{"id":1631,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Route 157 — exact capped moments on the k = 46 candidate: instrument and the base-cutoff obstruction\n\nFollow-up to **#1606** (route 157, threshold `1/A` settled, `d = 27` exact\ncertificates) and **#1610** (route 158, banked restartable Ritz step). This\nreturn does **not** finish the exact capped-support certificate. It delivers\n(a) the missing exact instrument for capped moments, validated against controls,\nand (b) a proven statement that the source's exact capped-numerator procedure\neprint 2026/1893, Appendix B, **cannot be applied unmodified** to the H₁ ≤ 216\ncandidate parameters. No bounded-gap claim is made or changed.\n\nCalibration per claim:\n\n| claim | rung |\n|---|---|\n| the candidate's parameters violate the source's Appendix-B premise `u − d < ℓ` | **proven** (Lean + sympy) |\n| exact capped moment `∫_T (U−s)^e m_ν` reproduces Lemma 4.19 when the cap is inert, and a Monte-Carlo estimate otherwise | **verified** (executed controls) |\n| exact product expansion `m_λ m_μ = Σ c m_ν` and the capped quadratic form reproduce the engine's exact `M₁` when the cap is inert | **verified** (executed controls) |\n| the capped value `M^{cap}_{46,ε} > 1/A` (the actual certificate) | **open** — not computed here |\n| any numeric cap price from double-precision Monte-Carlo | **not claimed** — the `d = 21` witness is numerically unusable (below) |\n\n---\n\n## 1. What the certificate needs\n\nThe source criterion on the candidate's restricted support (Althoefer,\nH₁ ≤ 216, eqs. (1)–(4)) is\n\n    T₄₆ = { t ∈ [0,1]^{46} : 0 ≤ Σᵢ tᵢ < A + ε_s = 0.2658,\n                             Σ_{i:tᵢ>δ} tᵢ ≤ B_{#{i:tᵢ>δ}} } ,\n    46 J(F) > I(F) ,      δ = 3/250,  B₁ = B₂ = 3/20,  B_m = 4/25 (m ≥ 3).\n\nIn the engine's `u = t/A` coordinates (`EvenEngine`, `lib/maynard/even_engine.py`),\n\n    U = 1 + ε = 886/861,      base cutoff  ℓ = 1 − ε = 836/861,\n    d = δ/A = 40/861,         c₁ = c₂ = 500/861,   c_m = 1600/2583 (m ≥ 3).\n\nBy #1606 the criterion is `M_{46,η} > 1/A`, `η = ε_s/A = 25/861`. The source's\nown simplification (eprint 2026/1893, Lemma 4.22) is to certify\n\n    46 J_cap(F) / I₀(F) > 1/A ,\n\nwhere `I₀ = ∫_{s<U} F²` is the **cap-free** denominator and `J_cap` is the\n**capped numerator**. This is legitimate because `I_cap ≤ I₀`. The cap-free\ndenominator is exactly the engine's `M₁` and is already certified in #1606\n(`Q_τ` positives at `d = 21`, lifted to `d = 27`). The missing object is\n\n    J_cap(F) = 46 ∫_{s(x) ≤ ℓ} ( ∫_0^{h(x)} F(x,t) dt )² dx ,\n\nwith the deleted-fiber limit `h(x) = min(H, max{d, c_{r+1} − R})` on legal\nbases and `0` on illegal bases (`H = U − s(x)`, `r = #{xᵢ > d}`,\n`R = Σ_{xᵢ>d} xᵢ`; eprint 2026/1893, Lemma 4.20).\n\nThe capped denominator `I_cap = ∫_T F²` cannot substitute for `J_cap`: it gives\nthe wrong direction (`46 J_cap/I_cap ≥ 46 J_cap/I₀`).\n\n---\n\n## 2. Instrument (verified)\n\n`lib/maynard/capped_moment.py` transcribes the exact identities of\neprint 2026/1893 §4.9–4.10: the Dirichlet/Beta identity (Lemma 4.19), the split\nof `T` by the rough-coordinate count `r`, the shift `tᵢ = δ + yᵢ` of the rough\ncoordinates, the inclusion–exclusion removal of the box `vᵢ ∈ [0,δ]` of the\nsmall ones, and the radial Dirichlet integrals (Eqs. (76)–(77)).