[{"type":"thinking","thinking":"The bib has all the keys I need. For NoLowZeros provenance there's no Gram/Turing/Edwards entry. Options:\n- Keep it self-contained: describe the provenance as the zero-location stage of Platt–Trudgian's computation (their paper does describe locating zeros via Riemann–Siegel + Turing's method). To be precise and avoid overclaiming, I can write: \"its provenance is the zero-location stage of the Platt--Trudgian computation, which isolates every zero up to the verification height by the Riemann--Siegel formula (the lowest at $\\gamma_1 = 14.134725\\ldots$) and confirms the total count by Turing's method \\cite{platt2021riemann}.\"\n\nThat's accurate: Platt–Trudgian's method isolates zeros in intervals and uses Turing's method to certify completeness. The lowest zero at 14.134725… is found by that same isolation (and is classical). This makes NoLowZeros a *separate* certificate from RHUpTo with explicit provenance, exactly what the user asked.\n\nNow — one nuance: does the Platt–Trudgian isolation literally start at height 0? Their verification: \"we isolated the first 12,363,153,437,138 zeros\" or similar (they state the count of zeros up to 3·10¹²). Yes, Platt–Trudgian: \"we have verified RH up to height 3·10¹², i.e. that the first ~1.2363×10¹³ nontrivial zeros lie on the critical line.\" The isolation covers from the bottom. So \"no zeros with |γ| < 14\" follows: all zeros in that range are on the line and the first one is at 14.134…. Good.\n\nNow let me write all the edits.\n\n### Edit 1 — Item 8 (affiliation)\nLine 29: `\\author{Dan T Nguyen\\thanks{Research Ai}}` → `\\thanks{Independent researcher}`\n\n### Edit 2 — Item 3 (abstract reclassification) + \"those five\" fix\nLines 46-56:\nCurrent: \"Moreover the full reduction is machine-verified in Lean~4 (package \\texttt{NguyenRH1}): the theorem is proved from exactly seven named hypotheses --- the two computational certificates above and five declared analytic inputs (Trudgian's explicit zero-counting bounds in their unit-window, two-unit, and global forms, a Platt-class zero-location certificate, and one short-segment interval constant), with the traced contour bound $|S_f(n)| \\le B(n)$ on $10^5 \\le n \\le (T_0-1)^2$ itself machine-proved from those five --- with axiom profile exactly the three standard axioms (\\texttt{propext}, \\texttt{Classical.choice}, \\texttt{Quot.sound}) and no \\texttt{sorry} anywhere in the development's Li chain.\"\n\nNew: \"Moreover the full reduction is machine-verified in Lean~4 (package \\texttt{NguyenRH1}): the theorem is proved from exactly seven explicit perimeter inputs, itemized with provenance in Appendix~B --- three explicit published zero-counting bounds of Trudgian and four computational certificates (the Platt--Trudgian RH verification, Johansson's rigorous Li table, a zero-location certificate for the low zeros, and one short-segment interval constant) --- with the traced contour bound $|S_f(n)| \\le B(n)$ on $10^5 \\le n \\le (T_0-1)^2$ itself machine-proved from six of the seven (all but the Li table), and with axiom profile exactly the three standard axioms (\\texttt{propext}, \\texttt{Classical.choice}, \\texttt{Quot.sound}) and no \\texttt{sorry} anywhere in the development's Li chain.\"\n\n### Edit 3 — Item 7 (Oesterlé) in intro (lines 104-112)\nCurrent: \"The statement has an unpublished precursor (Oesterlé \\cite{oesterle2000}, as cited in \\cite{voros2017li}: RH to height $T_0$ implies $\\lambda_n \\ge 0$ for $n \\le T_0^2$) and a published weaker form (Lagarias's window with a $\\log^2$ denominator, Remark~(i)); the explicit, fully constant-traced, machine-verified assembly --- Lagarias's contour argument \\cite{lagarias2007li} with Trudgian's explicit zero-counting bounds \\cite{trudgian2014}, an elementary archimedean lower bound, a proved margin convexity, and the finite verifications --- we have found nowhere else in the literature.