[{"type":"thinking","thinking":"The user has 4 new items (well, 3 actionable + 1 praise):\n\n**Item A (the big one): Step 4 margin stale value.** Step 4 says margin(8×10⁴) = 15 764, but §3.1 says final corrected margins are 13 160 / 88 911 / 2.458×10²⁶. Inconsistency. Fix: main value → 13 160, footnote preserves the trail 16 020 → 15 764 → 13 160 with explanations. Even better: use the exact certified lower bound the Lean theorem consumes.\n\nLet me think about what the Lean theorem actually consumes. From NGUYEN.md session-040 notes:\n- ii′-6-a2: \"base points re-certified (Bn(8e4) ∈ [273k, 332k], Bn(10⁵) ≤ 352k ⟹ margin(8e4) ≥ 4 001, hbase2 66 001 < 80 501)\".\n\nWait — that's after the dog-leg redefinition. Let me reconstruct the margin trail:\n1. 16 020 — E-028 with the Trudgian F + S_b majorant (original D-037 assembly).\n2. 15 764 — with the mildly enlarged FAlt majorant (F-A/F-B corrected Bn, session 038, ii′-6-a).\n3. 13 160 — with the dog-leg repair (D-074, session 040, ii′-6-a2). E-034 re-validated: margins 13 160 / 88 911 / 2.458e26.\n\nAnd the certified rational interval lower bounds the Lean consumes: hbase : margin(8×10⁴) ≥ 4 001 (session-040 re-certification: Bn(8e4) ∈ [273k, 332k] ⟹ margin ≥ 4 001); hbase2 : margin(10⁵) − margin(8×10⁴) ... session 040: \"hbase2 66 001 < 80 501\". Hmm, what's the exact statement? Session 038: \"base points re-certified (329,000/350,000; margins ≥ 7,001/82,501)\". Session 040 ii′-6-a2: \"Bn(8e4) ∈ [273k, 332k], Bn(10⁵) ≤ 352k ⟹ margin(8e4) ≥ 4 001, hbase2 66 001 < 80 501\".\n\nHmm — 66 001 < 80 501? That looks like hbase2 is stated as Bn(10⁵) − Bn(8e4) < something, or margin(10⁵) > margin(8e4) via... Let me check LiBasePoints.lean for the exact statements of the two base-point lemmas. The Lean theorem names from the paper footnote: `marginTilde_FAlt_80000_pos`. Let me grep LiBasePoints.lean for the two theorems to quote exact certified values.\n\nAlso the margins: 13 160 at 8×10⁴ — from E-034 (computed with mpmath at Lean constants). And the Lean-certified lower bound is ≥ 4 001 (rational interval arithmetic). So the footnote trail: 16 020 (E-028, F+S_b) → 15 764 (FAlt mildly enlarged, negative-weight term dropped) → 13 160 (dog-leg repair D-074: right-tail charge replaced by 2·DL(n)) → certified ≥ 4 001 (LiBasePoints rational interval bounds, what the machine consumes).\n\nWait, I need to double-check which corrections separate 15 764 → 13 160. The 15 764 was after ii′-6-a (session 038: F-A bridge 4F, F-B Z2 two-unit count, redefined Bn). The 13 160 is after ii′-6-a2 (session 040: dog-leg, horizontal share 2(e⁴+1)(7Ĥ) + 2·DogLegRight). Also F-E was absorbed without changing Bn. So the 15 764 → 13 160 drop is the dog-leg charge (2·DL(n) ≈ 2×(√n+1)/4×13.19 ≈ (√n+1)×6.6 ≈ 1 870 at n=8e4... hmm √80000 ≈ 282.8, (√n+1)/4 ≈ 70.96, ×(e^{5/2}+1 ≈ 13.1825) ≈ 935.5, ×2 sides ≈ 1 871. Plus 2×8.38 ≈ 17. Plus IIMaj terms... roughly 1 888. And 15 764 − 13 160 = 2 604. Hmm, not exactly — the DL replaces the old right-tail charge (e⁴+1-based piece) too. The pre-dog-leg B(n) had 2(e⁴+1)·7H(T) covering both horizontals fully; post-dog-leg the horizontal share is 2(e⁴+1)(7Ĥ) + 2·DogLegRight — where DogLegRight = ((√n+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/4. Hmm wait, per DL-5/6: dogLegUpper per side = VLeft + 8.38 + IIMaj/4, and Bn carries 2·DogLegRight total? Let me look at the current Bn def (from the earlier grep):\n\n```\ndef Bn (F : ℝ → ℝ) (n : ℝ) : ℝ :=\n  let T:= Real.sqrt n + 1\n  4 * F T + 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) +\n    40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T) +\n    2 * DogLegRight n + IIMaj n + 1\n```\n\nand the paper's display: 4F(T_n) + 4T_n[(2/3)Z_2(T_n) + 10.5 log + 48] + 40M_0 + 2(e⁴+1)·7H(T_n) + 2·DL(n) + I_Maj(n) + 1. Consistent. The horizontal term 2(e⁴+1)·7H(T) covers the mid-segments (both sides), 2·DL(n) covers the right tails.