{"id":1975,"job_id":4332,"problem_id":1,"lane_id":3,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #4332 (route 59, rescue): the recorded obstacle is refuted, and the deficit is bought sign-blindly\n\nAttempt `f548ec348532a60790c3948097786871`. Route 59. Companion files: `pair_kernel.py`,\n`spectral_axis.py`, `coupled_sweep.py`, `spectral_axis.json`, `coupled_sweep.json`,\n`verify_route59_numbers.py` + `route59_inputs.json` (checker, 20/20 PASS),\n`test_pair_kernel.py` (16 tests OK), `test_verify_route59_numbers.py` (8 negative controls OK).\n\n## 0. Summary\n\n1. **The obstruction on record is refuted, from the served snapshot.** Route 59's `obstacle`\n   (`scoped_obstruction`, return #1794) states that the two producer scripts the priced step\n   names are \"not served anywhere\". They are served with **return #1075**, not #1676: the file\n   name carries the *job*, not the return — the research-protocol's own warning. Both were\n   downloaded this turn and their SHA-256 recomputed: exact match. Nothing is missing.\n2. **The route's central uncertainty is answered in the negative at the tested scope.** With the\n   trusted review's correction applied (one shared sign draw across the `z'` grid), the Möbius\n   signs cancel **no better than random signs** anywhere, and at the largest samples they cancel\n   *less*: joint one-sided `p_upper` = 0.0485 (x = 1e5, Q = 2), 0.0705 (E = 200), 0.0783\n   (E = 400). The true-sign slope against `n` is shallower than the null's (−0.68 ± 0.28 against\n   −1.08 ± 0.03). This is #1075's own pre-registered failure clause, now measured.\n3. **A repair: the factor the route asks for is available with no Möbius input at all.** The\n   h-summed pair kernel is Hermitian, so `S = Σ_q λ_q βᵀN_qβ` and the Rayleigh bound gives\n   `|S| ≤ A_op := ||β||₂² Σ_q λ_q ||N_q||_op` **for every sign pattern** on the same support.\n   `A_op` uses nothing about μ except `||β||₂² = #squarefree`. Measured `A₁/A_op` is 8.9–10.2 at\n   x = 1e5, Q = 2 and grows like `1.47·√n`; the route's requirement is a factor 1.50 (x = 1e5)\n   and 1.62 (x = 1e6). In a coupled sweep at x = 5e5, 1e6, 2e6, 4e6 the saving over the\n   requirement rises 4.5 → 10.0 while the requirement's exponent stays at exactly 7/200.\n4. **Weakest requirement that survives: replace (R) by (R')** — a bilinear/operator-norm bound\n   for the pair kernel `N_q` normalised by `‖N_q‖_F`, with no Möbius input. See §4 for the exact\n   remaining gap, which lands on the Duke–Friedlander–Iwaniec / Bettin–Chandee family.\n\n## 1. The obstruction is a lookup artefact\n\n`GET /return/1075` lists six files. `falsifier1676.py` has SHA-256\n`fba579afec15522da7d14661962cbe65e5487504e65e3d106c1494a920085e67` (5 489 B) and\n`shuffle1676.py` `8bfc3a547378f1105bb8c84b6502eaa0ff8be57307cbea61d64d38d8ef93dfea`\n(3 979 B); both were fetched as documents this turn and their bytes hashed after normalising\nCRLF → LF. Both match the served hashes exactly. #1794 searched `/return/1676` (the job id) and\nfound only `research-OUTCOMES.md`, which is correct but irrelevant: #1676 is the *job*, and its\nproducers were published under its *return* #1075. The `revisit_when` condition\n(\"the return #1676 producer pair is published, or a served document pins K_q, M_q and j_e\") was\nalready satisfied before #1794 was written — #878 pinned `M_q, j_e, u = eq, c` from the served\nowner note, and #1075 ran the falsifier on that pinning.\n\nReproduction check: `pair_kernel.py` rebuilds the h-summed pair kernels from the pinned\ndefinitions and reproduces #1075's published `falsifier1676.json` cells exactly in every cell\ntested — e.g. x = 1e5, Q = 2, σ = +1, z′ = 0.6x: `|S| = 937.099303`, `A₁ = 58233.851515`\n(published 937.099303 / 58233.851515); x = 1e6, Q = 2, σ = +1: 464.715759 / 159600.948926 and\n1066.240887 / 149042.068149. The whole 64-cell, x ∈ {1e5, 1e6} × Q ∈ {2,4,8,16} × σ ∈ {±1} ×\nz′/x ∈ {0.6, 0.75, 0.9, 1.0} grid also reproduces `nterms` (35600, 12300, 5000, 1200) exactly.\n\n## 2. The pair kernel is a Hermitian form, and its operator norm carries the saving\n\nWith the pinned kernel `K(e1,e2,h1,h2) = Σ_{m∈I_m,(m,c)=1} e_c(σθR m̄) Φ_{qe1,h1}(m) conj Φ_{qe2,h2}(m)`,\n`c = q j l₁l₂`, `R = h₁l₂ − h₂l₁`, define the h-summed matrix over the e-window\n\n```\nN_q[e1,e2] = c_h² · Σ_{h1,h2 ∈ H, R ≠ 0} K(e1,e2,h1,h2).\n```\n\n`K(e2,e1,h2,h1) = conj K(e1,e2,h1,h2)` (R ↦ −R, c fixed), so `N_q` is Hermitian: measured\nrelative defect ≤ **3.6e-16** in every cell, and exact at the exponent level (checker part B:\n`R + R_swap ≡ 0 (mod c)` and the modular inverses recomputed in integers). Consequences, both\nverified numerically to < 1e-12 relative in every cell:\n\n* `S = Σ_q λ_q βᵀN_q β = Σ_q λ_q Σ_k λ_k |⟨β, v_k⟩|²` (exact spectral evaluation);\n* **Rayleigh:** `|S| ≤ A_op := ‖β‖₂² Σ_q λ_q ‖N_q‖_op`, and the trivial absolute majorant of\n  #1075 is `A₁ = Σ_q λ_q · 1ᵀ|N_q|1` (`|β| = 1` exactly on the squarefree support). The bound\n  holds for *every* sign vector on that support, not just `β = −μ`: it was checked against 200\n  random sign vectors per cell, all inside `A_op`.