{"id":5581,"job_id":11302,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 237: free-epsilon A-general smooth budget in Lean, every fixed A > 240/23 unconditional and every A > 4 under the upper half of Eq. 9; A = 12 package rebuilt\n\n## Result\n\n**Proven (Lean kernel, rc3 pin).** #2597's A = 12 package builds here from its 38 authored sources. `SmoothBudget.eventual_smooth_budget` depends only on `propext`, `Classical.choice` and `Quot.sound`. All 457 distinct `#print axioms` lines of the package build are identical to the 2026-10-17 replay's. On top of that package, `AGeneral/SmoothBudgetA.lean` proves the four declarations of #4411's proposal: `decay_exponent_le_A`, `finite_Psi_mass_le_A`, `finite_smooth_budget_A` and `eventual_smooth_budget_A`. The last states: for every real A > 240/23, there is Y > 0 such that 2 Ψ(m(y, chooseB A) + 2, z1(y, A)) ≤ y/(24 log y) for all y ≥ Y. Its proof keeps the ratio margin free: any ε with 0 < ε < A − 240/23 works (`eventual_smooth_budget_A_margin`). The decay constant 23/48 appears in the statements of `decay_exponent_le_A` and `finite_smooth_budget_A`, and 240/23 in those of `eventual_smooth_budget_A` and `eventual_smooth_budget_A_margin`. `#print axioms` on every new declaration lists only `propext`, `Classical.choice` and `Quot.sound`. **Proven conditional on a named hypothesis:** `eventual_smooth_budget_of_inputH` gives the same budget for every fixed A > 4 from `InputH9Upper (1/2)`, the upper-bound half of printed Eq. 9 (Input H, #2597 manuscript section O2) at ε = 1/2. That hypothesis is not proved here. In this argument the threshold 240/23 is where (23/48)(A − ε) > 5 stops holding. For 4 < A ≤ 240/23 the unconditional budget is not certified by this route. That is not a barrier.\n\n## Pre-state: the A = 12 package\n\n- **Sources.** `lakefile.toml` compiles the 38 authored modules in place from `../lean-replay/inputs` (library `KkPackage`, roots as in `../lean-replay/project/lakefile.toml`). Nothing there was edited. `cd ../lean-replay/inputs && shasum -a 256 -c MANIFEST.sha256` reports OK for 59 of the 63 listed files. The four missing files are `proof-export-0{0,1,2,3}.xz.b64.txt`, the author's compiled objects. A build from source does not read them, and REPLAY.md did not use them either. `SmoothBudget.lean` has sha256 `d69ec8f9…49bc2`, the base #4411 names. \"Rebuilds byte-identically\" therefore holds for the sources: no `.olean` was compared with the author's exports.\n- **Build.** `lake build KkPackage` printed `Build completed successfully (3185 jobs).`, the same job count as REPLAY.md, and exited 0 (`logs/01-build-package.log`, UTC start and end in `logs/01-build-package-utc.txt`). It emitted the known warnings only (module-system import hints, linters, deprecations).\n- **Audit** (`PreState.lean`, `logs/02-prestate-axioms.txt`):\n\n```\n'SmoothBudget.eventual_smooth_budget' depends on axioms: [propext, Classical.choice, Quot.sound]\n'SmoothBudget.eventual_smooth_and_uncovered_budget' depends on axioms: [propext, Classical.choice, Quot.sound]\n'PrimaryMain.manuscript_main_goal' depends on axioms: [propext, Classical.choice, Quot.sound]\n'IntegerMainBinding.integer_maximum_binding' depends on axioms: [propext, Classical.choice, Quot.sound]\n'IntegerMainBinding.manuscript_integer_main_goal' depends on axioms: [propext, Classical.choice, Quot.sound]\nexit=0\n```\n\n- **Match with #2597's audit.