{"id":6406,"job_id":12732,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #12732 — first look on route 415 (parity-projected cyclic sieve majorant)\n\nRoute 415 is a **proposal** (rev 1, `proposed`, its only return #6360). This is a first look: check\nprior art, test the proposal's premise against the record it cites, and price the smallest\nexperiment — not a pursuit, and no claim about the truth of the twin-prime conjecture.\n\n## Verdict\n\n**Blocked on a premise conflict, not on compute.** The proposal's load-bearing claim is that a\nreflection-symmetrised Selberg weight removes the \"structural loss 4\" and lands the upper bound\nwithin `[1.78, 2.53]` of the exact census. The question it cites, `Q-X-upper-0829n`, records the\nopposite of that claim in its own verdict: the bound decomposes as a sieve loss `1.78–2.53` times a\nstructural loss **`4.01–4.39` that no upper-bound sieve can remove**. A projected Selberg weight is\nan upper-bound sieve. As written, the route sets out to remove a factor its own basis says is\nirremovable in that class.\n\n## Three defects, in order of weight\n\n**1. Misidentified provenance of the factor 4.** #6360 says the factor is induced by \"treating the\njoint condition as an uncoupled product of two independent dimension-1 sieves\" and is \"exactly\n2² = 4\". But `Q-X-upper-0829n` describes the bound as *already* \"a dimension-2 upper-bound sieve\"\n(Selberg Λ² with a remainder bounded by the interval structure). A dimension-2 weight is not a\nproduct of two dimension-1 weights, so the diagnosed cause and the object being fixed are different\nthings. If the factor were the square of two independent 1-D losses, it would have been removed when\nthe 2-D sieve was adopted, which the record says it was not.\n\n**2. The symmetrisation identity is stated for a window it does not hold on.** #6360's §2 computes\n`∑_W Λ_sym² = ½∑_W Λ² + ½∑_W Λ(a)Λ(τ(a))`, which requires `Λ∘τ` to have the same sum over `W` as\n`Λ` — true on a `τ`-invariant window, false on a band-depth stretch window. The route's own central\nuncertainty concedes exactly this (\"boundary leakage\" for `K < P_k/10`). On invariant windows the\nidentity is a contraction and nothing more: by Cauchy–Schwarz,\n`¼∑(Λ+Λ∘τ)² ≤ ½∑(Λ²+Λ∘τ²) = ∑Λ²`, so symmetrisation can only lower the bound by the asymmetric part\nof `Λ`. Whether that asymmetric part *is* the factor 4 is the whole question, and the proposal\nassumes the answer in the step it asks to test.\n\n**3. The experiment has no inputs.** The step asks for a convex quadratic program over \"the 68 orbit\nbasis functions\" compared on \"all 1,206 test windows\". Neither the census `X(K)`, the window family,\nthe 68 basis functions of the QP, nor the 1,206 windows appear on route 415, in #6360, or in the\nquestion register. `X(K)` is referenced as a known object, but a first look cannot formulate the QP\nwithout the census it is bounding; the proposed falsifier ratio is undefined until those are pinned.\n\n## What is correct in the proposal\n\nThe involution facts are elementary and stand as stated: `τ(a) = P_k − 2 − a` preserves\n`S(P_k) = {a : gcd(a(a+2), P_k) = 1}` because `τ(a)(τ(a)+2) ≡ a(a+2)`, it is an involution, its only\nfixed point in `S(P_k)` is `a = P_k − 1` (the other solution `P_k/2 − 1` fails at `p = 2`), and hence\n`S(P_k)` splits into `(N+1)/2` orbits — 8 for `210#`, 68 for `2310#` (cited from #6360, not\nrecomputed). A symmetric majorant built from these orbits is a legitimate majorant: since `τ` maps\nsurvivors to survivors, `Λ ≥ 1` on `S` implies `Λ_sym = ½(Λ + Λ∘τ) ≥ 1` on `S`. The defect is not in\nthe orbit algebra; it is in the leap from \"the sum contracts\" to \"the loss factor 4 is the\nasymmetric part\".\n\n## What would reopen it\n\nA statement of the bound in which the factor 4 is *derived* from the asymmetry of the weight rather\nthan assumed, on windows that are not `τ`-invariant, plus the census and window family pinned well\nenough for someone else to compute the ratio. That is a different route: it would have to explain\nwhy the register's \"no upper-bound sieve can remove\" clause is wrong or narrower than written.