{"id":1796,"job_id":2032,"problem_id":1,"lane_id":1,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2032 (pursue route 29): the (D1) deficit 7/200 is a saving over the record's mass (ℓ¹) bound, not over the ℓ² trivial bound. In ℓ² the requirement is x^(−23/100), and #1080's sixth-moment saving x^(−7/150) falls short by x^(11/60)\n\n**Outcome: blocked.** The step's failure clause fires. The carried identification is wrong. #1080's margin x^(−7/600) is withdrawn, and the corrected requirement is named below.\n\nCaveats first. This is exponent arithmetic on the record's (D1) proof and on theorem statements quoted in #626/#629/#1080. I did not re-read the papers' full texts. The result concerns only the (D1) (m,q)-Cauchy arrangement at the top sector, on the binding band j = (e₁,e₂) = 1. Nothing here bounds G2 or changes the global margin, which remains OPEN.\n\n## 1. What the 7/200 is a saving over (re-derived from the record)\n\nSource: `research/structured-dispersion-estimate.md` §2 (3)–(4) and §4 steps 1–6 (served sha f6b0203a…). Top sector: a = 14/25, b = 1/2, σ = 1/20, α = a+b−1 = 3/50, E = b−σ = 9/20.\n- (6) gives |T|² ≤ (MQ)·Σ_q λ(q)𝔐_q. The budget for Σ_q λ𝔐_q is 2 − (a+σ) = 139/100. The Weil majorant is E³Q^{3/2} = x^{57/40}, so the deficit is **7/200** (regression of the record chain).\n- On the band j = 1, for each coprime pair (e₁,e₂) with c = q e₁ e₂ ~ x^{19/20}, step 4 bounds the (h₁,h₂)-sum by Σ|c_{h₁}c_{h₂}|·√c·⟨G^{1/2}⟩ = C²√c·x^{o(1)} (Lemma H, q < A). Summing over E² pairs gives 𝔐_q ≪ E³q^{1/2}. The requirement is therefore\n  **|B(e₁,e₂)| ≪ C²√c·x^{−7/200−δ}, where C² = ‖α‖₁ and ‖Ĝ‖₁ = x^{o(1)}.** That is a saving over the **mass-normalised (ℓ¹)** Weil bound. (Exact identity: D = a+3b−2−σ/2 = (α−σ/2)+(2b−1). The \"7/200 = exponent of A/√q\" identification of #626 holds only because b = 1/2 at this box.)\n\n## 2. Shape match (re-derived; holds)\n\nFor (m,c) = 1 and periodicity mod c, K = Σ_{m∈I,(m,c)=1} e_c(σθR m̄)F(m) = c⁻¹ Σ_{k mod c} F̂_c(k) S(σθR, −k; c) with F̂_c(k) = Σ_m F(m)e_c(km). So the completed object is B = Σ_R Σ_k α_R β_k S(σθR,k;c) at fixed c, with β_k = F̂_c(−k)/c. This agrees with #626. One point #626 left implicit: F = Φ_{qe₁,h₁}Φ̄_{qe₂,h₂} depends on (h₁,h₂), not only on R. At the top band v = 1, so the phase hz/(gmu) is O(1). Taylor-expanding e(·) to O(log x) terms separates it at x^{o(1)} cost. The separated coefficients have the same support as α_R. So this is not where the route fails.\n\n## 3. The normalisation step (fails)\n\n**Lemma (proven, one line).** If (e₁,e₂) = 1 and A < min(e₁,e₂), then (h₁,h₂) ↦ R = h₁e₂ − h₂e₁ is injective on [A,2A]². Proof: equal images give (h₁−h₁′)e₂ = (h₂−h₂′)e₁, so e₁ | h₁−h₁′ with |h₁−h₁′| ≤ A < e₁. Here α = 3/50 < E = 9/20. Finite check: all coprime e₁,e₂ < 40 with A < min; negative control (3,4,4) collides.\n\nSo α_R is supported on A² = x^{3/25} points of its range x^{51/100}. ‖α‖₁ = C², and ‖α‖₂ = Σ|c_h|² ≥ ‖α‖₁/|H| (= C²/A at |c_h| = C/A). β is spread: ‖β‖₁ = x^{o(1)} and ‖β‖₂ ≍ N′^{−1/2} with N′ = c/M = x^{39/100}. The ℓ² trivial bound ‖α‖₂‖β‖₂ min(c, √(M′N′c)) is therefore **C²√c·x^{39/200}**: it exceeds the record's own bound by x^{39/200} = √M′/A. Every bound of the form ‖α‖₂‖β‖₂·c·(saving) must first recover this sparsity loss.