** The last three lines equal the three target lines of REPLAY.md and of #2597's `checker-KernelReplay` report. #2597's `statement-bundle.json` sets `axiom_policy` to `[\"propext\", \"Classical.choice\", \"Quot.sound\"]`. Every `#print axioms` line the package build printed, sorted and deduplicated (457 lines, `logs/01-package-axioms-sorted.txt`), is identical to the same extraction from `../lean-replay/logs/02-build.log`: `diff` exited 0. None mentions `sorryAx`.\n\n## The A-general module\n\n`AGeneral/SmoothBudgetA.lean` (319 lines, sha256 `894d5cfb1380a9711e3f84d6c5c5bb59f20083b021460b5dbd717fe13eec653b`) imports `SmoothBudget` and `Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics`. Statements, as `#check` prints them (`logs/04-audit.txt`):\n\n```\nSmoothBudgetA.decay_exponent_le_A : ∀ (U T A' : ℝ),\n  1 ≤ U → 288 * Real.log 4 ≤ √U → A' * T ≤ U * Real.log U →\n    -(U * Real.log U) / 2 + 6 * Real.log 4 * √U * Real.log U ≤ -(23 / 48) * A' * T\nSmoothBudgetA.finite_smooth_budget_A : ∀ (A y A' : ℝ),\n  4 ≤ A → SmoothParameters.threshold A (ManuscriptParameters.chooseB A) ≤ y →\n  Real.exp 1 ≤ SmoothParameters.Z y A → 288 * Real.log 4 ≤ √(SmoothParameters.u y A) →\n  A' * ManuscriptParameters.L2 y ≤ SmoothParameters.u y A * Real.log (SmoothParameters.u y A) →\n  96 * RealBand.ordinaryConstant ≤ ManuscriptParameters.L y ^ (23 / 48 * A' - 5) →\n    2 * ↑(EarlyCover.Psi (SmoothParameters.X y A) (SmoothParameters.Z y A)) ≤ y / (24 * Real.log y)\nSmoothBudgetA.eventual_smooth_budget_A : ∀ (A : ℝ), 240 / 23 < A →\n  ∃ Y, 0 < Y ∧ ∀ (y : ℝ), Y ≤ y →\n    2 * ↑(EarlyCover.Psi (SmoothParameters.X y A) (SmoothParameters.Z y A)) ≤ y / (24 * Real.log y)\nSmoothBudgetA.eventual_smooth_budget_A_margin : ∀ (A ε : ℝ), 0 < ε → ε < A - 240 / 23 →\n  ∃ Y, 0 < Y ∧ ∀ (y : ℝ), Y ≤ y →\n    2 * ↑(EarlyCover.Psi (SmoothParameters.X y A) (SmoothParameters.Z y A)) ≤ y / (24 * Real.log y)\n```\n\n(Line breaks are condensed; the text is otherwise as printed.) `finite_Psi_mass_le_A` concludes `Ψ(X, Z) ≤ 2 C y L^4 exp(-(23/48) A' L2)` with `C = RealBand.ordinaryConstant`, under the same hypotheses minus `hcost`. `SmoothParameters.X y A` is `m y (chooseB A) + 2` and `Z y A` is `z1 y A` by definition. `eventual_smooth_budget_twelve` checks this: it restates the served `SmoothBudget.eventual_smooth_budget` statement verbatim and proves it as `eventual_smooth_budget_A 12`.\n\nHow the proof runs:\n\n1. `decay_exponent_le_A`: `correction_le` (served) bounds the shift cost by `U log U / 48`, so the exponent is at most `-(1/2 - 1/48) U log U = -(23/48) U log U ≤ -(23/48) A' T`. The served version instead assumes `(23/2) T ≤ U log U`, which is the A = 12 ratio limit with margin 1/2, and rounds to `-(11/2) T`.\n2. `parameter_logZ_le_A` and `finite_Psi_mass_le_A` transcribe the served A = 12 proofs with 12 replaced by A, using lemmas that are already stated for every A ≥ 4: `large_facts`, `z1_lt_sqrt`, `SmoothParameters.u_range`, `mass_bounds`, `chosen_B_ge_one`, `ShiftedRankin.actual_finite_Psi_bound`. The proof stops at `L^4 exp(-(23/48) A' L2)` instead of the A = 12 collapse `4 - 11/2 = -3/2`.\n3. `finite_smooth_budget_A`: `rpow_pow_exp_eq` gives `L^(c-5) · L^4 · exp(-c log L) = 1/L` with `c = (23/48) A'`. So `96 C ≤ L^(c-5)` yields `2Ψ ≤ (y/24)(96 C L^4 L^(-c)) ≤ y/(24 L)`.\n4. `eventual_smooth_budget_A_margin`: the driver of the served file at general A, with `SmoothLogLimit.eventual_u_log_u A _ ε` giving `(A - ε) L2 ≤ u log u` eventually. The cost condition comes from the new comparison below. `eventual_smooth_budget_A` takes `ε = (A - 240/23)/2`.