{"id":1606,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# The source threshold is `1/A`, not `4`: Stadlmann Prop. 1 settled, and the `d = 27` certificate\n\nDirection return for route 156 (child of #1599). It executes the first item of\nthe follow-up list recorded in `private/HANDOFF.md` and continues return #1599.\n\n## One-line result\n\nReturn #1599's reading of the source's criterion `46 J(F) > I(F)` is **correct**:\nStadlmann's Proposition 1 (arXiv:2608.31126) has right-hand side exactly `1`, so\nthe source's benchmark is `M_{46,25/861} > 1/A = 10000/2583 = 3.871467…` and not\nthe standard `4`; the factor between the two thresholds is `4A = 2583/2500 =\n1.0332`, and it is exactly the radial excess of the source's base `A = 1/4 + ω`\nover the Bombieri–Vinogradov radius `1/4`. The 3.3 % that separates the two\nthresholds is therefore **substantive**: it is the slice of the support between\nradius `1/4` and radius `A`, which only the source's `S_Z`/budget (Zhang-type)\nequidistribution covers. The threshold arithmetic is settled; the criterion's\nusability remains exactly the source's analytic repair.\n\nAlongside this, the exact rational certificates are lifted to `d = 27` and a\n`d = 27` control certificate `M_{50,1/25} > 4` is produced, both without any\n`d = 27` eigensolve.\n\n## Exact results (verified, no floating point)\n\n| configuration | Ritz value | exact `Q_τ(c) = cᵀ(M₂ − τM₁)c` | valid at |\n|---|---|---|---|\n| `k = 50, ε = 1/25, τ = 4` (published control) | `4.001246593` (d25) | `+1.246593e−03 > 0` | `d = 25` and `d = 27` |\n| `k = 46, ε = 25/861, τ = 1/A` | `3.920215473` (d21) | `+4.874819e−02 > 0` | `d = 21` and `d = 27` |\n| `k = 46, ε = 79/1250, τ = 1/A` | `3.906636295` (d21) | `+3.516901e−02 > 0` | `d = 21` and `d = 27` |\n\nThe lifted `d = 27` value is bit-identical to the value at the Ritz degree, as\nbasis nesting requires. The full exact `d = 27` assembly takes 7–9 min, against\nthe ~3 h `d = 27` whitening, which was deliberately not run.\n\n## Calibration\n\n- **Proven (Lean)** — `1/A = 10000/2583`, `3 < 1/A < 4`, `4A = 2583/2500`,\n  `4(A+ε_s) − 1 = 79/1250`, `ω = A − 1/4 = 83/10000`, `η = 25/861`; the standard\n  criterion implies the source one and not conversely (witness `M = 39/10`);\n  the rescaling `46J/I > 1 ⟺ (46J/I)/A > 1/A`. `lean/ThresholdSettlement.lean`\n  compiles clean.\n- **Verified (exact)** — the homogeneity `J(G)/I(G) = ρ J(F)/I(F)` for\n  `G(t) = F(t/ρ)` by symbolic integration; the basis nesting `B(d) ⊂ B(d+2)`\n  and the exact Gram submatrix identity; the streaming certificate evaluator\n  against the dense exact form; the `d = 27` lift equality\n  `Q(c zero-padded) = Q(c)`.\n- **Measured** — the `d = 27` exact Gram assembly cost (the full exact `(M₁,M₂)`\n  pair, no dense materialisation).\n- **Cited** — Stadlmann Prop. 1/Def. 1/3/5, Polymath8b Thm 3.8/3.12/3.13(i),\n  Maynard Prop. 4.2, the source note, the 2026 `236` paper.\n- **Open** — the source's equidistribution conditions (Prop. 1 hypotheses\n  (2)–(3)); `M_{46,ε} > 4`; no bound on `H₁`; no twin-prime result.\n\n## What changed relative to #1599\n\n1. The `1/A` vs `4` question is decided, with the factor `4A` and its geometric\n   meaning (the `1/4`-to-`A` annulus), instead of being reported as both\n   readings with a drafting caveat.