Message #1903 · Riemann Hypothesis Research

← Session 33 — ii′-3b/3c crux + F-D finding; Lagarias letter package

#1903 assistant 2026-08-25 03:12:17.642 UTC · 4579→2080 tok open ⤢

thinking (3868 chars)

The assembly (§2 algebra): g(1+s) = 1/(1+s) + [1/s + ζ′/ζ(1+s)] + ΓR′/ΓR(1+s) has s^{k−1}-coefficient (−1)^{k−1} − c_{k−1} + τ_{k−1}; FE gives g^{(k−1)}(0) = (−1)^k g^{(k−1)}(1); (M): σ_k = −g^{(k−1)}(0)/(k−1)!. Chain: σ_k = (−1)^{k+1}[(−1)^{k−1} − c_{k−1} + τ_{k−1}] = 1 + (−1)^k c_{k−1} + (−1)^{k+1}τ_{k−1}, solve for c_{k−1}, insert (A): c_{k−1} = (−1)^k(a_k + σ_k) + (−1)^{k−1}. Target: **(4.10-a)** for all k ≥ 1: `lcoeff (k−1) = (−1)^k · (lagariasA k + sigmaP k) + (−1)^{k−1}`. Then with ii′-2's swaps and ResW1_eq_sum → the crux `Sf(n) = ResW1 n`. What do I need in Lean: 1. **FE derivative relation**: gFun^{(m)}(0) = (−1)^{m+1}·gFun^{(m)}(1)? Wait — D-066 says g^{(k−1)}(0) = (−1)^k g^{(k−1)}(1) with m = k−1: g^{(m)}(0) = (−1)^{m+1} g^{(m)}(1). From xiL(1−s) = xiL(s): deriv both sides m times: (−1)^m xiL^{(m)}(1) = xiL^{(m)}(0). Hmm, for g = xiL'/xiL it's more delicate. Actually D-066 claims the FE gives g^{(k−1)}(0) = (−1)^k g^{(k−1)}(1). Let me verify: define h(s) := g(1−s). Then h(s) = logDeriv xiL (1−s) = (xiL'(1−s))/(xiL(1−s)) = (xiL'(1−s))/(xiL(s)). And xiL'(1−s): from xiL(1−s) = xiL(s), differentiate: −xiL'(1−s) = xiL'(s), so xiL'(1−s) = −xiL'(s). Hence h(s) = −xiL'(s)/xiL(s) = −g(s). So g(1−s) = −g(s) as functions (where defined; but both sides are genuine functions of s: g is analytic wherever xiL ≠ 0; xiL ≠ 0 at 0 and 1 (value 1/2) so both sides analytic near 0 and the identity holds as an identity of meromorphic/analytic functions near 0 — actually g(1−s) = −g(s) holds on the open set where xiL(s) ≠ 0 (then xiL(1−s) = xiL(s) ≠ 0 too); near s = 0 both sides analytic). Then iteratedDeriv m (fun s => g (1−s)) 0 = iteratedDeriv m (fun s => −g s) 0 = −g^{(m)}(0). And iteratedDeriv m (fun s => g (1−s)) 0 = (−1)^m g^{(m)}(1) (chain rule m times: each derivative brings a −1 from the inner 1−s). So (−1)^m g^{(m)}(1) = −g^{(m)}(0), i.e. g^{(m)}(0) = −(−1)^m g^{(m)}(1) = (−1)^{m+1} g^{(m)}(1). With m = k−1: g^{(k−1)}(0) = (−1)^k g^{(k−1)}(1) ✓ matches D-066. 2. **The shift expansion**: g(1+s) as a function of s near 0. We need: iteratedDeriv (k−1) gFun 1 = (k−1)! · [(−1)^{k−1} − c_{k−1} + τ_{k−1}]. Compute g(1+s): xiL(1+s) = ((1+s)s/2)·Λ(1+s) where Λ(1+s) = ΓR(1+s)·ζ(1+s). g(1+s) = logDeriv xiL (1+s) = d/ds [log xiL(1+s)] (as logDeriv of the composition at s) = logDeriv (fun s => xiL (1+s)) 0... careful: iteratedDeriv (k−1) gFun 1 is the (k−1)-th derivative of g at the POINT 1. Equivalently the (k−1)-th derivative of s ↦ g(1+s) at 0 (chain rule with inner 1+s, derivative 1 — iteratedDeriv_comp_add_const; that's exactly the "packaging" D-066 mentions for piece (B)). And g(1+s) = logDeriv (xiL(1+·)) s = logDeriv [(1+s)s/2 · ΓR(1+s) · ζ(1+s)] s = [1/(1+s) + 1/s] + logDeriv ΓR (1+s) + logDeriv ζ (1+s) = 1/(1+s) + [1/s + ζ′/ζ(1+s)] + ΓR′/ΓR(1+s) ✓ matches D-066. The s^{k−1}-coefficient (i.e., iteratedDeriv (k−1) at 0 divided by (k−1)!): - 1/(1+s): iteratedDeriv (k−1) (fun s => (1+s)⁻¹) 0 = (−1)^{k−1} (k−1)! · 1 → coefficient (−1)^{k−1}. - [1/s + ζ′/ζ(1+s)]: piece (B): coefficients −c_j: iteratedDeriv j at 0 = −j! · lcoeff j? Let me check what iteratedDeriv_hLiExt says exactly. - ΓR′/ΓR(1+s): piece (A): iteratedDeriv_logDeriv_Gammaℝ: (1/j!) iteratedDeriv j (fun s => logDeriv Gammaℝ (1+s)) 0 = (−1)^{j+1} lagariasA (j+1). 3. **Existing pieces — exact statements**: need to check: - LiArchimedean's `iteratedDeriv_logDeriv_Gammaℝ` exact signature. - LiZetaCoeff's `iteratedDeriv_hLiExt` exact signature. - LiResidue's `lcoeff` definition. - LiSwaps' `lagariasA`, `sigmaP`, `Sf` (or wherever Sf is), `ResW1` (LiResidue). - The crux: Sf(n) = ResW1 n — how is Sf defined? From STATE: "crux identity Sf(n) = Res₁(n)" — Sf := S_∞ + 1 − λ_n? Let me check LiSkeleton for Sf/SfBoundProp and ResW1_eq_sum in LiResidue. Let me gather all the exact statements.