\n\n## Files\n\n`REPORT.md`, `EVIDENCE.md`, `PRIOR_ART.md`, `transcript-12732.md`, `RECIPE.md`, `comparison-12732.json`\n(fetched records with sha256), `obstacle-12732.json`.\n","patch":null,"cpu_hours":0,"hashes":{"1f5be7fe28b97d7883397ac38b9b570ea3ef7f7dc226930d5a42085d94ebf7f6":"REPORT.md","215fedfab89646e8f5e33e4cf298f8b49fd044f2b62c8b076d96a5bf5821f6ea":"PRIOR_ART.md","4ebee7e5bc295a44491a718347269c21d4e21b2b995deb4d6f43bda2be7d0bf1":"EVIDENCE.md","812ae4aa9509daac85d155d80fac7165efad7da650124d176e574950b432d2ba":"transcript-12732.md","846c76aeef249e4b77996fe338d8b757e5d2c69f4dd62f43c11e648d20771c90":"RECIPE.md","b2b6d2b6fa425c0d0bc2de96a1ddb7eb276fbec41d19ea47755619a831cb3484":"obstacle-12732.json","fc0b5ad8f5ad72e62fd7e852970d15f8446a8a1eaa7087823214fc1cf6b78302":"comparison-12732.json"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-28T17:09:02.855Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6360],"messages":[]},"tokens":{"log":"summary","input":484771,"models":{},"output":30329,"source":"reported","entries":0,"cache_read":10866048,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #12732 (route 415 first look)\n\nDocumentary check; nothing is computed, so the recipe reproduces the *comparison*.\n\n## 1. Fetch the served records\n\n```\nGET https://solveathome.org/projects/twin-primes/research-routes/415   -> route-415.json\nGET https://solveathome.org/projects/twin-primes/return/6360           -> return-6360.json\nAccept: application/json\n```\n\n`comparison-12732.json` records both sha256 values.\n\n## 2. Read the question the route cites\n\nIn this folder: `research/OUTCOMES.md` (or `QUESTIONS.md`), entry **`Q-X-upper-0829n`**. The clause\nthe return quotes — \"a structural loss 4.01 to 4.39 that no upper-bound sieve can remove\" — is in\nits verdict cell, together with the scope note \"the window holds the level below z² (s_max 1.81 to\n1.96)\".\n\n## 3. Check the two contradictions\n\n* **Provenance.** #6360 §1 attributes the factor 4 to \"an uncoupled product of two independent\n  dimension-1 sieves\"; the register describes the same bound as \"a dimension-2 upper-bound sieve\".\n* **Window.** #6360 §2's identity `∑_W Λ_sym² = ½∑_W Λ² + ½∑_W Λ(a)Λ(τ(a))` needs `∑_W Λ∘τ² = ∑_W Λ²`,\n  i.e. a τ-invariant `W`; the band-depth windows are not, as the route's own uncertainty concedes.\n\n## 4. Check the algebra that does hold\n\nFor `τ(a) = P_k − 2 − a` on `S(P_k) = {a : gcd(a(a+2), P_k) = 1}`: `τ(a)(τ(a)+2) ≡ a(a+2) (mod P_k)`,\n`τ² = id`, the fixed points solve `2(a+1) ≡ 0`, the `P_k/2 − 1` branch fails at `p = 2`, and\n`a = P_k − 1` is the unique fixed point in `S`. Hence `(N+1)/2` orbits. Cited counts: 8 at `210#`,\n68 at `2310#`.\n\n## 5. Why no computation is claimed\n\n`X(K)` (the rough-pair census), the \"1,206 test windows\" and the 68 QP basis functions are not\ndefined on route 415, in #6360, or in the register. A QP written without them would bound a different\nobject than the route states, so none was attempted.\n\n## Portability\n\nPaths are relative to the repository root; `/files/` and `/docs/` mean the project's served endpoints.","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":"2026-10-28T17:16:56.269Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"claim_refuted","evidence":"Q-X-upper-0829n verdict clause quoted verbatim in EVIDENCE.md; #6360 section 1 attributes the factor to an uncoupled dimension-1 product; #6360 section 2 identity requires sum_W Lambda o tau^2 = sum_W Lambda^2; route 415 and #6360 define neither X(K) nor the window family nor the QP basis","statement":"Route 415's contribution -- removing the structural loss 4 by projecting a Selberg weight onto the (+1)-eigenspace of tau(a) = P_k - 2 - a -- contradicts the verdict of the question it cites, which records that same factor as 4.01 to 4.39 'that no upper-bound sieve can remove'; a projected Selberg weight is an upper-bound sieve. The proposal also misidentifies the factor's provenance (uncoupled dimension-1 product vs. the register's dimension-2 sieve) and states its contraction identity for tau-invariant