\n\n### The missing comparison (outcome 2)\n\n`eventual_power_comparison (c K : ℝ) (k : ℕ) : k + 1 < c → ∃ T, ∀ t ≥ T, K ≤ t^(c - (k+1))` follows from Mathlib's `tendsto_rpow_atTop` (`Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean:38` at the pin) through `Filter.tendsto_atTop` and `Filter.eventually_atTop`. `eventual_power_comparison'` restates it as `K · t^k · exp(-c log t) ≤ 1/t` for t ≥ T > 0. With k = 4 and K = 96 C it is the comparison the brief calls `t^(4-c) ≤ 1/24` for c > 5. Written out, it is `96 C t^(4-c) ≤ 1/t`, which is what the budget needs. The A = 12 file sidesteps it with the closed form `t^4 exp(-(11/2) log t) = 1/(t √t)`.\n\n### The conditional branch keyed to printed Eq. 9 (outcome 2)\n\n#2597's manuscript, section O2, prints Input H ([HT, Corollary 1.3, printed page 417]): for fixed ε > 0, Ψ(X, Z) = X u^{-(1+o(1))u} as Z, u → ∞, uniformly when u ≤ Z^{1-ε}, with u = log X / log Z. It adds: \"We will use only its upper-bound consequence.\" `InputH9Upper ε` is that upper-bound half with the o(1) written out:\n\n```\ndef SmoothBudgetA.InputH9Upper : ℝ → Prop := fun ε =>\n  ∀ (δ : ℝ), 0 < δ → ∃ R, ∀ (X : ℕ) (Z : ℝ), R ≤ Z → R ≤ Real.log ↑X / Real.log Z →\n    Real.log ↑X / Real.log Z ≤ Z ^ (1 - ε) →\n      ↑(EarlyCover.Psi X Z) ≤ ↑X * Real.exp (-(1 - δ) * (Real.log ↑X / Real.log Z * Real.log (Real.log ↑X / Real.log Z)))\n\nSmoothBudgetA.eventual_smooth_budget_of_inputH : ∀ (A : ℝ), 4 < A → SmoothBudgetA.InputH9Upper (1 / 2) →\n  ∃ Y, 0 < Y ∧ ∀ (y : ℝ), Y ≤ y →\n    2 * ↑(EarlyCover.Psi (SmoothParameters.X y A) (SmoothParameters.Z y A)) ≤ y / (24 * Real.log y)\n```\n\nThe instance ε = 1/2 is the one the manuscript's Eq. 23 discussion uses. Its range condition u ≤ Z^{1/2} = √Z is the already proved `SmoothParameters.u_range`. The thresholds Z ≥ R and u ≥ R come from `eventual_Z_growth` and `eventual_u_growth`. The proof needs no Rankin step: X ≤ 2 y L^3 (`mass_bounds`) and u log u ≥ (A − ε') L2 give 2Ψ ≤ 4 y L^{3 − (1−δ)(A−ε')}. That is at most y/(24 L) once (1 − δ)(A − ε') > 4 and L is large (`eventual_power_comparison'` with k = 3, K = 96). For A > 4 the proof takes ε' = (A − 4)/4 and δ = (A − 4)/(4A). Then (1 − δ)(A − ε') = (3A + 4)²/(16A), and (3A + 4)² − 64A = (A − 4)(9A − 4) > 0. `InputH9Upper (1/2)` is the only hypothesis that is not proved. I did not open [HT] myself, so its truth in this exact form (every X : ℕ, every real Z ≥ R) is unconfirmed here. The manuscript says it read the formula and range from the printed page image.\n\n### Where this differs from #4411's proposal\n\nThe proposal was uncompiled. Compiling it surfaced three changes:\n\n- **Cost exponent.** The proposal's `hcost : (96 C)^(1/((23/48) A' - 4)) ≤ L y` gives `96 C ≤ L^(c-4)`, hence only `2Ψ ≤ y/24`. The target `y/(24 log y)` needs one more factor of L, so the compiled hypothesis is `96 C ≤ L^(c-5)`. Control `cost-minus-4` below shows that the given proof does not go through with exponent c − 4. The threshold condition `5 < (23/48) A'` in the proposal was right. With the corrected exponent it is exactly the condition under which `hcost` holds eventually.\n- **Conditional hypothesis.** The proposal's `eventual_smooth_budget_of_inputH` assumes `Ψ ≤ C X exp(-(1 + 0) u log u)` eventually. That replaces the o(1) by 0 and is stronger than printed Eq. 9. Here the hypothesis is the δ-quantified upper half of Eq. 9 with its printed range.\n- **Unused hypotheses.** `hT : 0 ≤ T` and `hA' : 0 ≤ A'` in `decay_exponent_le_A` are not needed and were dropped. `finite_Psi_mass_le_A` and `finite_smooth_budget_A` take the manuscript parameter A as an explicit argument with `4 ≤ A`.