\n\nFor the footnote I don't need to decompose the 2 604 difference precisely — just say: the third value reflects the dog-leg repair (D-074): the broken right-horizontal charge \"3.3\" replaced by the zero-free dog-leg charge 2·DL(n), plus the F-E honest right-vertical kernel charge (absorbed, no B(n) change)... wait F-E didn't change Bn. The drop 15 764 → 13 160 is purely from the dog-leg redefinition (ii′-6-a2 redefined Bn: horizontal share 2(e⁴+1)(7Ĥ) + 2·DogLegRight replacing... what was there before? Before ii′-6-a2, the Bn after ii′-6-a had the horizontal as 2(e⁴+1)(7Ĥ) + IIMaj_c? Hmm — actually pre-dog-leg (session-038) Bn was:\n\n\"Bn_c(n) := 4F + 4T[(2/3)Z2 + 10.5 log + 48] + 40·1.3352 + 2(e^{16/9}+1)(7Ĥ + 5.94) + IIMaj_c + 1\" — no wait, that's the c=1.5 paused-fallback formula. The session-038 Bn (before F-D was known broken... no — session 038 ii′-6-a redefined Bn honestly with 4F and (2/3)Z2hat, but the horizontal piece still used the old broken \"3.3\"? No — let me re-read.\n\nFrom NGUYEN.md ii′-6-a entry (session 038): \"redefine Bn honestly (2F → 4F, (2/3)Ĉ₀ → (2/3)Z2hat); fold validated (E-030 V2/V3/V4)\". This was BEFORE the F-D discovery (F-D was found at session 038 END, during ii′-6-c design-check). So the session-038 Bn still had the broken right-tail \"3.3\"-style charge? Hmm — \"2(e⁴+1)(7Ĥ + 5.94)\"? \n\nActually the F-D finding: \"D-036/D-037/R-004's B(n) inherits it (charge '3.3' off by ~10⁷×; margin collapses for n ≳ 3×10⁴)\". Wait — but margins 15 764 at 8×10⁴ were computed AFTER ii′-6-a redefinition... and 13 160 after the dog-leg. If the broken charge survived into ii′-6-a's Bn, then 15 764 was computed with the broken charge — but margin(8e4) was still positive 15 764 because at n=8e4 the broken envelope hadn't collapsed the margin yet (collapse for n ≳ 3×10⁴ per F-D... hmm, that contradicts).\n\nLet me re-read F-D: \"the (6.12) stepping in Lagarias's Thm 6.1 is broken at T = √n (right-tail ~e^{0.066√n}, ~5.9×10¹⁰ envelope at n = 10⁵ measured by E-031); D-036/D-037/R-004's B(n) inherits it (charge '3.3' off by ~10⁷×; margin collapses for n ≳ 3×10⁴). Theorem statement survives (verified to 5×10⁵; sides cancel by conjugation + phase winding)\".\n\nHmm — \"margin collapses for n ≳ 3×10⁴\" — that refers to when the envelope is honestly computed the old proof gives an exploding envelope, so the OLD PROOF collapses. But the 15 764 was \"computed\" — wait, the 15 764 margin (session 038, E-030) was computed with the redefined honest Bn that still contained the broken charge? No...\n\nActually let me re-read the ii′-6-a session-038 entry: \"Bn redefined honestly (4F, (2/3)Z2hat); marginTilde_convex re-proved; base points re-certified (329,000/350,000; margins ≥ 7,001/82,501); endpoint margins re-run (14,682/90,653/2.458×10²⁶)\". So session-038's endpoint margins were 14 682 / 90 653 / 2.458e26 — computed by E-030 with the honest Bn (4F, Z2hat) — but this Bn still had the F-D-broken right-tail charge \"3.3\"? The B(n) display in the paper (pre-dog-leg, session 042 version) had \"2(e⁴+1)·7H(T) + I_Maj(n)\" — hmm, the session-042 paper already had the DL(n) term because the dog-leg repair (session 040) preceded session 042's paper revision.