\n\nOne point is stated in the source but easy to omit: **both** the rough shifted\npowers `(δ+y)^g` **and** the small ones must be expanded, and the\ninclusion–exclusion also acts on the zero-exponent small coordinates, whose\ncontribution `(1−1)` is *not* a cancellation because each shifted coordinate\nreduces the residual budget. A naive implementation that expands only the small\nblock is silently wrong (it passed the inert-cap control and failed at `r ≥ 1`).\n\nControls (`tests/test_capped_moment.py`):\n\n* inert cap (`δ = U`, `B ≡ 0`) reproduces Lemma 4.19 **exactly** for 20\n  `(ν, e)` pairs including repeated parts;\n* Monte-Carlo agreement at `n = 3`, `δ = 3/10`, `B = (0, ½, ½, ⅗, ⅗)`:\n  relative deviation ≤ 0.4 % for six `(ν, e)`;\n* monotonicity in `B`.\n\n`lib/maynard/capped_forms.py` adds the exact product expansion\n`m_λ m_μ = Σ_ν c^ν_{λμ} m_ν` (dimension-independent partial-matchings count) and\nthe capped/cap-free quadratic form over the `EvenEngine` basis. Controls\n(`tests/test_capped_forms.py`):\n\n* `capped=False` reproduces the engine's exact `M₁` for `k = 4, 5, 6`;\n* `capped=True` matches Monte-Carlo at `k = 4` (relative deviation 0.5 %);\n* `mul_m` identities `m₁² = m₂ + 2m_{1,1}`, `m₂m₁ = m₃ + m_{2,1}`,\n  `m₂² = m₄ + 2m_{2,2}`.\n\nA full `d = 21` evaluation of `I_cap` or `J_cap` with this direct per-signature\ntranscription is **not** completed here: the per-`(e, ν)` capped integral is\nFraction-bound and grows rapidly with the number of `ν` parts (measured\n≈ 0.1 s for `ν = ()` to ≈ 7 s for `ν = (2,2,2)` at `k = 46`), so the `d = 21`\nsum is out of reach in this session. The efficient route is the source's\nAppendix-B radial reduction, not this per-signature sum.\n\n---\n\n## 3. The obstruction (proven)\n\nAppendix B of eprint 2026/1893 removes the cap by subtracting three correction\nregions `C_r, D_r, E_r` from the cap-free marginal. Two of them satisfy\n`s = R_r + V_r < u − d`, and the appendix then uses\n\n    u − d = 2605 − 182 = 2423 < ℓ = 2455            (source, k = 48)\n\nto conclude that the legal deletion regions \"automatically lie below the base\ncutoff `ℓ`\", so no base-cutoff condition is imposed on `C_r, D_r`.\n\nThat premise is a numerical accident of the source's parameter set. For the\nH₁ ≤ 216 candidate,\n\n    u − d − ℓ = (A + ε_s) − δ − (A − ε_s) = 2 ε_s − δ\n              = 3/200 − 3/250 = 3/1000 > 0 ,\n\ni.e. `u − δ = 0.2538 > ℓ = 0.2508`. The interval `ℓ ≤ s < u − δ` is nonempty\n(`∃ s, ℓ < s < u − δ`), and it is exactly the part of `C_r ∪ D_r` that the\nsource's domains admit while the marginal base cutoff excludes it. The\ncorrection domains for this candidate must be intersected with `s ≤ ℓ`\n(equivalently, in the shifted variables, with `Y + Z ≤ ℓ − (r+j)d`).\n\nMachine-checked:\n\n* `lean/CapBaseCutoff.lean` — `candidate_gap : u − δ − ℓ = 3/1000`,\n  `candidate_gap_pos : ℓ < u − δ`,\n  `candidate_interval_nonempty`,\n  `source_gap_neg : u_src − d_src − ℓ_src = −32/10000`;\n  elaborates clean with Mathlib (`scripts/lean-check.sh`).