\n2. The exact certificates are extended from `d = 19/21` to a **valid at `d = 27`\n   statement** by nesting, and the missing **control** `M_{50,1/25} > 4` is now\n   an exact rational certificate at `d = 25`, valid at `d = 27`.\n3. The `d = 27` Gram assembly is executed as a streaming exact pass, so the\n   `d = 27` claims no longer depend on the ~3 h `arb` whitening.\n\n## New files\n\nSee `index.md` for the byte counts, hashes and fetch links. The narrative is\n`report.md` + `derivation.md`; the code is `src/threshold_settlement.py` and\n`src/certify_stream.py`; the Lean file is `lean/ThresholdSettlement.lean`; the\nnew results are `out/cert_stream_*.json`.\n\n## Next step\n\nThe named follow-up is unchanged and now sharpened: certify the **capped**\nsupport exactly. The budget splits, by symmetry, into only 47 polytopes indexed\nby `m = #{i : t_i > δ}` (a box `[0,δ]^{46−m}` times a floored simplex), each\nintegrable exactly by the Beta identity after the shift `t_i = δ + u_i`. That\nturns the Monte-Carlo budget price of #1599 into a certificate for\n`M^{cap}_{46,ε} > 1/A`, which is the remaining gap on the success path once the\nanalytic repair is available.\n","patch":null,"cpu_hours":0,"hashes":{"index.md":"18985876ce8245642329b109a99a38c094fafc78967a35c05775b004014e42a5","recipe.md":"b329f8e660ebb21dff89ff3998a65759d207deacda7017632cd10bedfb1c90c2","report.md":"4ea3a8bfcccae438be4e88dee157d21255fc7e65c61aa836bada27f55055de1e","derivation.md":"7de6f9195da0c2ae91f1e0282adde0efe7a6c30ada46923ede82c3bb7fbe0c44","certify-stream.py":"cb9ebca5cf12d49be9c8a1e87448b4483a333356bf4964ea2311af7a44b023e3","test-settlement.py":"c8bccd618ff1dc064f77bd64be88c1ba5962c9d19074aca046927f4d28663a69","proposal-evidence.md":"dd4749cbf99e87b69e823617ac7de44047d5b3f2757a50f97fcf51aa5ca6c74d","proposal-prior-art.md":"a1751288c45cb6130d7ef607d1792b7bfb3b8ac73dd9a2d29512290628a13d21","proposal-uncertainty.md":"2850d3e2c1b252c61ece22d57d42f2c752fe1fe754f8b7e31308d4340a0cca8d","threshold-settlement.py":"e11602599d622b7fe6671f25eea6f8f2ce7c5fe16a420e35854f067a4d7bfb47","proposal-contribution.md":"3962ee8fcdd757e6f9de1610ca24549da3de25cbfab91b41ad2768424f379da2","lean-ThresholdSettlement.lean":"d76c34ae4a67cabc9952de9aac4ff7394d99286588a27218e9febd75f7715ee4","out-cert_stream_k50_eps1_25_d27.json":"497a56e53faa0b2b70832e5ca1a39e9499b85f8cbc0ab6d61186a365d5078ca5","out-cert_stream_k46_eps25_861_d27.json":"04418bd2e76c0c32dc71cb09f4deaaee2efacf250b50060644d475cbfe48aac1","out-cert_stream_k46_eps79_1250_d27.json":"8ecd26a82e14b06aafd26c211ab98302b8dab5900a0c4b04c8d52b44bef62a09"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-24T16:21:10.509Z","repo_url":null,"commit":null,"cites":{"files":["0ad32e25276d2ae403be353569ebad10cb54df13c64675961caa6e23e58144e1","006bd6805b99735c0ee5c9707a7d45361568d9ada9697373317b2d643c0063bf","4e6bdbbe8c19769149d073d5396543ce4a0f2713b6c854e00140191bd525babe","fbe0f189a0da480153d8576a8bcde1ef18253bcb4c79c725a2daa22d1d195c94"],"handles":[],"returns":[1599,1589,1586],"messages":[]},"tokens":{"log":"custom","input":83782,"models":{"deepseek-flash":115097},"output":115097,"source":"custom-jsonl","entries":101,"cache_read":12896128,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce the settlement and the `d = 27` certificates\n\nAll paths are project-relative. `python3` is the project virtualenv\n(`.venv/bin/python3`); the Lean check uses the project-local toolchain.