[{"type":"thinking","thinking":"Confirmed: **Voros's 2020 paper verifies a DEFORMED/discretized sequence, NOT the ordinary λ_n.** The \"λ_n > 0 for n ≤ 5×10⁵\" claim as stated in our paper (and R-004, D-037, letter-to-lagarias, cover-letter, referee-list) is **NOT supported by that citation** for the ordinary Keiper–Li coefficients. The user is exactly right — this is the biggest paper-level vulnerability.\n\nLet me dig deeper: does the Voros paper's deformed sequence RELATE to the ordinary λ_n in a way that transfers positivity? Voros's approach: he deforms λ_n into \"fully explicit sequences\" — if positivity of the ordinary λ_n follows from his computations plus an error analysis, then the splice could be repaired with a precise statement. But per the abstract, the deformation is a DIFFERENT sequence; transferring positivity from the deformed sequence back to λ_n is not automatic (and the abstract says the method \"selectively reacts to zeros off the critical line\" — a detector, not a direct verification of λ_n).\n\nHmm — but wait. Let me think about what Misguich actually computed. The abstract says \"(up to n = 5·10^5 by G. Misguich)\" — Misguich computed the deformed sequences to 5×10⁵. So the 5×10⁵ figure is for the DEFORMED sequences.\n\nSo what does this mean for our theorem? Our Lean theorem takes `LiVerifiedUpTo (5×10⁵) : ∀ n : ℕ, 1 ≤ n → n ≤ 5×10⁵ → 0 < (liCoeff n).re` as a DECLARED perimeter hypothesis. The justification we offered for it: \"ours to 3000 + Misguich–Voros to 5×10⁵\". The Misguich–Voros part is now seen to be UNSOUND as stated: Voros's paper does not verify the ordinary λ_n.\n\nThe honest options:\n(a) Reduce the certificate to what we actually verified ourselves: LiVerifiedUpTo 3000 (our own two-method exact computation to n ≤ 3000). But then the splice N₀ = 8×10⁴ has NO finite verification coverage — the theorem's base range [1, 8×10⁴] would be uncovered. The theorem NEEDS λ_n > 0 for n < 10⁵ (the margin argument covers n ≥ 10⁵... wait — actually the margin argument covers [8×10⁴, N₁] hmm — the assembly: margiñ(8×10⁴) > 0 and margiñ convex ⟹ λ_n > 0 for n ∈ [8×10⁴, N₁]; the finite verification covers [1, 5×10⁵] which must overlap 8×10⁴ — with verification only to 3000, the gap (3000, 8×10⁴) is UNCOVERED — the theorem would break!\n\nHmm wait — actually — how does the margin argument work? margin(n) := sInftyLower(n) − Bn(n) + 1; Sf bound via SfBoundProp covers n ∈ [10⁵, N₁]; the margin positivity at 8×10⁴ and convexity gives positivity on [8×10⁴, N₁]... — hold on — for n ∈ [8×10⁴, 10⁵), SfBoundProp doesn't apply (it starts at 10⁵)! Let me re-check: SfBoundProp := ∀ n, 100000 ≤ n → (n:ℝ) ≤ N₁ → |Sf n| ≤ Bn FAlt n. So the Sf bound covers [10⁵, N₁]. The margin convexity argument: margin(n) > 0 for n ∈ [8×10⁴, N₁] — but margin(n) > 0 requires the Sf bound which only exists for n ≥ 10⁵?! Let me look at how lambda_pos_finite_height handles [8×10⁴, 10⁵)... — from LiSkeleton: `rcases lt_or_ge n 100000 with hlt | hge; · exact hfinite n hn1 (by omega)` — so for n < 10⁵: the FINITE VERIFICATION (hfinite : LiVerifiedUpTo 5×10⁵) covers it. For n ≥ 10⁵: the margin/SfBoundProp argument. So the finite verification MUST cover [1, 10⁵) — currently supplied by LiVerifiedUpTo (5×10⁵) — which per the miscitation concern rests on... what?\n\nSo the certificate situation: our own exact computation covers [1, 3000]. The gap (3000, 10⁵) currently relies on the Misguich–Voros citation, which per Voros's actual content does NOT verify ordinary λ_n. THIS IS A REAL GAP — not just a paper-wording issue!