\n\n| quantity (x-units) | value |\n|---|---|\n| required saving over the mass bound (record) | 7/200 |\n| sparsity loss of the ℓ² trivial bound | 39/200 |\n| **required saving over the ℓ² trivial bound** | **23/100** (= c^{23/95}; 23/45 of the full √(M′N′)) |\n| #1080 sixth moment, d = e₁, q prime (reproduced: 14/285 c-units) | 7/150 → **short by 11/60** |\n| #1080's bound relative to the record's C²√c | worse by x^{89/600} |\n| #1067 Thm 5.2 / #629 Thm 5.5 / BP optimum c^{1/32} / Pascadi c^{1/12} | short by 701/3200, 677/3000, 641/3200, 181/1200 |\n\nThe square-root-cancellation heuristic ‖α‖₂‖β‖₂√c would beat the mass bound by x^{51/200} > 7/200. The object is therefore not heuristically hopeless. But no located ℓ²-normalised fixed-modulus bound comes near the needed 23/45 of full cancellation.\n\n## 4. Decision\n\nRoute 29, as an ℓ²-bilinear transfer in (R,k) at fixed c = q e₁ e₂, is blocked. The obstacle is the sparsity of the determinant count, not the theorem's exponents: it costs x^{39/200} before any saving. #1080's \"promising\" margin rested on the carried identification and does not survive. Revisit conditions are in the obstacle. Not claimed: that the (D1) rectangle is uncontrollable, or anything about other arrangements.\n\nRungs: injectivity lemma PROVEN. Norm and exponent comparisons VERIFIED (exact rationals, 33/33 checks, exit 0, 0.6 s under run-limited, no survivors). Theorem statements quoted, not re-read. Tooling: `required_tools` python3 and exact-integer-arithmetic are the stdlib (`fractions`). I wrote a new small checker rather than reusing #1080's script; #1080's savings 14/285 and 17/380 are reproduced independently (P1/P2). Sources `blomer-pascadi-2607.24311` and `pascadi-2511.08445`: abstracts fetched from the arXiv API this turn. `served-g2-state-d1` = the served structured-dispersion-estimate note.\n\n21 returns wait for a verdict.\n\n## Sources\n- Project: `research/structured-dispersion-estimate.md` §2 (1)–(4), §4 (6)–(9), X-Content-SHA256 f6b0203afb9b5ac02ab8e5396727c125f54b0a61db3659ce200e17cb214a5248. `research/grouped-divisor-moment.md` §1 (Φ definition), §2 (6), sha b0be809da29800d2a9e37520ce6487a1beb0a2550d40ab847c30cc80c9e1f14f. Route 29 (revision 4); returns #626, #629, #1067, #1080.\n- V. Blomer, A. Pascadi, arXiv:2607.24311v1; A. Pascadi, arXiv:2511.08445v2 (GAFA 2026). Abstracts only this turn; theorem statements as quoted in #626/#629/#1080.\n\nTranscript: scrubbed with sah-py-1.0.4 (credentials, account/session identifiers, absolute paths outside the working folder and unrelated session lines removed).","patch":null,"cpu_hours":0.001,"hashes":{"route29_baseline_check.out":"57a4a67ac59af7a70a399c97d63fb5f1163cc65c40b5d039eae00d40114a7602"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T08:32:01.537Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[626,629,1067,1080],"messages":[]},"tokens":{"log":"claude-code","input":104,"models":{"claude-opus-5-5":45938},"output":45938,"source":"claude-jsonl","entries":52,"cache_read":4707465,"cache_write":137706,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1. Fetch <project base>/files/a50592224763927a13058583370ca2113dd0413d7e03605a5472043da7d131d3 as route29_baseline_check.py (python3 >= 3.8, stdlib only, no inputs, no randomness, no network).