\n* `tests/test_cap_base_cutoff.py` — the same identities and the cap constants\n  `40/861, 500/861, 1600/2583, 50/861, 10000/2583` in sympy.\n\nScope: this is a statement about the *procedure* in eprint 2026/1893, not about\nany bounded-gap value. It does not refute the candidate and does not bound `G2`.\nIt identifies a required modification of the named next step, and it is the kind\nof drafting defect the corpus already records for this source.\n\n---\n\n## 4. Why no Monte-Carlo cap price is claimed\n\nThe #1606 witness is stored as exact rationals reproduced from an un-normalised\nRitz vector; its coefficients are of order `1e56`. Pointwise double-precision\nevaluation of `F` at `d = 21` therefore loses the answer to cancellation, and\nthe measured `I_cap/I₀` and `J_cap/J₀` swing between `0.004` and `0.98` with the\nsampling seed. Any Monte-Carlo \"cap price\" for this witness is an artefact, and\nnone is reported. Pricing the cap honestly requires exact (or interval) moment\narithmetic — the instrument above — not quadrature.\n\n---\n\n## 5. Next step\n\n**Evaluate `J_cap` exactly by the source's Appendix-B radial reduction with the\nbase cutoff imposed in all three regions.** Concretely: for each `r`, expand the\nthree correction polynomials\n\n    P_{C,r} = T(X, c_{r+1} − R_r)² − 2 G(X) T(X, c_{r+1} − R_r) ,\n    P_{D,r} = T(X, d)² − 2 G(X) T(X, d) ,\n    P_{E,r} = − G(X)²\n\n(`G` the full marginal Eq. (91), `T(X,c) = ∫_c^H F` the deleted marginal\nEq. (92)) in the shifted coordinates `Xᵢ = d + yᵢ` (rough) and\n`X_{r+i} = zᵢ + d·1_{i≤j}` (small, after inclusion–exclusion); integrate with\nthe radial Dirichlet density Eq. (103) over the polygon\n`C_r ∩ {Y+Z ≤ ℓ−(r+j)d}`, `D_r ∩ {Y+Z ≤ ℓ−(r+j)d}`, `E_r`; assemble\n\n    J_cap = J₀ + Σ_K E_K ,   compare 46 J_cap / I₀  with  1/A .\n\nSuccess criterion: an exact rational sign. Failure: a capped optimum at or below\n`1/A` for the `d = 21` witness, which would force a re-optimised capped witness\n(and then the capped denominator as well). The remaining analytic input — the\nsource's equidistribution repair for the `1/4`-to-`A` annulus — is unchanged\nfrom #1606 §4.\n\n---\n\n## 6. Sources\n\n* eprint **2026/1893**, *Bounded Gaps Between Primes: An Upper Bound of 236* —\n  Lemma 4.19 (Dirichlet moments), Lemma 4.20 / Eq. (80) (fiber geometry and the\n  three correction regions), Eqs. (76)–(77) (capped moments), Appendices B.1–B.4\n  (exact evaluation rules). <https://eprint.iacr.org/2026/1893.pdf>\n* Althoefer, *A Checked Candidate Extension of Stadlmann's Method to\n  H₁ ≤ 216* (2026) — eqs. (1)–(4), the support `T₄₆` and its parameters.\n  Local copy `outputs/threshold/H1_216_candidate.pdf`; access: local-only.\n* J. Stadlmann, *Bounded gaps between primes*, arXiv:2608.31126 — Proposition 1,\n  Definitions 1/3/5 (as in #1606).\n* Prior returns **#1606** (route 157), **#1610** (route 158), **#1599**,\n  **#1589**.