\n\nTimeline: session 038 (ii′-6-a, F-D found at end) → session 039 (ii′-6-R repair design) → session 040 (ii′-6-DL dog-leg, ii′-6-a2 Bn redefined with DogLegRight, base re-certified) → session 041 (ii′-6-c..d fold, ii′-7 closure) → session 042 (paper revision).\n\nE-034 (session 040, DL-0): \"margins 14 419 / 90 317 / 2.458e26 at 8×10⁴/10⁵/N₁\" with C_zeta3 = 0.165. Then DL-2 re-validated at C=0.5: \"margins 13 165/88 916/2.458e26\". DL-3 at Lean constants: \"margins 13 160/88 911/2.458e26\". DL-6 at IIMaj/4: re-validated (those are the final: 13 160 / 88 911 / 2.458e26).\n\nSo final margins: 13 160 / 88 911 / 2.458e26 (E-034 at final Lean constants).\n\nSession-038 (pre-dog-leg, honest Bn with 4F/Z2hat but old right-tail): 14 682 / 90 653. The drop 14 682 → 13 160 = 1 522 from the dog-leg charge.\n\nAnd 15 764: where does that come from? From the session-042 footnote: \"16 020 with the Trudgian F + S_b zero-count majorant (E-028); 15 764 with the mildly enlarged majorant consumed by the machine verification\". Hmm wait — 15 764 is session-012 era (FAlt majorant, before ii′-6-a's 4F/Z2hat corrections). Session-038 re-run gave 14 682. So the trail is:\n\n1. 16 020 — E-028 (session 009): original D-037 B(n) with Trudgian F+S_b majorant, 2F bridge, C₀ near count, broken right-tail \"3.3\".\n2. 15 764 — session 012: FAlt mildly-enlarged majorant (negative-weight term dropped, cost ≈0.9√n) — the majorant the machine consumes for convexity; still the pre-ii′-6-a account (2F, C₀).\n3. 14 682 — session 038 (E-030): honest B(n) — F-A bridge 4F, F-B two-unit Z₂ count (both *increase* B(n), decreasing margin).\n4. 13 160 — session 040 (E-034): F-D dog-leg repair — right-tail charge replaced by 2·DL(n) (further decrease).\n5. ≥ 4 001 — the certified rational interval lower bound (LiBasePoints, session-040 re-certification) that the machine discharge actually consumes. (Session-038's was ≥ 7 001; the dog-leg Bn re-certification gives ≥ 4 001.)\n\nWait — session-040 ii′-6-a2 says: \"base points re-certified (Bn(8e4) ∈ [273k, 332k], Bn(10⁵) ≤ 352k ⟹ margin(8e4) ≥ 4 001, hbase2 66 001 < 80 501)\". Hmm: Bn(8e4) ∈ [273k, 332k] — certified interval for Bn at 8×10⁴. margin(8e4) = sInftyLower(8e4) − Bn(8e4) + 1. sInftyLower(80000) ≈ ½·80000·log(80000) + (C₁−γ/2)·80000 − 4 ≈ 40000×11.2898 + ... ≈ 451 592 − 115 ... ≈ 336k? So margin ≥ 336k − 332k ≈ 4 001. OK.\n\nAnd hbase2: margin(10⁵) > margin(8×10⁴) — certified as \"66 001 < 80 501\"? Probably margin(10⁵) ≥ 80 501... no wait, 88 911 is the computed margin(10⁵). Hmm, \"hbase2 66 001 < 80 501\": maybe Bn(10⁵) − Bn(8e4) bounds: margin(10⁵) − margin(8e4) = [sInftyLower(10⁵) − sInftyLower(8e4)] − [Bn(10⁵) − Bn(8e4)]. sInftyLower diff ≈ ½(10⁵·11.513 − 8e4·11.290) + (C₁−γ/2)·2e4 ≈ ½(1 151 292 − 903 182) − 115×0.2e4? ≈ 124 055 − 2 300 ≈ 121 755? Hmm not matching. Whatever — I should read LiBasePoints.lean for the exact theorem statements rather than reconstruct. The footnote should quote the exact certified values.\n\nLet me grep LiBasePoints.lean for the two base theorems.\n\n**Item B: exact source locations in Trudgian [10] for the three ZeroCount Props.** I need Trudgian 2014 \"An improved upper bound for the argument of the Riemann zeta function on the critical line\" (Math. Comp. 83:1053–1061). What's in it: Theorem 1: |S(T)| ≤ 0.112 log T + 0.278 log log T + 2.510 for T ≥ e (or similar). Actually Trudgian 2014's main result: |S(T)| ≤ 0.112 log T + 0.278 log log T + 2.510 for T ≥ e. Hmm — the constants 0.433/0.886/2.9 in C0hat look like they come from the zero-count corollary: N(T+1) − N(T) bounded via Backlund–Rosser + Trudgian's S(T) bound.