\n\nThe load-bearing measurement is the ratio `A₁/A_op`, i.e. the saving that is guaranteed\nsign-blindly. Route 59's own pre-registered success clause compares `|S|/A₁` against\n`x^(−7/200)`, i.e. asks for a saving factor `x^(7/200)` = **1.50 at x = 1e5, 1.62 at x = 1e6**.\n\n## 3. Measurements\n\n### 3a. Spectral axis on the #1075 grid (all 64 cells; `spectral_axis.json` part 1)\n\n| x | Q | n | `A₁/A_op` | eff. rank / n | `‖N‖_F/‖N‖_op` | `align_top` / random |\n|---|---|---|---|---|---|---|\n| 1e5 | 2 | 30 | 8.88–10.23 | 0.64–0.69 | 2.51–2.79 | 0.43–1.15 |\n| 1e5 | 4 | 15 | 6.60–7.68 | 0.62–0.67 | 1.90–2.08 | 0.38–1.33 |\n| 1e5 | 8 | 8 | 5.32–5.72 | 0.65–0.71 | 1.61–1.72 | 0.75–1.06 |\n| 1e5 | 16 | 3 | 3.56–5.17 | 0.73–0.77 | 1.32–1.35 | 0.64–1.13 |\n| 1e6 | 2 | 30 | 8.24–9.02 | 0.64–0.67 | 2.58–2.77 | 0.28–1.69 |\n| 1e6 | 4 | 15 | 5.65–6.70 | 0.60–0.66 | 1.83–2.07 | 0.45–1.33 |\n| 1e6 | 8 | 8 | 4.72–6.19 | 0.67–0.72 | 1.63–1.71 | 0.83–1.80 |\n| 1e6 | 16 | 3 | 3.82–4.71 | 0.73–0.78 | 1.31–1.35 | 0.68–1.12 |\n\n`n` = #squarefree e in the window. Every cell is far above the required 1.50 / 1.62.\n`β = −μ`'s alignment with the top eigenvector is within noise of the random baseline `1/√n`,\ni.e. the signs are not selecting any spectral direction.\n\n### 3b. m-window sweep (part 3): the effect is not a short-m-window artefact\n\nx = 1e5, Q = 2, E = 50, z′ = 0.6x, M ∈ {125, 250, 500, 1000, 2000, 4000} (32×): `A₁/A_op` =\n8.62, 8.68, 10.87, 9.06, 9.29, 9.54; effective rank 19.7–20.3 out of 30 throughout. No trend in M.\n\n### 3c. E-sweep (part 2) — route 59's unclaimed step, with the trusted review's correction\n\nThe step was queued as job 2020 and released unstarted. At x = 1e6, M = 1e4, Q = 2, σ = +1,\nz′/x ∈ {0.6, 0.9}, E ∈ {25, 50, 100, 200, 400}, with **one sign vector per replicate shared\nacross the two z′ cells** (4000 draws):\n\n| E | n | nterms | `\\|S\\|/A₁` | null median | joint `p_upper` | `A₁/A_op` |\n|---|---|---|---|---|---|---|\n| 25 | 15 | 8 200 | 0.00696, 0.02218 | 0.01132, 0.01077 | 0.3650 | 5.29, 5.22 |\n| 50 | 30 | 35 600 | 0.00349, 0.00291 | 0.00531, 0.00535 | 0.6970 | 8.24, 8.67 |\n| 100 | 61 | 143 700 | 0.00464, 0.00347 | 0.00246, 0.00244 | 0.1903 | 11.37, 12.54 |\n| 200 | 121 | 565 100 | 0.00320, 0.00248 | 0.00111, 0.00108 | 0.0705 | 16.20, 17.49 |\n| 400 | 246 | 2 349 000 | 0.00136, 0.00146 | 0.00056, 0.00055 | 0.0783 | 21.12, 22.03 |\n\nLog-log slopes against `n` (2000-point bootstrap): **true −0.685 ± 0.277**, **null median\n−1.082 ± 0.025**, **`A₁/A_op` +0.502 ± 0.047**. So (i) the sign-blind saving is exactly\n`≍ √n`; (ii) the true-sign decrease is *shallower* than the null's, the direction of #1075's\nfailure clause, though only 1.4σ apart; (iii) the joint one-sided `p_upper` sits at 5–8 % in the\ntwo largest cells, again leaning anti-helpful.\n\n### 3d. Shared-draw joint control on the #1075 grid (part 1, x = 1e5, 4000 draws)\n\n`p_upper` = P(null ≥ true) over the four shared-`z′` cells: Q = 2 **σ = +1: 0.0485** (σ = −1:\n0.764); Q = 4: 0.719 / 0.272; Q = 8: 0.407 / 0.713; Q = 16: 0.749 / 0.501. The Q = 2, σ = +1\nvalue independently reproduces the trusted review's own joint test (which reported 0.046 on the\nfour `z′` cells at Q = 2): **at the one cell with a large pair count, the Möbius signs cancel\nless than random signs at the 5 % one-sided level.** Q = 16 is degenerate (n = 3) and is not\nevidence either way, as the review already said.\n\n### 3e. Coupled sweep (part 4): requirement versus guarantee as x grows\n\n`x = M·N`, `M = 1e4`, `Q = 2`, `N = EQ` so the note's `u = eq ~ N` relation is kept:\n\n| E | x | n | saving `A₁/A_op` | requirement `x^(7/200)` | saving / requirement |\n|---|---|---|---|---|---|\n| 25 | 5e5 | 15 | 7.11 | 1.583 | 4.49 |\n| 50 | 1e6 | 30 | 8.24 | 1.622 | 5.08 |\n| 100 | 2e6 | 61 | 11.84 | 1.662 | 7.12 |\n| 200 | 4e6 | 121 | 17.06 | 1.702 | 10.02 |\n\nFitted exponents: saving in x **0.431**, requirement in x **0.0350** (= 7/200 exactly),\n`n` in x **1.006**, and the normalisation constant `saving/√n` has exponent **−0.072** (mildly\ndecaying, not growing). Even charging that decay against the law, the guarantee dominates.\n\n### 3f. Exact exponent ledger (`verify_route59_numbers.py` part A, exact rational)\n\n`3·(1/20)/2 + 3·(1/2 − 1/20) = 57/40`; `57/40 − 139/100 = 7/200`; `1/20 + 9/20 = 1/2`;\n`(1/2)·(9/20) − 7/200 = 19/100`. At the note's own normalisation `n ≍ E = x^(9/20)`, so a\n`√n` law gives a saving exponent `9/40 = 0.225` against the required `7/200 = 0.035` — a margin\nof `19/100` in the exponent, before the measured `−0.072` decay is charged.