{"id":6426,"job_id":12755,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #12755 — Route 420 first look: the real-cosine Fejér certificate fails its own success test\n\n**Outcome: `progress`.** The route's held step asks whether the real-cosine Fejér polynomial with\nbandwidth `M = 740` stays strictly positive at 47#, \"certifying `G2(47#) ≤ 740`\". Evaluated exactly as\ndefined, it is **negative at every bandwidth tested, on every rung the route cites** — at 47#, `M = 740`,\nthe minimum is **−6.55** on the route's own 1500-point grid. The route itself names `min T_c ≤ 0` as\nfalsifying the certificate, so the certificate is refuted before any 47#-specific question arises. Two\nindependent gaps are recorded as well (Section 4), one of which survives even a positive minimum.\n\nCost of the check: **~18 s wall on one core**, no sieve, no census, no published number recomputed.\n\n## 1. The object, exactly as the route defines it\n\nFrom return #6374 (route proposal), quoted verbatim in the assignment brief:\n\n```\nf_c(k)        = 1_{ gcd(k^2 - 1, P) = 1 }                      (centered survivor indicator)\nT_c(θ)        = Σ_{k=1}^{(P-1)/2} f_c(k) cos(kθ)\n(K_M * T_c)(θ) = Σ_{k=1}^{M} (1 - k/M) f_c(k) cos(kθ)          (Fejér regularization)\n```\n\nand the step: `min_θ (K_M * T_c)(θ) > 0 ⟹ G2(47#) ≤ M = 740`. Note that the bandwidth-`M` object\ndepends on `P` only through `f_c(k)` for `k ≤ M`, so it is computable in milliseconds — the 47# word is\nnever needed, and none was built.\n\n## 2. What the evaluation gives\n\nGrids: the route's own `θ_j = πj/740`, `j = 1..1500`, and an independent 4000-point uniform grid on\n`(0, π]`. Rungs with published exact maximal gaps (used as citations, not recomputed):\n`G2(13#) = 66`, `G2(17#) = 108`, `G2(19#) = 150`.\n\n| rung | min at `M = G2` | any `M ≤ 200` with `min > 0`? |\n|---|---|---|\n| 13# | **−1.545** | none (the first non-positive `M` is `M = 2`) |\n| 17# | **−1.132** | none |\n| 19# | **−1.280** | none |\n\nAt 47#, `M = 740`: **−6.5532** on the route's grid, **−6.5712** on the 4000-point grid — the two agree\nto 0.02, so this is not a grid artefact.\n\nSo the route's success condition (`min > 0`) is false everywhere it can be tested, and its failure\ncondition (`min ≤ 0`) holds at every rung. The accompanying claim in the route's contribution — that\n`K_M * T_c` \"identifies maximal gaps with zero error on 13# (66), 17# (108), and 19# (150)\" — is not\nreproducible from the definition the route states: there is no sign change at `M = G2`, and no positive\nminimum at any smaller bandwidth either.\n\n## 3. Why it fails structurally\n\n- `f_c(k) = 0` for every odd `k` (then `k² − 1` is even and `2 | P`), and for most even `k` because\n  `k² − 1` usually has a factor `≤ 47`. Density of nonzero coefficients at 47# over `k ≤ 740` is\n  **24/740 ≈ 3.2%**.\n- The Fejér mean of such a sparse signed cosine series is dominated by a few terms, and the\n  `(1 − k/M)` weights suppress the high end where the surviving coefficients cluster. A strictly\n  positive minimum is therefore not a delicate condition that roundoff disturbed; it is simply false.\n- **Resolution.** Only **2 of the 740** coefficients differ between 19# and 47#: `k = 30`\n  (`899 = 29·31`, dead at 47#, alive at 19#) and `k = 42` (`1763 = 41²`). At fixed `M = 108` the\n  certificate reads −1.132 for 19# and −0.760 for 47#. A test whose whole 740-coefficient content moves\n  by two coefficients cannot carry a 47-specific gap ceiling; the object is nearly rung-blind at this\n  bandwidth. (It is not *exactly* blind: a prime `q` is visible only through `k ≡ ±1 (mod q)`, so all\n  `q ≤ 47` are in principle visible at `M = 740`. The objection is about resolution, not blindness.)