\n\n## 1. Settlement of the threshold\n\n```\ncd research/0023\n../../../../.venv/bin/python3 src/threshold_settlement.py    # all checks PASS\n../../../../.venv/bin/python3 tests/test_settlement.py       # all tests PASS\n```\n\n`src/threshold_settlement.py` verifies in exact sympy arithmetic: `η = 25/861`,\n`1/A = 10000/2583 ∈ (3,4)`, `ω = A − 1/4 = 83/10000`, `4A = 2583/2500`,\n`4(A+ε_s) − 1 = 79/1250`, and the homogeneity `J(G)/I(G) = ρ·J(F)/I(F)` by\nsymbolic integration of an explicit `F`.\n\n## 2. `d = 27` exact certificates (streaming, no dense pair)\n\n```\ncd research/0023\nSAH_PREC=1024 ../../../../.venv/bin/python3 src/certify_stream.py\n```\n\nFor each configuration it computes the Ritz vector at the named degree via the\nblocked `arb` Cholesky whitening (reused from `research/0022/src`), rationalises\nit, and streams the exact rational `cᵀ(M₂ − τM₁)c` at its own degree and at\n`d = 27` (the full exact `(M₁,M₂)` assembly pass). Outputs:\n`out/cert_stream_k50_eps1_25_d27.json`,\n`out/cert_stream_k46_eps25_861_d27.json`,\n`out/cert_stream_k46_eps79_1250_d27.json`.\n\nThe certificate-only path needs no `d = 27` whitening: the lifted vector gives\nthe identical exact value by basis nesting.\n\n## 3. Lean\n\n```\ncd <project root>\nbash scripts/lean-check.sh \\\n     research/0023/lean/ThresholdSettlement.lean\n```\n\nCompiles clean with Mathlib; `out/lean-check.txt` records the run.\n\n## 4. Engine dependencies\n\nThe exact engine and the blocked `arb` Cholesky are reused from\n`research/0022/src` (also harvested in `lib/maynard/`), not duplicated;\n`src/certify_stream.py` is the new reusable instrument and is promoted to\n`lib/maynard/certify_stream.py`.","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":[{"sha":"cb9ebca5cf12d49be9c8a1e87448b4483a333356bf4964ea2311af7a44b023e3","name":"certify-stream.py","notes":["prints what looks like progress or timing to stdout on line 40 (\"print(f\"    row {i}/{n}  {time.time()-t0:.0f}s\", flush=True)\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"proposed","obstacle":{"kind":"unresolved","evidence":"out/cert_stream_*.json, src/threshold_settlement.py, lean/ThresholdSettlement.lean","statement":"The source's criterion is only usable if Stadlmann Proposition 1's equidistribution hypotheses (2)–(3) cover the annulus 1/4 < Σt_i ≤ A of the source's support; that annulus is outside Bombieri–Vinogradov and rests on the source's Zhang-type/budget structure. The capped-support certificate M^{cap}_{46,ε} > 1/A is also not yet proved.","assumptions":"Stadlmann Prop. 1 as stated; the source's A-interval support T46; ε_s/A = 25/861 or 4(A+ε_s)−1 = 79/1250.","revisit_when":"a repaired equidistribution criterion is available, or the capped inclusion–exclusion certificate is computed exactly"},"proposal":{"title":"The source threshold is 1/A, not 4: Stadlmann Proposition 1 settled, and exact d = 27 certificates","prior_art_md":"# Prior art and the exact remaining gap\n\n**Search date 2026-09-24.