\n\nLet me think about what D-036/D-037 say. The project's derivation D-036: \"Explicit |ξ′/ξ|/|L′/L| contour bounds (Lagarias Lemma 6.1; Davenport Ch. 15) + explicit N(T) bound (Backlund–Rosser/Trudgian)\". The C0hat = 0.433·log(t+2) + 0.886·loglog(t+2) + 2.9 — this is the unit-window zero count majorant derived from Trudgian's S(T) bound + the Backlund–Rosser formula N(T) = 1 + π⁻¹ϑ(T) + S(T) (ϑ = Riemann–Siegel theta). \n\nTrudgian 2014 Theorem 1 (I recall): |S(T)| ≤ 0.112 log T + 0.278 log log T + 2.510 for T ≥ e. Hmm, but 0.886/0.433... The two-unit form 0.6·L + 0.886·LL + 2.7 — hmm.\n\nI don't remember the exact theorem/corollary numbers. I need to check. Do we have Trudgian's paper locally? Check research/paper/ PDFs: 1-s2.0-S0022314X99923922-main.pdf (that's Bombieri–Lagarias, JNT 1999), lagarias-li-coefficients-automorphic-0404394.pdf, 2602.04022v1.pdf, 2606.09096v1.pdf (Suzuki), Anthropic paper, proof-riemann-2017-04.pdf, ssrn-4909306.pdf. No Trudgian PDF locally, it seems. \n\nBut the derivations D-036/D-037/D-069 should record exactly which theorem/corollary of Trudgian was used. D-069 is the session-035 design doc for the perimeter additions (ZeroCountProp2 + NoLowZeros) — it should have the exact provenance. Let me grep the derivations.\n\nAlso NGUYEN.md says: \"Trudgian 2014 N(T)/S(T) inside the trusted perimeter\" and \"ZeroCountProp2 (the two-unit Trudgian specialization 0.6·L + 0.886·LL + 2.7)\". D-069 Finding 1: \"D-056's piece 2 charges the near sum one C₀(t) for a TWO-unit window — the honest two-unit Trudgian form makes the assembly fit 10.5/48\".\n\nSo D-069 should state where 0.6/0.886/2.7 comes from. And D-036 or D-037 states where 0.433/0.886/2.9 (C0hat) comes from. Let me grep the derivation files.\n\nIf the exact theorem numbers are documented, add them to the table. If not, I need to derive/verify: \n- Trudgian 2014, Theorem 1: |S(T)| ≤ 0.112 log T + 0.278 log log T + 2.510 (T ≥ e). \n- Corollary 1 (or Theorem 2): zeros count? \n\nActually I recall Trudgian 2014 Math. Comp.: \"An improved upper bound for the argument...\" — Theorem: |S(T)| ≤ 0.112 log T + 0.278 log log T + 2.510 for T ≥ e. And there's a corollary about N(T)? The zero-count window bounds are then elementary: N(T+1)−N(T) = [ϑ(T+1)−ϑ(T)]/π + S(T+1)−S(T), with |ϑ′(t)| ≤ (1/2)log(t/2π) (bounds on ϑ′). \n\nThe cleanest honest approach for the paper table: check D-036/D-037/D-069 for the documented provenance and use that. If the archive says \"Trudgian Thm 1\" — cite it. Let me grep.\n\n**Item C: γ₁ = 14.134725 citation.** Check whether platt2021riemann literally supports \"the lowest zero at 14.134725…\". Platt–Trudgian's paper (Bull. LMS 53:792–797, 2021) — its main theorem: RH true up to 3·10¹². They describe the method: isolated zeros using rigorous Riemann–Siegel via Platt's implementation, Turing's method. Does the paper state γ₁ = 14.134725…? The paper might mention the count of zeros (12 363 153 437 138 zeros? or similar). The lowest zero value is classical and appears e.g. in Platt's thesis. To be safe: keep [9] for the method/verification, and add a classical primary source for γ₁. Options:\n- Gram, J.-P. (1903). \"Note sur les zéros de la fonction ζ(s) de Riemann.\" Acta Math. 27: 289–304. — the classical first-zeros computation.\n- Or Edwards, \"Riemann's Zeta Function\" (1974/2001 Dover), which tabulates the first zeros.\n- Or Platt's thesis (2017, \"Computing the Riemann Zeta Function using the Riemann–Siegel formula\", PhD thesis, University of Bristol).\n- Or better: Platt 2017 \"Isolating some non-trivial zeros of zeta\" (Math. Comp. 86(307):2449–2467) — rigorously isolates zeros; its abstract/intro would list γ₁ to high precision. That's a strong primary source: rigorous interval isolation of zeros including the first.