[{"type":"thinking","thinking":"Confirmed: `liCoeff` is defined as the unordered `tsum` over the subtype of nontrivial zeros with the summand `(zeroMult ρ : ℂ) * (1 - (1 - 1/ρ)^n)`.\n\nNow the critical question: is this family `Summable` (absolutely)? My asymptotic analysis says NO for n ≥ 1: |1−(1−1/ρ)ⁿ| ≈ n/|γ| for |γ| ≫ n (on the line), and Σ 1/|γ| diverges.\n\nWait — actually, let me be careful. For OFF-LINE zeros (σ ≠ 1/2), the pairing argument changes. But the DIVERGENCE of Σ‖summand‖ only needs the on-line zeros (or any zeros): |1−e^{inθ_γ}| with θ_γ ≈ 1/γ — for zeros with γ ∈ [n·2^k, n·2^{k+1}], |1−e^{in/γ}| ≈ n/γ ≈ 2^{−k}. Number of such zeros ≈ n·2^k·log — so each dyadic block contributes ≈ n·log·constant to Σ‖summand‖ — diverges as we sum over k. YES, definitively NOT Summable.\n\nBUT: the conjugate-pair structure: the subtype {ρ // IsNontrivialZero ρ} — is it closed under conjugation? Yes (ζ real-valued on real axis ⟹ zeros come in conjugate pairs — provable via `riemannZeta` real on ℝ and Schwarz reflection; Zeta23 has zeta_reflect_zero for reflect ρ = 1−ρ̄, and conjugation separately). So the family is NOT summable but is \"pairwise summable\".\n\nSo `liCoeff n` as defined = 0 (junk) IF Lean can't prove Summable... no wait — tsum is DEFINED as: if Summable f then the sum else 0. It's not that \"Lean can't prove\" — tsum is defined by classical choice; if the family is in fact not summable (mathematically), then `tsum f = 0` is TRUE (provable from `tsum_eq_zero_of_not_summable`). So `liCoeff n = 0` would be PROVABLE for each n ≥ 1!! That makes:\n- `LiVerifiedUpTo N`: `0 < (liCoeff n).re` = `0 < 0` — FALSE (provably false).\n- `SfBoundProp`: |SInfty n + 1 − 0| ≤ Bn — for large n, SInfty n + 1 ~ ½n log n vs Bn ~ C√n log²n — FALSE for large n (provably, eventually).\n\nSo SfBoundProp as currently encoded is almost certainly FALSE (for n ≥ 1 large), and LiVerifiedUpTo false. The theorem lambda_pos_finite_height_verified is vacuously true (false hypotheses). This is a latent DEFECT in the session-012/013 encoding — not caught because liCoeff is never evaluated and the audit checks axioms, not truth of hypotheses.\n\nHmm wait — wait. Let me double check the asymptotic |1−(1−1/ρ)ⁿ| ~ n/|γ| requires n/|γ| small, i.e. |γ| ≫ n. For FIXED n and |γ| → ∞, yes. The summand as a function of ρ for fixed n: f(ρ) = mult·(1−(1−1/ρ)ⁿ). As |ρ| → ∞: (1−1/ρ)ⁿ → 1 (since 1/ρ → 0), so f(ρ) → 0 like n·(1/ρ)·mult: indeed |f(ρ)| ~ n·mult/|ρ|. And Σ mult/|ρ| = Σ over zeros — with multiplicity — of 1/|ρ| diverges (standard: Σ 1/|ρ| diverges even without multiplicity since N(T) ~ T log T). So not Summable. CONFIRMED (mathematically, with certainty).\n\nHmm, but hold on — one more subtlety: `Summable f` for f : ι → ℂ where ι = {ρ // IsNontrivialZero ρ}. If the subtype is EMPTY (no proof of zero existence in the system), then f IS summable (empty sum) and liCoeff = 0. Either way (empty or nonempty), liCoeff n = 0 for n ≥ 1?? NO WAIT — if nonempty and not summable, tsum = 0. If empty, tsum = 0. If nonempty and summable — impossible per above. So liCoeff n = 0 for all n ≥ 1, PROVABLY (well, provably modulo proving whether the subtype is nonempty and non-summability — either way the value is 0, but the PROOF differs; the VALUE is 0 regardless).\n\nHmm, actually for n = 0: 1−(1−1/ρ)⁰ = 0, so liCoeff 0 = 0 correctly. For n ≥ 1: liCoeff n = 0 junk.