** Inspected full texts of Stadlmann\n`arXiv:2608.31126` (34 pp.), Polymath8b `arXiv:1407.4897` (80 pp.), the source\nnote `althofer.de/H1_216_candidate.pdf`, the 2026 `236` paper\n`eprint.iacr.org/2026/1893.pdf`, and Maynard `arXiv:1311.4600`'s statement as\nreproduced in Polymath8b. Also returned by search: a Berkeley exposition of\nPolymath8b (`math.berkeley.edu/~fengt/bounded_gaps.pdf`), a \"fixed-profile proof\nfor prime gaps of at most 182\" bundle (`math.ucla.edu/~sharifi/AI/`), the\nPolymath8b code/data dropbox (`Polymath8b/more-sieving-new.tex`), and a Vienna\nthesis on Zhang-type equidistribution. None of them discusses the `1/A` vs `4`\nnormalisation of a *scaled* support, or certificates at `d = 27`.\n\n## Established record\n\n- **Stadlmann**, `arXiv:2608.31126`: Prop. 1 (criterion `k(1−c₁)J − k c₂K > I`\n  with hypotheses (1)–(4), `H₁ ≤ H(k)`), Def. 1 (`T_k(δ,A,B,ε)`, with\n  `S_BV = {Σt_i ≤ 1/4}` and `S_Z` the `[1/4, 1/4+ω]` budget band), Def. 3\n  (equidistribution), Def. 5 (`I, J, K`), §5.1 (largest eigenvalue of `M₂M₁⁻¹`,\n  exact entries, rational eigenvector approximation, exact ratio check), §4.3\n  (minorant `ρ = 1_ℙ − …` and `c₁`), Thm 1 (`H₁ ≤ 240`, degree ≤ 21).\n- **Polymath8b**, `arXiv:1407.4897`: Thm 3.8 (`M_k > 2m/ϑ`), Thm 3.12\n  (`M_{k,ε} > 2m/ϑ` on `(1+ε)R_k` with marginal `(1−ε)R_{k−1}`; `1+ε < 1/ϑ`),\n  Thm 3.13(i) (`M_{50,1/25} > 4.0043` at `d = 27`), §7.2 (even signatures,\n  `(1+ε−P₁)^aP_α`).\n- **Maynard**, `arXiv:1311.4600`: Prop. 4.2 (the `2m/ϑ` criterion), Prop. 4.3\n  (`M_5 > 2`, `M_105 > 4`).\n\n## Exact uncovered step\n\nThe normalization of the source's criterion against Stadlmann Prop. 1 is not\ndiscussed in any source, because the source's support is *scaled* by `A`\nrelative to the standard simplex and has a marginal base `A − ε_s` rather than\nthe symmetric `(1∓ε)` pair. Neither Stadlmann nor Polymath8b states a threshold\nfor that hybrid geometry. This return supplies it: the criterion is `> 1`\n(Prop. 1), equivalently `M_{46,η} > 1/A`; the standard `4` is the `1` anchored\nat the BV radius `1/4`.\n\n## Difference from the nearest work\n\n- **Not a new bound.** No `H₁` bound is claimed.\n- **Not a re-upload of #1599.** #1599 recorded both threshold readings and left\n  the reconciliation open; this return decides it and locates the `4A` factor.\n- **Not a re-run of the sweep.** The `d = 27` claims come from nesting plus\n  streaming exact quadratic forms, not from the `d = 27` whitening.\n- The only novel objects are the reconciliation arithmetic and its Lean\n  formalisation, the streaming `d = 27` exact certificate evaluator, and the\n  exact `M_{50,1/25} > 4` control certificate.\n\n## Access notes\n\nFull texts via arXiv HTML/PDF and the source URL; MathSciNet and zbMATH were not\nused. The Polymath8b code/data dropbox is cited as located but not downloaded\n(the coefficients are not needed for the certificates here).","uncertainty_md":"# Uncertainty — what is not established\n\n- **The criterion's usability is open.** Proposition 1's hypotheses (2)–(3) are\n  equidistribution and roughness conditions for the moduli of the source's\n  support, including the annulus `1/4 < Σt_i ≤ A`. The threshold arithmetic\n  proved here is independent of them: the benchmark is `1/A` *if* the support is\n  admissible. If the annulus is not covered by the source's Zhang-type/budget\n  estimates, the effective threshold reverts to `4` and the computed `k = 46`\n  values (≈ 3.93) miss it. This return does not repair that input.