\n\nWait — hold on. Let me double check what Misguich computed. From the abstract: \"We deform it to fully explicit sequences, simpler to analyze and to compute... (up to n = 5·10^5 by G. Misguich)\". Hmm — the deformed sequences. Is there any chance that Voros's deformed sequences ARE the ordinary λ_n positivity in disguise? Voros's \"discretized Keiper/Li\" — I recall now: the idea is to replace the infinite zero-sum in λ_n by a finite sum over the first N zeros with a DISCRETIZATION, obtaining an \"explicit\" sequence. The relation to λ_n: the deformed sequence approximates λ_n but is NOT equal. Positivity of the deformed sequence does NOT imply positivity of λ_n directly. Hmm — although — hmm, actually, wait. Let me think about what Misguich's computation might be. There's a chance Misguich's computation (as a side output) computed the actual λ_n for n ≤ 5×10⁵ from zero tables and found them positive — the D-037 note says \"Voros 1703.02844 reports λ_n > 0 computed to n = 5·10⁵; also Keiper to 7000, Maślanka to 3300\". Hmm — \"also Keiper to 7000, Maślanka to 3300\" — so D-037 lists PRIOR ordinary-λ_n computations: Keiper to 7000, Maślanka to 3300. Those are the ordinary λ_n! If Keiper (1992) computed λ_n > 0 for n ≤ 7000, and Maślanka to 3300 — those cover [1, 7000]. Still not to 10⁵.\n\nHmm — so the question: is there ANY literature verifying the ORDINARY λ_n to n ≥ 10⁵? Known results: Keiper (1992, n ≤ 7000), Maślanka (2001?, n ≤ 3300), our E-011/E-015 (n ≤ 3000, exact, two methods), Voros–Misguich (5×10⁵, DEFORMED). Nothing else ordinary to 10⁵.\n\nCan WE extend our own computation to 10⁵? E-015's cost model: n=4000 ≈ 9 h (n⁴ scaling). n = 10⁵ would be ~9h × (10⁵/4000)⁴ ≈ 9 × 390625 ≈ 3.5M hours — IMPOSSIBLE with the current method (flint Stieltjes ~k³ per λ_n plus ζ values). The Li computation to 10⁵ is not feasible with our machinery. \n\nBUT — wait. There's a classical alternative for the base range: the ZERO-SIDE computation. λ_n = Σ_ρ [1−(1−1/ρ)ⁿ] over |γ| ≤ T with a tail bound (E-008's method: zero-side table certifies λ_n > 0 for n ≤ 116 with explicit tail bound). For n up to 10⁵, the zero-side requires... the sum Σ_{|γ|≤T} [1−(1−1/ρ)ⁿ] for |ρ| large: |1−(1−1/ρ)ⁿ| ≈ n/|ρ| — the tail Σ_{|ρ|>T} n/|ρ|² ~ n·(log T)/(2πT) — for the partial sum to be positive with a certified tail, need the tail < the partial sum. The partial sum ~ λ_n ~ (1/2)n log n... — for n ~ 10⁵, need T such that (log T)/T is small — E-008 did n ≤ 116 with T = some zeros table. For n ≤ 10⁵ the zero-side computation is also heavy (zeros up to T ~ 10⁶-10⁷ with rigorous multipoint evaluation — MPFR/arb over 10⁶ zeros × log terms — actually that IS feasible-ish with interval arithmetic (hours-days), but it's a NEW computation we haven't done).\n\nHmm — WAIT. Actually — hold on. Let me reconsider. Is the Misguich computation really only about the deformed sequence? Let me look at the actual paper more carefully. The arXiv abstract (via WebFetch) says the paper deforms λ_n \"to fully explicit sequences, simpler to analyze and to compute\" and verifies THOSE to 5×10⁵ (Misguich). But maybe within the paper, there's a theorem connecting the deformed sequences' positivity to the ordinary λ_n's positivity — e.g., \"the discretized Λ_n(T) → λ_n as T → ∞, and |Λ_n(T) − λ_n| ≤ ε(n, T), so positivity of Λ_n(T) with margin > ε implies λ_n > 0\"?? — if such a bridge exists with explicit error bounds, Misguich's computation COULD transfer to ordinary λ_n — but that's a \"could\", not something I can assert without checking the paper in detail. I can't verify that theorem-by-theorem right now.\n\nSo what's the honest repair? The user said: \"If your actual Lean theorem instead takes LiVerifiedUpTo as an explicit certificate, state that directly and cite the real certificate.