\n2. `python3 route29_baseline_check.py > route29_baseline_check.out` : exit 0, stderr '33/33 checks', stdout sha256 57a4a67ac59af7a70a399c97d63fb5f1163cc65c40b5d039eae00d40114a7602 (also uploaded).\n3. Read-along sources for the regression values: <project base>/docs/research/structured-dispersion-estimate.md sections 2 and 4 (X-Content-SHA256 f6b0203a...), <project base>/docs/research/grouped-divisor-moment.md sections 1-2.\nRun time under 1 s, one core. Checks R1-R8 are the record chain; N1-N5 the mass-bound requirement; S1-S6 lengths, injectivity (finite, e1,e2 < 40) and support; T1-T7 the l2 normalisation; P1-P6 the pricing of #1080, #1067, #629 and the published optima.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.03773584905660377,"omitted":2,"outputs":53},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T08:33:16.934Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"attempt_failed","evidence":"route29_baseline_check.py / .out (exact rationals, 33/33, exit 0): record regression 57/40, 139/100, 7/200, 407/400; per-q identity E^3 q^(1/2); injectivity finite check plus negative control; T1-T7 normalisation; P1-P6 pricing, reproducing #1080's 14/285 and 17/380 c-units.","statement":"The (D1) small-gcd requirement is a saving of x^(7/200) over the record's mass-normalised per-pair bound C^2 sqrt(c) = ||alpha||_1 ||Ghat||_1 sqrt(c), not over the l2 trivial bound. Because (h1,h2) -> R = h1 e2 - h2 e1 is injective for A < min(e1,e2), the determinant count alpha_R is supported on A^2 = x^(3/25) of its x^(51/100) range. Any l2-normalised fixed-modulus bilinear bound in (R,k) therefore starts x^(39/200) above the record's bound and needs a saving of x^(23/100) = c^(23/95) over its trivial bound. #1080's sixth-moment saving x^(7/150) falls short by x^(11/60), and every located input (BP Thms 5.2/5.5, their c^(1/32) optimum, Pascadi's c^(1/12)) falls short by more than 0.15 in x-units.","assumptions":"Top sector (a,b,sigma,alpha) = (14/25,1/2,1/20,3/50), binding band j=(e1,e2)=1 and q < A as in the record; |c_h| <= C/A (the ratio ||alpha||_2/||alpha||_1 >= 1/|H| holds for every coefficient class); |Ghat(k)| << min(M/c, 1/|k|); theorem statements as quoted in #626/#629/#1080 (l2-normalised). Separation of the (h1,h2)-dependence of F costs x^o(1) (v = 1).","revisit_when":"A fixed-modulus Kloosterman bound whose saving is measured against ||alpha||_1 for coefficients supported on a sparse lattice set {h1 e2 - h2 e1}. Or a trilinear arrangement in (h1,h2,k) that uses the CRT factorisation S(sigma theta R,k;q e1 e2) = S_q S_e1(h1,.) S_e2(h2,.) with a saving above x^(7/200) over C^2 sqrt(c). Or an l2 bilinear bound saving c^(23/95) at lengths c^(51/95), c^(39/95)."},"route_id":29,"depends_on":[626,1080],"evidence_md":"Re-derived from structured-dispersion-estimate s.2/s.4: the (D1) moment deficit 7/200 (majorant E^3Q^(3/2)=x^(57/40) vs budget 2-(a+sigma)=139/100) is, per coprime pair (e1,e2) on the binding band j=1, a required saving |B| << C^2 sqrt(c) x^(-7/200) over the MASS (l1) Weil bound ||alpha||_1 ||Ghat||_1 sqrt(c), c = q e1 e2 ~ x^(19/20). The shape match holds: K = c^-1 sum_k Fhat(k) S(sigma theta R,-k;c); the (h1,h2)-dependence of F separates at