\n\nActually the cleanest: since NoLowZeros is about \"no zeros with |γ| < 14\" — this is literally a byproduct of any RH verification that starts from 0. The user asks: make sure the sentence is literally supportable from [9] or a dataset referenced by [9]; if not, add the primary source for the first-zero computation.\n\nSafest fix: add a footnote/parenthetical citing the classical source. I'd add Gram 1903 as the classical primary source for γ₁ = 14.134725 (Gram computed the first 10+ zeros? He computed the first few: 14.134, 21.022, 25.011...). Actually Gram 1903 computed the first 10 zeros? He computed 14.134..., 21.022..., 25.011... — yes, Gram computed the first ~10-15 zeros (later extended by Backlund, Hutchinson). Hutchinson 1925 computed the first 138 zeros.\n\nAlternatively cite the LMFDB or Odlyzko's tables — but for a print reference, Gram (1903) is the classical one, and Platt's 2017 Math. Comp. paper is the rigorous modern one. I'll add: \"(classical since Gram \\cite{gram1903}; rigorously isolated to high precision in \\cite{platt2017isolating})\". Hmm, but adding two references for one number may be overkill. The user said: \"If not, add the appropriate primary source for the first-zero computation.\"\n\nDecision: add ONE primary source — Platt 2017 \"Isolating some non-trivial zeros of zeta\" (Math. Comp. 86:2449–2467) is the most rigorous modern primary source from the same author as [9], and it literally isolates γ₁ (I believe it isolates zeros to many digits; the first zero 14.134725141734693790... appears). But am I sure the paper states the first zero? It isolates \"some non-trivial zeros\" — including low ones? IIRC the paper isolates zeros in specific ranges (e.g., around large heights for Landau-type computations?) Hmm. Actually Platt 2017 Math. Comp. \"Isolating some non-trivial zeros of zeta\" — I believe it rigorously located zeros in certain ranges. Not 100% sure it includes γ₁.\n\nGram 1903 definitely has the first zero (he found 14.134725? — Gram computed 14.134... to several digits; the exact digits 14.134725 came later? Gram's value: 14.1347...). The value 14.1347251417... is now standard; e.g., in Edwards' book and in the Riemann–Siegel literature.\n\nAlternative that is bulletproof: cite the Platt–Trudgian paper for the verification method AND note γ₁'s value is classical, citing Edwards' book (which prints 14.134725...). Edwards, H.M., \"Riemann's Zeta Function\", Academic Press 1974 / Dover 2001 — page 96 or so has the first zeros? Edwards tabulates? Hmm, Edwards does discuss Gram's computation (§6.4?) and gives values.\n\nOr Odlyzko: \"Tables of zeros of the Riemann zeta function\" — online tables; the first zero file starts at 14.134725141734693790457251983562470270784257115699243... Citing Odlyzko's published table paper: \"The 10^20-th zero of the Riemann zeta function and 70 million of its neighbors\" (1989, ATT preprint) — not the first zero though. \n\nSimplest bulletproof: keep [9] and add a plain statement that the value is classical and appears in the verification's zero-isolation data — plus a footnote pointing to LMFDB? The user wants \"a specific certified dataset referenced by [9]\" or \"the appropriate primary source\". \n\nGiven uncertainty about what [9] literally prints, the safest formulation WITHOUT claiming verification: \"finding the lowest zero at $\\gamma_1 = 14.134725\\ldots$ (a value classical since Gram's 1903 computation \\cite{gram1903} and reproduced by every modern verification)\". And add gram1903 to the bib. Gram's paper: J.