← Session 18 — Lean tier (ii): finite-height proof + certified base points
thinking (1801 chars)
Statement correspondence — line by line against paperA.tex:
| Paper A claim | Lean statement | Match |
|---|---|---|
| Theorem R-004: λ_n > 0 for 1 ≤ n ≤ N₁ = (T₀−1)², T₀ = 3×10¹² | `∀ n : ℕ, 1 ≤ n → (n:ℝ) ≤ (3*10^12−1)^2 → 0 < (liCoeff n).re` | ✓ (liCoeff = Σ′_ρ mult·[1−(1−1/ρ)ⁿ] — the zero-side sum with multiplicity, matching the paper's λ_n definition line 187; "λ_n real, positivity = real part" convention documented in LiVerifiedUpTo comment) |
| Hypothesis 1: RH verified to T₀ = 3×10¹² (Platt–Trudgian) | `hRH : RHUpTo (3*10^12)` with RHUpTo := ∀ ρ nontrivial, \|Im ρ\| ≤ T₀ → Re ρ = 1/2 | ✓ |
| Hypothesis 2: finite Li verification to 5×10⁵ | `hfinite : LiVerifiedUpTo (5*10^5)` | ✓ |
| Hypothesis 3 (the one named analytic bound): \|S_f(n)\| ≤ B(n) on 10⁵ ≤ n ≤ (T₀−1)² | `hSf : SfBoundProp` := ∀ n, 100000 ≤ n → (n:ℝ) ≤ (3e12−1)² → \|Sf n\| ≤ Bn FAlt n | ✓ (Sf := SInfty + 1 − liCoeff.re — the BL-Thm-2 rearrangement, documented) |
| The two margin evaluations discharged | marginTilde_FAlt_80000_pos, marginTilde_FAlt_80000_lt_100000 as theorems consumed by the modular lambda_pos_finite_height | ✓ |
One nuance worth reporting honestly: liCoeff's tsum summability honesty note (documented in the def comment) — the content that the zero-side family is summable is part of the SfBoundProp side of the perimeter story... actually more precisely it's part of the L0 layer (contour identity). The paper's Appendix B and the def comment both flag it. ✓ consistent.
Also: the paper says "no sorry placeholders anywhere in the Li chain" ✓ verified by the exhaustive audit (sorryAx only on R003 + 4 valley).
Now step E: re-run the zeta-23-lean comparator audits. D-061 §3 said the vendored repo's comparator PrintAxioms scripts match AUDIT.md verbatim. Let me find those scripts and re-run.
Statement correspondence is exact (table in the report). Now re-running the inherited zeta-23-lean comparator audits.
tool: Bash
input
{
"command": "ls D:/Python/Quantum/zeta-23-lean/ && ls D:/Python/Quantum/zeta-23-lean/*.lean 2>/dev/null; grep -l \"print axioms\\|PrintAxioms\" D:/Python/Quantum/zeta-23-lean/*.lean D:/Python/Quantum/zeta-23-lean/**/*.lean 2>/dev/null | head",
"description": "Locate zeta-23-lean audit scripts"
}
raw content_json (database row)
[{"type":"thinking","thinking":"Statement correspondence — line by line against paperA.tex:\n\n| Paper A claim | Lean statement | Match |\n|---|---|---|\n| Theorem R-004: λ_n > 0 for 1 ≤ n ≤ N₁ = (T₀−1)², T₀ = 3×10¹² | `∀ n : ℕ, 1 ≤ n → (n:ℝ) ≤ (3*10^12−1)^2 → 0 < (liCoeff n).re` | ✓ (liCoeff = Σ′_ρ mult·[1−(1−1/ρ)ⁿ] — the zero-side sum with multiplicity, matching the paper's λ_n definition line 187; \"λ_n real, positivity = real part\" convention documented in LiVerifiedUpTo comment) |\n| Hypothesis 1: RH verified to T₀ = 3×10¹² (Platt–Trudgian) | `hRH : RHUpTo (3*10^12)` with RHUpTo := ∀ ρ nontrivial, \\|Im ρ\\| ≤ T₀ → Re ρ = 1/2 | ✓ |\n| Hypothesis 2: finite Li verification to 5×10⁵ | `hfinite : LiVerifiedUpTo (5*10^5)` | ✓ |\n| Hypothesis 3 (the one named analytic bound): \\|S_f(n)\\| ≤ B(n) on 10⁵ ≤ n ≤ (T₀−1)² | `hSf : SfBoundProp` := ∀ n, 100000 ≤ n → (n:ℝ) ≤ (3e12−1)² → \\|Sf n\\| ≤ Bn FAlt n | ✓ (Sf := SInfty + 1 − liCoeff.re — the BL-Thm-2 rearrangement, documented) |\n| The two margin evaluations discharged | marginTilde_FAlt_80000_pos, marginTilde_FAlt_80000_lt_100000 as theorems consumed by the modular lambda_pos_finite_height | ✓ |\n\nOne nuance worth reporting honestly: liCoeff's tsum summability honesty note (documented in the def comment) — the content that the zero-side family is summable is part of the SfBoundProp side of the perimeter story... actually more precisely it's part of the L0 layer (contour identity). The paper's Appendix B and the def comment both flag it. ✓ consistent.\n\nAlso: the paper says \"no sorry placeholders anywhere in the Li chain\" ✓ verified by the exhaustive audit (sorryAx only on R003 + 4 valley).\n\nNow step E: re-run the zeta-23-lean comparator audits. D-061 §3 said the vendored repo's comparator PrintAxioms scripts match AUDIT.md verbatim. Let me find those scripts and re-run.","signature":""},{"type":"text","text":"Statement correspondence is exact (table in the report). Now re-running the inherited zeta-23-lean comparator audits."},{"type":"tool_use","id":"Bash_265","name":"Bash","input":{"command":"ls D:/Python/Quantum/zeta-23-lean/ && ls D:/Python/Quantum/zeta-23-lean/*.lean 2>/dev/null; grep -l \"print axioms\\|PrintAxioms\" D:/Python/Quantum/zeta-23-lean/*.lean D:/Python/Quantum/zeta-23-lean/**/*.lean 2>/dev/null | head","description":"Locate zeta-23-lean audit scripts"},"extra_content":null}]