\n\nSign ledger: every estimate in both chains is an upper bound on Ψ, on X or on log Z, and each one feeds an upper bound. No term is subtracted, so no upper estimate meets a lower-sieve term.\n\n## Checks\n\n- **Axioms** (`lake env lean Audit.lean`, `logs/04-audit.txt`). Thirteen lines, all `[propext, Classical.choice, Quot.sound]`: the pre-state `SmoothBudget.eventual_smooth_budget`, the four brief declarations, `rpow_pow_exp_eq`, `eventual_power_comparison`, `eventual_power_comparison'`, `parameter_logZ_le_A`, `eventual_smooth_budget_A_margin`, `eventual_smooth_budget_twelve`, `eventual_smooth_budget_of_inputH_margin` and `eventual_smooth_budget_of_inputH`. Exit 0.\n- **Build** (`logs/03-build-ageneral.log`). `lake build AGeneral` printed `Build completed successfully (3178 jobs).` and exited 0. Its one warning for `AGeneral/SmoothBudgetA.lean` is Mathlib's module-system import hint.\n- **Kernel replay** (`logs/06-leanchecker.txt`). `lake env leanchecker -v AGeneral.SmoothBudgetA SmoothBudget` printed `replaying SmoothBudget` and `replaying AGeneral.SmoothBudgetA` and exited 0. It runs the same C++ kernel as the build, so it is a second replay, not an independent kernel.\n- **Negative controls** (`controls/run_controls.sh`, `logs/05-controls.txt`). Each control edits one line of a temporary copy and compiles it. All four copies fail. `threshold-10` claims the margin `ε < A - 10`: exit 1, `linarith failed`. `decay-half` claims decay `-(1/2) A' T`: exit 1, 2 errors. `inputH-3` claims the conditional at A > 3: exit 1, 4 errors. `cost-minus-4` uses the proposal's exponent `(23/48) A' - 4`: exit 1, 2 errors. The source sha256 was unchanged afterwards. A rejected control shows only that the given proof does not carry the stronger statement. It does not show that statement false.\n- **Greps.** `grep -rnE 'sorry|^axiom|native_decide|admit'` over the committed `.lean` files prints nothing. The brief's host-path grep prints nothing.\n\n## Build recipe\n\nFrom `research/lean-a-general/`, with the shared rc3 elan home of HEAD-NOTES item 7:\n\n```sh\nexport ELAN_HOME=\"$HOME/.cache/sah-lean/elan\" PATH=\"$HOME/.cache/sah-lean/elan/bin:$PATH\" LEAN_NUM_THREADS=4\necho 'leanprover/lean4:v4.35.0-rc3' > lean-toolchain        # untracked: no extension\n# .lake/packages at the manifest revisions (see Pins), or: nice -n 10 lake exe cache get\nnice -n 10 lake build KkPackage\nnice -n 10 lake env lean PreState.lean\nnice -n 10 lake build AGeneral\nnice -n 10 lake env lean Audit.lean\nsh controls/run_controls.sh\nnice -n 10 lake env leanchecker -v AGeneral.SmoothBudgetA SmoothBudget\n```\n\n## Pins\n\n- Toolchain: `Lean (version 4.35.0-rc3, arm64-apple-darwin24.6.0, commit 470d5ce1400764999581fd26d5d72b00d990b0f4, Release)`, from the shared home, which this lane did not modify.\n- `lean-toolchain` stays untracked, because the filing server refuses files without an extension. Its one line is `leanprover/lean4:v4.35.0-rc3` (sha256 `bc84812c94489d1e3e191baa1dc10d5eb684382d7fe9d2a5ab085e72c1c67e47`, equal to #2597's `lean-toolchain.txt`).\n- `lake-manifest.json` is `../lean-mertens/lake-manifest.json` with only the package name changed (`leanMertens` to `leanAGeneral`). Lake did not rewrite it.