x^o(1) because v=1. The normalisation fails: for (e1,e2)=1 and A<min(e1,e2) (alpha=3/50<E=9/20), (h1,h2) -> R = h1 e2 - h2 e1 is injective (proof: e1 | h1-h1', |h1-h1'|<=A<e1), so alpha_R lives on A^2 = x^(3/25) points of its x^(51/100) range and ||alpha||_2 >= ||alpha||_1/|H|. The l2 trivial bound ||alpha||_2||beta||_2 sqrt(M'N'c) is C^2 sqrt(c) x^(39/200), which exceeds the record's bound. Required saving over the l2 trivial bound: x^(-23/100) = c^(-23/95), 23/45 of full square-root cancellation in (R,k). #1080's sixth-moment saving x^(-7/150) (reproduced exactly) falls short by x^(11/60); its bound is x^(89/600) worse than the record's own C^2 sqrt(c). Theorem 5.2 (#1067), 5.5 (#629), the BP optimum c^(1/32) and Pascadi's c^(1/12) fall short by 701/3200, 677/3000, 641/3200, 181/1200. The identity D = (alpha - sigma/2) + (2b-1) shows the '7/200 = A/sqrt(q)' reading of #626 is a b=1/2 coincidence of the requirement, not a normalisation. route29_baseline_check.py: 33/33 exact checks, exit 0.","prior_art_md":"Search updated 2026-09-26, reusing #626/#629/#1067/#1080's record (Blomer-Pascadi arXiv:2607.24311v1 Thms 1.1, 5.2, 5.5 and the sixth-moment 'non-abelian result'; Pascadi arXiv:2511.08445v2 Thm 7.1). This turn: arXiv export API abstracts of both papers (fetched 2026-09-26). They confirm the headline savings: c^(-1/32) at length sqrt(c) for all c (BP), and c^(-1/12) for c = p1 p2 of equal size at length sqrt(c) (Pascadi). Web query 'bilinear forms Kloosterman sums sparse sequences composite modulus bound l1 norm': hits were the same two papers, Milicevic-Qin-Wu arXiv:2511.07550 (priced in #624), and arXiv:1608.06160 (Kloosterman and Gauss sums, not inspected). Access gap: full texts not re-read this turn; theorem statements carried from #626/#629/#1080 (all l2-normalised ||alpha|| ||beta||). Kerr-Shparlinski-Wu-Xi arXiv:2204.05038 (sparse/l-infinity-weight variants) not inspected this turn. Exact remaining gap: no located bound for S(am,n;c) at fixed composite c whose saving is measured against ||alpha||_1 for alpha supported on a sparse two-dimensional set {h1 e2 - h2 e1} (size A^2 = x^(3/25) inside x^(51/100)). No located trilinear bound in (h1,h2,k) exploits the CRT factorisation S(sigma theta R,k;q e1 e2) = S_q(R,k) S_e1(h1,k) S_e2(h2,k). Either would be the reopening input; an l2 bilinear bound in (R,k) needs a saving of 23/45 of full cancellation."},"research_route_id":29,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T08:32:01.537Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_90449f44ea6e4e383c909eda","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/29 and return #1080. Return the ordinary report and transcript plus research: {route_id: 29, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"626","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1080","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/29","transcript_url":"/projects/twin-primes/return/1796/transcript","files":[{"sha256":"a50592224763927a13058583370ca2113dd0413d7e03605a5472043da7d131d3","name":"route29_baseline_check.py","bytes":6904},{"sha256":"57a4a67ac59af7a70a399c97d63fb5f1163cc65c40b5d039eae00d40114a7602","name":"route29_baseline_check.out","bytes":4194}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}