-P. Gram, \"Note sur les zéros de la fonction ζ(s) de Riemann\", Acta Math. 27 (1903), 289–304. Gram computed the first 10 zeros? He computed zeros 1–10? Records: Gram (1903) computed the first 10 zeros to ~6 digits? Hmm — Gram found 14.134725, 21.022040, 25.010858 (first few). I'm fairly confident Gram 1903 contains the first zero at 14.134725 (to 6 decimals: 14.134725). Yes — Gram's table: 14.134725, 21.022040, 25.010856, 30.424876, 32.935057, ... — these are Gram's values (later confirmed). Good, gram1903 supports γ₁ = 14.134725.\n\nBut wait — does the user's demand require the \"no zeros below 14\" claim's provenance, or just the γ₁ value? They said: make sure the sentence \"finding the lowest zero at γ₁ = 14.134725… [9]\" is literally supportable from [9] or a certified dataset referenced by [9]; if not, add the primary source for the first-zero computation. So: γ₁'s value needs a literal source. Gram 1903 is the primary classical source. The no-zeros-below-14 claim follows from RHUpTo + γ₁ being the least positive ordinate... wait no — NoLowZeros says no zeros (anywhere in the strip) with |γ| ≤ 14. RHUpTo (zeros with |γ| ≤ 3e12 are on the line) + the zero-location computation (the zeros on the line up to 3e12 are exactly those isolated, starting at 14.134725) ⟹ no zeros with |γ| < 14. The Turing-method completeness (no missed zeros) is what makes the count exact. So the honest provenance sentence: the Platt–Trudgian computation isolates each zero and confirms completeness by Turing's method; its first isolated zero is at 14.134725…, a value classical since Gram \\cite{gram1903}.\n\nHmm — do I know that Platt–Trudgian's [9] says they isolated from the bottom? Their abstract: \"We verify the Riemann hypothesis up to height 3·10¹², i.e., all zeros with imaginary part in (0, 3·10¹²] are on the critical line.\" The method section describes using Platt's rigorous ζ evaluation + Turing's method over the range. I'm fairly confident the paper describes starting at 0. To be careful, I can phrase: \"the Platt–Trudgian verification \\cite{platt2021riemann} is by zero isolation over the full initial range (Riemann–Siegel evaluations, completeness by Turing's method); the lowest zero lies at γ₁ = 14.134725…, a value classical since Gram \\cite{gram1903}.\"\n\nI think adding gram1903 to the bib and this phrasing solves item C cleanly.\n\n**Item D: bibliography [4] wording.** references.bib johansson2015 note: \"Section 4.2: rigorous interval-arithmetic computation of the ordinary Keiper--Li coefficients to $n = 10^5$, arXiv:1309.2877\" → change to \"Section 4.2: rigorous interval-arithmetic computation of the Keiper coefficients $\\Lambda_n$ (the normalization of Step~5) to $n = 10^5$\" — wait, bib note can't reference \"Step 5\" of the paper meaningfully... Just: \"Section 4.2: rigorous interval-arithmetic computation of the Keiper coefficients $\\Lambda_n$ to $n = 10^5$, arXiv:1309.2877\". The user suggested exactly: \"Section 4.2: rigorous interval-arithmetic computation of the Keiper coefficients Λ_n to n=10⁵.\" Use that verbatim-ish.\n\nNow item B details — the Trudgian source locations. Let me grep D-036/D-037/D-069 for what they record. Files: research/derivations/D-036-*.md, D-037-*.md, D-069-*.md. Let me search.\n\nPlan:\n1. Grep derivations for Trudgian theorem references (0.112, 0.278, 2.510, Theorem, Corollary).\n2. Grep LiBasePoints.lean for exact certified base-point statements.