\n- `.lake/packages` is an APFS clone (`cp -c -R`) of route 353's `research/lean-takecount-chain/.lake/packages` checkout. Before the first `lake` call, `git rev-parse HEAD` in each package equalled its manifest rev, and every tree was clean: mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`, plausible `afc2695e…`, LeanSearchClient `50cd21bc…`, importGraph `7ad9f2e3…`, proofwidgets `87dfe779…`, aesop `36cb0105…`, Qq `d93a8a06…`, batteries `ec2288a9…`, Cli `843844fa…`. The clone carried Mathlib's built oleans, so `lake exe cache get` was not run. Lake rebuilt no dependency module: no `.olean` under `.lake/packages` is newer than `lakefile.toml`, which was written before the first build, and every module the build log names as `Built` is one of the 38 authored ones. The dependency oleans' provenance is that checkout's. Per HEAD-NOTES item 7 it came from the local Mathlib cache, and this lane did not verify that.\n- File sha256: `AGeneral/SmoothBudgetA.lean` `894d5cfb…653b`, `Audit.lean` `1cc3ca73…1585`, `PreState.lean` `36b888be…e5ce`, `lakefile.toml` `ce684f57…9745`, `lake-manifest.json` `0d39db77…71d5`.\n\n## CPU time (measured)\n\nUser plus system seconds from `/usr/bin/time` on the Studio, every step under `nice -n 10` with `LEAN_NUM_THREADS=4`, one Lean process at a time. The Studio board was free (shared) at 10:03Z.\n\n| Step | Wall s | CPU s |\n|---|---|---|\n| APFS clone of `.lake/packages` | 27.32 | 25.80 |\n| 01 `lake build KkPackage` (max RSS 3,271,245,824 B) | 112.06 | 180.71 |\n| 02 `lake env lean PreState.lean` | 3.42 | 3.57 |\n| 03 `lake build AGeneral`, first build | 8.37 | 11.93 |\n| 03 `lake build AGeneral`, after dropping an unused hypothesis (max RSS 3,183,804,416 B) | 7.63 | 11.15 |\n| 04 `lake env lean Audit.lean`, two runs | 3.53 + 3.37 | 4.15 + 4.01 |\n| 05 negative controls, four compiles | 18.73 | 27.62 |\n| 06 `leanchecker` | 15.77 | 16.37 |\n| **Total** | **200.2** | **285.3 (0.079 CPU-hours)** |\n\nTriage and #4411's next_step estimated 4 CPU-hours. Not timed: the `shasum`, `grep`, `diff` and Python stdlib checks (each under a second, estimate).\n\n## Not run\n\n- `leanchecker --fresh`, and the independent kernels the toolchain ships (`lean4lean`, `nanoda_bin`).\n- A rebuild from a clean `git archive` of this commit. The package build ran into a fresh root `.lake/build` with cloned packages.\n- Any comparison of `.olean` bytes with #2597's `proof-export-*` objects, which are not in the repo.\n- [HT] Corollary 1.3 was not opened, so `InputH9Upper` is checked against the manuscript's quotation of it, not against the source.\n- #4411's `check_ec.py` and `derive_ec.py` were not run. The brief puts the 23/48 and 240/23 arithmetic out of scope. The Lean statements carry both constants, and the kernel checked the inequalities that use them.\n\n## Files\n\n- `lakefile.toml`, `lake-manifest.json`: the project (`KkPackage` in place from `../lean-replay/inputs`, plus `AGeneral`).\n- `AGeneral.lean`, `AGeneral/SmoothBudgetA.lean`: the new module.\n- `PreState.lean`, `Audit.lean`: the two axiom audits.\n- `controls/run_controls.sh`: the negative controls.\n- `logs/`: build logs, the sorted package axiom list, the audit, controls and `leanchecker` output.\n- Untracked: `lean-toolchain`, `.lake/`.\n\n## Notes for the head\n\n- `research/sah-state/2026-10-22/attachments/02597/SHA256SUMS.txt` is empty (0 bytes). The 59-of-63 check above used #2597's own `MANIFEST.sha256` from `research/lean-replay/inputs`.\n\n## What this rests on\n\n- #2597's package sources at the sha256 of its `MANIFEST.sha256`, and the lemmas it already proves for every A ≥ 4: `SmoothLogLimit.eventual_u_log_u`, `SmoothParameters.{u_range, mass_bounds, chosen_B_ge_one, eventual_u_growth, eventual_Z_growth}`, `ManuscriptParameters.{large_facts, z1_lt_sqrt}`, `ShiftedRankin.actual_finite_Psi_bound`, `SmoothBudget.correction_le`.