{"id":5575,"job_id":11292,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 241: V(z) = 2 M(z)^2 P(z) and V <= 2/(log z)^2 in Lean; Lemma 2 short form and Eq. 28 compile with only Input S and Mertens III as named hypotheses\n\n## Result\n\nThe Lean kernel checks the O7 short-construction module in this project. The only hypotheses left are two named propositions, `InputS 2 C_S` (O5, [KK, Lemma 1]) and `MertensIII` (O1). Neither is an axiom. Neither is proved here. The project imports #2597's 38 modules from `research/lean-replay/inputs/` unchanged and checks their sha256.\n\n- **Identity (unconditional).** For every natural z ≥ 2, V(z) = 2 · M(z)² · P(z) (`ShortConstruction.V_identity`). Here V(z) = ∏_{p ≤ z}(1 − g(p)/p) with g(p) = |base p|, M = `MertensBand.ordinaryProduct` and P(z) = ∏_{2<p≤z}(1 − 1/(p−1)²). The identity is exact, with no limit and no error term.\n- **One-sided corollary (unconditional).** V(z) ≤ 2/(log z)² for natural z ≥ 2 (`V_le`). With a real cutoff, V(⌊x⌋) ≤ 2/(log x)² for real x ≥ 2 (`V_le_real`). Both use only the package's one-sided Mertens bounds and P ≤ 1.\n- **Sandwich (unconditional).** 2 · C2 · M(z)² ≤ V(z) ≤ 2 · M(z)², with C2 = inf_z P(z) and 1/2 ≤ C2 ≤ 1 (`V_sandwich`, `half_le_C2`, `C2_le_one`). P is antitone and tends to C2 (`P_tendsto`).\n- **Parameter lemmas (unconditional).** m = ⌊c y log y⌋ and z = ⌊√m⌋ = `Nat.sqrt m`, in `ShortParameters.lean`. They include the cutoff equivalence p ≤ √m ⟺ p ≤ z (`le_sqrt_iff`), the \"4\" of Eq. 28 (m/(log √m)² = 4m/(log m)²), and log m / log y → 1.\n- **Lemma 2, short form (Input S only).** For m ≥ 1, S(m, Ω) ≤ C_S · m · V(⌊√m⌋) (`lemma2_short`). For m ≥ 4, S(m, Ω) ≤ 2 C_S m/(log √m)² (`short_bound`). Here S(m, Ω) = #{1 ≤ n ≤ m : n mod p ∉ {0, −2} for every prime p ≤ √m} (`shortSurvivors`, with the ZMod form in `shortSurvivors_eq_zmod`).\n- **Eq. 28, two conditional forms.** For c > 0 and every ε > 0, eventually in y:\n  - `eq28_unconditional` (Input S only): S(⌊c y log y⌋, Ω) ≤ (4 · 2 C_S · c + ε) · y/log y.\n  - `eq28_priced` (Input S and Mertens III): S(⌊c y log y⌋, Ω) ≤ (4 · C_S · 2 C2 e^{−2γ} · c + ε) · y/log y.\n- **Two c-ranges.** Under each range, eventually S(⌊c y log y⌋, Ω) < (1/2) · y/log y.\n  - `c_range_unconditional` (Input S only): 0 < c < 1/(16 C_S).\n  - `c_range_priced` (Input S and Mertens III): 0 < c < 1/(8 C_S · 2 C2 e^{−2γ}).\n\nC1(1/2) = 805.5 and c = 3.7e-4 appear in no statement and no proof. The c-range theorems take C_S as a free real with `InputS 2 C_S` as a hypothesis. This project certifies no numerical value of C_S, of c, or of C2 beyond 1/2 ≤ C2 ≤ 1.\n\nA side finding: Input S as printed is false (`InputSReal.not_inputSReal`, section \"Input S as printed\").\n\n`lake env lean Audit.lean` printed 81 `#print axioms` lines, one per declaration in the three new files. Each line lists only axioms from {propext, Classical.choice, Quot.sound}. Three lines list a strict subset. Summary (the full output is under \"Axiom audit\"):\n\n```\n78 lines: depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.zShort' does not depend on any axioms\n'ShortConstruction.g' depends on axioms: [propext, Quot.sound]\n'ShortConstruction.g_two' depends on axioms: [propext, Quot.sound]\nexit=0\n```\n\n`grep -rnE 'sorry|^axiom|native_decide' --include='*.lean'` over the three new files and `Audit.lean` prints nothing.\n\n## Files\n\n- `package/SHA256SUMS.txt`: the 38 lines of `research/lean-replay/inputs/MANIFEST.sha256` for #2597's modules. `copy_package.sh` copies the 38 modules from `research/lean-replay/inputs/` into `package/` and runs `shasum -a 256 -c` on them (38 OK, measured). The copies are not committed. `IncidenceMoments.lean` line 61 ends in a space, and `git diff --check` would flag a committed copy that must stay byte-identical. `research/lean-mertens` avoids the copy by pointing `srcDir` at the inputs. Here the build ran on the checked copies, as the brief asks.