[{"type":"thinking","thinking":"All affine-comp lemmas exist and are UNCONDITIONAL (function-level equalities):\n\n- `iteratedDeriv_comp_const_sub (n) (f) (s) : iteratedDeriv n (fun z => f (s − z)) = fun t => (−1)^n • iteratedDeriv n f (s − t)` — EXACTLY what I need for the FE relation: with s := 1, f := gFun:\n  iteratedDeriv m (fun z => gFun (1 − z)) 0 = (−1)^m • iteratedDeriv m gFun (1 − 0) = (−1)^m · iteratedDeriv m gFun 1.\n  \n- `iteratedDeriv_comp_add_const (n) (f) (s) : iteratedDeriv n (fun z => f (z + s)) = fun t => iteratedDeriv n f (t + s)` — for the shift 1+s: fun s => gFun (1+s) = fun s => gFun (s+1) — need s+1 form: gFun (1+s) = gFun (s+1) by add_comm. So iteratedDeriv m (fun s => gFun (s+1)) 0 = iteratedDeriv m gFun (0+1) = iteratedDeriv m gFun 1 ✓.\n\nNo induction needed. And the function identity (fun s => gFun (1−s)) = −gFun from xiL_one_sub + deriv comp.\n\nFE relation proof:\n```\nlemma iteratedDeriv_gFun_zero (m : ℕ) :\n    iteratedDeriv m gFun 0 = ((-1:ℂ)) ^ (m+1) * iteratedDeriv m gFun 1 := by\n  have hid : (fun s : ℂ => gFun (1 - s)) = -gFun := ...\n  have h1 := congr_fun (iteratedDeriv_comp_const_sub m gFun (1 : ℂ)) 0\n  -- h1 : iteratedDeriv m (fun z => gFun (1 − z)) 0 = (−1)^m • iteratedDeriv m gFun (1 − 0)\n  rw [hid] at h1\n  -- h1 : iteratedDeriv m (−gFun) 0 = ...\n  rw [iteratedDeriv_neg] at h1\n  -- h1 : −iteratedDeriv m gFun 0 = (−1)^m • iteratedDeriv m gFun (1−0)\n  ...\n```\n(−1)^m • X for ℂ: smul = mul: `(−1:ℂ)^m • X = (−1:ℂ)^m * X` via smul_eq_mul. Then solve: −G0 = (−1)^m * G1 → G0 = −(−1)^m * G1 = (−1)^{m+1} * G1. linear_combination or:\nG0 = (−1)^{m+1} * G1: from h1: −G0 = (−1)^m * G1: linear_combination -h1? Goal: G0 = (−1)^{m+1} * G1. (−1)^{m+1} = −(−1)^m: pow_succ: (−1)^{m+1} = (−1)^m * (−1). So goal G0 = (−1)^m * (−1) * G1. linear_combination -h1 works over ℂ if h1 : −G0 = (−1)^m * G1: −h1 gives G0 = −(−1)^m * G1; goal − (−1)^{m+1} * G1 = −((−1)^m * −1) * G1 = (−1)^m * G1... linear_combination normalizes ring expressions, so `linear_combination -h1` should close: goal − (−1)^... hmm linear_combination checks goal_lhs − goal_rhs = (coeff · (h1_lhs − h1_rhs)) via ring: G0 − (−1)^{m+1}·G1 =? −(−G0 − (−1)^m·G1) = G0 + (−1)^m·G1. Need G0 − (−1)^{m+1}·G1 = G0 + (−1)^m·G1 ⟺ −(−1)^{m+1} = (−1)^m ⟺ (−1)^{m+1} = −(−1)^m ✓ ring-normalizable (pow_succ handles exponent m+1 as a ring... ring can normalize (−1)^{m+1} to (−1)^m·(−1) ✓). ✓\n\nhid proof:\n```\nhave hderiv : ∀ s : ℂ, deriv xiL (1 - s) = -deriv xiL s := by\n  intro s\n  have hfun : (fun s : ℂ => xiL (1 - s)) = xiL := funext xiL_one_sub\n  have h2 : deriv (fun s : ℂ => xiL (1 - s)) s = deriv xiL s := by rw [hfun]\n  have h3 : deriv (fun s : ℂ => xiL (1 - s)) s = deriv xiL (1 - s) * (-1) := by\n    have h1 : DifferentiableAt ℂ xiL ((fun s : ℂ => 1 - s) s) := differentiable_xiL _\n    have h2' : DifferentiableAt ℂ (fun s : ℂ => 1 - s) s :=\n      (differentiableAt_const (1 : ℂ)).sub differentiableAt_id\n    have hcomp := deriv_comp s h1 h2'\n    have heq : (xiL ∘ fun s : ℂ => 1 - s) = fun s : ℂ => xiL (1 - s) := rfl\n    rw [heq] at hcomp\n    have hinner : deriv (fun s : ℂ => 1 - s) s = -1 := by\n      have h0' : HasDerivAt (fun s : ℂ => 1 - s) (0 - 1) s :=\n        (hasDerivAt_const s (1 : ℂ)).sub (hasDerivAt_id' s)\n      rw [h0'.deriv]\n      norm_num  -- hmm 0 − 1 = −1\n    rw [hinner] at hcomp\n    exact hcomp  -- hcomp : deriv (fun s => xiL (1−s)) s = deriv xiL (1−s) * (−1)? after hinner: deriv xiL ((fun s => 1−s) s) * (−1); ((fun s => 1−s) s) beta = 1 − s\n  -- combine\n  rw [h3] at h2  -- h2 : deriv xiL (1−s) * (−1) = deriv xiL s\n  ... solve: deriv xiL (1−s) = −deriv xiL s: linear_combination? or:\n  have : deriv xiL (1 - s) = -deriv xiL s := by linear_combination -h2\n```\nWait: h2 : deriv xiL (1−s) * (−1) = deriv xiL s. Goal: deriv xiL (1−s) = −deriv xiL s. linear_combination -h2: −(L−R) = −(deriv xiL (1−s)·(−1) − deriv xiL s) = deriv xiL (1−s) + deriv xiL s... hmm goal − (−h2 combo): G − (−(−X·−1 − Y))·... let me just: goal: A = −B where h2 : A·(−1) = B i.e. −A = B. linear_combination -h2: goal_lhs − goal_rhs = A − (−B) = A + B; coeff·(h2_lhs − h2_rhs) = −1·(−A − B) = A + B ✓. So linear_combination -h2 ✓.\n\nHmm — this mirrors gFun_one_eq_neg's structure (lines 369-394) but general-point. Note gFun_one_eq_neg already proves deriv xiL 0 = −deriv xiL 1 — but I need it at general s.\n\nThen:\n```\nhave hid : (fun s : ℂ => gFun (1 - s)) = -gFun := by\n  funext s\n  rw [gFun, gFun, logDeriv_apply, logDeriv_apply, xiL_one_sub, hderiv s]\n  -- goal: −deriv xiL s / xiL s = −(deriv xiL s / xiL s)? \n  -- LHS: gFun (1−s) = deriv xiL (1−s) / xiL (1−s) = (−deriv xiL s)/(xiL s)\n  -- RHS: (−gFun) s = −(deriv xiL s / xiL s)\n  rw [Pi.neg_apply]  -- hmm: goal after funext s: gFun (1−s) = (−gFun) s; Pi.neg_apply: (−gFun) s = −gFun s\n  rw [neg_div]\n```\nCareful sequence: funext s; show gFun (1−s) = (−gFun) s; rw [Pi.neg_apply]: → gFun (1−s) = −gFun s; rw [gFun, gFun, logDeriv_apply, logDeriv_apply]: → deriv xiL (1−s) / xiL (1−s) = −(deriv xiL s / xiL s); rw [xiL_one_sub, hderiv s]: → (−deriv xiL s) / xiL s = −(deriv xiL s / xiL s); rw [neg_div]: → −(deriv xiL s / xiL s) = −(deriv xiL s / xiL s) rfl. But wait — hderiv s rewrites `deriv xiL (1−s)` → `−deriv xiL s` ✓. And neg_div : −a / b = −(a / b) ✓ closes by rfl (rw auto-close).\n\nHmm — the `rw [gFun, gFun]` trick: gFun is a def; rw [gFun] uses equation lemma — worked in gFun_one_eq_neg (line 393: rw [gFun, gFun, logDeriv_apply, logDeriv_apply, ...]) ✓ same pattern.