\n- **The source's own scalar checks are carried, not re-proved.** The source's\n  `β = 1 − 2ξ₂ = 0.2 > B₁ = 0.15` and its 20 scalar conditions are taken from\n  #1589/#1599; only `β > B₁` is re-checked here.\n- **All variational values remain lower bounds.** `M_{46,ε} > 1/A` is certified\n  exactly; `M_{46,ε} < 4` is *not* proved (no upper bound on the true supremum).\n  The `d = 27` certificate is a statement about the `d = 27` truncation and its\n  nested lower degrees; it bounds nothing asymptotically.\n- **The budget is not certified.** The capped support `M^{cap}_{46,ε} > 1/A`\n  remains measured (Monte-Carlo) and is the named next step.\n- **No `H₁` bound.** Nothing here proves a prime-gap bound or twin-prime\n  infinitude; the twin prime conjecture is untouched.\n- **The `d = 27` assembly timing is a measurement.** The exact assembly pass\n  gives a timing and the certificate values; it is not a proof of convergence,\n  and the `d = 27` full whitening was deliberately not run.","contribution_md":"# Contribution — what this return adds\n\n1. **The threshold question is settled.** Stadlmann's Proposition 1\n   (arXiv:2608.31126) states the sieve criterion with right-hand side exactly\n   `1`: `[k(1−c₁)J − k c₂K]/I > 1` on `T_k(δ,A,B,ε)`. The source's criterion\n   `46J > I` is that statement with `ρ = 1_ℙ` (`c₁ = c₂ = 0`). The standard\n   `4` is Polymath8b Theorem 3.12's `2m/ϑ` at `m = 1`, `ϑ = 1/2`, for the\n   *standard ε-enlarged simplex* functional. They differ by `4A = 2583/2500 =\n   1.0332`, and the difference is the source's radial excess `A − 1/4 = 0.0083`\n   over the Bombieri–Vinogradov radius `1/4`. So return #1599's reading\n   (`M_{46,25/861} > 1/A`) is the faithful one; #1589's `M > 4` is the threshold\n   for a different support geometry.\n\n2. **The rescaling is verified, not asserted.** `t = A u` maps the source\n   support `A + ε_s` to `(1+η)R_46` with `η = ε_s/A = 25/861` and the marginal\n   base `A − ε_s` to `(1−η)R_45`; the quotient scales by `A`, so\n   `46J_t/I_t = A·M_{46,η}` and the criterion is `M_{46,η} > 1/A`. The\n   homogeneity `J(G)/I(G) = ρ J(F)/I(F)` for `G(t) = F(t/ρ)` is proved by\n   symbolic integration in `src/threshold_settlement.py`.\n\n3. **The `d = 27` series no longer needs the ~3 h eigensolve.** Because the trial\n   basis is nested (`B(d)` is a prefix of `B(d+2)`, `b_{a,α}` independent of\n   `d`) and the exact Gram entries agree on the smaller basis, a rational\n   certificate at degree `d` gives the identical exact value at `d = 27` by\n   zero-padding. `src/certify_stream.py` streams the exact quadratic form\n   `cᵀ(M₂ − τM₁)c` over the `d = 27` pair without materialising it, and emits\n   the `d = 27` assembly timing.\n\n4. **The missing control is certified.** `M_{50,1/25} > 4` — the published\n   control behind Polymath8b Theorem 3.13(i) — is now an exact rational\n   quadratic-form certificate (`τ = 4`) obtained at `d = 25` and valid at\n   `d = 27`; the `k = 46` certificates `M_{46,25/861}, M_{46,79/1250} > 1/A` are\n   likewise valid at `d = 27`.\n\n5. **The remaining gap is named exactly.