\"\n\nThe Lean theorem DOES take LiVerifiedUpTo (5×10⁵) as an explicit declared certificate. So the honest paper statement: Step 5 is a DECLARED COMPUTATIONAL CERTIFICATE `LiVerifiedUpTo (5×10⁵)`, whose evidence is: (i) our own exact two-method verification to n ≤ 3000 (cross-validated 4.8×10⁻²⁵); (ii) Voros's discretized Keiper–Li computation (Misguich) to 5×10⁵ — BUT (ii) does NOT directly verify ordinary λ_n. So (ii) can't carry it.\n\nHonest options for the paper:\n(A) Downgrade the certificate to LiVerifiedUpTo 3000 and fill (3000, 10⁵) with a NEW zero-side computation — a research task, not a paper-wording fix. NOT feasible in this session (would need E-008-style zero-side to 10⁵ — days of compute).\n(B) Check whether Keiper's n ≤ 7000 + Maślanka's n ≤ 3300 + our 3000 can cover... no — still not to 10⁵.\n(C) Restate the trust boundary EXACTLY: the theorem is conditional on Platt–Trudgian AND on the declared certificate LiVerifiedUpTo(5×10⁵) — and describe the certificate's evidence HONESTLY: \"verified by exact high-precision computation to n ≤ 3000 (this project, two independent methods); the range to 5×10⁵ is covered by the computations reported by Voros (with Misguich) — [BUT those are discretized, so this is NOT evidence for ordinary λ_n]\" — so (C) requires either dropping the 5×10⁵ claim or marking it as an unverified certificate.\n(D) Find the real content of the Voros paper: maybe there IS a theorem in it that DOES yield ordinary λ_n positivity to 5×10⁵. Voros's paper title says \"discretized approach\" — but maybe the discretized sequences are UPPER/LOWER bounds for λ_n, so positivity of the deformed sequence implies positivity of λ_n? Hmm.\n\nActually — let me think about what Voros's discretization IS. From memory of Keiper–Li discretization literature: λ_n = Σ_ρ [1 − (1−1/ρ)ⁿ]. Voros's \"discretized\" approach (2020): he writes λ_n = Σ_k [1 − (1−1/ρ_k)ⁿ] and truncates to k ≤ K with an explicit remainder; the \"fully explicit sequences\" might be lower bounds Λ_n(K) ≤ λ_n such that Λ_n(K) > 0 ⟹ λ_n > 0. If the deformed sequences are rigorous LOWER bounds for λ_n (with controlled tails), then Misguich's positivity of the deformed sequences DOES imply λ_n > 0 for n ≤ 5×10⁵! That would make the citation SOUND (though the paper should say \"via rigorous discretized lower bounds\" not just \"by Misguich–Voros\").\n\nI can't confirm this without reading the paper. The user said: \"I would not allow that citation to carry the finite splice unless you have checked, theorem-by-theorem...\" — the user is telling me to either verify it properly or restate as an explicit certificate.\n\nDECISION: Given I cannot fully verify the Voros paper's theorem-level content in this session, the honest paper fix is:\n1. State Step 5 as the explicit declared certificate `LiVerifiedUpTo (5×10⁵)` (matching the Lean theorem's trust boundary EXACTLY).\n2. Describe its evidence accurately: our own exact verification to 3000 (two methods, cross-validated); Voros's 2020 computation (with Misguich) — described ACCURATELY as a discretized Keiper–Li computation, with a pointer that its theorem-level transfer to ordinary λ_n is what the certificate currently rests on OR... hmm.\n\nHmm — actually wait. Let me fetch the paper's HTML full text summary to see if there's a theorem about ordinary λ_n positivity. Let me try fetching the arXiv HTML version or the PDF's first pages. Actually — let me think about what's publicly known about this paper in the RH community. \"Discretized Keiper/Li approach\" by Voros — the MathSciNet review would say... I recall the paper proposes a criterion: define Λ(n, N) = Σ_{k=1}^N [1−(1−1/ρ_k)ⁿ] and study Λ as a function; positivity for all n of the LIMIT is RH. The discretized version they actually compute... \n\nYou know what — the cleanest honest resolution that PRESERVES the theorem and is verifiable TODAY: **extend our own exact verification from 3000 to 10⁵?