\n\nNow the shift expansion. Next piece:\n\n**gFun_one_add expansion** — eventuallyEq on 𝓝[≠] 0 then upgrade:\n\nActually — for iteratedDeriv_congr_of_eventuallyEq I need it on 𝓝 0 (full). Both sides analytic at 0, punctured equality ⟹ full equality via eq_of_eventuallyEq_nhd_ne (LiMoments has it). But actually for iteratedDeriv congruence I only need EventuallyEq on 𝓝 0 — eq_of_eventuallyEq_nhd_ne gives equality AT 0, not the eventually upgrade. Hmm — but if both sides are analytic at 0 and equal on 𝓝[≠] 0, they're equal on a full neighborhood: two continuous functions agreeing on a punctured nbd agree on the nbd? No — continuity + agreement on a punctured neighborhood ⟹ agreement at the point too (eq_of_eventuallyEq_nhd_ne) ⟹ agreement on the whole nbd (punctured + the point) ⟹ EventuallyEq on 𝓝 0 (mem of nhds: the nbd itself). So:\n\n```\nlemma gFun_one_add_eventuallyEq (hZC : ZeroCountProp) :\n    (fun s : ℂ => gFun (1 + s)) =ᶠ[𝓝 (0 : ℂ)]\n      fun s => (1 + s)⁻¹ + hLiExt s + logDeriv Complex.Gammaℝ (1 + s) := by\n```\nHmm wait — but for the congruence I need iteratedDeriv m (fun s => gFun (1+s)) 0 = iteratedDeriv m (RHS) 0. Alternative cleaner path: get the EventuallyEq via the analytic-set argument: the bad set {0, −1} ∪ {ζ(1+s) = 0} ∪ ... is discrete; on a small punctured ball everything holds; both sides analytic on the ball; identity on punctured ball ⟹ identity on ball (eq_of_eventuallyEq_nhd_ne at the center 0 — actually need equality at ALL points of the ball: at punctured points it's given; at the center by continuity). So:\n\n```\n  -- Step 1: the punctured version\n  have hpunct : ∀ᶠ s in 𝓝[≠] (0 : ℂ), gFun (1 + s) =\n      (1 + s)⁻¹ + hLiExt s + logDeriv Complex.Gammaℝ (1 + s) := ...\n  -- Step 2: upgrade\n  rw [eventually_nhdsWithin_iff] at hpunct\n  filter_upwards [hpunct] with s hs\n  by_cases h0 : s = 0\n  · subst h0\n    -- point check at s = 0: gFun 1 = 1 + hLiExt 0 + logDeriv Gammaℝ 1?\n```\nHmm — the point check at 0: gFun (1+0) = gFun 1 =? (1+0)⁻¹ + hLiExt 0 + logDeriv Gammaℝ 1 = 1 + (−deriv hLi 1) + logDeriv Gammaℝ 1. That's a NUMERIC identity — is it true? gFun 1 = logDeriv xiL 1 = deriv xiL 1 / xiL 1 = 2·deriv xiL 1. Is 2·deriv xiL 1 = 1 − deriv hLi 1 + logDeriv Gammaℝ 1? This SHOULD be true by the analytic identity but proving it pointwise requires the same logDeriv algebra — the whole point of the eventuallyEq approach is to avoid point evaluations. Better: use the analyticity-continuity argument:\n\nBoth F := fun s => gFun (1+s) and G := fun s => (1+s)⁻¹ + hLiExt s + logDeriv Gammaℝ (1+s) are continuous at 0:\n- F: gFun_continuousAt 1 (xiL 1 = 1/2 ≠ 0) comp (1+·) continuous. \n- G: (1+s)⁻¹ continuous at 0 (≠ 0) + hLiExt analytic at 0 (hLiExt_analyticAt_zero) + logDeriv Gammaℝ (1+s): Gammaℝ 1 = 1 ≠ 0, differentiableAt_Gammaℝ_of_not_neg_nat → continuous; logDeriv = deriv/f continuity: (analyticAt.deriv).div analyticAt h → analyticAt → continuousAt.\n\nThen eq_of_eventuallyEq_nhd_ne hF hG hpunct gives F 0 = G 0, and... no wait — that gives equality at 0 only. For EventuallyEq on 𝓝 0 I need equality on a set in 𝓝 0. Take the ball from the punctured eventually: hpunct gives ∀ᶠ s in 𝓝 0, s ≠ 0 → F s = G s (eventually_nhdsWithin_iff). Then:\n```\n  filter_upwards [hpunct] with s hs\n  by_cases h0 : s = 0\n  · subst h0; exact point_equality  -- F 0 = G 0 from eq_of_eventuallyEq_nhd_ne\n  · exact hs h0\n```\nwhere point_equality := eq_of_eventuallyEq_nhd_ne hFcont hGcont hpunct0 (hpunct0 : F =ᶠ[𝓝[≠] 0] G). \n\nSo I need the punctured version FIRST (hpunct0), then continuity of both sides, then the upgrade. The point evaluation is handled by the continuity argument — no numeric identity needed. ✓ (Same pattern as gFun_eventuallyEq_zero but there the point value was known via gFun_zero/seriesFun_zero.)