\n\nCONSEQUENCE: The entire session-012 encoding has liCoeff ≡ 0. `LiVerifiedUpTo` is false. `SfBoundProp` is false (for large n). `lambda_pos_finite_height_verified` is vacuous. **The paper's Lean headline is currently encoding-empty.** The fix is exactly L7 done right: define liCoeff via the PAIRED sum (absolutely summable) or prove the pairing reformulation.\n\nHmm, hold on. Let me reconsider whether it's really false. SfBoundProp: |Sf n| ≤ Bn FAlt n where Sf n = SInfty n + 1 − (liCoeff n).re = SInfty n + 1 (with junk liCoeff). SInfty n ≥ sInftyLower n = ½n log n + ... (proved). Bn FAlt n with T = √n+1: dominant term 4T·(2/3)C0hat(T) ≈ 4√n·(0.433·0.5 log n) ≈ 0.87√n log n; plus 2F(T) ≈ (√n/π)·0.5·log n ≈ 0.16√n log n; total Bn ~ 1.03√n log n + lower. vs Sf ≈ ½n log n. Ratio ~ n/√n = √n → ∞. So indeed FALSE for n ≥ ~10⁶-ish. And the range starts at 10⁵ — borderline: at n=10⁵, SInfty ≥ ½·10⁵·11.5 − ... ≈ 5.7×10⁵; Bn(10⁵) ≤ 347 000 (certified). 5.7×10⁵ > 3.47×10⁵ — FALSE already at n = 10⁵! (Modulo the actual SInfty value: sInftyLower(10⁵) ≥ 432 500 certified > Bn ≤ 347 000. So |Sf| = SInfty + 1 ≥ 432 501 > 347 000 ≥ Bn — PROVABLY FALSE at n = 10⁵ with junk liCoeff!) \n\nWow. OK so this is definite: with the current encoding, SfBoundProp is provably false. Good thing tier ii′ exists — this is exactly what closing it fixes. But ALSO important: this defect affects the PAPER's claim \"machine-verified modulo SfBoundProp\" — because the hypothesis as encoded is false, the \"verification\" is vacuous. The mathematical content (D-037 chain) is fine; the ENCODING of liCoeff is wrong. The honest fix (paired sum) restores everything: with the correct liCoeff, LiVerifiedUpTo is the true computational fact, SfBoundProp is the true analytic bound, and the theorem means what the paper says.\n\nPriority: this becomes ii′-0's #1 design item — the liCoeff redefinition. It's \"L7\" but bigger than a note: it's the correctness of the base definition. The fix plan:\n- Define `liCoeffPaired n := ∑' ρ : {ρ // IsNontrivialZero ρ ∧ 0 < ρ.im}, mult·[(1−(1−1/ρ)ⁿ) + (1−(1−1/ρ̄)ⁿ)]` (upper-half zeros; need: zeros with Im = 0 — none exist (ζ has no real zeros in (0,1) — provable? ζ on (0,1) is real and negative... ζ(s) < 0 for 0 ≤ s < 1 (real, since the pole at 1 and ζ(0) = −1/2; needs proof — Mathlib might have `riemannZeta_neg_of...`? There's deriv_riemannZeta_zero etc. Actually ζ(s) ≠ 0 for real s ∈ (0,1): since the Dirichlet eta η(s) > 0 for s > 0 real (alternating series) and ζ = η/(1−2^{1−s}) with denominator < 0 for s ∈ (0,1) — so ζ < 0 there. Provable via Mathlib's eta? There's `riemannZeta_neg_nat...` hmm, need to check. Alternatively avoid the issue: sum over Im > 0 PLUS explicitly handle Im = 0 (empty by a lemma) — or sum over Im ≥ 0 with pairing formula adjusted. Simplest: prove no zeros on ℝ in (0,1) — actually easier: prove no REAL zeros at all in the strip via ζ(s) ∈ ℝ for s ∈ ℝ and the eta argument. Might already be in Zeta23 (IsNontrivialZero requires 0 < re < 1; real zeros would be in (0,1); the PNT zero-free Re ≥ 1 covers the boundary only. The eta argument is elementary; likely a few dozen lines).\n- Prove Summable of the paired family: pair summand = mult·[2 − (1−1/ρ)ⁿ − (1−1/ρ̄)ⁿ]; with ρ̄ also a zero of same mult. Asymptotic: ≈ mult·n²/|ρ|²-ish for large γ (second-order cancellation) — rigorous bound: |1−(1−w)ⁿ + ... | hmm need |pair| ≤ C(n)·mult/|ρ|² for |ρ| ≥ some explicit function of n (e.g. |ρ| ≥ 2n). Then Σ mult/|ρ|² < ∞ via Zeta23's `zero_sum_inv_sq` (WeilEF.ZeroSummability — check its exact statement) — if that gives Σ over the subtype of (mult)·|ρ|⁻² summable, we get Summable of pairs over upper zeros by comparison.