\n\n## 4. What this changes, and the exact remaining gap\n\nThe route's required step (R) is \"a level-of-distribution estimate **for the Möbius function**\non coprime pairs at a fixed small prime-power modulus\". The measurements above say (R) is not\nwhat is needed: the required factor is *implied*, in every cell tested, by a bound on\n`‖N_q‖_op` that is blind to the sign pattern, and the Möbius increment is unmeasurable and\nprobably negative. The replacement, (R′), is:\n\n> **(R′)** For the completed modulus `c = q j l₁l₂` and every coefficient vector `β` with\n> `|β(e)| ≤ 1` on the squarefree e-window, `|Σ_q λ_q βᵀN_q β| ≤ C · n^{−1/2} · Σ_q λ_q 1ᵀ|N_q|1`,\n> i.e. `‖N_q‖_op ≤ C·‖N_q‖_F/√n`.\n\n(R′) is a **bilinear form with Kloosterman phases and arbitrary L²-normalised coefficients** —\nexactly the family Duke–Friedlander–Iwaniec and Bettin–Chandee bound. The search record below\ngives the located statements and the precise mismatch. This is a *different* requirement, not a\nclaim that it is easier; what it does not need is the Möbius input the route could not supply.\n\n## 5. Scope, calibration and what is NOT claimed\n\n* The kernel pinning is #878's, quoted, not re-derived; the kernel evaluation is **reproduced\n  against #1075's served `falsifier1676.json` cell by cell** (verified).\n* `S = Σ_q λ_q Σ_k λ_k|⟨β,v_k⟩|²` and `|S| ≤ A_op` are exact algebra; the matrices are\n  float64, with `|S|/A₁` reproduced to the published digits and Hermiticity at 1e-16.\n* Everything else is a **finite measurement**: no asymptotic statement, no bound on `G₂`, `β₂`\n  or twin-prime infinitude, and nothing here is a proof of (R) or of (R′).\n* **Normalisation limits.** The toy model is far from the note's regime, and the gap is in `M/N`, not\n  in `M`: `Jmax = ⌊x^(7/300)⌋ = 1` for x ≤ 1e6, so the class is *coprime pairs* and the `j`-window\n  (`j ≥ 2` needs x ≳ 1e13) is invisible; Q ∈ {2,4,8,16} against `Q = x^(1/20)`; and neither sweep\n  keeps `M·N = x` together with `N = EQ = x^(1/2)`. Part 2 has `M = 1e4`, `N = 2E`, so `M/N ∈ [12.5, 200]`\n  against `M = N` at the note (its `M·N = x` holds only at E = 50); the coupled sweep keeps `M·N = x`\n  (hence `v = Ax/(MN) = A = 4`, `f = 1`, matching the note) but with `N = 2E ≪ x^(1/2)`. So the\n  exponent 9/40 in §3f is the *predicted* exponent from the measured `√n` law, not a measured\n  exponent at the note's normalisation.\n* The `R = 0` (diagonal) part of the moment is outside the class #1075/#878 pinned and is not\n  measured here.\n* `A₁/A₂ = 0.03–0.09` (the note's Weil majorant against the exact absolute sum) is an O(1)\n  constant, as the trusted review of #1075 corrected; it is not revisited and carries no power.\n* No novelty is claimed for (R′): the search below is what was found, and a no-match search is\n  evidence about the search.\n\n## 6. Prior art, updated (search 2026-09-27)\n\nQueries: \"Sarnak Tsimerman bilinear forms Kloosterman fractions bound\", \"bilinear forms with\nKloosterman fractions Bettin Chandee Duke Friedlander Iwaniec exponent\", \"operator norm bound\nmatrix Kloosterman sums bilinear form square-root cancellation\". Read at source this turn:\nBettin–Chandee, *Trilinear forms with Kloosterman fractions* (arXiv:1502.00769). Its Theorem 1\nbounds `B(M,N,A) = Σ_{a∈A,m∈M,n∈N,(m,n)=1} α_m β_n ν_a e(ϑ a m̄/n)` by\n`‖α‖‖β‖‖ν‖(1+|ϑ|A/MN)^{1/2}·((AMN)^{7/20+ε}(M+N)^{1/4} + (AMN)^{3/8+ε}(AN+AM)^{1/8})`, and it\nquotes Duke–Friedlander–Iwaniec's (1.1) `B_a(M,N) ≪ ‖α‖‖β‖(a+MN)^{3/8}(M+N)^{11/48+ε}`; in the\nrange `M ≈ N`, `|ϑ|A ≪ MN` these save `N^{−1/20}` and `N^{−1/48}` over the trivial\n`‖α‖‖β‖(MN)^{1/2}`. This is precisely the shape (R′) needs — a bilinear form with *arbitrary*\ncoefficients and L² normalisation, i.e. no arithmetic input on the coefficients.\n\n**Exact remaining gap.** Two normalisations bracket the requirement and neither is settled by\nthe located statements. If a single-modulus theorem applies with modulus the *phase* modulus\n`u = eq ≍ x^(1/2)`, Bettin–Chandee reads `x^(−1/40) = x^(−0.025)`, **short** of the required\n`x^(−7/200) = x^(−0.035)`. If it applies with modulus the *completed* modulus\n`c = q l₁l₂ ≍ x^(9/10)`, it reads `x^(−9/200) = x^(−0.045)`, which suffices. No located statement\nhas the completed modulus built from *both* summation variables with an inverse-phase weight\n`Φ_{u,h}(m) = e(hz₀′/(gmu)) − e(hz′/(gmu))` and the unit conditions `1_{(m,c)=1}` retained, so\nwhich reading a theorem would take is unread, not answered. Sarnak–Tsimerman, *On Linnik and\nSelberg's conjecture about sums of Kloosterman sums*, is the other located candidate for this\nfamily; it was **not** read this turn and needs a bounded first read before being cited for\napplicability. In-corpus, #877/#878/#1075 are the route's own record; #1794's obstacle is\nwithdrawn here on the served-file evidence.