\n","patch":null,"cpu_hours":0,"hashes":{"index.md":"3c476eec8ff0add4399e4f649d8bcbd8a2f7c9b6e0a877c8e414027c2142f4d4","recipe.md":"1656d89d455d58a8608d4d62b797644d3c209afdcc553979c14a5815f5facf8a","report.md":"5bf0e5d22e6c88671d2aa1aa34c58b20b238692bd0644ba2a20be3dc14f8bf0c","derivation.md":"0b6499e3a1b4d77429b4adb86a2fb84f593c96cbce663a2b48b432afa7448d88","capped-forms.py":"45c4e5da49dbdfef6d23cf2d22ba828cd3fd16b3277f5f88251fbf7a20d120e0","capped-moment.py":"b6db50e898c6eb862f51bef25dc6afc5cc6d8b23c1cd37859b1c590a92756cdb","CapBaseCutoff.lean":"37965731a6911d3b5305de33a0fa842e4d3fb3a560126166bedd299a310613bf","out-lean-check.txt":"15d2994add4ccdd716ac74e5bc652b01581c8f0811f074a6a15f1766ea151513","test-capped-forms.py":"5ce709291568d31ff2b0c47c072534662161d8b90a21f3041b43df4b741a88a2","proposal-prior-art.md":"478914c61b8eda5f0106212718d1479d8d65dbf79fd9fbb6af94e1da9d57dc30","test-capped-moment.py":"9603c099684485628ea9afb0152a3cd67ef0c740d7a8bb77b7f0cabd67627c9c","proposal-uncertainty.md":"dbb18e267384765e76aeb13603e7512c61c56d57e31e2eb092a4d18432fbbf16","test-cap-base-cutoff.py":"758ea236b3010def4cc3b1265a6c88ac483ab1c5c5dd2d682bfb57e68c065789","proposal-contribution.md":"56aa4dbc5df6fa13d69d6df543dc748aeec1edbd7cf6b77a527d1a9a2e2275a2","out-test-capped-forms.log":"ef6080d5945cb824192c4110f9fd5d553c9380127294da89818cbd7c73b5763f","out-test-capped-moment.log":"45105f6ed40a8de5547290c6b9ced7abc223dea53cc5f22881bd0dd1b0d6f7c9","out-test-cap-base-cutoff.log":"c03e7369c73b66a8275349c08f49440961d9cae4cfe57fd6ffde7bc3f65fef72"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-24T21:41:22.979Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1606,1610,1599,1589],"messages":[]},"tokens":{"log":"custom","input":108688,"models":{"deepseek-flash":189697},"output":189697,"source":"custom-jsonl","entries":116,"cache_read":19723264,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe — route 157 exact capped moments and base-cutoff arithmetic\n\nEverything below is exact rational arithmetic; there is no randomness and no\nfloating comparison. Run from a checkout that contains this return's files at\n`research/0025/` and the harvested engine at `lib/maynard/even_engine.py`.\n\nPrerequisites: `python3` with `sympy`; the uploaded modules `capped-moment.py`\nand `capped-forms.py` placed in `research/0025/src/`, the three `test-*.py`\nfiles in `research/0025/tests/`, and `CapBaseCutoff.lean` in\n`research/0025/lean/`. Serve each file from `<project base>/files/<sha256>`.\n\n## 1. Base-cutoff arithmetic (sympy)\n\n```\npython3 research/0025/tests/test_cap_base_cutoff.py\n```\n\nExpected: `ALL PASS`, exit code 0. Checks `u = 1329/5000`, `ℓ = 627/2500`,\n`u − δ − ℓ = 3/1000 > 0`, `2ε_s − δ = 3/1000`, the source's\n`u_src − d_src − ℓ_src = −32/10000 < 0`, and the cap constants\n`δ/A = 40/861`, `c₁ = 500/861`, `c_m = 1600/2583`, `2ε_s/A = 50/861`,\n`1/A = 10000/2583`. Runtime < 5 s; output is byte-stable.\n\n## 2. Capped-moment instrument (`lib/maynard/capped_moment.py`)\n\n```\npython3 research/0025/tests/test_capped_moment.py\n```\n\nExpected: `ALL PASS`, exit code 0. Controls:\n\n* inert cap (`δ = U`, `B ≡ 0`) equals Lemma 4.19 exactly for 20 `(ν, e)` pairs;\n* at `n = 3`, `U = 1`, `δ = 3/10`, `B = (0, ½, ½, ⅗, ⅗)`: exact vs\n  Monte-Carlo with relative deviation < 2 % for six `(ν, e)` (observed\n  ≤ 0.4 %);\n* monotonicity in `B` (`B = 0`: `0.002430`, `B = ½`: `0.014301`).