\"\n\nNew: \"The statement has an unpublished precursor: Voros \\cite{voros2017li} reports an unpublished result of Oesterlé \\cite{oesterle2000} that RH to height $T_0$ implies $\\lambda_n \\ge 0$ for $n \\le T_0^2$ (the typescript is uncirculated; we rely on Voros's citation and have not independently verified it). A published weaker form is Lagarias's window with a $\\log^2$ denominator (Remark~(i)). The novelty claimed here is not the $T^2$ scale itself but the explicit, fully constant-traced, \\emph{strict}-positivity assembly --- Lagarias's contour argument \\cite{lagarias2007li} with Trudgian's explicit zero-counting bounds \\cite{trudgian2014}, an elementary archimedean lower bound, a proved margin convexity, the dog-leg repair of Section~\\ref{subsec:dogleg}, and the finite verifications --- and its machine verification; that assembly we have found nowhere else in the literature.\"\n\n### Edit 4 — Item 6 (Voros sentence §3, lines 148-151)\nCurrent: \"Voros \\cite{voros2006sharpenings} showed that ``$\\lambda_n > 0$ for all $n \\geq N_0$'' is already equivalent to RH for any $N_0$, so no partial infinite-range statement exists; the maximal content of the Li route short of RH is an \\emph{explicit finite-height} statement.\"\n\nNew: \"Voros \\cite{voros2006sharpenings} showed that ``$\\lambda_n > 0$ for all $n \\geq N_0$'' is already equivalent to RH for any $N_0$: eventual positivity of all sufficiently large $\\lambda_n$ would itself imply RH, so a result short of RH cannot assert an infinite positive tail. The natural partial statement on this route is an \\emph{explicit finite-height} one.\"\n\n### Edit 5 — Item 5 (sentence after Theorem 2, after line 164 \"\\end{theorem}\")\nInsert: \"Theorem~\\ref{thm:finite-height-li} is stated with the two computational assumptions; the formal theorem consumes five further explicit perimeter inputs --- the zero-counting bounds, a zero-location certificate, and one interval constant --- which are established published results or stand-alone certificates, instantiated with provenance in Appendix~B. They enter as hypotheses of the Lean reduction, not as additional conjectural assumptions.\"\n\n### Edit 6 — Item 2 (T-notation, Step 2 display definitions, line 202)\nCurrent: \"$T = \\sqrt n + 1$, $F$ the honest zero-count majorant (the formal development's mildly enlarged Trudgian shape), $Z_2(t)$ the explicit two-unit-interval zero count (including the $\\log\\log$ term), $H(T) = 3C_0(T)^2 + 10.5\\log(T{+}2) + 48$ with $C_0$ the unit-interval count, ...\"\n\nNew: \"Here $T_n := \\sqrt n + 1$ is the deterministic majorant height: the contour itself runs at a zero-avoiding height $\\tau_n \\in [\\sqrt n, \\sqrt n+1]$, chosen by a pigeonhole on the unit-window zero count with ordinate clearance $1/(9\\hat C_0(\\sqrt n+2))$ (\\texttt{exists\\_zero\\_avoiding\\_T\\_wide}), and every majorant, being nondecreasing in the height, is evaluated at $T_n$; $F$ the honest zero-count majorant (the formal development's mildly enlarged Trudgian shape), $Z_2(t)$ the explicit two-unit-interval zero count (including the $\\log\\log$ term), $H(T) = 3C_0(T)^2 + 10.5\\log(T{+}2) + 48$ with $C_0$ the unit-interval count, ...\"\n\nWait — the display itself uses T: \"B(n) = 4F(T) + 4T[...] + ... + 2(e^4+1)·7H(T) + ...