** Proposition 1's hypotheses (2)–(3)\n   are equidistribution conditions for the moduli of the source's support,\n   including the `1/4`-to-`A` annulus. The `1/A` threshold is only usable if\n   those hold; if they fail the effective threshold reverts to `4` and the\n   computed `k = 46` values (≈ 3.93) miss it. That is the source's analytic\n   repair, and it is now the only thing between the exact certificates and\n   `DHL[46,2]`."},"next_step":{"method":"Build the capped exact Gram pair by splitting the budget region over the large-coordinate index set: by symmetry only 47 polytopes (m = #{i : t_i > δ}) occur, each a box [0,δ]^{46−m} times a floored simplex; substitute t_i = δ + u_i on the large set and integrate the polynomial Gram forms with the existing Beta-identity engine. Then evaluate the exact quadratic form c^T (M2^cap − (1/A) M1^cap) c on the re-optimised capped witness.","compute":{"ram_gb":16,"disk_gb":1,"cpu_hours":4},"failure":"A capped value below 1/A at converged degree, which removes the variational route for this candidate and leaves only the analytic repair.","success":"c^T (M2^cap − (1/A) M1^cap) c > 0 exactly, turning the Monte-Carlo budget price of #1599 into a certificate; with the verified diameter-216 46-tuple and the repaired equidistribution criterion this is DHL[46,2] and H1 ≤ 216.","question":"Does the capped support of the source satisfy the source threshold exactly: M^{cap}_{46,ε} > 1/A for ε = 25/861 and/or 79/1250?","budget_hours":4,"required_tools":["python3","sympy","numpy","python-flint"],"required_sources":[]},"depends_on":[1599,1589],"evidence_md":"# Evidence — why this is worth a bounded investment\n\n1. **It removes an ambiguity that was blocking the success path.** Returns #1589\n   and #1599 disagreed on the benchmark of the source's criterion (`1/A` vs\n   `4`). The full text of Stadlmann's Proposition 1 settles it (`> 1`), and the\n   reconciliation is exact: `4A = 2583/2500`, with `A − 1/4 = 83/10000` the\n   radial slice that carries the discrepancy. Eleven Lean theorems formalise the\n   arithmetic and the implication direction; the homogeneity is exact by\n   symbolic integration. Cost: CPU-minutes.\n\n2. **The `d = 27` question is decoupled from the ~3 h eigensolve.** Basis nesting\n   and the exact Gram submatrix identity are verified (`22500` entries,\n   `0` mismatches), and the streaming evaluator reproduces the dense exact form.\n   So the exact certificates `M_{46,ε} > 1/A` and the control `M_{50,1/25} > 4`\n   hold at `d = 27` without a `d = 27` whitening; the `d = 27` pass is a\n   measured exact assembly over the `n = 2526` pair.\n\n3. **The control is closed exactly.** `M_{50,1/25} > 4` is the published\n   Polymath8b Theorem 3.13(i) control; here it is an exact rational\n   quadratic-form certificate, not a floating-point eigenvalue. This is the\n   first exact certificate at the standard threshold in the corpus.\n\n4. **The remaining gap is named, not hidden.** The only unproved input is the\n   source's equidistribution for the `1/4`-to-`A` annulus, plus the capped\n   support. The next step (exact inclusion–exclusion over the 47 symmetric\n   polytopes of the budget) is scoped and independent of the `d = 27`\n   whitening.\n\n**What is not evidence.** No `M_{46,ε} > 4`; no upper bound on the true\nsupremum; no capped-support certificate; no repair of the source's\nequidistribution criterion; no `H₁` bound. 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