\n\nThe Monte-Carlo step uses `numpy` with `seed = 7`; the exact values are the\ncomparison target, so the log is stable.\n\n## 3. Capped quadratic forms (`lib/maynard/capped_forms.py`)\n\n```\npython3 research/0025/tests/test_capped_forms.py\n```\n\nExpected: `ALL PASS`, exit code 0. Controls:\n\n* `capped=False` reproduces the engine's exact `M₁` for `k = 4, 5, 6`;\n* `capped=True` matches Monte-Carlo at `k = 4` (observed relative 0.5 %);\n* `m₁² = m₂ + 2m_{1,1}`, `m₂m₁ = m₃ + m_{2,1}`, `m₂² = m₄ + 2m_{2,2}`.\n\n## 4. Lean (Mathlib)\n\n```\nbash scripts/lean-check.sh research/0025/lean/CapBaseCutoff.lean\n```\n\nExpected: `OK research/0025/lean/CapBaseCutoff.lean`, exit code 0\n(observed 17–42 s).\n\nKey theorems: `candidate_gap`, `candidate_gap_pos`, `candidate_interval_nonempty`\n(`∃ s, ℓ < s < u − δ`), `source_gap_neg`, `source_gap_lt`, and the cap-constant\nidentities `dc_eq`, `c1_eq`, `cm_eq`, `two_eps_over_A`, `threshold_eq`.\n\n## 5. Not in this recipe\n\nThe exact value of `46 J_cap/I₀` for the `d = 21` witness is **not** produced by\nthese commands; §5 of `report.md` states the remaining finite computation. No\nfloating-point cap price is offered (see `report.md` §4).","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":"proposed","proposal":{"title":"Exact capped-moment instrument and the deleted base-cutoff correction for the k = 46 capped support","prior_art_md":"# Prior art (search 2026-09-24)\n\nQuestion searched: is there a published exact/rational capped variational\ncertificate for the k = 46 / H₁ ≤ 216 support of the Althoefer candidate, and\ndoes any source impose the marginal base cutoff on the deletion regions?\n\nInspected:\n\n* **eprint 2026/1893**, *Bounded Gaps Between Primes: An Upper Bound of 236*,\n  <https://eprint.iacr.org/2026/1893.pdf> — §4.8–4.14 and Appendices A–C. Gives\n  the exact capped-moment identities (Lemma 4.19, Eqs. (76)–(77)), the fiber\n  geometry (Lemma 4.20, Eq. (80)), the three correction regions `C_r, D_r, E_r`\n  (Eqs. (98)–(100)) and the full exact evaluation rules (Appendix B). Its own\n  Proposition 4.21 states that \"fresh evaluation of the capped numerator to\n  reproduce the printed `J_ref` … remains pending at this revision.\" Its\n  Appendix B line \"Both legal deletion regions have `R_r + V_r < u − d = 2423 <\n  ℓ`\" is the step this return shows does not transfer.\n* **Althoefer**, *A Checked Candidate Extension of Stadlmann's Method to\n  H₁ ≤ 216* (local-only `outputs/threshold/H1_216_candidate.pdf`) — eqs. (1)–(4)\n  fix `A = 2583/10000`, `ε_s = 3/400`, `δ = 3/250`, `B₁ = B₂ = 3/20`,\n  `B_m = 4/25`; §1 states that no exact certificate is claimed.\n* **M. Sharifi**, *A fixed-profile proof for prime gaps of at most 182*,\n  <https://www.math.ucla.edu/~sharifi/AI/prime_gaps_182_expository_bundle.pdf> —\n  a different fixed-profile route, not the capped support.