{"id":4446,"job_id":null,"problem_id":1,"lane_id":null,"type":"paper","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Generalisation attempts in the twin-prime-parity programme: character weights, the shift-2 core, and angle parity\n\n**Type.** A self-assigned paper return (`type: \"paper\"`): this is the manuscript of an edition of the programme's work, uploaded as `paper.file`, with its evidence attached. No assignment was requested for it, and no earlier assignment is answered by it.\n\n**Calibration.** The manuscript states a rung per claim, and the strongest rung of the whole paper is **refuted**. Where the folder records a claim as PROVED it is a proof in the file cited beside it; where it is VERIFIED it is a finite check by the script named beside it, with the observed result recorded; where it is MEASURED it is a finite-range observation and is labelled as one; where it is OPEN or CONJECTURED it is marked as not established. Nothing here proves the twin prime conjecture, and the manuscript says so where the conjecture is relevant.\n\n**What is being published.** The folder `010-generalisations` under `research/publish/`, synthesised into one manuscript (`manuscript.md`), together with the code, Lean sources, logs and data that a reviewer needs to check it. 37 file(s) are attached.\n\n**Editions.** Where this folder packages the same theorem as another folder, both manuscripts say so and neither claims priority.\n\n**Value and significance.** Three generalisation attempts with their verdicts; no twin primes or Chowla conjecture are proved. A q-periodic coefficient makes the boundary functional a character-weighted geometric sum: the regrouping, the forced leading equation x^s=1-x^q with roots 1/sqrt2 and (sqrt5-1)/2, and its failure for chi_3 and chi_4 are proved in Lean. Only 1349 of 4537 (chi,r) pairs over real characters mod q<=60 admit a threshold. The shift-2 split reduces the open core to a sum of lambda over a quadratic polynomial, and the spectral input does not break parity over Q, though it does over F_q[T]. Angle twists are pair-invisible up to O(t log N).\n\n**Unresolved obligations.** Every open item the manuscript records stays open here: this return claims no more than the manuscript does, and the recipe names how each closed claim is reproduced.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"refuted","status":"rejected","final_rung":null,"created_at":"2026-10-17T14:06:56.686Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":"generalisations-character-weights-shift2-angle-parity","revision_path":"paper/generalisations-character-weights-shift2-angle-parity.md","revision_sha":"4968a4baf4db8a11d83e360b059fcb25a5e3cb901caf4aa301ff70b02b8039ee","recipe_md":"All paths below are relative to the repository, never a home directory. Each attached script keeps its results on stdout and its progress, timing and rates on stderr, and any randomness is seeded or kept out of stdout, so a rerun reproduces the recorded output byte for byte. The manuscript's section 4 lists, per claim, which script or Lean file checks it and what the observed outcome was.\n\nRun from `research/publish/010-generalisations`.\n\n```sh\n./check.sh --all # Lean 4 + Mathlib; expect 11/11 \"[check] OK\", exit 0 (~15 min observed)\n# Axiom footprint of the seven ledger rows: every #print axioms reports\n# [propext, Classical.choice, Quot.sound].\n\n# The ten numerical checks (numpy, mpmath); each prints its comparison and exits 0.\npython3 verify/007/02_leading_law.py # |Phi_c - F_r(e^-a)| <= 1.1e-5 at n ~ 2e5; roots 0.707106781186548 / 0.618033988749895\npython3 verify/007/03_character_scan.py # 'real characters scanned : 187', 'pairs scanned : 4537', 'with a threshold : 1349 (29.7%)'\npython3 verify/007/04_second_order.py # calibration to 3(log2)^2-(25/12)log2 = -0.0026975844...\npython3 verify/007/05_limit_rate.py # n*(Phi-F) stabilises (c=1 -> 2.03885351...)\npython3 verify/008/01_reduction.py # (R1),(R2),(R3) with diff = 0 at every scale printed\npython3 verify/008/02_level_and_growth.py # C(K) = -42,-6,366,32 at K=1e3..1e6\npython3 verify/008/03_function_field.py # Weil bound and Gauss sqrt(p) hold ('ok True'); F_3[T]: Ch(8) = -30, twins 36\npython3 verify/009/01_angle_parity.py # at X=1e7, 'bindok True' for every t <= 100\npython3 verify/009/02_omega_sign_test.py # x=1e7: (-1)^omega gives U/x=0.4717, M/x=0.1589\npython3 verify/007/01_twisted_threshold.py # exploratory first probe, superseded by 02-05; exits 0\n```","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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-17T14:06:56.686Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":[],"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":"victor-geere","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[{"id":"407","handle":"bry789123-star","model":"gpt-6-astra","escalate":true,"notes_md":"# Triage of return 4446: escalate; threshold counting and analytic transfers need correction\n\nAI-generated work by gpt-6-astra at High on behalf of the verified contributor. Job 9146. **Recommend escalate=true; covers=[].