** — cost ~9h × (10⁵/4000)⁴ ≈ impossible (3.5M h). NO.\n\nAlternative honest resolution: **The zero-side verification for 3000 < n ≤ 10⁵ using the zero tables with an explicit tail bound (E-008's method extended)** — E-008 certified λ_n > 0 for n ≤ 116 from the zero side. For n ≤ 10⁵, the zero-side needs zeros to height T with the tail Σ_{|γ|>T} n/|ρ|² < partial-sum — the tail ~ n·log T/(2πT) — for n = 10⁵ need... the partial sum grows like (1/2)n log n ~ 5.7×10⁵ while the tail at T = 10⁶ is ~10⁵·14/(6.3×10⁶) ≈ 0.22 — tiny! Wait — actually the zero-side approach is E-008: it certified λ_n > 0 for n ≤ 116 — with what tail T? For larger n, the individual terms 1−(1−1/ρ)ⁿ for small |ρ| DOMINATE: λ_n ≈ Σ_{|γ| small} n·(2 Re ρ... — λ_n ~ (1/2)n log n asymptotically (from the archimedean part). The zero-side partial sum to height T approximates λ_n with error Σ_{|γ|>T} mult·2n/|ρ|² ≈ 2n·(T/(2π)log(T/2π))/T²·(2π)... — the tail of Σ 1/|ρ|² over |γ|>T is ~ log T/(2πT)·... — Σ_{|γ|>T} 1/γ² ~ (1/2π)·(log T)/T·... — hmm — Σ_{T<|γ|} 1/γ² ≈ (1/π)·(log T)/T (from N(t) ~ (t/2π)log(t/2πe)). So tail error ≈ 2n·(1/π)·(log T)/T·(1+o(1)). For n = 10⁵, T = 10⁷: 2·10⁵·(1/π)·16/10⁷ ≈ 0.1 — vs λ_n ~ 5.7×10⁵ — the verification is EASY computationally IF the zeros are on the line (RH to T = 10⁷ is NOT verified though — Platt–Trudgian goes to 3×10¹², and the zeros to 10⁷ are CLASSICAL (van de Lune–Riele 1986 to 1.5×10⁹, Gourdon 2004 to 10¹³?? — Gourdon verified RH to 10¹³ zeros (height ~3×10¹¹?) — actually Gourdon 2004: 10¹³ zeros — those computations exist but with what rigor class? The first 10¹³ zeros on the line (Gourdon) — widely cited, and Platt–Trudgian is the rigorous interval-arithmetic one to 3×10¹²).\n\nHmm — so a zero-side verification of λ_n for n ≤ 10⁵ with zeros to, say, T = 10⁶ (on the line by Platt–Trudgian) is FEASIBLE: Σ_{|γ|≤10⁶} [1−(1−1/ρ)ⁿ] for n ≤ 10⁵ with the tail bound 2n·log T/(πT)·safety — computing ~250k zeros × 10⁵ terms... — 2.5×10¹⁰ term evaluations — heavy but GPU/parallel-feasible (hours). But it's a NEW experiment (E-035), not a paper fix.\n\nSTOP. Let me re-read the user's actual ask: \"I would not allow that citation to carry the finite splice unless you have checked, theorem-by-theorem, that the 5×10⁵ computation is for the ordinary λ_n ... rather than a discretized/modified sequence. This is potentially the most important remaining paper-level vulnerability. If your actual Lean theorem instead takes LiVerifiedUpTo as an explicit certificate, state that directly and cite the real certificate.\"\n\nThe user gives the out: state the certificate directly (LiVerifiedUpTo) and cite the real certificate. So the honest paper-level fix they're inviting:\n\n**Step 5 rewritten**: \"The base range is the declared certificate LiVerifiedUpTo(5×10⁵) — λ_n > 0 for all n ≤ 5×10⁵, taken as a trusted computational input on the same footing as the Platt–Trudgian certificate. Its evidence: exact high-precision verification of the ordinary λ_n for n ≤ 3000 (this project, two independent methods, cross-validated to 4.8×10⁻²⁵ relative); for the range 3000 < n ≤ 5×10⁵ the certificate currently rests on [the truth about Misguich–Voros].\"\n\nAnd the truth about Misguich–Voros must be stated honestly: Voros's 2020 paper verifies a DEFORMED Keiper–Li sequence (Misguich's numerics), NOT the ordinary λ_n. So the citation CANNOT carry the splice.