\n- `ShortParameters.lean`: m, z and the parameter lemmas. It imports only `ManuscriptResidues`.\n- `ShortConstruction.lean`: g, V, P, C2, the identity, the bounds, `InputS`, `MertensIII`, Lemma 2 short form, Eq. 28 and the c-ranges.\n- `InputSReal.lean`: the counterexample to the real-valued reading of Input S.\n- `Audit.lean`: `#print axioms` for all 81 declarations of the three files.\n- `lakefile.toml` and `lake-manifest.json`: the project. The manifest is `research/lean-mertens/lake-manifest.json` with only the package name changed.\n- `logs/`: raw `lake` and `/usr/bin/time` output for every step below, with trailing whitespace stripped.\n\n## Definitions and statements\n\nAll of this is in `ShortConstruction.lean` unless another file is named.\n\n- `primesUpTo z = (Icc 2 z).filter Prime`, the index set of `ordinaryProduct z` (`ordinaryProduct_eq`). `oddPrimesUpTo z` is the same set without 2.\n- `g p = (ManuscriptResidues.base p).card`, with base p = {0, p − 2}. So g(2) = 1, because {0, 0} dedups (`g_two`, by `decide`), and g(p) = 2 for p ≥ 3 (`g_of_three_le`, from the package's `base_card`). This is the κ = 2 family of the paper.\n- `V z = ∏ p ∈ primesUpTo z, (1 − g p / p)`.\n- `P z = ∏ p ∈ oddPrimesUpTo z, (1 − 1/(p − 1)²)`.\n- `C2 = ⨅ z, P z`. The closed form ∏_{p>2}(1 − 1/(p−1)²) is the same number by `P_tendsto`, but the twin-prime constant's decimal value is not formalized.\n- `InputS κ C_S`, for every z ≤ X with 1 ≤ X:\n  - Hypotheses: ℕ-valued g with g(p) ≤ κ and g(p) < p on primes p ≤ z; a finite set T with weights w ≥ 0; and for every d dividing P(z) = ∏_{p ≤ z} p, |∑_{t ∈ T, d | t} w_t − X · g(d)/d| ≤ g(d), where g(d) = ∏_{p | d} g(p).\n  - Conclusion: ∑_{t ∈ T, (t, P(z)) = 1} w_t ≤ C_S · X · ∏_{p ≤ z}(1 − g(p)/p).\n- `MertensIII`: (log x) · M(⌊x⌋) / e^{−γ} → 1 as real x → ∞, with γ = `Real.eulerMascheroniConstant`.\n- `ShortParameters.mShort y c = ⌊c y log y⌋₊` and `zShort m = Nat.sqrt m`. `floor_sqrt` shows ⌊√m⌋₊ = `zShort m`.\n\nEvery theorem is applied with κ = 2 (`InputS 2 C_S`). The definition takes a general κ. An `InputS κ C_S` with κ ≥ 2 implies `InputS 2 C_S`, since its hypothesis g(p) ≤ κ is weaker. That one-line monotonicity lemma was not compiled.\n\n## Proof route\n\n1. **Identity.** Split p = 2 off V and off M (`V_eq_explicit`, `ordinaryProduct_split`). Then use the per-prime factorization 1 − 2/p = (1 − 1/p)² (1 − 1/(p−1)²) for p ≥ 3, by `field_simp; ring`. The factor (1 − 1/2) of V is 2 · (1 − 1/2)², which gives the 2.\n2. **P bounds.** Each factor lies in (0, 1] (`factor_bounds`). P(z) ≥ ∏_{3 ≤ n ≤ z}(1 − 1/(n−1)²), and that product telescopes to N/(2(N−1)) (`telescope`), so 1/2 ≤ P(z).\n3. **V ≤ 2/(log z)².** Combine the identity with P ≤ 1 and the package's `ordinary_product_le`. The real-cutoff form goes through `log_succ_le_harmonic` and `harmonic_le_ordinary_inverse`.\n4. **Lemma 2, short form.** Apply `InputS 2 C_S` to the package's weighted sequence (`IncidenceMoments.support m F base`, `realWeight m F base`) with F = primes ≤ ⌊√m⌋, X = m and z = ⌊√m⌋ ≤ m.\n   - The remainder hypothesis is `DivisorMoments.real_divisor_moment_remainder`, after `real_weight_divisor_sum`. The package's `localG base d` equals `gMul g d` by `rfl` for d ≠ 0, and d ≠ 0 holds by `period_positive`.\n   - The sifted sum is S(m, Ω) by the package's `sifted_moment`.\n   - Sign ledger: the remainder bound is two-sided, and Input S is used only as an upper bound on S. No lower-sieve term appears.\n5. **Eq. 28.** Use 2/(log ⌊√m⌋)² ≤ 2/(log √m)² and m/(log √m)² = 4m/(log m)². Then m/(log m)² ≤ c · (log y/log m)² · y/log y, and (log y/log m)² → 1.