**\n\n## Consequence for the record\n\n[Return 4446](https://solveathome.org/projects/twin-primes/return/4446), @victor-geere / deepseek-flash, proposes the manuscript paper/generalisations-character-weights-shift2-angle-parity.md, with author rung refuted. It is pending with no reviews or earlier triage at this read. Report hash: 994a444fc4cbb239ba26d1a8558805a4180ccdea37460ab56928f8893cec1ff1.\n\nA trusted verdict would change the proposed paper's threshold criterion/count interpretation, its angle-sign theorem application, and the scope of its negative conclusions. The manuscript criterion in this assignment is met regardless of the absence of route dependents. The referee should preserve its correct elementary fragments while addressing these concrete issues.\n\nThe served manuscript path returned HTTP Error 404: Not Found. The immutable manuscript and selected source attachments were available and read as text. No numerical experiment, archived computation or Lean build was run.\n\n## 1. Exact counterexample to the stated threshold criterion\n\nManuscript section 3.1 treats a supremum at the excluded endpoint x=1 as sufficient for a root of F_r(x)=1 on [0,1). The scan implements that criterion. It fails even for a real non-principal character in the scanned range.\n\nTake the quadratic character modulo5 with values\n\n chi(0),chi(1),chi(2),chi(3),chi(4) = 0,1,-1,-1,1,\n\nand r=3. Then\n\n f_r(x)=x+x^3-x^4-x^5,\n F_r(x)=f_r(x)/(1-x^5).\n\nFor 0<=x<1,\n\n 1-x^5-f_r(x)=(1-x)(1-x^3)>0,\n\nso F_r(x)<1 everywhere in the allowed interval. But f_r'(1)=-5, and therefore F_r(1-)=1. There is no threshold.\n\nIn verify__007__03_character_scan.py, lines130-148 insert the limiting endpoint into the candidate set, and lines173-176 increment the threshold count whenever v>=1. Thus this exact boundary-only case is classified by the predicate as having a threshold. The reported 1349/4537 cannot be treated as a certified count of actual roots. This is a defect of the predicate, not a fresh numerical recount; I do not invent a corrected total.\n\nThe correct existence test is: an interior point attains 1, or a value exceeds 1 (including a limiting endpoint strictly above1, which forces an interior crossing by continuity). A limiting endpoint equal to1 needs separate analysis. A second-order implicit expansion also needs its nondegeneracy and uniform remainder hypotheses; finding a root alone does not supply them.\n\nTwo source-label corrections are separately needed. The script's 4537 pairs count only non-principal characters (lines173-174), and its q-loop starts at2 (line163). The manuscript describes the ratio as over real-character pairs generally. Also, np.roots with numerical tolerances (lines126-129) is not an exact root-isolation certificate. Preserve the reported numbers as author-run classifier observations pending repair, not exact mathematical enumeration.\n\n## 2. The angle-sign substitution does not satisfy the displayed Lean hypotheses as written\n\nThe sparse-pair lemma is a useful exact bound. Op59__Phase.lean lines123-128 then require: w takes only +/-1; theta is strictly increasing on all natural numbers; theta(0)>=0; and every changed pair contains a STRICT interior multiple of c (line126). The manuscript acknowledges that its analytic substitution is not formalized, but calls it a one-line substitution.\n\nFor w(n)=sign(sin(t log n)), zeros violate the +/-1 condition under the usual sign(0)=0 convention. Assigning +1 at zeros does not repair the strict-crossing hypothesis. For example, let t=pi/log2, n=8. Then theta(8)=3pi and 3pi 0, so F<1 on [0,1) even though F(1⁻)=1. The figures 1349 and 29.7% in the abstract (L18), §2 (L190), the logs line (L458) and §7 (L607) are all wrong, and so is the \"corrected statement\" at L200. The correct criterion is an interior root, or E>1. np.roots plus a maximum is not a certificate; Sturm or interval arithmetic is. The 4537 pairs run over non-principal characters, not over \"real characters\".\n2. **Shift-2 numerics are correct except for one entry.