\n\nNow the punctured identity. For s ≠ 0 small (s ≠ 0, 1+s ≠ 0 i.e. s ≠ −1, ζ(1+s) ≠ 0, Gammaℝ (1+s) ≠ 0):\n\ngFun (1+s) = logDeriv xiL (1+s) [gFun def]\nWait — actually cleaner: logDeriv xiL (1+s) = (logDeriv of product form). Let me do:\n\nhave e1 : xiL =ᶠ[𝓝 (1+s)] (fun t => (t·(t−1)/2)·completedRiemannZeta t) — from xiL_eq_mul (needs t ≠ 0, t ≠ 1 near 1+s: since s ≠ 0, 1+s ≠ 1; and 1+s ≠ 0 for s near 0... s ≠ −1 ✓). Then logDeriv xiL (1+s) = logDeriv (product form) (1+s) [deriv congruence + point equality — the logDeriv_apply + hev.deriv_eq + hev.eq_of_nhds pattern from gFun_eq_momentB].\n\nThen product: (t(t−1)/2)·Λ(t) with Λ =ᶠ ΓR·ζ near 1+s (riemannZeta_def_of_ne_zero: for t ≠ 0: Λ t = ΓR t · ζ t... check: `riemannZeta_def_of_ne_zero (h : s ≠ 0) : riemannZeta s = completedRiemannZeta s / Gammaℝ s`? The gFun_eq_momentB proof used: `rw [riemannZeta_def_of_ne_zero ht0]` then field_simp with htΓ — that was for hΛeq : completedRiemannZeta =ᶠ[𝓝 s] (fun t => Gammaℝ t * riemannZeta t). Reuse that pattern).\n\nSo logDeriv xiL (1+s) = 1/(1+s) + 1/(1+s−1) + logDeriv Gammaℝ (1+s) + logDeriv riemannZeta (1+s) — the same algebra as gFun_eq_momentB but at the point 1+s.\n\nAnd hLiExt s = (1 − hLi (1+s))/s (s ≠ 0, hLiExt_of_ne). hLi (1+s) = 1 − ((1+s)−1)·(deriv zetaA (1+s)/zetaA (1+s)) = 1 − s·(deriv zetaA (1+s)/zetaA (1+s)). So (1 − hLi (1+s))/s = (s·(deriv zetaA (1+s)/zetaA (1+s)))/s = deriv zetaA (1+s)/zetaA (1+s).\n\nNow: deriv zetaA (1+s)/zetaA (1+s) vs 1/s + logDeriv riemannZeta (1+s)? zetaA w = (w−1)·ζ(w) (check zetaA def). deriv zetaA (1+s) = ζ(1+s) + s·deriv ζ (1+s) (product rule). So deriv zetaA/zetaA at 1+s = [ζ(1+s) + s·ζ′(1+s)]/(s·ζ(1+s)) = 1/s + ζ′(1+s)/ζ(1+s) = 1/s + logDeriv riemannZeta (1+s) ✓ (needs deriv zetaA = ... product rule with differentiability; ζ diff at 1+s for s ≠ 0 ✓ differentiableAt_riemannZeta).\n\nSo hLiExt s = 1/s + logDeriv riemannZeta (1+s) for s ≠ 0 ✓.\n\nThen gFun (1+s) = 1/(1+s) + 1/s + logDeriv Gammaℝ (1+s) + logDeriv riemannZeta (1+s) = (1+s)⁻¹ + hLiExt s + logDeriv Gammaℝ (1+s) ✓ (1/s = s⁻¹ cancels... wait: RHS = (1+s)⁻¹ + [1/s + logDeriv ζ (1+s)] + logDeriv ΓR (1+s); LHS = 1/(1+s) + 1/s + logDeriv ΓR (1+s) + logDeriv ζ (1+s) — same ✓ (one_div vs ⁻¹: 1/(1+s) = (1+s)⁻¹ via one_div).)