\" with \"$T = \\sqrt n + 1$\" after. If I rename the majorant height to T_n, the display should use T_n for consistency. Let me edit the display: \"4F(T_n) + 4T_n[...] + 2(e^4+1)·7H(T_n)\" and then the text after.\n\nThe display:\n\\[\n\\begin{split}\nB(n) =\\;& 4F(T) + 4T\\bigl[\\tfrac23 Z_2(T) + 10.5\\log(T{+}2) + 48\\bigr] \\\\\n& + 40M_0 + 2(e^4{+}1)\\,7H(T) + 2\\,\\mathrm{DL}(n) + I_{\\mathrm{Maj}}(n) + 1,\n\\end{split}\n\\]\n$T = \\sqrt n + 1$, ...\n\nNew display: replace T → T_n in the three spots (4F(T_n), 4T_n[...Z_2(T_n)...], H(T_n)). And H(T) definition keeps T as dummy variable (fine — H is a function).\n\nThen: \"$T_n := \\sqrt n + 1$ (a deterministic majorant height; the actual contour height is a zero-avoiding $\\tau_n \\in [\\sqrt n, \\sqrt n+1]$ chosen by pigeonhole with clearance $1/(9\\hat C_0(\\sqrt n+2))$, and all majorants are nondecreasing in the height, so evaluation at $T_n$ majorizes the charges at $\\tau_n$), $F$ the honest zero-count majorant ...\"\n\nHmm — hat on C_0: the paper uses $C_0$ not $\\hat C_0$. Keep $C_0$. Clearance: $1/(9 C_0(\\sqrt n+2))$.\n\n### Edit 7 — Item 2 (Step 3, lines 226-230)\nCurrent: \"\\emph{Step 3 (bridge).} For $n \\le (T_0-1)^2$ every zero with $|\\gamma| \\le \\sqrt n + 1$ is on the line, so $|1-(1-1/\\rho)^n| \\le 2$ per zero and $|\\lambda_n(T)| \\le 4F(T)$ (the residue sum runs over both signs of $\\gamma$); all other contour terms are \\emph{unconditional}.\"\n\nNew: \"\\emph{Step 3 (bridge).} For $n \\le (T_0-1)^2$ we have $\\tau_n \\le \\sqrt n + 1 \\le T_0$, so by assumption~(i) every zero with $|\\gamma| \\le \\tau_n$ is on the line; hence $|1-(1-1/\\rho)^n| \\le 2$ per zero and $|\\lambda_n(\\tau_n)| \\le 4F(T_n)$ (the residue sum runs over both signs of $\\gamma$, the count majorized at $T_n$); all other contour terms are \\emph{unconditional}.\"\n\n### Edit 8 — Item 2 (§3.1 dog-leg, lines 284-305)\nCurrent: \"Keep the cut height $T = \\sqrt n$ (so the bridge and the endpoint $N_1 = (T_0-1)^2$ are untouched) and deform only the right horizontal segment $[3, 2\\sqrt n]$ at height $T$ into a \\emph{dog-leg}: lift vertically from $3 + iT$ to $3 + iU$ with $U = \\tfrac32 \\sqrt n$ (the left vertical), traverse horizontally at height $U$ (the high segment), and drop back at $2\\sqrt n$ (the right vertical).\"\n\nNew: \"Keep the residue cut at the actual contour height $\\tau_n \\in [\\sqrt n, \\sqrt n+1]$ (so the bridge and the endpoint $N_1 = (T_0-1)^2$ are untouched) and deform only the right horizontal segment $[3, 2\\sqrt n]$ at height $\\tau_n$ into a \\emph{dog-leg}: lift vertically from $3 + i\\tau_n$ to $3 + iU_n$ with $U_n := \\tfrac32\\tau_n$ (the left vertical), traverse horizontally at height $U_n$ (the high segment), and drop back at $2\\sqrt n$ (the right vertical).\"\n\nThen bounds: \"Each dog-leg piece has an honest bound: the left vertical has length $\\tau_n/2 \\le (\\sqrt n+1)/2$ and contributes $\\le \\tfrac{\\sqrt n+1}{4}(e^{5/2}+1)$; the high segment contracts geometrically (the repaired stepping ratio $q = e^{4/9}/2 < 1$, certified) and contributes $\\le 8.38$; and the right vertical is exponentially small, $\\le \\tfrac14 I_{\\mathrm{Maj}}(n)$.\"\n\nAlso earlier in §3.1: \"when $T \\asymp \\sqrt n$\" — generic T, fine.\n\nAlso: \"the per-side bound at $T = \\sqrt n$ appears to hold empirically by a strong hidden cancellation\" — this is about the exact T = √n (E-032). Fine as-is; maybe clarify \"at the exact height $T = \\sqrt n$\". Keep.\n\nNote the footnote in Step 2 refers to \"Section~\\ref{subsec:dogleg}\" — fine.\n\nAlso the correction-note footnote (i) says \"The bridge is $4F$, not $2N$\" — fine.\n\n### Edit 9 — Item 1 (Step 5, lines 253-263)\nCurrent: \"\\emph{Step 5 (finite verification; the declared certificate \\texttt{LiVerifiedUpTo($10^5$)}).