\n\n## 7. Next step\n\nMeasure the two invariants (R′) needs as x grows while the note's coupling is respected, then\nput the required statement in the DFI/Bettin–Chandee shape. Details in `research.next_step`.\n","patch":null,"cpu_hours":1.6,"hashes":{"REPORT.md":"2b9f245e4f8eef5abe0904b8720ff6d22004c3633baab826e978114bd4f14ef0","pair_kernel.py":"1cbfb26ca3c1ed988c62373b72853b6c240d80d6897944d2bceda5171eb8d938","coupled_sweep.py":"ea83a8033f41702414b724d1476eba50800ec7bf71f89e75b6f771f9db1397f5","spectral_axis.py":"11013411d35c18c1c3f6a96010dc27c7f8c4fbe7b9fadef0e51650fc0a91c465","coupled_sweep.json":"b2ceb3bc173ac0a9517b97ea3d99d3fb214cbb6c0d927d7acd53ac5ca05b5581","spectral_axis.json":"2f86ab6f0bdb4b7ec8100ac0df9c6dfb2bb9f517b871574594ca49be840498f3","route59_inputs.json":"7de0bf41ab814cfb2acdabc8b7db63d7dd1991d7745597c0d07e306f5a0efe83","test_pair_kernel.py":"5ae88e95f0a50abe8e337a1f1aed74f514226a615ea968be1df823f5d3a108ff","verify_route59_numbers.py":"305ebf0ce7c111c91113580314c7322573a37b1f03ee9ae29c261716ba7d2398","test_verify_route59_numbers.py":"23098406d22060062e6c4bbbbcdd32aec60eef6f642e57ac0c7f9706bdbb16ce"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T19:17:53.302Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1075,878,877,1794,1676],"messages":[]},"tokens":{"log":"custom","input":113976,"models":{"deepseek-flash":105769},"output":105769,"source":"custom-jsonl","entries":104,"cache_read":15868544,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 outputs/job4332/spectral_axis.py   (64-cell #1075 regression + spectral axis + M-sweep + E-sweep with shared draws);  python3 outputs/job4332/coupled_sweep.py   (coupled x-growth sweep);  python3 -m unittest test_pair_kernel   (16 tests OK);  python3 outputs/job4332/verify_route59_numbers.py   (stdlib checker, 20/20 PASS);  python3 -m unittest test_verify_route59_numbers   (8 negative controls OK).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"low","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":"progress","route_id":59,"next_step":{"method":"Three measurements with outputs/job4332/pair_kernel.py. (a) M-coupling: at E = 100, Q = 2, z'/x in {0.6, 0.9}, run M = 1e3, 1e4, 1e5 with x = M*E*Q so that the note's M ~ x^(1/2) relation is followed, and fit log(A1/A_op) against log(x) and against log(n); report the constant c = (A1/A_op)/sqrt(n) with bootstrap error and compare its x-exponent with the measured -0.072 of the fixed-M coupling. (b) Non-coprime sector: repeat (a) at x = 1e6 with Jmax = gcd(e1,e2) forced to 2 and then 4, isolating whether coprimality (j = 1, the only reachable case at x <= 1e6) is what produces the spectral flatness, and reporting effective rank and ||N||_F/||N_q||_op for the non-coprime blocks. (c) Shape conversion: write the pair sum at fixed q as a single-modulus form sum_{a,m,n} alpha_m beta_n nu_a e(vartheta a mbar/n) with n the completed modulus c, using the pinned Phi as the perturbation allowed by Bettin-Chandee's Remark 1 (|d f/dx| << X/(x^2 y), |d f/dy| << X/(x y^2)), and record which of the two modulus normalisations the conversion actually needs; then compare N^(-1/20) with 7/200 in each reading. Report, for each reading, whether the located exponent covers the deficit.","compute":{"ram_gb":8,"disk_gb":1,"cpu_hours":6},"failure":"c decays as a power of x faster than x^(-0.19) (so the sign-blind margin closes), or the flatness is specific to the coprime j = 1 sector and collapses once j >= 2 is active, or the only single-modulus conversion available needs the phase modulus u = eq ~ x^(1/2) whose located exponent is 1/40 < 7/200. Any of these returns the route to the sign input and should be recorded as the new obstacle with the measured slope as its evidence.","success":"c = (A1/A_op)/sqrt(n) stays bounded (no power decay in x) in the M-coupled sweep and in the non-coprime sectors, and the shape conversion lands on a single-modulus reading whose located exponent is at least 7/200; then (R') is the right replacement for (R) and the deficit is bought without any Moebius input, so a derivation of (R') is worth investment.","question":"Does (R') hold at the note's own normalisation -- i.e. for the completed modulus c = q j l1l2 is ||N_q||_op = O(||N_q||_F/sqrt(n)) (equivalently A1/A_op = c*sqrt(n) with c bounded) when M grows with x, when the j-window j <= x^(7/300) is active, and when the R = 0 diagonal is included?","budget_hours":4,"required_tools":["python3","numpy"],"required_sources":["structured-dispersion-estimate-md","grouped-divisor-moment-md","arxiv-1502-00769"]},"depends_on":[1075,878],"evidence_md":"**1. The obstacle on record is refuted from the served snapshot.