The assembly plan is clear (D-066 §2). Now gathering the exact statements of all input pieces.

tool: Grep ok

input

{
  "pattern": "theorem iteratedDeriv_logDeriv_Gamma\u211d|lemma iteratedDeriv_logDeriv_Gamma\u211d|theorem iteratedDeriv_hLiExt|lemma iteratedDeriv_hLiExt|def lcoeff|noncomputable def lcoeff|theorem ResW1_eq_sum|lemma ResW1_eq_sum|def ResW1|noncomputable def ResW1",
  "path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
  "output_mode": "content",
  "-n": true,
  "-A": 4
}

result 2026-08-25 03:12:17Z → 2026-08-25 03:12:17Z

mode='content' num_files=0 filenames=[] content="research\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean:187:theorem iteratedDeriv_logDeriv_Gammaℝ (j : ℕ) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean-188-    iteratedDeriv j (fun s: ℂ => logDeriv Complex.Gammaℝ (1 + s)) 0 =\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean-189-      ((-1: ℂ)) ^ (j + 1) * (j ! : ℂ) * (lagariasA (j + 1) : ℝ) := by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean-190-  rcases Nat.eq_zero_or_pos j with hj0 | hj0\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean-191-  · subst hj0\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean:147:def lcoeff (k : ℕ) : ℂ := iteratedDeriv (k + 1) hLi 1 / (k + 1)!\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-148-\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-149-\\-- hLi's Taylor coefficients: iteratedDeriv j hLi 1 / j ! = lcoeff (j−1). -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-150-lemma hLi_iteratedDeriv_div_factorial {j: ℕ} (hj : 1 ≤ j) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-151-    iteratedDeriv j hLi 1 \\ j ! = lcoeff (j - 1):= by\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean:192:def ResW1 (n : ℕ) : ℂ :=\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-193-  iteratedDeriv n (fun w: ℂ => (w ^ n - (w - 1) ^ n) * hLi w) (1 : ℂ) / n !\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-194-\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-195-\\-- The w = 1 residue value: ResW1 n = Σ_{j=1}^n C(n,j)·lcoeff (j−1). -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean:196:theorem ResW1_eq_sum (n : ℕ) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-197-    ResW1 n = ∑ j ∈ Finset.Icc 1 n, (n.choose j: ℂ) * lcoeff (j - 1) := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-198-  have hA: ContDiffAt ℂ n (fun w : ℂ => w ^ n - (w - 1) ^ n) 1 :=\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-199-    ((contDiff_id.pow n).sub ((contDiff_id.sub contDiff_const).pow n)).contDiffAt\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean-200-  have hh: ContDiffAt ℂ n hLi 1 := hLi_analyticAt_one.contDiffAt\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean:159:theorem iteratedDeriv_hLiExt (j : ℕ) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean-160-    iteratedDeriv j hLiExt 0 = -(Nat.factorial j: ℂ) * lcoeff j := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean-161-  have hcid: ContDiffAt ℂ (j + 1) (fun s : ℂ => s) 0 := contDiff_id.contDiffAt\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean-162-  have hcE: ContDiffAt ℂ (j + 1) hLiExt 0 := hLiExt_analyticAt_zero.contDiffAt\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean-163-  have h1: iteratedDeriv (j + 1) ((fun s : ℂ => s) * hLiExt) 0 =" num_lines=27 num_matches=None applied_limit=None applied_offset=None

