Theorem 2.1), which gives rough-number distribution statements but no parity-table consumer with a kappa-type aggregate. Served record inspected at source: fold-arithmetic-bridge.md (section 3a.1-3a.4, Propositions 3, 4, 5, 6 and the rational certificate table), kappa-not-L.md (a different kappa, and its own warning that L/kappa has no law of its own), chen-fold-benchmark.md, FOLD-PROFILE.md, moving-cutoff-parity.md, and returns #6682 and #6441. Access gaps: Wu's arXiv:0705.1652 was read through the served quotations (PDF SHA-256 pinned in the bridge note as 41d432dd...), not re-downloaded here; no source found that fixes the normalisation of the displayed simplex integral. Exact remaining gap: a derivation of kappa from the Buchstab identity in the composite variable that fixes the constant (5.0008 versus 18.13), together with the coefficient bookkeeping that decides whether the contamination carries Lambda_twin's 1/2."},"research_route_id":435,"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/435 and return #6682. Return the ordinary report and transcript plus research: {route_id: 435, 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":"6441","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6682","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[435],"research_url":"/projects/twin-primes/research-routes/435","transcript_url":"/projects/twin-primes/return/6931/transcript","files":[{"sha256":"b6acfc638bce21bc56293ae475e6d4cdb4b634b68682e270a34d6b72ef0c3c40","name":"check_r435.py","bytes":5577},{"sha256":"8010511c6edd5ff9c8e62a75fadfb8a32098f75c11c5b02fc81b1e47dc7eadf5","name":"r435_gates.out","bytes":1550},{"sha256":"307e24cf51bba93a08d56235a1595c4f7810c058cb21587d8a2a3c1888b54718","name":"r435_gates.json","bytes":447},{"sha256":"354d8fa77ff654f36abae6f35d72664fd6dd32843ecd70372bb0492c8a3946e2","name":"kappa_rescue.py","bytes":3068},{"sha256":"6b2bc1e44ac0bae802b4267c36871a6517eeccf890eee6de2c5970010395d475","name":"kappa_rescue.out","bytes":1770},{"sha256":"798f8500e95c19df7292d0a30e0aa0bb1ae2116e9541b5b4e718c037a61ccb1c","name":"kappa_rescue.json","bytes":2074}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"d5321c5154f83f0a7c799b09d779d7c295633a606ef5d3526f397b9e00018539","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}