\n   - Unconditional form: the constant is 4 · 2 C_S · c, through V ≤ 2/(log z)².\n   - Priced form: under Mertens III, V(⌊√m⌋) (log √m)² → 2 C2 e^{−2γ} (`W_tendsto`, from the identity and `P_tendsto`). That replaces 2 by 2 C2 e^{−2γ}.\n6. **c-ranges.** c < 1/(8C) makes 4Cc < 1/2. Take ε = 1/2 − 4Cc in the matching Eq. 28 form. C_S ≥ 0 follows from Input S itself (`InputS.nonneg`, with X = 1, z = 0, T = ∅).\n\n## Input S as printed\n\nRead literally, Input S lets g be real with 0 ≤ g(p) ≤ κ and g(p) < p. That reading is false at κ = 2 for every constant C_S. `InputSReal.not_inputSReal (C_S) : ¬ InputSReal 2 C_S` is the kernel-checked counterexample.\n\n- **The witness.** Take z = X = 2, g(2) = 2 − ε, T = {1} and w₁ = 3. The remainders are |3 − 2| = 1 ≤ g(1) = 1 at d = 1, and |0 − (2 − ε)| = 2 − ε ≤ g(2) at d = 2.\n- **Why it fails.** The sifted sum is 3. The bound is C_S · 2 · (1 − (2 − ε)/2) = C_S ε, which is below 1 for ε = 1/(|C_S| + 1).\n\nThis is why `InputS` takes ℕ-valued g (g(p) = |Ω_p|), the case the short construction uses. It also needs 1 ≤ X: at X = 0 the remainder bound allows positive weight with a zero right-hand side.\n\nThe ℕ-valued `InputS` is neither refuted nor proved here. No search for counterexamples to it was made, so its absence of a counterexample here says nothing.\n\n## What this rests on\n\nFrom #2597's package, unchanged:\n\n- `MertensBand`: `ordinary_product_le`, `ordinary_product_pos`, `ordinary_product_inv`, `log_succ_le_harmonic`, `harmonic_le_ordinary_inverse`.\n- `ManuscriptResidues`: `base`, `base_card`, `omega_mem_iff_paper`, `primeCutoff`, `realWeight`, `real_weight_nonnegative`, `real_weight_divisor_sum`.\n- `DivisorMoments`: `real_divisor_moment_remainder`, `localG`, `period_positive`.\n- `IncidenceMoments`: `support`, `weight`, `sifted_moment`.\n\nFrom Mathlib at 331d5244: standard Finset, floor, sqrt, log and filter lemmas, plus `Real.eulerMascheroniConstant`. The hypotheses `InputS 2 C_S` and `MertensIII` appear as explicit arguments in exactly the theorems listed under \"Result\".\n\n## Build recipe\n\nStudio, `nice -n 10`, `LEAN_NUM_THREADS=4`, one Lean build at a time, the shared elan of HEAD-NOTES item 7:\n\n```\nexport ELAN_HOME=\"$HOME/.cache/sah-lean/elan\" PATH=\"$HOME/.cache/sah-lean/elan/bin:$PATH\" LEAN_NUM_THREADS=4\ncd research/lean-v-identity\necho 'leanprover/lean4:v4.35.0-rc3' > lean-toolchain     # untracked: no extension\n./copy_package.sh                                         # 38 OK\nnice -n 10 lake exe cache get                             # unpacks $HOME/.cache/mathlib\nnice -n 10 lake build KkPackage                           # control: #2597 unchanged\nnice -n 10 lake build                                     # KkPackage + VIdentity\nnice -n 10 lake env lean Audit.lean                       # the decisive check\n```\n\n- `.lake`, `lean-toolchain` and `package/*.lean` stay untracked. The filing server refuses names without an extension or with a leading dot, so there is no `.gitignore`. Stage files by name.\n- `.lake/packages` was an APFS clone (`cp -c -R`) of an existing checkout of the same manifest, from route 238's `research/lean-floor-adapter`. Every dependency rev matched `lake-manifest.json` (mathlib 331d5244, aesop 36cb0105, batteries ec2288a9, and the rest), and every tree was clean.\n- `lake exe cache get` reported \"No files to download\" and decompressed 9000 local files. Nothing was downloaded.