} The base range $n < 10^5$ is covered by rigorous high-precision computation of the ordinary Keiper--Li coefficients: for all $n \\le 10^5$ by Johansson \\cite{johansson2015} (interval-arithmetic error bounds throughout; the full table $\\lambda_0, \\dots, \\lambda_{100000}$ all positive, matching the RH-true asymptotic, with $\\lambda_{100000} = 4.6258\\ldots$ certified to $\\sim 2900$ digits), and independently for all $n \\le 3000$ by this project (two methods --- the arithmetic side via Bombieri--Lagarias Thm~2 and the zero side --- agreeing to $4.8\\times10^{-25}$ relative, and $\\lambda_1$ agreeing with its closed form to better than $10^{-28}$ relative). The crossover $N_0 = 8\\times 10^4$ is covered by a factor $1.25$.\"\n\nNew: \"\\emph{Step 5 (finite verification; the declared certificate \\texttt{LiVerifiedUpTo($10^5$)}).} The base range $n < 10^5$ is covered by rigorous high-precision computation of the ordinary Keiper--Li coefficients. A normalization remark is required here. Johansson \\cite{johansson2015} tabulates the \\emph{Keiper} normalization, defined by\n\\[\n\\log\\xi\\!\\left(\\frac1{1-x}\\right) = -\\log 2 + \\sum_{n \\ge 1} \\Lambda_n x^n;\n\\]\nthis differs from the Li normalization used in this paper by the positive factor $\\lambda_n = n\\Lambda_n$ for all $n \\ge 1$ (equate coefficients in $x\\frac{d}{dx}\\log\\xi(1/(1-x))$ and use the symmetry $\\rho \\leftrightarrow 1-\\rho$ of the zero set), so positivity is identical in the two conventions. Johansson's table (interval-arithmetic error bounds throughout) gives $\\Lambda_1, \\dots, \\Lambda_{100000}$ all positive, matching the RH-true asymptotic, with $\\Lambda_{100000} = 4.6258\\ldots$ certified to $\\sim 2900$ digits --- i.e.\\ $\\lambda_{100000} = 10^5\\Lambda_{100000} \\approx 462\\,580.78$ in the present normalization. Independently, this project verifies $\\lambda_n > 0$ for all $n \\le 3000$ in the Li normalization (two methods --- the arithmetic side via Bombieri--Lagarias Thm~2 and the zero side --- agreeing to $4.8\\times10^{-25}$ relative, and $\\lambda_1$ agreeing with its closed form to better than $10^{-28}$ relative). The crossover $N_0 = 8\\times 10^4$ is covered by a factor $1.25$.\"\n\nWait — should I double check \"Λ_100000 = 4.6258…\" → λ = 462580.78. Yes.\n\nAlso the existing footnote about Misguich–Voros stays.\n\nAlso Appendix A first item: add \"(Li normalization)\" — minor: \"λ_n > 0 verified for n ≤ 3000 from the arithmetic side\" → add \"(the Li normalization of Section 3)\". I'll add a short parenthetical.\n\n### Edit 10 — Item 3+4 (Appendix B instantiation → provenance table, lines 432-446)\nCurrent paragraph: \"The seven hypotheses instantiate from cited results and certificates as follows. \\texttt{RHUpTo} comes from ... in the same trust class.\"\n\nNew: \"All seven hypotheses are trusted inputs --- assumed, not proved, in the development; everything else in the proof is machine-checked. Their provenance is as follows.\n\n[table]\n\n\\texttt{NoLowZeros} is logically separate from \\texttt{RHUpTo}: it asserts a location fact about the low zeros, not RH on an initial segment. Its provenance is the zero-location stage of the Platt--Trudgian computation, which isolates every zero up to the verification height by the Riemann--Siegel formula and confirms the total count by Turing's method, finding the lowest zero at $\\gamma_1 = 14.134725\\ldots$ \\cite{platt2021riemann}; hence no nontrivial zero has $|\\gamma| < 14$.\"\n\nTable (4 columns): Hypothesis | Statement | Source | Status.