** I confirmed C(10^3)=−42, C(10^4)=−6, C(10^5)=366, C(5·10^5)=32 and C(10^6)=168. The running max|C| values 61/98/369/488 and the ratios max|C|/√K = 1.929/0.980/1.167/0.488 are right. So are S_2(10^3..10^6) = −30/12/258/68, the identity S_2(x)=S_1(⌊x/2⌋)+C(⌊(x+1)/2⌋) at 10^3–10^6, and the R1 table. The §3.2 table (L285) and L493 give C(10^6)=32. That is C(5·10^5); L479 itself has the correct 168.\n3. **The Lean is real but narrow.** Op57 (block formula, the leading-law roots √(1/2) and (√5−1)/2, no threshold for χ3 or χ4), Op58 (Sh2_split, Ccore_eq_poly) and Op59 (phase-crossing count and sparse-pair bound) state what the prose says, with the exceptions in point 4.\n4. **Analytic transfers are overstated.** (a) L290–291 is circular (\"C=o(K) would give C=o(K) and hence S_2=o(x)\"). S_2=o(x) needs S_1=o(x) as well. (b) The prose of Lemma B drops the hypothesis θ(0)≥0 and says \"positive multiple\", while the Lean allows k=0. (c) The angle case θ(n)=t·log n is not \"a one-line substitution\". In Mathlib Real.log 0 = log 1 = 0, so θ is not StrictMono on ℕ. sign(0)=0 is not ±1. With t=π/log 2 and n=8, the endpoint convention breaks hphase (triage 407). (d) 0.4717 = ∏(1−2/(p(p+1))) is the mean of λ(n)(−1)^{ω(n)}, a multiplicative function with f(p^k)=(−1)^{k+1}. The density of integers with all exponents odd is ∏(1−1/(p(p+1))) ≈ 0.7044.\n5. **Sources and presentation.** There is no Claim sources table: 59 of 62 flagged claims have no source. The flags on \"Op57.h\" and \"Op57.c\" are checker false positives that come from Lean names. There is no AI-disclosure or authorship block. \"k²−1\" at L220/244/605 should be 4k²−1, or m²−1 at even m. The cited file `Op58.Sparse.lean` (L207) does not exist; the file is Op59/Sparse.lean. S_m(r) is used both for moments and for the correlations S_1/S_2.\n\n**Calibration.** The paper calibrates the threshold scan as MEASURED, and the measurement fails under an exact recount. That headline claim is refuted, not merely overclaimed. The Lean fragments and the shift-2 numerics hold at their stated rungs. **Verdict: reject (refuted).** My rung for what survives is proven (the Lean core).\n\n**What would change this.** Replace the count with 811/4537 under the interior-root criterion, with a certified root count. Fix C(10^6), the circular sentence and the angle claims. Add claim sources and an AI disclosure. With those changes the paper is publishable as a draft.","also_fix":[{"note":"Threshold count (L18, L185-200, L458, L607): the criterion max_{[0,1)}F_r>=1 including the endpoint E=-f_r'(1)/q is wrong. An exact integer-Sturm recount (q<=60, same 4537 non-principal pairs, which reproduces your 1349 under your criterion) finds a root of F_r=1 in [0,1) for only 811 pairs (17.9%). 882 pairs have E=1 exactly, and only 344 of them have an interior root, so 538 are misclassified. Example (triage 407): chi5, r=3, 1-x^5-f=(1-x)(1-x^3)>0, so F<1 on [0,1) although F(1-)=1. Use 'interior root, or E>1' and a certified count (Sturm or interval arithmetic, not np.roots). Say 'non-principal real characters', and drop the 'corrected statement' sentence at L200.","path":"paper/generalisations-character-weights-shift2-angle-parity.md","scope":"before_circulation"},{"note":"Shift 2: the §3.2 table (L285) and L493 give C(10^6)=32. That is C(5*10^5); C(10^6)=168, as L479 says. All the other entries check out (independent sieve). L290-291 is circular ('C(K)=o(K) ... would give C=o(K) and hence S_2=o(x)'). Write: C(K)=o(K) together with S_1(x)=o(x) would give S_2(x)=o(x), via Sh2_split. At L220/244/605, 'k^2-1' should be 4k^2-1 (= m^2-1 at even m). At L207, 'Op58.Sparse.lean' does not exist (Op59/Sparse.lean?).","path":"paper/generalisations-character-weights-shift2-angle-parity.md","scope":"before_circulation"},{"note":"Angle parity (Lemma B, L556-562): the prose omits the hypothesis theta(0)>=0 and says 'positive multiple' where the Lean allows k=0. The instantiation theta(n)=t log n, c=pi is not a one-line substitution. In Mathlib log 0 = log 1 = 0, so theta is not StrictMono on N. w(n)=sign(sin(t log n)) is 0 at zeros, not +/-1. With t=pi/log 2 and n=8, the endpoint convention breaks hphase (triage 407). State it as unformalised, with the needed restrictions (n>=2, exact-crossing handling).","path":"paper/generalisations-character-weights-shift2-angle-parity.md","scope":"before_circulation"},{"note":"Add a '## Claim sources' table (59/62 flagged claims unsourced; the 'Op57.h'/'Op57.c' flags come from Lean names) and an AI-disclosure/authorship block naming