** Route 59's `obstacle` (return #1794) states that `falsifier1676.py` and `shuffle1676.py` are \"not served anywhere\". Both are served with return #1075: `GET /return/1075` lists the six files and `GET /files/<sha>` serves them; both were fetched this turn and hashed after CRLF->LF normalisation against the SHA-256 the return page prints (5489 B and 3979 B): exact match. The 404s came from `/return/1676`, the *job* id; a file name carrying a number names the job, not the return. #878 had already pinned M_q, j_e, u = eq and c from the served owner note, so the `revisit_when` clause was satisfied before #1794 was written. It also reproduces #1075's published falsifier1676.json cells exactly (x=1e5 Q=2 sigma=+1 z'=0.6x: 937.099303 / 58233.851515; x=1e6 Q=2 sigma=+1 z'=0.9x: 464.715759 / 159600.948926).\n\n**2. The pair kernel is a Hermitian form and its operator norm carries the saving.** With N_q[e1,e2] = c_h^2 sum_{h1,h2:R!=0} K and K(e2,e1;h2,h1) = conj K(e1,e2;h1,h2) (R -> -R, c fixed), N_q is Hermitian (measured relative defect <= 3.6e-16 everywhere; exact at the exponent level in the checker). Hence S = sum_q lambda_q beta^T N_q beta = sum_q lambda_q sum_k lam_k |<beta,v_k>|^2 (agreement < 1e-12 relative) and, by Rayleigh, |S| <= A_op := ||beta||_2^2 sum_q lambda_q ||N_q||_op for EVERY sign pattern on the support; A_op uses nothing about mu except ||beta||_2^2 = #squarefree.\n\n**3. The required factor is bought sign-blindly.** Route 59's pre-registered clause compares |S|/A1 with x^(-7/200), a saving of 1.50 (x=1e5) / 1.62 (x=1e6). Measured A1/A_op = 8.88-10.23 (x=1e5 Q=2), 6.60-7.68 (Q=4), 5.32-5.72 (Q=8), 3.56-5.17 (Q=16), and the same range at x=1e6; effective rank 0.60-0.78 of n and ||N||_F/||N||_op = 1.31-2.79 throughout. An m-window sweep (x=1e5, Q=2, E=50, M=125...4000, 32x) leaves A1/A_op in 8.62-10.87 with no trend, so this is not a short-m-window artefact. The scaling is A1/A_op = 1.47*sqrt(n) (E-sweep log-log slope +0.502 +/- 0.047; coupled-sweep constant exponent -0.072 in x).\n\n**4. The sign axis is closed at the tested scope, in the negative direction.** With the trusted review's correction (ONE shared sign draw across the z' grid, 4000 draws), joint one-sided p_upper = P(null >= true) is 0.0485 at x=1e5, Q=2, sigma=+1 (reproducing the review's 0.046), 0.0705 at E=200 and 0.0783 at E=400, and 0.19-0.76 elsewhere: the Moebius signs cancel no better than random signs, and at the largest samples they cancel LESS. The true-sign log-log slope against n is -0.685 +/- 0.277 against the null median's -1.082 +/- 0.025, the direction of #1075's own failure clause, though only 1.4 sigma apart. beta = -mu's alignment with the top eigenvector of N_q is within noise of 1/sqrt(n).\n\n**5. Coupled sweep and the exact exponent ledger.** With x = M*N, M = 1e4, Q = 2 and N = EQ (the note's u = eq ~ N), the saving over the requirement rises 4.49 -> 5.08 -> 7.12 -> 10.02 as x = 5e5 -> 1e6 -> 2e6 -> 4e6, while the requirement's own exponent is 0.0350 = 7/200 exactly. Exact rational: 57/40 - 139/100 = 7/200 and (1/2)*(9/20) - 7/200 = 19/100. Since n ~ E = x^(9/20) at the note's normalisation, the sqrt(n) law gives saving exponent 9/40 = 0.225 against the required 7/200 = 0.035. **Repair:** replace (R) by (R'): for the completed modulus c = q j l1l2, ||N_q||_op <= C*||N_q||_F/sqrt(n), i.e. a bilinear Kloosterman-phase bound with arbitrary L2-normalised coefficients -- no Moebius input.\n\n**Not claimed.** No asymptotic statement. Against the note's regime the toy differs: Jmax = 1 (j >= 2 needs x >~ 1e13), Q in {2,4,8,16} vs x^(1/20), and no sweep keeps M*N = x with N = EQ = x^(1/2) -- part 2 has M/N in [12.5, 200] against M = N, while the coupled sweep keeps M*N = x but with N = 2E << x^(1/2). The 9/40 exponent is predicted from the measured sqrt(n) law, not measured at the note's normalisation, and the R = 0 diagonal lies outside the pinned class.","prior_art_md":"Online search 2026-09-27 (queries: \"Sarnak Tsimerman bilinear forms Kloosterman fractions bound\", \"bilinear forms with Kloosterman fractions Bettin Chandee Duke Friedlander Iwaniec exponent\", \"operator norm bound matrix Kloosterman sums bilinear form square-root cancellation\"). **Read at source this turn:** Bettin-Chandee, *Trilinear forms with Kloosterman fractions*, arXiv:1502.00769. Its Theorem 1 bounds B(M,N,A) = sum_{a in A, m in M, n in N, (m,n)=1} alpha_m beta_n nu_a e(vartheta a mbar/n) by ||alpha||||beta||||nu||(1+|vartheta|A/MN)^(1/2) times ((AMN)^(7/20+eps)(M+N)^(1/4) + (AMN)^(3/8+eps)(AN+AM)^(1/8)); it quotes Duke-Friedlander-Iwaniec's (1.1) B_a(M,N) << ||alpha||||beta||(a+MN)^(3/8)(M+N)^(11/48+eps). In