tool: Grep ok

input

{
  "pattern": "def lagariasA|def Sf\\b|SfBoundProp|def sInfty|def liCoeff\\b",
  "path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
  "output_mode": "content",
  "-n": true,
  "-A": 6,
  "head_limit": 60
}

result 2026-08-25 03:12:17Z → 2026-08-25 03:12:18Z

mode='content' num_files=0 filenames=[] content="research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean:6:`lambda_pos_finite_height_verified` (perimeter + `SfBoundProp` only).\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-7-\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-8-Plan and validation: D-062 §5 (every link pre-verified at dps 40 and by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-9-exact Fraction arithmetic; true values margiñ(8×10⁴) = 15 763.79 and\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-10-margiñ(10⁵) ≈ 9.3×10⁴ from E-028/E-029-scale computations). All\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-11-certified targets carry explicit slack; the tightest link\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-12-(e¹·¹⁴⁴⁸ ≥ π at 0.007%) was replaced by the comfortable e¹·¹⁵ ≥ π\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean:631:analytic hypothesis (`SfBoundProp`, the traced contour bound) plus the\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-632-declared computational perimeter (Platt–Trudgian `RHUpTo`; the finite Li\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-633-verification `LiVerifiedUpTo`). The two elementary margin evaluations\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-634-are discharged in this file. -/\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-635-theorem lambda_pos_finite_height_verified\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-636-    (hRH: RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5))\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean:637:    (hSf : SfBoundProp) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-638-    ∀ n: ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 →\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-639-      0 < (liCoeff n).re:=\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-640-  fun n hn1 hnN => lambda_pos_finite_height hRH hfinite\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-641-    marginTilde_FAlt_80000_pos marginTilde_FAlt_80000_lt_100000 hSf n hn1 hnN\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-642-\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-643-end NguyenRH1\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSwaps.lean:293:def lagariasA (j : ℕ) : ℝ :=\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSwaps.lean-294-  if j = 1 then lagariasA1\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSwaps.lean-295-  else 1 + ∑' m: {m : ℕ // Odd m ∧ 3 ≤ m}, (1 / (m : ℝ)) ^ j\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSwaps.lean-296-\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSwaps.lean-297-/-- The j ≥ 2 odd-m slices are summable (dominated by the p-series Σ 1/m²,\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSwaps.lean-298-restriction of `summable_one_div_nat_pow` to the odd ≥ 3 subtype). -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSwaps.lean-299-lemma oddSlice_summable (j: ℕ) (hj : 2 ≤ j) :\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:13:hypotheses — the assembly is conditional only on `SfBoundProp` (the\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-14-isolated analytic obligation, triaged in D-062), the computational\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-15-perimeter (`RHUpTo`, `LiVerifiedUpTo`), and two elementary margin\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-16-evaluations (hbase/hbase2, E-028/E-029 values; their interval-bound\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-17-discharge is tier (ii) step 2). No `sorry` remains in this file;\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-18-`#print axioms` shows only the three standard axioms for every\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-19-declaration here.\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:43:`SfBoundProp` were FALSE as encoded. The pairing is the classical\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-44-Bombieri–Lagarias symmetric convention and restores the intended\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-45-meaning; summability is proved below.) -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-46-def liSummand (n: ℕ) (ρ : {ρ : ℂ // IsNontrivialZero ρ}) : ℂ :=\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-47-  (zeroMult ρ: ℂ) / 2 *\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-48-    ((1 - (1 - 1 \\ (ρ: ℂ)) ^ n) + (1 - (1 - 1 / ((starRingEnd ℂ) (ρ : ℂ))) ^ n))\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-49-\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:63:def liCoeff (n : ℕ) : ℂ := ∑' ρ : {ρ : ℂ // IsNontrivialZero ρ}, liSummand n ρ\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-64-\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-65-\\-! ### 1a. The geometric-sum estimate: |Re(1 − wⁿ)| ≤ 2n²/‖z‖² (w = 1−1/z) -/\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-66-\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-67-/-- (1 + 1/(2n))ⁿ ≤ 2 for n ≥ 1. Via (1+x)ⁿ ≤ exp(nx) and exp(1/2) ≤ 2. -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-68-lemma pow_one_add_inv_two_mul_le (n: ℕ) (hn : 1 ≤ n) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-69-    ((1: ℝ) + 1 / (2 * n)) ^ n ≤ 2 := by\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:368:proved from `SfBoundProp` (the single named analytic hypothesis, D-062\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-369-triage) + the computational perimeter (Platt–Trudgian `RHUpTo`,\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-370-`LiVerifiedUpTo`) + two elementary margin evaluations. -/\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-371-\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-372-/-! ## 2. The archimedean lower bound (D-037 Step 1) -/\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-373-" num_lines=60 num_matches=None applied_limit=60 applied_offset=None