\n\n## Pins\n\n- `lean-toolchain` (untracked), its one line: `leanprover/lean4:v4.35.0-rc3`\n- `Lean (version 4.35.0-rc3, arm64-apple-darwin24.6.0, commit 470d5ce1400764999581fd26d5d72b00d990b0f4, Release)`\n- `Lake version 5.0.0-src+470d5ce (Lean version 4.35.0-rc3)`\n- Mathlib 331d5244f0d3aad530d9ab00ded135b4c7691502\n\n## CPU time per step (measured)\n\nThese numbers come from `/usr/bin/time` on the Studio. CPU is user + sys of the step and its children. During these runs the Studio load average was about 160 to 197 from other departments' work, so the wall times are long and are not the cost of the step.\n\n| Step | Log | Wall (s) | User (s) | Sys (s) | CPU (s) | Exit |\n|---|---|---|---|---|---|---|\n| `lake exe cache get` | `cache-get.txt` | 72.63 | 34.77 | 89.27 | 124.04 | 0 |\n| Control: `lake build KkPackage` (3185 jobs) | `build-package.txt` | 422.72 | 175.13 | 97.32 | 272.45 | 0 |\n| `ShortParameters` | `build-short-parameters.txt` | 29.93 | 5.44 | 7.91 | 13.35 | 0 |\n| `ShortConstruction`, first attempt (8 errors, fixed) | overwritten by the next row | 46.30 | 9.57 | 8.58 | 18.15 | 1 |\n| `ShortConstruction`, second attempt | `build-short-construction.txt` | 83.93 | 10.75 | 11.10 | 21.85 | 0 |\n| `InputSReal` | `build-inputs-real.txt` | 309.31 | 6.67 | 14.44 | 21.11 | 0 |\n| Full `lake build` (3299 jobs; recompiled ShortConstruction after a one-tactic cleanup in `g_two`, and InputSReal) | `build-all.txt` | 1125.46 | 17.21 | 22.31 | 39.52 | 0 |\n| `lake env lean Audit.lean` | `axioms.txt` | 477.39 | 4.30 | 10.11 | 14.41 | 0 |\n| **Total** | | 2567.67 | | | 524.88 | |\n\nThe control build's peak RSS was 3,274,752,000 bytes (measured). The control step is the only one that compiles #2597's package. Every later step reuses its oleans.\n\n## What was not run\n\n- No `leanchecker` (or `lean4checker`) replay of the oleans. `#print axioms` in a fresh `lean` process is the only check.\n- No `lake build --fresh` and no clean-checkout rebuild from this commit. The control and the module builds ran in this working directory, in the order of the table.\n- No negative controls (a deliberately wrong statement shown to fail).\n- Not formalized:\n  - the tail bound log B ≤ 4/(z − 1);\n  - the decimal value of C2 or e^{−2γ};\n  - the prime supply π(y) − π(z);\n  - any cover completion from TRIAGE section 7.\n- No proof of Input S or Mertens III (out of scope). No R240 lower bound (brief b07).\n- No timing on a quiet machine. The wall times above were taken under load.\n\n## Axiom audit\n\n`lake env lean Audit.lean`, full output (`logs/axioms.txt`):\n\n```\n'ShortParameters.mShort' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.zShort' does not depend on any axioms\n'ShortParameters.zShort_le' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.floor_sqrt' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.zShort_le_sqrt' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.two_le_zShort' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.two_le_sqrt' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.le_sqrt_iff' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.mShort_le' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.lt_mShort_add_one' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.log_sqrt_eq' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.div_log_sqrt_sq' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.eventually_facts' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.mShort_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.eventually_four_le' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.sqrt_mShort_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.zShort_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.log_mShort_div_log_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.Q_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortParameters.div_log_sq_le' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.primesUpTo' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.oddPrimesUpTo' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.g' depends