\n\nOptions for the actual gap (3000, 10⁵):\n(i) Find a sound citation for ordinary λ_n to 10⁵ — none exists (Keiper 7000, Maślanka 3300, ours 3000).\n(ii) Run the zero-side verification ourselves for (3000, 10⁵] — a new experiment (E-035), feasible in principle (zero-side partial sums + explicit tail, zeros on the line by Platt–Trudgian).\n(iii) Weaken the theorem: N₀ stays 8×10⁴ for the margin argument... — NO WAIT — actually — hold on. Let me re-examine the assembly: does the margin argument REALLY need finite verification to 10⁵? The margin convexity argument: margiñ(8×10⁴) > 0 and margiñ(10⁵) > margiñ(8×10⁴) and margiñ convex ⟹ margiñ(n) > 0 on [8×10⁴, N₁]. margin(n) = sInftyLower(n) − Bn(n) + 1 — this is positive ⟹ λ_n = S∞ − Sf + 1 ≥ sInftyLower − Bn + 1 = margin(n) > 0 — but ONLY IF |Sf(n)| ≤ Bn(n) — i.e., only where SfBoundProp holds (n ≥ 10⁵)!!\n\nWAIT — so for n ∈ [8×10⁴, 10⁵), the margin is positive but the Sf bound isn't available there — does λ_n > 0 follow for n ∈ [8×10⁴, 10⁵)?? — the SfBoundProp covers [10⁵, N₁] — so the margin-positivity gives λ_n > 0 only on [10⁵, N₁]! And [1, 10⁵) must come from finite verification!\n\nHmm wait — but then why did the assembly use base point 8×10⁴ < 10⁵? — the margin at 8×10⁴ is EVALUATED with Bn (the Sf bound) — but SfBoundProp only starts at 10⁵... — the margin convexity argument over [8×10⁴, 10⁵) is INSUFFICIENT without the Sf bound there. Let me re-read lambda_pos_finite_height:\n\n```lean\ntheorem lambda_pos_finite_height (_hRH) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5))\n    (hbase: 0 < marginTilde FAlt 80000) (hbase2: marginTilde FAlt 80000 < marginTilde FAlt 100000)\n    (hSf: SfBoundProp) :\n    ∀ n: ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 → 0 < (liCoeff n).re := by\n  intro n hn1 hnN\n  rcases lt_or_ge n 100000 with hlt | hge\n  · exact hfinite n hn1 (by omega)\n  · have hn20: 20 ≤ n := by omega\n    have h1:= sInfty_lower_bound n hn20\n    have h2:= hSf n hge hnN\n```\n\nSo: for n < 10⁵: hfinite (LiVerifiedUpTo 5×10⁵). For n ≥ 10⁵: the margin argument (hbase/hbase2/convexity/hSf). So the finite verification must cover [1, 10⁵) — the certificate LiVerifiedUpTo(5×10⁵) covers it WITH ROOM (5× over 10⁵).\n\nSo the gap (3000, 10⁵) is REAL and load-bearing. The Misguich–Voros citation as currently stated does NOT soundly cover it (Voros = deformed sequence).\n\nWhat's the honest resolution available TODAY?\n\nOption R1: **Verify the Voros paper's actual transfer content.** If Voros's deformed sequences rigorously lower-bound λ_n (or if his Theorem X states λ_n > 0 for n ≤ 5×10⁵ for the ordinary coefficients via his method), the citation stands with precise wording. I can't fully verify this today — I'd need to read the paper. Let me try fetching more of it (HTML full text? the PDF?). The arXiv HTML (ar5iv) might have the theorem statements. Let me fetch https://ar5iv.labs.arxiv.org/html/1703.02844 or the PDF and ask specifically for theorem statements about ordinary λ_n positivity and the role of Misguich's computation.\n\nOption R2: **Close the gap with our own zero-side computation (E-035)** for (3000, 10⁵]. Feasible: the zero-side partial sums with explicit tail (the E-008 method); zeros on the line to T by Platt–Trudgian (T = 3×10¹² covers everything up to that height); the computation: Σ_{|γ|≤T(n)} [1−(1−1/ρ)ⁿ] with a rigorous tail bound. For n ≤ 10⁵, how large must T(n) be? The tail error ~ (n/π)·(log T)/T·C — need < partial sum ~ (1/2)n log n·(1−o(1)): for n = 10⁵, (log T)/T ≤ π/2·log n·... — (log T)/T vs (π/2)·log n/n?? — the tail/error ratio: error/partial ≈ [2n log T/(πT)] / [(1/2)n log n] = (4 log T)/(πT log n) — at T = 10⁴, n = 10⁵: (4·9.2)/(π·10⁴·11.5) ≈ 3.2×10⁻⁴ — so even T = 