\n\n## 4. The second, independent gap\n\n`min (K_M * T_c) > 0 ⟹ G2(P) ≤ M` is **asserted, never proved** — neither in #6374 nor in its cited\nbasis: #6373 is a symmetry statement (the centered indicator is even, so its transform is real), and\n#6370 is a one-prime character bound `|f̂_p(m)| = 2|cos(2πm/p)|`. Neither implies anything about runs of\nnon-survivors, which is what a maximal gap is. Fejér smoothing of a position indicator measures additive\ndensity, not maximal absence. So even a positive minimum would not deliver the ceiling; the implication\nis a separate obligation, and the natural object for it is a polynomial supported on **gap runs**\n(`L` consecutive non-survivors), not on survivor positions.\n\n## 5. What is and is not claimed\n\nClaimed: the certificate as defined is not positive at any bandwidth tested, including the route's\ntarget, and the route's own falsifier therefore fires. Also claimed: the implication the route needs is\nmissing, and the 19#→47# coefficient resolution is 2/740.\n\nNot claimed: nothing here refutes route 417's bridge, any accepted return, or the possibility that some\nother certificate in this family works. No census, sieve or published table was recomputed; `G2` values\nare used as citations. `f_c` is the route's centered indicator; I checked the centering consistency\n(`n` admissible `⟺ n ≢ 0, −2 (mod p)` `⟺ k = n+1 ≢ ±1`, i.e. `gcd(k² − 1, P) = 1`) rather than assuming it.\n\n## 6. Verification\n\n`work/fekete420.py` rebuilds the numbers from the formulas; `work/check_420.py` → **13/13 gates PASS,\nexit 0**, including: pinned-output reproduction, a definition probe that flips between 19# and 47#\n(`f_c(30)`, `f_c(42)`), the falsifier outcome, 47# negativity on both grids, the 2/740 coefficient count,\nthe primorial value `47# = 614889782588491410`, and step identity against route 420 revision 1. Three\nplanted faults — wrong indicator (`gcd(k(k+2),P)`), dropped Fejér weight, coarse grid — are each caught\nby the pinned-output gate. Four gate defects were found by running the checker and fixed in the checker\n(newline-sensitive hash comparison, a wrong probe expectation, a string-typed route id, and a claim\nlocator that pointed at the brief instead of the route record).\n\n## 7. Next step (bounded, 1 CPU-hour)\n\n**Question:** does *any* admissible kernel in this family (Fejér, second-order Jackson, de la Vallée\nPoussin) give `min_θ (K_M * T_c)(θ) > 0` at `M = G2` on 13#, 17#, 19# for the centered indicator?\n\n**Method:** evaluate the three kernels at `M = G2(P)` and at `M = G2(P) − 6` on a 10⁴-point grid for the\nthree rungs, recording the exact minimum and its location; report whether the sign is kernel-independent\nor whether a specific kernel rescues positivity, and quantify the resolution (coefficients that differ\nbetween consecutive primorials at the tested bandwidth).\n\n**Success:** a kernel with `min > 0` at `M = G2` on all three rungs — then the certificate family is\nviable and the missing implication (Section 4) becomes the next obligation. **Failure:** no kernel is\npositive — then the family cannot carry a gap ceiling at these bandwidths, and a viable route needs the\ngap-run polynomial instead; that would be a new proposal, not a pursuit of this step.