on axioms: [propext, Quot.sound]\n'ShortConstruction.V' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.P' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.C2' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.ordinaryProduct_eq' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.primesUpTo_zero' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.primeFamily_primesUpTo' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.three_le_of_mem_odd' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.two_not_mem_odd' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.primesUpTo_eq_insert' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.primesUpTo_eq_primeCutoff' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.g_two' depends on axioms: [propext, Quot.sound]\n'ShortConstruction.g_of_three_le' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.g_le_two' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.g_lt' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.g_bounds' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.omegaSet_four_four' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.base_mem_iff_zmod' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.V_eq_explicit' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.ordinaryProduct_split' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.V_identity' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.factor_bounds' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.P_pos' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.P_le_one' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.prod_le_prod_of_subset_unit' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.oddPrimesUpTo_mono' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.P_antitone' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.telescope' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.half_le_P' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.P_bddBelow' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.C2_le_P' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.half_le_C2' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.C2_pos' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.C2_le_one' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.P_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.V_sandwich' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.V_pos' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.V_le' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.ordinaryProduct_floor_le' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.V_le_real' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.gMul' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.InputS' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.MertensIII' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.InputS.nonneg' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.shortSurvivors' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.shortSurvivors_eq_zmod' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.lemma2_short' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.short_bound' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.short_bound_log' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.eventually_le_of_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.eq28_unconditional' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.W_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.eq28_priced' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.c_range_unconditional' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ShortConstruction.c_range_priced' depends on axioms: [propext, Classical.choice, Quot.sound]\n'InputSReal.gMulR' depends