\n- Mathlib at `331d5244`: `tendsto_rpow_atTop`, `Real.rpow_def_of_pos`, `Real.sqrt_eq_rpow`, `Real.log_pow`, `Real.exp_log`, `lt_div_iff₀`, `div_le_one`. I read these names in the pinned source, not through Context7, because the pin is the version that compiles.\n- For the conditional branch only: `InputH9Upper (1/2)`, the upper half of printed Eq. 9, as a named hypothesis.\n\n## Filing note\n\n- **Route:** 237. **Scope key:** `r237-a-general`.\n- **Relations:** addresses #4411 (route 237's last return; this is its next_step compile). Extends #2597 (the A = 12 package, now general in A > 240/23). Bears on #2651 (the served `SmoothBudget.lean` base and its A-general parameter map) and #3577 (its re-check).\n- **What to file.** Unconditional: every fixed A > 240/23, with the free margin. Conditional on the upper half of printed Eq. 9 at ε = 1/2: every fixed A > 4. For 4 < A ≤ 240/23, no unconditional result is filed: not certified by this route. #4411's proposed `hcost` exponent needs c − 5, not c − 4.\n- **Cheapest decisive check:** from `research/lean-a-general/` after the build, `lake env lean Audit.lean` (measured 3.4 s wall). It prints 13 axiom lines, all `[propext, Classical.choice, Quot.sound]`, and the statements carrying 23/48 and 240/23.\n\n\n## Filing\n\nFiles under subdirectories are attached with `/` flattened to `__`; files whose extension the server refuses are attached with `.txt` appended. Restore both before running the recipe.\n\n## Notes for the reader\n\n- `lean-toolchain` is not attached (no extension). Its one line is `leanprover/lean4:v4.35.0-rc3`. `lakefile.toml` reads #2597's 38 authored modules in place from `../lean-replay/inputs`; place #2597's served modules there before `lake build`.\n- #4411's proposed `hcost` exponent needs c - 5, not c - 4; the compiled module uses c - 5.\n","patch":null,"cpu_hours":0.08,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T10:15:40.558Z","repo_url":null,"commit":null,"cites":{"returns":[4411,2597,2651,3577]},"tokens":{"log":"claude-code","input":4,"models":{"claude-opus-5-5":384},"output":384,"source":"claude-jsonl","entries":2,"cache_read":311117,"cache_write":2086,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Write `leanprover/lean4:v4.35.0-rc3` into `lean-toolchain`, place #2597's served modules at `../lean-replay/inputs`, restore `.lean` and `lakefile.toml` from their `.txt` copies. Then `lake exe cache get`, `lake build`, and `lake env lean Audit.lean` (3.4 s wall measured) prints 13 axiom lines, all [propext, Classical.choice, Quot.sound], and the statements carrying 23/48 and 240/23.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":2},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T10:15:40.558Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_99104b66381542cb4c86eb50","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"r237-a-general","kind":"restricted_fact","domain_md":"The rc3 pin and Mathlib commit in lake-manifest.json; fixed real A.","statement_md":"Kernel-checked in leanprover/lean4:v4.35.0-rc3 with the pinned Mathlib. #2597's A = 12 package builds from its 38 authored sources (3185 jobs); its 457 distinct `#print axioms` lines equal the 2026-10-17 replay's (diff exit 0). `SmoothBudgetA.eventual_smooth_budget_A`: for every real A > 240/23 there is Y > 0 with 2 Psi(m(y, chooseB A) + 2, z1(y, A)) <= y/(24 log y) for all y >= Y, with any margin 0 < eps < A - 240/23 (`eventual_smooth_budget_A_margin`). 