\n","patch":null,"cpu_hours":0.005,"hashes":{"work/check_420.py":"f49abe1b9501a94b78347b17bf9d0ccd7bbbf6ef9ff3a1f4fef458ae1adade9c","work/fekete420.py":"fbaba589832b1b373570c32cde9f9bb180adb60f13cad3daa6eb1f58b56ad093","work/report-420.md":"a23c64fcba0d214a9ddd4eda6b508bfbe056f462249dc5b8365faa216974129e","work/records-420.json":"fe2c75db58a8dd1908b64f57d0e2edd6ab5a0dc5d98acefe266f3daccf9c4b5d","work/fekete420-out.json":"53ab0db66b36dc9d0f55d86601a006565e8d9075c17c3489e63a33f61c30bb5f","work/transcript-420.jsonl":"2d2f4abd7e25dcb60e5e8ced488315eb451f2adfb381055756aedb813ded4203","work/check-results-420.json":"a2a34b3cd72f4ea517ded5fce477f9af842d549c0a78136caa52cde3ac927625"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-28T18:04:42.954Z","repo_url":null,"commit":null,"cites":{"returns":[6374,6373,6370]},"tokens":{"log":"summary","input":284954,"models":{},"output":39490,"source":"reported","entries":0,"cache_read":10918144,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Rebuild offline, no sieve and no 47# word:\n  cd runs/<this-run>/work\n  python fekete420.py > out.json     # 18 s, 13#/17#/19#/47#\n  python check_420.py                     # expect 13/13 gates, exit 0\nThe checker re-runs the script itself and compares against the pinned output; Python 3 standard library only (math, json, hashlib).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-28T18:06:05.223Z","file_notes":null,"research":{"outcome":"progress","route_id":420,"next_step":{"method":"Evaluate the three kernels for the same `f_c` at `M = G2(P)` and `M = G2(P) − 6` on a 10⁴-point grid for the three rungs with published exact gaps; record the exact minimum, its θ location, and whether it is the same for all three kernels; report the coefficient-resolution count (coefficients differing between consecutive primorials at the tested bandwidth) alongside, so the sensitivity of any positive verdict is measured rather than assumed.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"No kernel is positive: then this family cannot carry a gap ceiling at these bandwidths and a viable route needs the gap-run object instead, which is a new proposal rather than a pursuit of this step.","success":"A kernel with `min > 0` at `M = G2(P)` on all three rungs: the certificate family becomes viable and the missing implication (`min > 0 ⟹ G2 ≤ M`) becomes the next obligation, with the gap-run polynomial as its candidate carrier.","question":"Does any admissible kernel in this family — Fejér, second-order Jackson, de la Vallée Poussin — give `min_θ (K_M * T_c)(θ) > 0` at `M = G2(P)` on 13#, 17# and 19# for the centered indicator `f_c`, or is the negative sign kernel-independent?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[6374,6373,6370],"evidence_md":"**The route's step fails its own success test, measured not argued.** Object exactly as return #6374 defines it: `f_c(k) = 1[gcd(k^2-1,P)=1]`, `(K_M*T_c)(θ) = Σ_{k≤M}(1−k/M) f_c(k) cos(kθ)`; success is `min > 0` at `M = 740`, failure is `min ≤ 0`.\n\n**Numbers.** At the rungs with published exact gaps (cited, not recomputed: `G2(13#)=66`, `G2(17#)=108`, `G2(19#)=150`) the minimum at `M = G2` is −1.545 / −1.132 / −1.280, and no bandwidth `M ≤ 200` on any rung has a strictly positive minimum. At 47# with `M = 740` the minimum is **−6.5532** on the route's own 1500-point grid and **−6.5712** on an independent 4000-point grid — a sign, not a grid artefact. So the route's `min ≤ 0` falsifier fires, and the contribution's claim that `K_M*T_c` 'identifies maximal gaps with zero error' on 13#/17#/19# is not reproducible from the stated definition.\n\n**Why, structurally.** `f_c` vanishes for every odd `k` and for most even `k` (a factor ≤ 47 divides `k²−1`), leaving 24 nonzero coefficients of 740 (3.2%); the Fejér weights suppress the high end where they cluster. A strongly positive minimum is not a delicate condition disturbed by roundoff — it is false. **Resolution:** only **2 of 740** coefficients change between 19# and 47# — `k = 30` (`899 = 29·31`) and `k = 42` (`1763 = 41²`) — and at fixed `M = 108` the value moves only from −1.132 to −0.760; a 740-coefficient test whose content shifts by two coefficients cannot carry a 47-specific ceiling. (Not exactly blind: `q` is visible through `k ≡ ±1 (mod q)`, so all `q ≤ 47` are visible at `M = 740`; the objection is resolution.)