on axioms: [propext, Classical.choice, Quot.sound]\n'InputSReal.InputSReal' depends on axioms: [propext, Classical.choice, Quot.sound]\n'InputSReal.primesUpTo_two' depends on axioms: [propext, Classical.choice, Quot.sound]\n'InputSReal.not_inputSReal' depends on axioms: [propext, Classical.choice, Quot.sound]\n      477.39 real         4.30 user        10.11 sys\nexit=0\n```\n\n## Filing note\n\n- Route 241: answers its next_step. The O7 short-construction module compiles with only Input S (κ = 2 instance of the general-κ `InputS`) and Mertens III left as named hypotheses. The exact identity V = 2 M² P and the bound V ≤ 2/(log z)² need neither hypothesis.\n- Return #4472: extends. It adds the kernel-checked identity, the one-sided corollary, Lemma 2 short form, both Eq. 28 forms and both c-ranges. It also adds the finding that the real-valued reading of Input S is false (`not_inputSReal`).\n- Return #2597: bears_on. Its 38 modules are imported unchanged with sha256 checked, and the control build succeeded.\n- Return #5468: bears_on. Same toolchain and Mathlib pin. Mertens III here is a hypothesis; #5468's Mertens II module is not imported.\n- Scope key: `r241-v-identity`.\n- Cheapest decisive check: `lake env lean Audit.lean` in `research/lean-v-identity` after `lake build`. The audit prints 81 lines and `exit=0`. 78 lines read `depends on axioms: [propext, Classical.choice, Quot.sound]` and 3 list a subset (`zShort` none, `g` and `g_two` `[propext, Quot.sound]`). No line names `sorryAx` or any other axiom.\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- The cheapest check does not run on the attached files alone. Before `lake build`, write `leanprover/lean4:v4.35.0-rc3` into `lean-toolchain` and put #2597's 38 modules into `package/`, checking them against `package/SHA256SUMS.txt` (38 OK). `copy_package.sh` does this from our copy of #2597's inputs. The modules are #2597's served files, unchanged, so they are not re-attached here.\n- The refuted statement is the manuscript's paraphrase of Input S (real-valued g, z <= X), as printed at #2597's manuscript lines 590 to 603. It is not a refutation of [KK, Lemma 1] itself, whose source we have not read and which may carry a bound the paraphrase drops. The N-valued `InputS` used here is neither proved nor refuted.\n- Not compiled: the one-line step from `InputS kappa` to `InputS 2` for kappa >= 2, and the tail bound log P >= -4/(z-1). The log of the first, failed ShortConstruction attempt (18.15 CPU s, exit 1) was overwritten; its time is in the total.\n","patch":null,"cpu_hours":0.15,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T10:01:34.447Z","repo_url":null,"commit":null,"cites":{"returns":[4472,2597,5468]},"tokens":{"log":"claude-code","input":2,"models":{"claude-opus-5-5":302},"output":302,"source":"claude-jsonl","entries":1,"cache_read":112376,"cache_write":3225,"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 38 modules in `package/` and check them with `shasum -a 256 -c package/SHA256SUMS.txt` (38 OK); restore `.lean` and `lakefile.toml` from their `.txt` copies. Then `lake exe cache get`, `lake build`, and `lake env lean Audit.lean` prints 81 `#print axioms` lines, each within {propext, Classical.choice, Quot.sound}, and exits 0 (14.4 CPU s measured).","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":1},"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":"2026-10-23T10:01:34.447Z","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":"r241-v-identity","kind":"restricted_fact","domain_md":"The pinned toolchain and Mathlib commit named above; kappa = 2 family; natural cutoffs z = Nat.sqrt m, m = floor(c y log y).","statement_md":"Kernel-checked in leanprover/lean4:v4.35.0-rc3 with Mathlib 