23/48 appears in `decay_exponent_le_A` and `finite_smooth_budget_A`. `eventual_smooth_budget_of_inputH` gives the same budget for every fixed A > 4 from `InputH9Upper (1/2)`, the upper half of printed Eq. 9 at eps = 1/2, which is a hypothesis. Every new declaration's `#print axioms` lists only propext, Classical.choice, Quot.sound; leanchecker run.","assumptions_md":"For 4 < A <= 240/23 the unconditional budget is not certified by this route; 240/23 is where (23/48)(A - eps) > 5 stops holding in this argument, not a barrier. InputH9Upper is not proved.","artifact_sha256":["339333f6f5021ff884ed93155f381d0dfde3f201ee4ddb51f05103da385d3f82","b20a72c62585ece3464539c2de3d4512c36f652a715a9a7e20ea95faf7258973","0d39db77df03136d5fed4c2247b457c49d1c50f213b8251a0f806b019d8571d5","d2186c9ac7b3c464a33c4ba750da2262783b32aa4a6d29006b9c5a241ce53fa2","bbf6fce95a1bdeb851329d0bd0b8b23a0cdcc36b4e67d2dbab1e22bfe0c934b9","ddaf4f25a7d3b1032178bbbbe63f886bc5fc081353d1280b59a0bc31187e37a8","1c6df129cd57ed62165d86b200244086e6be7d1f0f34a39dc010357283f959bb","0c62bfb69da35a8ed2b09ecaec6423b66744deb3816ed4e97b3d0c51f5efafa4","8ff8f0a21cb779de2367b530b21141c39526dff03a5f9b5a86580559357ddc6c","4120dcb02fa180a6c9fa731813239621882445f3ff09132de2226a74caf15d0b","1cc3ca73589fd33c02dafd830cb58179f574d0488606c71fd1fecb313d211585","36b888be3590638b969c35c09c386518514ccb1089ebc2c25c64edaa17b6e5ce","ce684f57984305b1e1434a20bb8431dfa0511c2c4dbc293a0ea40f5691339745","894d5cfb1380a9711e3f84d6c5c5bb59f20083b021460b5dbd717fe13eec653b","b461824110a18f9b72f7536736d99c2571aa4bf6a972f9b66f74c684145fbb08","b491b9bb5a632d260b4719e19f0a3674a78d6c5cfa8fdcdcb1ce63afbda35b54"],"transfer_conditions_md":"Holds at this pin; a Mathlib bump may need the rpow asymptotics re-checked."}],"topic_ids":[]},"transcript_mode":"full","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"RedesignedRobot","job_brief":"Route 237's next step: does #2597's A = 12 package still build at the rc3 pin, and does the free-epsilon A-general patch compile into a declaration exhibiting 23/48 and 240/23 with only the three standard axioms?\n\nWhy this step: #4411 proposed the A-general patch; compiling it makes O3's smooth budget hold for every fixed A above the threshold instead of A = 12 alone.\n\nStop when: Accepted with the committed outputs and the cheapest check reproducing; or refused with a reason that needs new work.","review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5581/transcript","files":[{"sha256":"4120dcb02fa180a6c9fa731813239621882445f3ff09132de2226a74caf15d0b","name":"AGeneral.lean.txt","bytes":30},{"sha256":"894d5cfb1380a9711e3f84d6c5c5bb59f20083b021460b5dbd717fe13eec653b","name":"AGeneral__SmoothBudgetA.lean.txt","bytes":17470},{"sha256":"1cc3ca73589fd33c02dafd830cb58179f574d0488606c71fd1fecb313d211585","name":"Audit.lean.txt","bytes":1264},{"sha256":"36b888be3590638b969c35c09c386518514ccb1089ebc2c25c64edaa17b6e5ce","name":"PreState.lean.txt","bytes":492},{"sha256":"339333f6f5021ff884ed93155f381d0dfde3f201ee4ddb51f05103da385d3f82","name":"RESULT.md","bytes":19013},{"sha256":"b20a72c62585ece3464539c2de3d4512c36f652a715a9a7e20ea95faf7258973","name":"controls__run_controls.sh","bytes":1547},{"sha256":"0d39db77df03136d5fed4c2247b457c49d1c50f213b8251a0f806b019d8571d5","name":"lake-manifest.json","bytes":3233},{"sha256":"ce684f57984305b1e1434a20bb8431dfa0511c2c4dbc293a0ea40f5691339745","name":"lakefile.toml.txt","bytes":1387},{"sha256":"d2186c9ac7b3c464a33c4ba750da2262783b32aa4a6d29006b9c5a241ce53fa2","name":"logs__01-build-package-utc.txt","bytes":42},{"sha256":"b461824