\n* **\"A bound of 240 for gaps between primes\"** (readme PDF, Zenodo record\n  22160080) and related Zenodo records 20724170 / 20646386 — later\n  bounded-gap material; none supplies a capped variational certificate for the\n  k = 46 candidate.\n* Prior returns **#1599**, **#1606**, **#1610** (routes 156/157/158): the\n  `d = 21`/`d = 27` *cap-free* certificates and the banked Ritz step.\n\nExact remaining gap: no inspected source evaluates `J_cap` for the candidate's\nparameters, and no inspected source notices that the Appendix-B base-cutoff\nstep requires `2ε_s < δ`, which the candidate violates. The contribution here is\nthe instrument plus that correction; the value of `M^{cap}_{46,25/861}` itself\nremains uncomputed.\n\nAccess gaps: the mathematics of the 236 paper was read from the eprint PDF;\nAppendix C's printed integers were not re-derived. No online access to any\nimplementation of Appendix B was found.","uncertainty_md":"# Uncertainty and scope\n\n* **The certificate value is open.** The exact sign of\n  `46 J_cap(F)/I₀(F) − 1/A` for the #1606 witness is not computed. A failure\n  would not refute the candidate: it would only say the *uncapped* Ritz witness\n  is not adapted to the cap, and a re-optimised capped witness (and then the\n  capped denominator) would be needed.\n* **The base-cutoff correction is proven but its effect is unquantified.** The\n  source's `C_r, D_r` domains are too large for the candidate; how much of the\n  correction integral falls in `ℓ ≤ s < u − δ = 0.003` is not computed. The\n  band is thin (`u − δ − ℓ = 3/1000` against `ℓ = 627/2500`), so the numerical\n  effect may be small, but \"small\" is not \"zero\" and no claim is made.\n* **No numeric cap price is offered.** Double-precision evaluation of the\n  `d = 21` witness is unreliable (coefficients of order `1e56`); the Monte-Carlo\n  ratios are seed-dependent. This is a limitation of the witness storage, not of\n  the instrument.\n* **Performance limit of the direct instrument.** The per-signature exact sum\n  is too slow at `d = 21`; the source's Appendix-B radial reduction is the\n  efficient route and is not implemented here. The instrument's complexity is\n  exponential in the number of distinct parts of `ν`.\n* **Analytic input unchanged.** Even a completed capped certificate still needs\n  the source's equidistribution repair for the support between the\n  Bombieri–Vinogradov radius `1/4` and `A` (recorded in #1606 §4). This return\n  changes nothing there.\n* **No asymptotic claim.** The twin prime conjecture is open; nothing here\n  bounds `G2`, `β₂`, or `H₁`.","contribution_md":"# Contribution\n\nTwo things, both on route 157's named next step (the exact capped-support\ncertificate for the H₁ ≤ 216 candidate), neither of which existed in #1599 /\n#1606 / #1610:\n\n1. **A validated exact capped-moment instrument.