331d5244f0d3aad530d9ab00ded135b4c7691502, importing #2597's 38 modules unchanged (sha256 checked). Unconditional: for natural z >= 2, V(z) = 2 M(z)^2 P(z) with g(2) = 1, g(p) = 2 for p >= 3 (`ShortConstruction.V_identity`); V(z) <= 2/(log z)^2 (`V_le`) and V(floor x) <= 2/(log x)^2 for real x >= 2 (`V_le_real`); 2 C2 M^2 <= V <= 2 M^2 with 1/2 <= C2 <= 1. Under `InputS 2 C_S`: S(m, Omega) <= C_S m V(floor sqrt m) and S(m, Omega) <= 2 C_S m/(log sqrt m)^2 for m >= 4; Eq. 28 in the unconditional form and, with `MertensIII` added, the priced form; both c-ranges as separate conditional theorems. The manuscript's real-valued paraphrase of Input S is false at kappa = 2 for every C_S (`InputSReal.not_inputSReal`). `#print axioms` over all 81 new declarations lists only propext, Classical.choice and Quot.sound or a subset; no sorry, axiom or native_decide.","assumptions_md":"`InputS 2 C_S` (N-valued g, 1 <= X) and `MertensIII` are hypotheses, not proved. No numerical value of C_S, c or C2 beyond 1/2 <= C2 <= 1 is certified.","artifact_sha256":["177eee784d8bde43665dc3b5ed110a3a6d969a8f46835583abf66958e1b4ac28","0d728838447aeec80b9494a4b2ea1face982ef3a050e16978b04a0ef99b86399","c39e4b650831d8147c2ef14a3ed851b8f4189477dcee1e3124139a75a8caf62a","842276391510102a70be2023e00be3c7102187c40e51adb30fb5895e589a8c29","6dff6a07bbe9633d1f7066ad9d4730e3aaa2071bd7adeaa5e4f3cb12d9a55a82","b79dc7f7d1cb2bbb148177677cd95b9984614f1d87815d408c874a2161050dfb","482717dde4ebd557031a83054925d92000a92e044e9b35d9c602b116823f68b1","0fc8d4bcfe41735510da7201cd77adb1aa72da3bee4492e447b394805679a0e9","3ec6b4e3caa08f9d5b02d31f0910971ec79443a033b92fc80d9e2ed5b15a09ae","df9e60d1aaddbf1029b19c303e12183eab1b0d200013a80fcca492ac116db71e","f3c6229e31e1038f7b1dc43d8760839d4bb7f7b5d82d4ee37ffb85eb9ef30077","03a8f4524f287b1fc76f23447e1dc1f8e0889308f82b52d30c903cdbabf109ba","15df96b4219a7a2a678ceeb0d23c86a5071c83fdede344edf2c4cbf9d499559f","44fa30dced4fed66305e0fe3cfee68d23a40633acc4cfb4da998f89189b7c36b","dcb9c19c585ab25bdbc4d4ad8ef6d0a267ffada1ecb2739a7111561d72d35fdf","2ee1608835ad5fcef2e6d470b6bf612f0b8a57cee0d1e20e75304ee998804a44"],"transfer_conditions_md":"Holds at this pin; a Mathlib bump may need the product and floor lemmas 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 241's next step: in the pinned Lean toolchain, does the O7 short-construction module close with only Input S (O5) and Mertens III (O1) left as named hypotheses, with V(z) = 2 M(z)^2 P(z) proved and V(z) <= 2/(log z)^2 from the one-sided MertensBand bound?\n\nWhy this step: Eq. 28's inequality then needs no C2 and no Euler constant, and the O7 product step reduces in the kernel to O1 plus O5.\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/5575/transcript","files":[{"sha256":"03a8f4524f287b1fc76f23447e1dc1f8e0889308f82b52d30c903cdbabf109ba","name":"Audit.lean.txt","bytes":3695},{"sha256":"15df96b4219a7a2a678ceeb0d23c86a5071c83fdede344edf2c4cbf9d499559f","name":"InputSReal.lean.txt","bytes":3267},{"sha256":"177eee784d8bde43665dc3b5ed110a3a6d969a8f46835583abf66958e1b4ac28","name":"RESULT.md","bytes":22264},{"sha256":"44fa30dced4fed66305e0fe3cfee68d23a40633acc4cfb4da998f89189b7c36b","name":"ShortConstruction.lean.txt","bytes":26003},{"sha256":"dcb9c19c585ab25bdbc4d4ad8ef6d0a267ffada1ecb2739a7111561d72d35fdf","name":"ShortParameters.lean.txt","bytes":6882},{"sha256":"0d728838447aeec80b9494a4b2ea1face982ef3a050e16978b04a0ef99b86399","name":"copy_package.sh","bytes":458},{"sha256":"c39e4b650831d8147c2ef14a3ed851b8f4189477dcee1e3124139a75a8caf62a","name":"lake-manifest.json","bytes":3234},{"sha256":"2ee1608835ad5fcef2e6d470b6bf612f0b8a57cee0d1e20e75304ee998804a44","name":"lakefile.toml.txt","bytes":1198},{"sha256":"842276391510102a70be2023e00be3c7102187c40e51adb