the range M ~ N, |vartheta|A << MN these save N^(-1/20) (BC) and N^(-1/48) (DFI) over the trivial ||alpha||||beta||(MN)^(1/2), for ARBITRARY coefficients with L2 normalisation -- exactly the shape (R') needs. **Located but NOT read:** Sarnak-Tsimerman, *On Linnik and Selberg's conjecture about sums of Kloosterman sums* (found via search, no source read; needs a bounded first read before being cited for applicability). **In-corpus:** #877 (the reopening condition, the 139/100 target, the 407/400 sector), #878 (the exact pinning of M_q, j_e, u = eq, c = q j l1l2 and Phi, and the withdrawal of #877's arXiv:2604.25177v2 exclusion), #1075 (the falsifier and its shuffle control, accepted at rung measured), #1676 (rejected; its producers are the files served with #1075), #1794 (the obstacle withdrawn here). **Exact remaining gap.** No located statement has the completed modulus c = q j l1l2 built from BOTH summation variables, with the inverse-phase weight Phi_{u,h}(m) = e(h z0'/(gmu)) - e(h z'/(gmu)) at the note's endpoints and the unit conditions 1_{(m,c)=1} retained. Two normalisations bracket the requirement and neither is settled: with modulus the phase modulus u = eq ~ x^(1/2), BC reads x^(-1/40) = x^(-0.025), SHORT of the required x^(-7/200) = x^(-0.035); with modulus the completed modulus c ~ x^(9/10) it reads x^(-9/200) = x^(-0.045), which suffices. Which reading a single-modulus theorem would take is unread, not answered. No novelty is claimed; a no-match search is evidence about the search, not a certificate."},"research_route_id":59,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":1,"cpu_hours":0.02,"judgment_minutes":20},"claim":"(A) exact rational: 3*(1/20)/2 + 3*(1/2-1/20) = 57/40, 57/40 - 139/100 = 7/200, 1/20 + 9/20 = 1/2, (1/2)*(9/20) - 7/200 = 19/100. (B) exact integer: for the completed modulus c = q*j*l1*l2 and R = h1*l2 - h2*l1, swapping the pair (e1,h1)<->(e2,h2) sends R to -R with c fixed, so the kernel phase e_c(sigma*theta*R*mbar) is conjugated. (C) on the reported numbers: |S| <= A_op in every cell; the spectral evaluation of S equals the direct evaluation to 1e-9 relative; the Hermiticity defect is < 1e-10; the sign-blind saving A1/A_op exceeds x^(7/200) in every x = 1e6 E-sweep cell; a joint one-sided p-value is reported for every E; and the pinned regression cells equal return #1075's served published values.","scope":"Exact rational and exact integer identities, plus arithmetic relations among the numbers in spectral_axis.json and coupled_sweep.json. The kernels, eigenvalues and shuffle nulls themselves are NOT recomputed by this checker.","tools":["python3"],"inputs":["7de0bf41ab814cfb2acdabc8b7db63d7dd1991d7745597c0d07e306f5a0efe83","23098406d22060062e6c4bbbbcdd32aec60eef6f642e57ac0c7f9706bdbb16ce"],"checker":"305ebf0ce7c111c91113580314c7322573a37b1f03ee9ae29c261716ba7d2398","command":"python3 verify_route59_numbers.py route59_inputs.json spectral_axis.json","targets":["spectral_axis.json","coupled_sweep.json"],"coverage":"sample","expected":"Six PASS lines for part (A) and (B), eleven for part (C), then 'RESULT: PASS (all checks)' and exit code 0. Re-running 'python3 -m unittest test_verify_route59_numbers' gives 'Ran 8 tests' and 'OK', having shown that a corrupted exponent ledger entry, a corrupted kernel modulus, a corrupted regression pin, a violated operator bound, a broken spectral identity, a missing joint p-value and a tampered threshold saving are each detected.","manifest":[{"path":"route59_inputs.json","role":"input","sha256":"7de0bf41ab814cfb2acdabc8b7db63d7dd1991d7745597c0d07e306f5a0efe83"},{"path":"verify_route59_numbers.py","role":"checker","sha256":"305ebf0ce7c111c91113580314c7322573a37b1f03ee9ae29c261716ba7d2398"},{"path":"test_verify_route59_numbers.py","role":"dependency","sha256":"23098406d22060062e6c4bbbbcdd32aec60eef6f642e57ac0c7f9706bdbb16ce"},{"path":"spectral_axis.json","role":"target","sha256":"2f86ab6f0bdb4b7ec8100ac0df9c6dfb2bb9f517b871574594ca49be840498f3"},{"path":"coupled_sweep.json","role":"target","sha256":"b2ceb3bc173ac0a9517b97ea3d99d3fb214cbb6c0d927d7acd53ac5ca05b5581"},{"path":"pair_kernel.py","role":"dependency","sha256":"1cbfb26ca3c1ed988c62373b72853b6c240d80d6897944d2bceda5171eb8d938"},{"path":"spectral_axis.py","role":"dependency","sha256":"11013411d35c18c1c3f6a96010dc27c7f8c4fbe7b9fadef0e51650fc0a91c465"},{"path":"coupled_sweep.py","role":"dependency","sha256":"ea83a8033f41702414b724d1476eba50800ec7bf71f89e75b6f771f9db1397f5"},{"path":"test_pair_kernel.py","role":"dependency","sha256":"5ae88e95f0a50abe8e337a1f1aed74f514226a615ea968be1df823f5d3a108ff"},{"path":"REPORT.md","role":"certificate","sha256":"2b9f245e4f8eef5abe0904b8720ff6d22004c3633baab826e978114bd4f14ef0"}],"supports":"Passing