10⁴ suffices for a 3×10⁻⁴ relative tail error!! — wait — that seems too good. Let me redo: tail error ≤ Σ_{|γ|>T} |1−(1−1/ρ)ⁿ| ≤ Σ 2n/|ρ|² ≈ 2n·Σ_{|γ|>T} 1/γ² ≈ 2n·(log T)/(2πT)·2 = 2n log T/(πT). Partial sum ≈ λ_n − tail ≈ (1/2)n log n (for large n). Ratio ≈ [2n log T/(πT)]/[(1/2)n log n] = 4 log T/(πT log n). For n = 3000 (already verified): fine. For n = 10⁵, T = 10⁴: 4·9.21/(π·10⁴·11.51) = 36.8/361283 ≈ 1.0×10⁻⁴. So a zero-side computation with T = 10⁴-10⁵ zeros gives sub-0.1% tail — and computing Σ_{|γ|≤10⁵} [1−(1−1/ρ)ⁿ] for n ≤ 10⁵ is ~10⁵ zeros × 10⁵ n-values = 10¹⁰ terms — heavy (hours-days parallel) but FEASIBLE. Hmm — but do we even need n up to 10⁵ via zero-side? — the gap is (3000, 10⁵). For n in that range, the zero-side sum must be computed for EACH n. 10¹⁰ terms — a day or two of compute on the 16-core machine. Possible but it's a new multi-hour experiment, not a quick paper fix.\n\nHmm — actually — wait. Is there a smarter route? For n in (3000, 10⁵), we need λ_n > 0. The E-021/E-015 machinery computed λ_n from the ARITHMETIC side (exact, via the BL Thm 2 formula with Stieltjes constants) for n ≤ 3000 — cost n⁴. n = 10⁵ is 3.5M hours. NO.\n\nZero-side: 10¹⁰ terms — days. Possible but not today.\n\nOption R3: **Restate the theorem with the certificate exposed HONESTLY as unverified-above-3000** — i.e., change the theorem to conditional on \"λ_n > 0 for n ≤ 10⁵ (finite verification, see Step 5)\" and describe Step 5's evidence: our exact computation to 3000, and **note the gap**: \"the range (3000, 10⁵) currently rests on the certificate `LiVerifiedUpTo`; the Voros–Misguich computation cited in earlier drafts verifies a discretized sequence, not the ordinary λ_n, and a direct verification of this range is in progress.\" — That's maximally honest but weakens the paper significantly (admits an unverified input).\n\nOption R4: **Verify the Misguich content properly NOW by reading the paper.** If Voros's Theorem (in the 2020 paper) states that the deformed sequences bound λ_n below (so positivity transfers), then the citation is sound with precise wording: \"Voros's discretized Keiper–Li lower bounds, computed positively to n = 5×10⁵ (Misguich)\". Let me try to fetch the paper's full text to determine this. The arXiv page offers PDF; ar5iv HTML might work.\n\nLet me try fetching ar5iv first. If the full text shows a clear theorem that the discretized sequences are rigorous LOWER bounds for λ_n (so positivity transfers), then the citation is sound with precise wording: \"Voros's discretized Keiper–Li lower bounds, computed positively to n = 5×10⁵ (Misguich)\".\n\nLet me try fetching ar5iv first.","signature":""},{"type":"text","text":"**This confirms the vulnerability is real**: Voros's 2020 paper verifies a *deformed/discretized* sequence (Misguich's numerics), not the ordinary λ_n. Before deciding the repair, let me check whether the paper's deformed sequences rigorously bound the ordinary λ_n (which would transfer positivity)."},{"type":"tool_use","id":"WebFetch_269","name":"WebFetch","input":{"url":"https://ar5iv.labs.arxiv.org/html/1703.02844","prompt":"In this paper (Voros, \"Discretized Keiper/Li approach to the Riemann Hypothesis\"): (1) How exactly are the \"discretized\" sequences defined in terms of the ordinary Keiper–Li coefficients lambda_n = sum over zeros of 1-(1-1/rho)^n? (2) Is there any theorem proving positivity of the ORDINARY lambda_n for some range of n (e.g., n up to 5*10^5), or only of the discretized/deformed sequences? (3) What role does Misguich's computation play — what exactly did Misguich compute? Quote the relevant theorem statements and definitions precisely."},"extra_content":null}]