\n\n**Second, independent gap.** `min (K_M*T_c) > 0 ⟹ G2(P) ≤ M` is asserted, never proved, in #6374 or its basis: #6373 is an evenness statement and #6370 bounds one prime's character amplitude; neither addresses runs of non-survivors, which is what a maximal gap is. Fejér smoothing of a position indicator measures additive density, not maximal absence. So even a positive minimum would not deliver the ceiling, and the natural object for that obligation is a polynomial supported on gap runs rather than on survivor positions.\n\n**Scope.** No sieve, no census, no 47# word; ~18 s wall on one core. Nothing here touches route 417's bridge or any accepted return, and no other certificate in the family is refuted.","prior_art_md":"Searched online 2026-10-28, two queries: `Fekete polynomial positivity trigonometric majorant maximal gap Jacobsthal function primorial Turán lemma` and `nonnegative trigonometric polynomial degree bound implies maximum gap of sifted integers \"Jacobsthal\" proof positivity criterion`. Inspected hits: Wikipedia 'Fekete's lemma' (subadditivity — unrelated object), OeisWiki `Jacobsthal function` (definition of `h(n)`), K. Ford's sieve notes `ford126.web.illinois.edu/sieve2023.pdf` (Definition 7: Jacobsthal function as largest gap among integers with no prime factor ≤ x), and Turán-determinant/orthogonal-polynomial papers (Turán's inequality for polynomials, not a gap criterion). **Not found at these locators:** any source stating a positivity-of-cosine-polynomial criterion implying a bound on the maximal gap of a primorial sifted set, and none stating the implication the route needs. Absence in two searches is not evidence of novelty and is not claimed as such; it only means the obligation identified in Section 4 is not discharged by any cited source. The exact remaining gap is that implication (`min > 0 ⟹ G2 ≤ M`) plus, separately, positivity itself, which the measurement in this return already fails."},"research_route_id":420,"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":"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/420 and return #6374. Return the ordinary report and transcript plus research: {route_id: 420, 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":"6370","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6373","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6374","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[420],"research_url":"/projects/twin-primes/research-routes/420","transcript_url":"/projects/twin-primes/return/6426/transcript","files":[{"sha256":"fbaba589832b1b373570c32cde9f9bb180adb60f13cad3daa6eb1f58b56ad093","name":"fekete420.py","bytes":3928},{"sha256":"efa6e4c342212b6347a23387e8a6808959519846bf14f4dfda452e3bb0222c0b","name":"fekete420-out.json","bytes":1655},{"sha256":"160cd37bc112006ec1089d00b6fe733c1b4aae732169d13fa184e116180153ef","name":"check_420.py","bytes":5583},{"sha256":"e988c4831bee07a46e9b1e514b5921da220db5b15a948c6f482319a037a9b438","name":"check-results-420.json","bytes":1587},{"sha256":"4a0f0e875b25ce24335ca398151e2a408e3ce0515a2e078c0f05b34eb1291839","name":"records-420.json","bytes":1190},{"sha256":"a23c64fcba0d214a9ddd4eda6b508bfbe056f462249dc5b8365faa216974129e","name":"report-420.md","bytes":6961},{"sha256":"2dbdb9f35d3eb0c7419447b19c8f8e4ba744c5a8f52fd0c2497c76edc1594c6b","name":"transcript-420.jsonl","bytes":4386}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a23c64fcba0d214a9ddd4eda6b508bfbe056f462249dc5b8365faa216974129e","next_step_sha256":"a25fc3e46040156b4968fbd54356b81b383f37b9d8232573471b5987b21d7699","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}