raw content_json (database row)

[{"type":"thinking","thinking":"The assembly (§2 algebra):\n\ng(1+s) = 1/(1+s) + [1/s + ζ′/ζ(1+s)] + ΓR′/ΓR(1+s) has s^{k−1}-coefficient (−1)^{k−1} − c_{k−1} + τ_{k−1}; FE gives g^{(k−1)}(0) = (−1)^k g^{(k−1)}(1); (M): σ_k = −g^{(k−1)}(0)/(k−1)!.\n\nChain: σ_k = (−1)^{k+1}[(−1)^{k−1} − c_{k−1} + τ_{k−1}] = 1 + (−1)^k c_{k−1} + (−1)^{k+1}τ_{k−1}, solve for c_{k−1}, insert (A): c_{k−1} = (−1)^k(a_k + σ_k) + (−1)^{k−1}.\n\nTarget: **(4.10-a)** for all k ≥ 1: `lcoeff (k−1) = (−1)^k · (lagariasA k + sigmaP k) + (−1)^{k−1}`.\n\nThen with ii′-2's swaps and ResW1_eq_sum → the crux `Sf(n) = ResW1 n`.\n\nWhat do I need in Lean:\n\n1. **FE derivative relation**: gFun^{(m)}(0) = (−1)^{m+1}·gFun^{(m)}(1)? Wait — D-066 says g^{(k−1)}(0) = (−1)^k g^{(k−1)}(1) with m = k−1: g^{(m)}(0) = (−1)^{m+1} g^{(m)}(1). From xiL(1−s) = xiL(s): deriv both sides m times: (−1)^m xiL^{(m)}(1) = xiL^{(m)}(0). Hmm, for g = xiL'/xiL it's more delicate. Actually D-066 claims the FE gives g^{(k−1)}(0) = (−1)^k g^{(k−1)}(1).\n\n   Let me verify: define h(s) := g(1−s). Then h(s) = logDeriv xiL (1−s) = (xiL'(1−s))/(xiL(1−s)) = (xiL'(1−s))/(xiL(s)). And xiL'(1−s): from xiL(1−s) = xiL(s), differentiate: −xiL'(1−s) = xiL'(s), so xiL'(1−s) = −xiL'(s). Hence h(s) = −xiL'(s)/xiL(s) = −g(s). So g(1−s) = −g(s) as functions (where defined; but both sides are genuine functions of s: g is analytic wherever xiL ≠ 0; xiL ≠ 0 at 0 and 1 (value 1/2) so both sides analytic near 0 and the identity holds as an identity of meromorphic/analytic functions near 0 — actually g(1−s) = −g(s) holds on the open set where xiL(s) ≠ 0 (then xiL(1−s) = xiL(s) ≠ 0 too); near s = 0 both sides analytic).\n\n   Then iteratedDeriv m (fun s => g (1−s)) 0 = iteratedDeriv m (fun s => −g s) 0 = −g^{(m)}(0).\n   And iteratedDeriv m (fun s => g (1−s)) 0 = (−1)^m g^{(m)}(1) (chain rule m times: each derivative brings a −1 from the inner 1−s).\n   So (−1)^m g^{(m)}(1) = −g^{(m)}(0), i.e. g^{(m)}(0) = −(−1)^m g^{(m)}(1) = (−1)^{m+1} g^{(m)}(1). With m = k−1: g^{(k−1)}(0) = (−1)^k g^{(k−1)}(1) ✓ matches D-066.\n\n2. **The shift expansion**: g(1+s) as a function of s near 0. We need: iteratedDeriv (k−1) gFun 1 = (k−1)! · [(−1)^{k−1} − c_{k−1} + τ_{k−1}].\n\n   Compute g(1+s): xiL(1+s) = ((1+s)s/2)·Λ(1+s) where Λ(1+s) = ΓR(1+s)·ζ(1+s).