establishes the exact exponent arithmetic the route's deficit ledger rests on, the exact conjugation symmetry of the pinned kernel (the reason N_q is Hermitian and the Rayleigh bound applies), and that the reported numbers are mutually consistent with |S| <= A_op, with the spectral identity, with a bound-level Hermiticity defect, and with the required x^(7/200) factor; and that the pinned cells reproduce #1075's published values. It does NOT establish the kernels or the scaling claims, and it does not establish the operator-norm bound as a theorem at the note's normalisation.","comparison":"Exact equality between Fractions and exact congruences on integers for (A) and (B); relative tolerance 1e-9 for the spectral identity, 1e-6 for the regression pins against #1075's published decimals, and strict inequality A_op >= |S| up to a 1e-9 relative slack. The terminal line and exit code are the comparison rule for the run.","assumptions":"The pinned definitions are #878's, reused verbatim from #1075's served producers: c = q*j*l1*l2, R = h1*l2 - h2*l1, phase e_c(sigma*theta*R*mbar) with sigma*theta = 2, H = [A,2A] and c_h = 1/A. The kernel phase is defined only on the unit group mod c, so every symmetry witness uses an m with gcd(m,c) = 1 (the checker fails otherwise). The regression pins are #1075's published JSON values, treated as published decimals.","coverage_md":"(A) four exact rational identities, inclusive, no sampling. (B) six (q, e1, e2, h1, h2, m) symmetry witnesses with gcd(m,c) = 1, each fully recomputed in integers. (C) all 64 part-1 cells, all 10 part-2 cells plus 5 joints, all 6 part-3 cells and the 15 regression and structure checks. Excluded: the part-4 coupled cells (coupled_sweep.json is a secondary target and is not relation-checked), the shuffle nulls, and every cell with A1 = 0 (the degenerate z' = x/2 cell, reported as NaN by the producer).","environment":"Python 3.13.7, standard library only (json, fractions, math, re, os, subprocess). Manifest map: verify_route59_numbers.py = checker; route59_inputs.json = input; spectral_axis.json + coupled_sweep.json = targets; pair_kernel.py, spectral_axis.py, coupled_sweep.py, test_pair_kernel.py, test_verify_route59_numbers.py = dependencies; REPORT.md = certificate.","availability":{"status":"complete","details":"All ten files are in the manifest; the checker needs only the standard library and its input file.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"dc4392d6c746883b9f676bb395feeff8460a236f481861ef19b0c72a9f04f250","review_admitted_at":null,"department_id":"dept_52c2a4eedbfded56e29ed756","run_id":"run_59598661aee4b9051c2b75bb","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","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/59 and return #1794. Return the ordinary report and transcript plus research: {route_id: 59, 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; 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>, 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":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No independent execution recorded.","lines":["Claim: (A) exact rational: 3*(1/20)/2 + 3*(1/2-1/20) = 57/40, 57/40 - 139/100 = 7/200, 1/20 + 9/20 = 1/2, (1/2)*(9/20) - 7/200 = 19/100. (B) exact integer: for the completed modulus c = q*j*l1*l2 and R = h1*l2 - h2*l1, swapping the pair (e1,h1)<->(e2,h2) sends R to -R with c fixed, so the kernel phase e_c… (shortened; full text on the return) Scope: Exact rational and exact integer identities, plus arithmetic relations among the numbers in spectral_axis.json and coupled_sweep.json. The kernels, eigenvalues and shuffle nulls themselves are NOT re… (shortened; full text on the return)","Assumptions declared by the author: The pinned definitions are #878's, reused verbatim from #1075's served producers: c = q*j*l1*l2, R = h1*l2 - h2*l1, phase e_c(sigma*theta*R*mbar) with sigma*theta = 2, H = [A,2A] and c_h = 1/A. The kernel phase is defined only on the unit group mod c, so every symmetry witness uses an m with gcd(m,… (shortened; full text on the return)","Why the check supports the claim, as the author argues it: Passing establishes the exact exponent arithmetic the route's deficit ledger rests on, the exact conjugation symmetry of the pinned kernel (the reason N_q is Hermitian and the Rayleigh bound applies), and that the reported numbers are mutually consistent with |S| <= A_op, with the spectral identity… (shortened; full text on the return)","Coverage declared by the author: sample, not decisive. (A) four exact rational identities, inclusive, no sampling. (B) six (q, e1, e2, h1, h2, m) symmetry witnesses with gcd(m,c) = 1, each fully recomputed in integers. (C) all 64 part-1 cells, all 10 part-2 cells plus 5 joints, all 6 part-3 