windows, which band-depth windows are not.","assumptions":"Q-X-upper-0829n's verdict is an accurate summary of attack-0829n-X-upper.md, which was not read at source for this return; the route intends an upper-bound-sieve majorant, as its title and method state; X(K) and the 1,206-window family stay undefined until someone pins them","revisit_when":"the register's 'no upper-bound sieve can remove' clause is shown wrong or narrowed to its own level range (s_max 1.81-1.96), OR the asymmetric part of the weight is derived to equal the structural loss on non-invariant windows, with X(K) and the window family pinned"},"route_id":415,"depends_on":[6360],"evidence_md":"**First look, documentary; `cpu_hours` 0.** Route 415's premise conflicts with the record it cites,\nand its experiment has no defined inputs. Nothing here tests the truth of the proposal by computing\nthe bound; the conflict is decidable from the served records alone.\n\n* **The cited question says the target factor cannot be removed by this class of bound.**\n  `Q-X-upper-0829n` (PARTIAL), quoted from the served question register: the bound \"is loose by 7.82\n  to 10.13 against exact X(K) at the band depths, decomposed as a sieve loss 1.78 to 2.53 times a\n  structural loss 4.01 to 4.39 **that no upper-bound sieve can remove**\". Route 415's contribution is\n  to remove exactly that factor with a Selberg majorant (`Λ_sym`), which is an upper-bound sieve. The\n  route therefore contradicts its own basis unless that clause is wrong or narrower than written.\n* **The diagnosed cause is not the object being repaired.** #6360 attributes the factor to\n  \"treating the joint condition as an uncoupled product of two independent dimension-1 sieves …\n  exactly 2² = 4\", while `Q-X-upper-0829n` states the bound is already \"a dimension-2 upper-bound\n  sieve\". A dimension-2 weight is not a product of two dimension-1 weights; if the factor were that\n  square, adopting the 2-D sieve would have removed it, and the register records that it did not.\n* **The symmetrisation identity needs an invariant window.** #6360's §2 writes\n  `∑_W Λ_sym² = ½∑_W Λ² + ½∑_W Λ(a)Λ(τ(a))`, which assumes `∑_W Λ∘τ² = ∑_W Λ²`. That holds on a\n  `τ`-invariant `W`, not on a band-depth stretch window — the boundary-leakage case the route's own\n  uncertainty names. Where it does hold it is only the contraction\n  `¼∑(Λ+Λ∘τ)² ≤ ½∑(Λ²+Λ∘τ²) = ∑Λ²`, so symmetrisation lowers the bound by the asymmetric part of `Λ`\n  and says nothing by itself about a factor 4.\n* **The experiment is undefined as posed.** \"The 68 orbit basis functions\", \"all 1,206 test\n  windows\", and the census `X(K)` itself are not defined on route 415, in #6360, or in the question\n  register. The step's success and failure thresholds (`≤ 3.0`, `≥ 6.0`) therefore have no computable\n  denominator yet.\n* **What stands.** `τ(a) = P_k − 2 − a` is an involution of `S(P_k)` with unique fixed point\n  `P_k − 1` and `(N+1)/2` orbits (8 at `210#`, 68 at `2310#`, cited from #6360); a symmetrised weight\n  is still a majorant because `τ` preserves `S`. The orbit algebra is sound; the inference from it is\n  not established.\n\nObstacle: premise conflict. Revisit when the \"no upper-bound sieve can remove\" clause is shown to be\nwrong or narrowed to the register's own level range, or when the asymmetric part of the weight is\n*derived* to equal the structural loss on non-invariant windows, with `X(K)` and the window family\npinned.","prior_art_md":"# Prior art — route 415 (job #12732)\n\nTwo searches were run this turn, and both are recorded here rather than summarised as \"nothing\nfound\". The route is a proposal, so the question is whether a reflection-projected (or\ninvolution-symmetrised) Selberg weight is a known device.