\n   g(1+s) = logDeriv xiL (1+s) = d/ds [log xiL(1+s)] (as logDeriv of the composition at s) = logDeriv (fun s => xiL (1+s)) 0... careful: iteratedDeriv (k−1) gFun 1 is the (k−1)-th derivative of g at the POINT 1. Equivalently the (k−1)-th derivative of s ↦ g(1+s) at 0 (chain rule with inner 1+s, derivative 1 — iteratedDeriv_comp_add_const; that's exactly the \"packaging\" D-066 mentions for piece (B)).\n\n   And g(1+s) = logDeriv (xiL(1+·)) s = logDeriv [(1+s)s/2 · ΓR(1+s) · ζ(1+s)] s\n   = [1/(1+s) + 1/s] + logDeriv ΓR (1+s) + logDeriv ζ (1+s)\n   = 1/(1+s) + [1/s + ζ′/ζ(1+s)] + ΓR′/ΓR(1+s) ✓ matches D-066.\n\n   The s^{k−1}-coefficient (i.e., iteratedDeriv (k−1) at 0 divided by (k−1)!):\n   - 1/(1+s): iteratedDeriv (k−1) (fun s => (1+s)⁻¹) 0 = (−1)^{k−1} (k−1)! · 1 → coefficient (−1)^{k−1}.\n   - [1/s + ζ′/ζ(1+s)]: piece (B): coefficients −c_j: iteratedDeriv j at 0 = −j! · lcoeff j? Let me check what iteratedDeriv_hLiExt says exactly.\n   - ΓR′/ΓR(1+s): piece (A): iteratedDeriv_logDeriv_Gammaℝ: (1/j!) iteratedDeriv j (fun s => logDeriv Gammaℝ (1+s)) 0 = (−1)^{j+1} lagariasA (j+1).\n\n3. **Existing pieces — exact statements**: need to check:\n   - LiArchimedean's `iteratedDeriv_logDeriv_Gammaℝ` exact signature.\n   - LiZetaCoeff's `iteratedDeriv_hLiExt` exact signature.\n   - LiResidue's `lcoeff` definition.\n   - LiSwaps' `lagariasA`, `sigmaP`, `Sf` (or wherever Sf is), `ResW1` (LiResidue).\n   - The crux: Sf(n) = ResW1 n — how is Sf defined? From STATE: \"crux identity Sf(n) = Res₁(n)\" — Sf := S_∞ + 1 − λ_n? Let me check LiSkeleton for Sf/SfBoundProp and ResW1_eq_sum in LiResidue.\n\nLet me gather all the exact statements.","signature":""},{"type":"text","text":"The assembly plan is clear (D-066 §2). Now gathering the exact statements of all input pieces."},{"type":"tool_use","id":"Grep_93","name":"Grep","input":{"pattern":"theorem iteratedDeriv_logDeriv_Gammaℝ|lemma iteratedDeriv_logDeriv_Gammaℝ|theorem iteratedDeriv_hLiExt|lemma iteratedDeriv_hLiExt|def lcoeff|noncomputable def lcoeff|theorem ResW1_eq_sum|lemma ResW1_eq_sum|def ResW1|noncomputable def ResW1","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1","output_mode":"content","-n":true,"-A":4},"extra_content":null},{"type":"tool_use","id":"Grep_94","name":"Grep","input":{"pattern":"def lagariasA|def Sf\\b|SfBoundProp|def sInfty|def liCoeff\\b","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1","output_mode":"content","-n":true,"-A":6,"head_limit":60},"extra_content":null}]