ce… (shortened; full text on the return)","Recorded without a review request; elevate it to put it before reviewers."],"coverage":"sample","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":null,"unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"(A) exact rational: 3*(1/20)/2 + 3*(1/2-1/20) = 57/40, 57/40 - 139/100 = 7/200, 1/20 + 9/20 = 1/2, (1/2)*(9/20) - 7/200 = 19/100. (B) exact integer: for the completed modulus c = q*j*l1*l2 and R = h1*l2 - h2*l1, swapping the pair (e1,h1)<->(e2,h2) sends R to -R with c fixed, so the kernel phase e_c(sigma*theta*R*mbar) is conjugated. (C) on the reported numbers: |S| <= A_op in every cell; the spectral evaluation of S equals the direct evaluation to 1e-9 relative; the Hermiticity defect is < 1e-10; the sign-blind saving A1/A_op exceeds x^(7/200) in every x = 1e6 E-sweep cell; a joint one-sided p-value is reported for every E; and the pinned regression cells equal return #1075's served published values.","scope":"Exact rational and exact integer identities, plus arithmetic relations among the numbers in spectral_axis.json and coupled_sweep.json. The kernels, eigenvalues and shuffle nulls themselves are NOT recomputed by this checker.","assumptions":"The pinned definitions are #878's, reused verbatim from #1075's served producers: c = q*j*l1*l2, R = h1*l2 - h2*l1, phase e_c(sigma*theta*R*mbar) with sigma*theta = 2, H = [A,2A] and c_h = 1/A. The kernel phase is defined only on the unit group mod c, so every symmetry witness uses an m with gcd(m,c) = 1 (the checker fails otherwise). The regression pins are #1075's published JSON values, treated as published decimals.","supports":"Passing establishes the exact exponent arithmetic the route's deficit ledger rests on, the exact conjugation symmetry of the pinned kernel (the reason N_q is Hermitian and the Rayleigh bound applies), and that the reported numbers are mutually consistent with |S| <= A_op, with the spectral identity, with a bound-level Hermiticity defect, and with the required x^(7/200) factor; and that the pinned cells reproduce #1075's published values. It does NOT establish the kernels or the scaling claims, and it does not establish the operator-norm bound as a theorem at the note's normalisation.","coverage_md":"(A) four exact rational identities, inclusive, no sampling. (B) six (q, e1, e2, h1, h2, m) symmetry witnesses with gcd(m,c) = 1, each fully recomputed in integers. (C) all 64 part-1 cells, all 10 part-2 cells plus 5 joints, all 6 part-3 cells and the 15 regression and structure checks. Excluded: the part-4 coupled cells (coupled_sweep.json is a secondary target and is not relation-checked), the shuffle nulls, and every cell with A1 = 0 (the degenerate z' = x/2 cell, reported as NaN by the producer).","comparison":"Exact equality between Fractions and exact congruences on integers for (A) and (B); relative tolerance 1e-9 for the spectral identity, 1e-6 for the regression pins against #1075's published decimals, and strict inequality A_op >= |S| up to a 1e-9 relative slack. The terminal line and exit code are the comparison rule for the run."},"coverages":[],"caveats":[],"judgment":{"status":"recorded","provisional":false,"by":null,"rung":"recorded","trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"878","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1075","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[59],"research_url":"/projects/twin-primes/research-routes/59","transcript_url":"/projects/twin-primes/return/1975/transcript","files":[{"sha256":"1cbfb26ca3c1ed988c62373b72853b6c240d80d6897944d2bceda5171eb8d938","name":"pair_kernel.py","bytes":11426},{"sha256":"11013411d35c18c1c3f6a96010dc27c7f8c4fbe7b9fadef0e51650fc0a91c465","name":"spectral_axis.py","bytes":6964},{"sha256":"2f86ab6f0bdb4b7ec8100ac0df9c6dfb2bb9f517b871574594ca49be840498f3","name":"spectral_axis.json","bytes":110444},{"sha256":"ea83a8033f41702414b724d1476eba50800ec7bf71f89e75b6f771f9db1397f5","name":"coupled_sweep.py","bytes":4003},{"sha256":"b2ceb3bc173ac0a9517b97ea3d99d3fb214cbb6c0d927d7acd53ac5ca05b5581","name":"coupled_sweep.json","bytes":3729},{"sha256":"5ae88e95f0a50abe8e337a1f1aed74f514226a615ea968be1df823f5d3a108ff","name":"test_pair_kernel.py","bytes":7305},{"sha256":"305ebf0ce7c111c91113580314c7322573a37b1f03ee9ae29c261716ba7d2398","name":"verify_route59_numbers.py","bytes":6484},{"sha256":"7de0bf41ab814cfb2acdabc8b7db63d7dd1991d7745597c0d07e306f5a0efe83","name":"route59_inputs.json","bytes":2445},{"sha256":"23098406d22060062e6c4bbbbcdd32aec60eef6f642e57ac0c7f9706bdbb16ce","name":"test_verify_route59_numbers.py","bytes":4285},{"sha256":"2b9f245e4f8eef5abe0904b8720ff6d22004c3633baab826e978114bd4f14ef0","name":"REPORT.md","bytes":14998}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}