\n- Then the BL identity Sf = SInfty + 1 − λ.re and the contour argument proceed with the paired liCoeff.\n- LiVerifiedUpTo with the new liCoeff: true (computational fact), and its relationship to E-011/E-015's λ_n: the paired sum equals the standard λ_n (this IS the standard definition of the Li coefficient as a sum over zeros, which converges only when paired — the literature defines λ_n = Σ_ρ [1−(1−1/ρ)ⁿ] with the understanding of symmetric pairing/principal value! e.g., Bombieri–Lagarias define it via the Hadamard-product Hadamard sum). So the new def matches the literature.\n\nAlso check: does the old `liCoeff` appear in LiBasePoints.lean and the final theorem — yes via Sf and hypotheses. Replacing the def: the frozen-file rule vs fixing a defect — the defect MUST be fixed for the paper's claim to be meaningful, so the freeze yields to the fix (the freeze protects against destabilizing edits, not against correctness fixes; and the fix makes Paper A MORE correct, keeping its statement intact). The paper text says \"Li coefficients\" — the meaning is unchanged; the encoding is repaired. I'll flag this prominently to the user in the session summary and in D-064.\n\nActually — wait. I should double-check my divergence claim once more, because it's the crux of a significant claim (that the current encoding is vacuous). Σ_{ρ} 1/|ρ|: N(T) ~ (T/2π)log(T/2π). Σ_{0<γ≤T} 1/γ ≈ ∫_e^T (1/2π)log(t/2π) dt/t ≈ (1/4π)(log T)². Diverges ✓. And with the summand ~ n·mult/|ρ| for large |ρ| ✓. And tsum of non-summable = 0 ✓ (tsum_def / `tsum_eq_zero_of_not_summable`). And Bn(10⁵) ≤ 347 000 < 432 501 ≤ SInfty(10⁵)+1 — the certified bounds from LiBasePoints (sInftyLower(10⁵) ≥ 432 500; Bn(10⁵) ≤ 347 000) — so SfBoundProp at n=10⁵ reads 432 501 ≤ 347 000 — false ✓✓. Certain.\n\nOne caveat: is `Summable` of the junk family maybe TRUE after all because the subtype could be... no, subtype is countable-ish (discrete zeros), summability over it is Σ‖f‖ < ∞, diverges. Done.\n\nAlso RHUpTo: fine as stated (vacuously true or not, it's a well-formed Prop about actual zeros — actually it's TRUE as a Prop if Platt–Trudgian holds; the encoding is fine).\n\nLiVerifiedUpTo with junk liCoeff: \"0 < (0).re\" false ✓.\n\nSo the three-hypothesis theorem currently has TWO false hypotheses (hfinite, hSf) — vacuous twice over. The theorem as a Lean object is fine (it's a function from hypotheses), but the paper's Appendix B claim must eventually be backed by the corrected defs. This is precisely the kind of thing the contour-closure project will fix end-to-end: paired liCoeff (summable, correct), SfBoundProp proved (true), LiVerifiedUpTo stays a hypothesis but now a TRUE one (and dischargeable in principle by computation, though 5×10⁵ is way beyond Lean compute — stays perimeter, as designed).\n\nGood. Now read D-036 and D-056 (the analytic content to formalize). These are long files; let me read them.","signature":""},{"type":"text","text":"The `liCoeff` encoding needs a careful look — this is important. Let me verify the summability situation and read the analytic sources."},{"type":"tool_use","id":"Read_29","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\derivations\\D-036-explicit-contour-constants.md"},"extra_content":null}]