\n\nRows:\n1. \\texttt{RHUpTo} & RH holds for all zeros with $|\\gamma| \\le 3\\times10^{12}$ & Platt--Trudgian \\cite{platt2021riemann} (interval arithmetic) & computational certificate \\\\\n2. \\texttt{LiVerifiedUpTo} & $\\lambda_n > 0$ for $1 \\le n \\le 10^5$ & Johansson \\cite{johansson2015} (Step 5 normalization); this project to $n \\le 3000$ (App.~A) & computational certificate \\\\\n3. \\texttt{ZeroCountProp} & unit-window zero count $\\le C_0(t{+}1)$ (mult-weighted) & Trudgian \\cite{trudgian2014} & published explicit bound \\\\\n4. \\texttt{ZeroCountProp2} & two-unit zero count $\\le 0.6L + 0.886\\log L + 2.7$ & Trudgian \\cite{trudgian2014} & published explicit bound \\\\\n5. \\texttt{ZeroCountGlobalProp} & global count $N(T) \\le F(T)$ majorant & Trudgian \\cite{trudgian2014} & published explicit bound \\\\\n6. \\texttt{NoLowZeros} & $|\\gamma| \\ge 14$ for every nontrivial zero & zero-location computation \\cite{platt2021riemann} (see text) & computational certificate \\\\\n7. \\texttt{MZeroProp} & short-segment constant $M_0 \\le 1.3352$ & this project (E-023, interval arithmetic) & computational certificate \\\\\n\nCareful with ZeroCountProp2: the form is \"0.6·L + 0.886·LL + 2.7\" where L = log(T+2)? and LL = log log. In the table: \"$\\le 0.6L + 0.886\\log L + 2.7$, $L = \\log(T{+}2)$\" — hmm, need to check what the paper/Lean says. NGUYEN.md: \"ZeroCountProp2 (two-unit count 0.6·L + 0.886·LL + 2.7)\". LL = log L presumably. To avoid getting the exact formula wrong in the table, keep it vaguer: \"two-unit zero count, explicit constants\" or check LiSkeleton for the exact def. Let me check the Lean def of ZeroCountProp2 quickly to get it right. Also ZeroCountProp exact form: \"mult-weighted ≤ Ĉ₀(t+1)\" and Ĉ₀(t) = ? (C0hat). And ZeroCountGlobalProp: N(T) ≤ FAlt(T)? Let me grep LiSkeleton for the Prop definitions to state them accurately in the table.\n\nAlso for MZeroProp: \"M₀ ≤ 1.3352\" — what is M₀ exactly? \"the short-segment constant\" — bound on |ζ′/ζ| on |t| < 10 region? D-036 |t|<10 gap closed (M₀ = 1.3352). It's ‖logDeriv ζ‖ on the short segment? Let me check the Prop def.\n\nThen Edit 11 — Appendix B final paragraph (lines 488-494): \"The finite-height Li theorem remains, by design, \\emph{conditional} on computations (Platt--Trudgian's interval arithmetic; the Li-coefficient verifications); the five analytic inputs are explicit published bounds of the same literature class \\cite{trudgian2014, platt2021riemann}. The formal reduction leaves exactly those as trusted inputs, ...\"\n\nNew: \"The finite-height Li theorem remains, by design, \\emph{conditional} on its perimeter (the four computational certificates: Platt--Trudgian's interval arithmetic, the Li-coefficient verifications, the zero-location certificate, the interval constant; and the three explicit published zero-counting bounds \\cite{trudgian2014}). The formal reduction leaves exactly those seven as trusted inputs, which we regard as the honest formal target: every other step of the proof is machine-checked.\"\n\nLet me grep the Lean Prop definitions for ZeroCountProp/ZeroCountProp2/ZeroCountGlobalProp/MZeroProp/NoLowZeros to state them precisely.","signature":""},{"type":"text","text":"All cite keys exist. Let me check the exact Lean definitions of the perimeter Props so the provenance table states them precisely."},{"type":"tool_use","id":"Grep_13","name":"Grep","input":{"pattern":"def ZeroCountProp|def ZeroCountProp2|def ZeroCountGlobalProp|def NoLowZeros|def MZeroProp|def RHUpTo|def LiVerifiedUpTo|def C0hat|def Z2hat","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1","output_mode":"content","-n":true,"-A":8},"extra_content":null}]