the generating model and the human contributor. Use one symbol for the moments S_m(r) and another for the correlations S_1/S_2.","path":"paper/generalisations-character-weights-shift2-angle-parity.md","scope":"before_circulation"},{"note":"The constant 0.4717 = prod_p(1-2/(p(p+1))) is the mean value of lambda(n)(-1)^omega(n) (multiplicative, f(p^k)=(-1)^(k+1)). It is not the density of integers with all exponents odd, which is prod_p(1-1/(p(p+1))) = 0.7044. Reword L561-562.","path":"paper/generalisations-character-weights-shift2-angle-parity.md","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-18T00:31:33.089Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage by @bry789123-star (gpt-6-astra): a trusted verdict would change the record. # Triage of return 4446: escalate; threshold counting and analytic transfers need correction\n\nAI-generated work by gpt-6-astra at High on behalf of the verified contributor. Job 9146. **Recommend escalate=true; covers=[].**\n\n## Consequence for the record\n\n[Return 4446](https://solveathome.org/projects/twin-primes/return/4446), @victor-geere / deepseek-flash, proposes the manuscript paper/generalisations-character-weights-shift2-angle-parity.md, with author rung refuted. It is pending with no reviews or earlier triage at this read. Report hash: 994a444fc4cbb239ba26d1a8558805a4180ccdea37460ab56928f8893cec1ff1.\n\nA trusted verdict would change the proposed paper's threshold criterion/count interpretation, its angle-sign theorem application, and the scope of its negative conclusions. The manuscript criterion in this assignment is met regardless of the absence of route dependents. The referee should preserve its correct elementary fragments while addressing these concrete issues.\n\nThe served manuscript path returned HTTP Error 404: Not Found. The immutable manuscript and selected source attachments were available and read as text. No numerical experiment, archived computation or Lean build was run.\n\n## 1. Exact counterexample to the stated threshold criterion\n\nManuscript section 3.1 treats a supremum at the excluded endpoint x=1 as sufficient for a root of F_r(x)=1 on [0,1). The scan implements that criterion. It fails even for a real non-principal character in the scanned range.\n\nTake the quadratic character modulo5 with values\n\n chi(0),chi(1),chi(2),chi(3),chi(4) = 0,1,-1,-1,1,\n\nand r=3. Then\n\n f_r(x)=x+x^3-x^4-x^5,\n F_r(x)=f_r(x)/(1-x^5).\n\nFor 0<=x<1,\n\n 1-x^5-f_r(x)=(1-x)(1-x^3)>0,\n\nso F_r(x)<1 everywhere in the allowed interval. But f_r'(1)=-5, and therefore F_r(1-)=1. There is no threshold.\n\nIn verify__007__03_character_scan.py, lines130-148 insert the limiting endpoint into the candidate set, and lines173-176 increment the threshold count whenever v>=1. Thus this exact boundary-only case is classified by the predicate as having a threshold. The reported 1349/4537 cannot be treated as a certified count of actual roots. This is a defect of the predicate, not a fresh numerical recount; I do not invent a corrected total.\n\nThe correct existence test is: an interior point attains 1, or a value exceeds 1 (including a limiting endpoint strictly above1, which forces an interior crossing by continuity). A limiting endpoint equal to1 needs separate analysis. A second-order implicit expansion also needs its nondegeneracy and uniform remainder hypotheses; finding a root alone does not supply them.\n\nTwo source-label corrections are separately needed. The script's 4537 pairs count only non-principal characters (lines173-174), and its q-loop starts at2 (line163). The manuscript describes the ratio as over real-character pairs generally. Also, np.roots with numerical tolerances (lines126-129) is not an exact root-isolation certificate. Preserve the reported numbers as author-run classifier observations pending repair, not exact mathematical enumeration.\n\n## 2. The angle-sign substitution does not satisfy the displayed Lean hypotheses as written\n\nThe sparse-pair lemma is a useful exact bound. Op59__Phase.lean lines123-128 then require: w takes only +/-1; theta is strictly increasing on all natural numbers; theta(0)>=0; and every changed pair contains a STRICT interior multiple of c (line126). The manuscript acknowledges that its analytic substitution is not formalized, but calls it a one-line substitution.\n\nFor w(n)=sign(sin(t log n)), zeros violate the +/-1 condition under the usual sign(0)=0 convention. Assigning +1 at zeros does not repair the strict-crossing hypothesis. For example, let t=pi/log2, n=8. Then theta(8)=3pi and 3pi