\n3. Make edits:\n   - Step 4 main value 15 764 → 13 160 + footnote trail 16 020 → 15 764 → 14 682? or → 13 160? The user wrote \"16 020 → 15 764 → 13 160\". Hmm — should I include 14 682 (the session-038 honest-Bn value)? The user's proposed trail is 16 020 → 15 764 → 13 160. But there WAS an intermediate 14 682 (F-A/F-B corrections). The user may not know/care about 14 682. For maximal honesty the trail could be 16 020 → 15 764 → 14 682 → 13 160 → (certified ≥ 4 001). But the user explicitly proposed \"16 020 → 15 764 → 13 160 and explain which correction caused each decrease.\" Hmm — 15 764 → 13 160 directly bundles F-A/F-B AND F-D. Explaining \"which correction caused each decrease\" with the 3-value trail: 16 020→15 764 (FAlt enlargement), 15 764→13 160 (the ii′-6 corrections F-A/F-B/F-D). But the four-value trail is more precise. I think the four-value trail is better and still matches the user's spirit (\"explain which correction caused each decrease\"). Let me verify the 14 682 number from E-030/STATE: session-038 entry says \"endpoint margins re-run (14,682/90,653/2.458×10²⁶)\". Yes. And NGUYEN.md journal-038/E-030 should confirm. So trail: 16 020 (E-028, F+S_b) → 15 764 (FAlt, negative-weight term dropped) → 14 682 (E-030: F-A bridge 4F + F-B two-unit Z₂) → 13 160 (E-034: F-D dog-leg charge) → certified ≥ 4 001 (LiBasePoints rational intervals; what the machine consumes).\n\n   Hmm wait — is 13 160 vs the certified ≥ 4 001 the right pairing? The paper's main Step-4 value should be the computed 13 160 (E-034, deterministic script), and the footnote ends with the certified ≥ 4 001. Also margin(10⁵) > margin(8×10⁴) — the second base point. And \"also computed to n = 4×10⁵ unit grid, N₁ log grid; margin(N₁) = 2.458e26\" — that stays; margin(10⁵) = 88 911 could be added to Step 4 main text: \"margin(8×10⁴) = 13 160 > 0 with margin(10⁵) = 88 911 > margin(8×10⁴)\". Currently Step 4 says \"margin(10⁵) > margin(8×10⁴)\" without a value — adding 88 911 improves consistency with §3.1's trio. Good.\n\n   Also the Remark (i)/(iii)? No margin values there. Appendix A item 3: \"adopted N₀ = 8×10⁴ with margin 16 020 and positive derivative (15 764 with the formal development's majorant — Step-4 footnote)\" — this is historical E-028 description; it already points to the Step-4 footnote. It's in the \"Margin assembly (E-028...)\" item describing that experiment — historically accurate (E-028 did produce 16 020/15 764). Maybe update the parenthetical: \"(15 764 with the formal development's majorant, and 13 160 after the ii′-6 corrections — Step-4 footnote)\". Yes, update for consistency.\n\n   - Table: add Trudgian theorem/equation locations (pending what the derivations record).\n   - NoLowZeros: add gram1903 + rephrase.\n   - references.bib: [4] note wording; add gram1903 entry.\n\n4. Recompile, verify, check pages.\n5. Commit.\n\nLet me start by grepping the derivations and LiBasePoints.","signature":""},{"type":"text","text":"Four items, all clear. Item A needs the exact certified base-point values from LiBasePoints.lean, and item B needs the Trudgian source locations as recorded in D-036/D-037/D-069. Checking those first."},{"type":"tool_use","id":"Grep_55","name":"Grep","input":{"pattern":"theorem.*80000|theorem.*100000|marginTilde_FAlt_80000|marginTilde_FAlt_100000|hbase|4 001|4001","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean","output_mode":"content","-n":true,"-B":2,"-A":4,"head_limit":50},"extra_content":null},{"type":"tool_use","id":"Grep_56","name":"Grep","input":{"pattern":"Trudgian|0\\.112|0\\.278|2\\.510|Theorem 1|Corollary","path":"D:\\Python\\Quantum\\research\\derivations","glob":"D-069*.md","output_mode":"content","-n":true,"head_limit":40},"extra_content":null}]