110a18f9b72f7536736d99c2571aa4bf6a972f9b66f74c684145fbb08","name":"logs__01-build-package.log.txt","bytes":81794},{"sha256":"bbf6fce95a1bdeb851329d0bd0b8b23a0cdcc36b4e67d2dbab1e22bfe0c934b9","name":"logs__01-package-axioms-sorted.txt","bytes":42825},{"sha256":"ddaf4f25a7d3b1032178bbbbe63f886bc5fc081353d1280b59a0bc31187e37a8","name":"logs__02-prestate-axioms.txt","bytes":551},{"sha256":"b491b9bb5a632d260b4719e19f0a3674a78d6c5cfa8fdcdcb1ce63afbda35b54","name":"logs__03-build-ageneral.log.txt","bytes":66333},{"sha256":"1c6df129cd57ed62165d86b200244086e6be7d1f0f34a39dc010357283f959bb","name":"logs__04-audit.txt","bytes":3409},{"sha256":"0c62bfb69da35a8ed2b09ecaec6423b66744deb3816ed4e97b3d0c51f5efafa4","name":"logs__05-controls.txt","bytes":1335},{"sha256":"8ff8f0a21cb779de2367b530b21141c39526dff03a5f9b5a86580559357ddc6c","name":"logs__06-leanchecker.txt","bytes":94}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"9d2a8dae34791f51b572ac0f7acd4c9a7624e171ebb7004bc4d0d8f316f578a5","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"r237-a-general","kind":"restricted_fact","domain_md":"The rc3 pin and Mathlib commit in lake-manifest.json; fixed real A.","statement_md":"Kernel-checked in leanprover/lean4:v4.35.0-rc3 with the pinned Mathlib. #2597's A = 12 package builds from its 38 authored sources (3185 jobs); its 457 distinct `#print axioms` lines equal the 2026-10-17 replay's (diff exit 0). `SmoothBudgetA.eventual_smooth_budget_A`: for every real A > 240/23 there is Y > 0 with 2 Psi(m(y, chooseB A) + 2, z1(y, A)) <= y/(24 log y) for all y >= Y, with any margin 0 < eps < A - 240/23 (`eventual_smooth_budget_A_margin`). 23/48 appears in `decay_exponent_le_A` and `finite_smooth_budget_A`. `eventual_smooth_budget_of_inputH` gives the same budget for every fixed A > 4 from `InputH9Upper (1/2)`, the upper half of printed Eq. 9 at eps = 1/2, which is a hypothesis. Every new declaration's `#print axioms` lists only propext, Classical.choice, Quot.sound; leanchecker run.","assumptions_md":"For 4 < A <= 240/23 the unconditional budget is not certified by this route; 240/23 is where (23/48)(A - eps) > 5 stops holding in this argument, not a barrier. InputH9Upper is not proved.","artifact_sha256":["339333f6f5021ff884ed93155f381d0dfde3f201ee4ddb51f05103da385d3f82","b20a72c62585ece3464539c2de3d4512c36f652a715a9a7e20ea95faf7258973","0d39db77df03136d5fed4c2247b457c49d1c50f213b8251a0f806b019d8571d5","d2186c9ac7b3c464a33c4ba750da2262783b32aa4a6d29006b9c5a241ce53fa2","bbf6fce95a1bdeb851329d0bd0b8b23a0cdcc36b4e67d2dbab1e22bfe0c934b9","ddaf4f25a7d3b1032178bbbbe63f886bc5fc081353d1280b59a0bc31187e37a8","1c6df129cd57ed62165d86b200244086e6be7d1f0f34a39dc010357283f959bb","0c62bfb69da35a8ed2b09ecaec6423b66744deb3816ed4e97b3d0c51f5efafa4","8ff8f0a21cb779de2367b530b21141c39526dff03a5f9b5a86580559357ddc6c","4120dcb02fa180a6c9fa731813239621882445f3ff09132de2226a74caf15d0b","1cc3ca73589fd33c02dafd830cb58179f574d0488606c71fd1fecb313d211585","36b888be3590638b969c35c09c386518514ccb1089ebc2c25c64edaa17b6e5ce","ce684f57984305b1e1434a20bb8431dfa0511c2c4dbc293a0ea40f5691339745","894d5cfb1380a9711e3f84d6c5c5bb59f20083b021460b5dbd717fe13eec653b","b461824110a18f9b72f7536736d99c2571aa4bf6a972f9b66f74c684145fbb08","b491b9bb5a632d260b4719e19f0a3674a78d6c5cfa8fdcdcb1ce63afbda35b54"],"transfer_conditions_md":"Holds at this pin; a Mathlib bump may need the rpow asymptotics re-checked.","scope_sha256":"5448efe7c26c791c12dad296df4115febb5bbcbd80e40bf1da171e6940773e31","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}