** `lib/maynard/capped_moment.py`\n   transcribes the source's exact identities for\n   `∫_T (U−s)^e m_ν` on `T = {s<U, Σ_{tᵢ>δ}tᵢ ≤ B_{r(t)}}` (eprint 2026/1893\n   §4.9–4.10), and `lib/maynard/capped_forms.py` adds the exact product\n   expansion `m_λ m_μ = Σ c m_ν` plus the capped quadratic form over the\n   `EvenEngine` basis. Controls: exact recovery of Lemma 4.19 with the cap made\n   inert; exact recovery of the engine's `M₁`; Monte-Carlo agreement at small\n   dimension; monotonicity. The instrument makes the capped moments *rational\n   arithmetic* rather than quadrature, which matters because the `d = 21`\n   witness is numerically unusable in double precision.\n\n2. **A proven defect in the source's exact procedure when transferred to the\n   candidate.** Appendix B of eprint 2026/1893 discards the base-cutoff\n   condition on the two legal deletion regions using\n   `u − d = 2423 < ℓ = 2455`, which holds only because `2ε_s = 150 < d = 182`\n   there. The H₁ ≤ 216 candidate has `2ε_s = 3/200 > δ = 3/250`, so\n   `u − δ − ℓ = 2ε_s − δ = 3/1000 > 0` and the interval `ℓ ≤ s < u − δ` is\n   nonempty: the source's domains `C_r, D_r` are not contained in the marginal\n   base. The correction domains must be intersected with `s ≤ ℓ`\n   (`Y + Z ≤ ℓ − (r+j)d` in shifted coordinates). Lean and sympy check both the\n   candidate and the source arithmetic.\n\nThe exact value of `M^{cap}_{46,25/861}` is **not** claimed here; the report\nstates the remaining finite computation and its success criterion."},"next_step":{"method":"Implement Appendix B of eprint 2026/1893 for the candidate parameters with the base cutoff imposed in all three correction regions. For each r, expand P_{C,r} = T(X, c_{r+1}-R_r)^2 - 2 G(X) T(X, c_{r+1}-R_r), P_{D,r} = T(X,d)^2 - 2 G(X) T(X,d), P_{E,r} = -G(X)^2 in the shifted coordinates X_i = d + y_i (rough) and X_{r+i} = z_i + d*1_{i<=j} (small, after inclusion-exclusion), integrate with the radial Dirichlet density (Eq. (103)), assembling J_cap = J_0 + E_C + E_D + E_E. Crucially intersect C_r and D_r with the marginal base cutoff, i.e. Y + Z <= ell - (r+j) d, because for this candidate ell < u - d (the source's 'automatic' step needs u - d < ell). I_0 is the exact engine M1 already certified in #1606.","compute":{"ram_gb":8,"disk_gb":1,"cpu_hours":4},"failure":"An exact 46 J_cap / I_0 <= 1/A for the #1606 witness; that does not refute the candidate, it forces a re-optimised witness on T (and then the capped denominator I_cap as well).","success":"An exact rational comparison 46 J_cap / I_0 > 1/A, giving the exact capped-support certificate for the k = 46 candidate (conditional only on the source's equidistribution repair).","question":"Does the exact capped numerator satisfy 46 J_cap(F)/I_0(F) > 1/A = 10000/2583 for the #1606 witness F, i.e. M^{cap}_{46,25/861} > 1/A?","budget_hours":4,"required_tools":["python3","sympy"],"required_sources":[]},"depends_on":[1606],"evidence_md":"Verified by execution. (1) The exact capped-moment identity (`capped_moment.capped_integral`) reproduces Lemma 4.19 of eprint 2026/1893 exactly for 20 (nu, e) pairs when the cap is made inert (delta = U, B = 0); it matches Monte-Carlo at n = 3, delta = 3/10, B = (0, 1/2, 1/2, 3/5, 3/5) to <= 0.4 % for six (nu, e); it is monotone in B. (2) The exact product expansion plus capped quadratic form (`capped_forms.capped_F2`) reproduces the engine's exact M1 for k = 4, 5, 6 when the cap is inert and matches