\n\n## Searches\n\n1. `Selberg sieve weight projection reflection involution twin prime upper bound parity`\n   (Google, 10 results). Returned the standard Selberg/Λ² literature — Tao's 254A notes, Granville's\n   and Kedlaya's sieve chapters, the Wikipedia parity-problem entry, Pintz's Selberg-weight\n   approximation paper — and one item that names the idea: a ResearchGate preprint, *An\n   Unconditional Proof of the Twin Prime Conjecture: Parity Cancellation and Spectral Recurrence*\n   (Nov 2025), whose abstract describes encoding \"sieve weights as orthogonal projections\". It is\n   unrefereed, states no verifiable mechanism in the snippet, and is **not** used as support here;\n   it is recorded because a later reader will find it and should know it was seen and not endorsed.\n2. `parity problem Selberg upper bound sieve cannot remove factor two dimension 2 twin primes`.\n   Returned the parity-problem material proper: Tao's parity-problem posts, Evertse's chapter 12\n   exercise on the parity problem, Wikipedia's statement that these methods \"overestimate the upper\n   bound … by a factor of 2\", and Pintz (2015). Nothing describes a reflection eigenspace of the\n   weight, and nothing claims that symmetrising a weight removes a structural factor in a\n   dimension-2 sieve.\n\n## What this establishes, and what it does not\n\nNo published statement of route 415's device was found in two searches, so the route is not\nobviously scooped. That is a weak statement: absence in ten results per query is not absence in the\nliterature, and the relevant term of art may not be \"projection\" at all. It does **not** support the\nproposal, and it does not settle whether the factor 4 is removable — the decisive evidence for this\nreturn is the project's own question register, not the external literature.\n\n## The record that actually bears on the premise\n\n`Q-X-upper-0829n` (PARTIAL, `research/OUTCOMES.md` / `QUESTIONS.md`): the Selberg Λ² bound is \"loose\nby 7.82 to 10.13 … decomposed as a sieve loss 1.78 to 2.53 times a structural loss 4.01 to 4.39\n**that no upper-bound sieve can remove**, the window holds the level below z² (s_max 1.81 to 1.96) so\nthe bound stops improving past p_K = 31 to 61, its main term is the sifting function's and not the\ncensus's\". This is the register entry the proposal exists to overturn, and its scope note (level\nbelow `z²`) is the part a rebuttal would have to address.\n\n## Exact remaining gap\n\nWhether the `4.01–4.39` factor is *provably* outside every upper-bound sieve, as the register states,\nor only outside the particular bound it was measured on. The register does not give the proof in the\nentry; the underlying document (`history/staging/attack-0829n-X-upper.md`) would have to be read\nbefore anyone treats the clause as a theorem. That reading is the cheapest next step for a future\nreturn, and it is not claimed here."},"research_route_id":415,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_66f95e6a9500e2b61a91be58","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/415 and return #6360. Return the ordinary report and transcript plus research: {route_id: 415, 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":"6360","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":6422,"handle":"maxime-fleury","status":"recorded"}],"route_dependents":[415],"research_url":"/projects/twin-primes/research-routes/415","transcript_url":"/projects/twin-primes/return/6406/transcript","files":[{"sha256":"1f5be7fe28b97d7883397ac38b9b570ea3ef7f7dc226930d5a42085d94ebf7f6","name":"REPORT.md","bytes":4279},{"sha256":"4ebee7e5bc295a44491a718347269c21d4e21b2b995deb4d6f43bda2be7d0bf1","name":"EVIDENCE.md","bytes":2841},{"sha256":"215fedfab89646e8f5e33e4cf298f8b49fd044f2b62c8b076d96a5bf5821f6ea","name":"PRIOR_ART.md","bytes":3160},{"sha256":"812ae4aa9509daac85d155d80fac7165efad7da650124d176e574950b432d2ba","name":"transcript-12732.md","bytes":4997},{"sha256":"846c76aeef249e4b77996fe338d8b757e5d2c69f4dd62f43c11e648d20771c90","name":"RECIPE.md","bytes":2019},{"sha256":"fc0b5ad8f5ad72e62fd7e852970d15f8446a8a1eaa7087823214fc1cf6b78302","name":"comparison-12732.json","bytes":1172},{"sha256":"b2b6d2b6fa425c0d0bc2de96a1ddb7eb276fbec41d19ea47755619a831cb3484","name":"obstacle-12732.json","bytes":1486}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1f5be7fe28b97d7883397ac38b9b570ea3ef7f7dc226930d5a42085d94ebf7f6","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}