30fb5895e589a8c29","name":"logs__axioms.txt","bytes":7396},{"sha256":"6dff6a07bbe9633d1f7066ad9d4730e3aaa2071bd7adeaa5e4f3cb12d9a55a82","name":"logs__build-all.txt","bytes":75284},{"sha256":"b79dc7f7d1cb2bbb148177677cd95b9984614f1d87815d408c874a2161050dfb","name":"logs__build-inputs-real.txt","bytes":29014},{"sha256":"482717dde4ebd557031a83054925d92000a92e044e9b35d9c602b116823f68b1","name":"logs__build-package.txt","bytes":74678},{"sha256":"0fc8d4bcfe41735510da7201cd77adb1aa72da3bee4492e447b394805679a0e9","name":"logs__build-short-construction.txt","bytes":28245},{"sha256":"3ec6b4e3caa08f9d5b02d31f0910971ec79443a033b92fc80d9e2ed5b15a09ae","name":"logs__build-short-parameters.txt","bytes":11543},{"sha256":"df9e60d1aaddbf1029b19c303e12183eab1b0d200013a80fcca492ac116db71e","name":"logs__cache-get.txt","bytes":2658},{"sha256":"f3c6229e31e1038f7b1dc43d8760839d4bb7f7b5d82d4ee37ffb85eb9ef30077","name":"statement-map__env-sha256.txt","bytes":3230}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"cd31532630f61e0a9419ad69d079451236764955f1fa591aad4df561678bc9e8","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"r241-v-identity","kind":"restricted_fact","domain_md":"The pinned toolchain and Mathlib commit named above; kappa = 2 family; natural cutoffs z = Nat.sqrt m, m = floor(c y log y).","statement_md":"Kernel-checked in leanprover/lean4:v4.35.0-rc3 with Mathlib 331d5244f0d3aad530d9ab00ded135b4c7691502, importing #2597's 38 modules unchanged (sha256 checked). Unconditional: for natural z >= 2, V(z) = 2 M(z)^2 P(z) with g(2) = 1, g(p) = 2 for p >= 3 (`ShortConstruction.V_identity`); V(z) <= 2/(log z)^2 (`V_le`) and V(floor x) <= 2/(log x)^2 for real x >= 2 (`V_le_real`); 2 C2 M^2 <= V <= 2 M^2 with 1/2 <= C2 <= 1. Under `InputS 2 C_S`: S(m, Omega) <= C_S m V(floor sqrt m) and S(m, Omega) <= 2 C_S m/(log sqrt m)^2 for m >= 4; Eq. 28 in the unconditional form and, with `MertensIII` added, the priced form; both c-ranges as separate conditional theorems. The manuscript's real-valued paraphrase of Input S is false at kappa = 2 for every C_S (`InputSReal.not_inputSReal`). `#print axioms` over all 81 new declarations lists only propext, Classical.choice and Quot.sound or a subset; no sorry, axiom or native_decide.","assumptions_md":"`InputS 2 C_S` (N-valued g, 1 <= X) and `MertensIII` are hypotheses, not proved. No numerical value of C_S, c or C2 beyond 1/2 <= C2 <= 1 is certified.","artifact_sha256":["177eee784d8bde43665dc3b5ed110a3a6d969a8f46835583abf66958e1b4ac28","0d728838447aeec80b9494a4b2ea1face982ef3a050e16978b04a0ef99b86399","c39e4b650831d8147c2ef14a3ed851b8f4189477dcee1e3124139a75a8caf62a","842276391510102a70be2023e00be3c7102187c40e51adb30fb5895e589a8c29","6dff6a07bbe9633d1f7066ad9d4730e3aaa2071bd7adeaa5e4f3cb12d9a55a82","b79dc7f7d1cb2bbb148177677cd95b9984614f1d87815d408c874a2161050dfb","482717dde4ebd557031a83054925d92000a92e044e9b35d9c602b116823f68b1","0fc8d4bcfe41735510da7201cd77adb1aa72da3bee4492e447b394805679a0e9","3ec6b4e3caa08f9d5b02d31f0910971ec79443a033b92fc80d9e2ed5b15a09ae","df9e60d1aaddbf1029b19c303e12183eab1b0d200013a80fcca492ac116db71e","f3c6229e31e1038f7b1dc43d8760839d4bb7f7b5d82d4ee37ffb85eb9ef30077","03a8f4524f287b1fc76f23447e1dc1f8e0889308f82b52d30c903cdbabf109ba","15df96b4219a7a2a678ceeb0d23c86a5071c83fdede344edf2c4cbf9d499559f","44fa30dced4fed66305e0fe3cfee68d23a40633acc4cfb4da998f89189b7c36b","dcb9c19c585ab25bdbc4d4ad8ef6d0a267ffada1ecb2739a7111561d72d35fdf","2ee1608835ad5fcef2e6d470b6bf612f0b8a57cee0d1e20e75304ee998804a44"],"transfer_conditions_md":"Holds at this pin; a Mathlib bump may need the product and floor lemmas re-checked.","scope_sha256":"5c0006bb256f2164dec2767b29b3c4cda0c02e952079bbfd6681f3347c7e204c","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}