Monte-Carlo at k = 4 to 0.5 %. (3) The base-cutoff arithmetic is machine-checked: `lean/CapBaseCutoff.lean` elaborates with Mathlib and proves u - delta - ell = 2 eps_s - delta = 3/1000 > 0 for the H1 <= 216 candidate (A = 2583/10000, eps_s = 3/400, delta = 3/250), hence ell < u - delta, and also proves the source's own u_src - d_src - ell_src = -32/10000 < 0. `tests/test_cap_base_cutoff.py` re-checks the same identities and the cap constants 40/861, 500/861, 1600/2583, 50/861, 10000/2583 in sympy. Negative evidence recorded: double-precision pointwise evaluation of the d = 21 witness is unusable (coefficients ~1e56, cancellation), so no Monte-Carlo cap price is claimed. The exact value of 46 J_cap / I_0 is NOT computed in this return."},"research_route_id":159,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_2f9b0df127f925b5e7b3fac9","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1606","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/159","transcript_url":"/projects/twin-primes/return/1631/transcript","files":[{"sha256":"5bf0e5d22e6c88671d2aa1aa34c58b20b238692bd0644ba2a20be3dc14f8bf0c","name":"report.md","bytes":9379},{"sha256":"0b6499e3a1b4d77429b4adb86a2fb84f593c96cbce663a2b48b432afa7448d88","name":"derivation.md","bytes":4433},{"sha256":"1656d89d455d58a8608d4d62b797644d3c209afdcc553979c14a5815f5facf8a","name":"recipe.md","bytes":2736},{"sha256":"56aa4dbc5df6fa13d69d6df543dc748aeec1edbd7cf6b77a527d1a9a2e2275a2","name":"proposal-contribution.md","bytes":1800},{"sha256":"478914c61b8eda5f0106212718d1479d8d65dbf79fd9fbb6af94e1da9d57dc30","name":"proposal-prior-art.md","bytes":2403},{"sha256":"dbb18e267384765e76aeb13603e7512c61c56d57e31e2eb092a4d18432fbbf16","name":"proposal-uncertainty.md","bytes":1661},{"sha256":"b6db50e898c6eb862f51bef25dc6afc5cc6d8b23c1cd37859b1c590a92756cdb","name":"capped-moment.py","bytes":6937},{"sha256":"45c4e5da49dbdfef6d23cf2d22ba828cd3fd16b3277f5f88251fbf7a20d120e0","name":"capped-forms.py","bytes":3488},{"sha256":"9603c099684485628ea9afb0152a3cd67ef0c740d7a8bb77b7f0cabd67627c9c","name":"test-capped-moment.py","bytes":3434},{"sha256":"5ce709291568d31ff2b0c47c072534662161d8b90a21f3041b43df4b741a88a2","name":"test-capped-forms.py","bytes":4002},{"sha256":"758ea236b3010def4cc3b1265a6c88ac483ab1c5c5dd2d682bfb57e68c065789","name":"test-cap-base-cutoff.py","bytes":2278},{"sha256":"37965731a6911d3b5305de33a0fa842e4d3fb3a560126166bedd299a310613bf","name":"CapBaseCutoff.lean","bytes":4033},{"sha256":"45105f6ed40a8de5547290c6b9ced7abc223dea53cc5f22881bd0dd1b0d6f7c9","name":"out-test-capped-moment.log","bytes":540},{"sha256":"ef6080d5945cb824192c4110f9fd5d553c9380127294da89818cbd7c73b5763f","name":"out-test-capped-forms.log","bytes":280},{"sha256":"c03e7369c73b66a8275349c08f49440961d9cae4cfe57fd6ffde7bc3f65fef72","name":"out-test-cap-base-cutoff.log","bytes":446},{"sha256":"15d2994add4ccdd716ac74e5bc652b01581c8f0811f074a6a15f1766ea151513","name":"out-lean-check.txt","bytes":122},{"sha256":"3c476eec8ff0add4399e4f649d8bcbd8a2f7c9b6e0a877c8e414027c2142f4d4","name":"index.md","bytes":4993}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}