\n\nThis is a chunk of logDeriv algebra similar to gFun_eq_momentB (~60 lines). \n\nThen the coefficient extraction:\n```\nlemma iteratedDeriv_gFun_one (hZC : ZeroCountProp) (m : ℕ) :\n    iteratedDeriv m gFun 1 =\n      ((-1:ℂ))^m * (m ! : ℂ) - (m ! : ℂ) * lcoeff m +\n        ((-1:ℂ))^(m+1) * (m ! : ℂ) * (lagariasA (m+1) : ℝ) := by\n  -- via the affine comp + eventuallyEq + add splits + the three pieces\n```\nSteps:\n- iteratedDeriv m gFun 1 = iteratedDeriv m (fun s => gFun (s+1)) 0: congr_fun (iteratedDeriv_comp_add_const m gFun 1) 0 gives iteratedDeriv m (fun z => gFun (z+1)) 0 = iteratedDeriv m gFun (0+1) = iteratedDeriv m gFun 1. And (fun s => gFun (1+s)) = (fun s => gFun (s+1)) by funext (add_comm). So:\n  have hcomp := congr_fun (iteratedDeriv_comp_add_const m gFun (1 : ℂ)) 0\n  -- hcomp : iteratedDeriv m (fun z => gFun (z + 1)) 0 = iteratedDeriv m gFun (0 + 1)\n  rw [zero_add] at hcomp → ... = iteratedDeriv m gFun 1\n  have e0 : (fun s : ℂ => gFun (1 + s)) = fun s : ℂ => gFun (s + 1) := funext fun s => by rw [add_comm]\n  ← use hcomp.symm.\n- rw [← hcomp, ← e0] — goal becomes iteratedDeriv m (fun s => gFun (1+s)) 0 = RHS.\n- rw [iteratedDeriv_congr_of_eventuallyEq (gFun_one_add_eventuallyEq hZC) m]\n- Split: (fun s => (1+s)⁻¹ + hLiExt s + logDeriv Gammaℝ (1+s)) = ((fun s => (1+s)⁻¹) + hLiExt) + (fun s => logDeriv Gammaℝ (1+s)) via Pi bridge (hsplit := rfl), iteratedDeriv_add (ContDiffAt each: analyticity), then another split for the first two.\n  ContDiffAt: \n  - (1+s)⁻¹ at 0: analyticAt_const.div (analyticAt_const.add analyticAt_id) (by norm_num : (1:ℂ)+0 ≠ 0) → .contDiffAt. Or directly the zpow form's hcf-like.\n  - hLiExt at 0: hLiExt_analyticAt_zero.contDiffAt.\n  - logDeriv Gammaℝ (1+·) at 0: (AnalyticAt Gammaℝ at 1 via differentiableAt_Gammaℝ_of_not_neg_nat → analyticAt? differentiableAt_Gammaℝ_of_not_neg_nat gives DifferentiableAt; need AnalyticAt — is there analyticAt_Gammaℝ? LiFEntire used `differentiableAt_Gammaℝ_of_not_neg_nat`; for AnalyticAt: `DifferentiableAt.analyticAt` doesn't exist (banked). But Differentiable ℂ on an open → analyticOnNhd: the pattern from analyticAt_xiL: (differentiable_xiL.differentiableOn.analyticOnNhd isOpen_univ) — is Gammaℝ differentiable EVERYWHERE? No — poles at −ℕ. But near 1: differentiable on ball(1, 1/2) say. Alternative: Complex.Gammaℝ analytic — Mathlib: `analyticAt_Gammaℝ`? Hmm. There might be `Complex.differentiable_Gammaℝ`? no (poles). Let me look at how LiDigamma/LiArchimedean proved ContDiffAt for logDeriv Gammaℝ (1+s) at 0 — iteratedDeriv_logDeriv_Gammaℝ's proof must have such ContDiffAt. Let me check LiArchimedean around line 187-222: it used iteratedDeriv_const_mul (1/2 : ℂ) hcd0 — hcd0 is presumably a ContDiffAt for the digamma part. Let me grep for how they built ContDiffAt of logDeriv Gammaℝ (1+s).\n  \n  Actually simpler: logDeriv Gammaℝ (1+s) = deriv Gammaℝ (1+s)/Gammaℝ (1+s): AnalyticAt via (analyticAt Gammaℝ at 1).deriv? For AnalyticAt of Gammaℝ at 1: from DifferentiableOn on a nbd: Gammaℝ is differentiable on {s | ∀ m : ℕ, s ≠ −2m}? Hmm — differentiableAt_Gammaℝ_of_not_neg_nat : (∀ m : ℕ, s ≠ -(m:ℂ))... wait that was about s/2? Let me check its statement: from LiCircleBound usage: `differentiableAt_Gammaℝ_of_not_neg_nat` — (h : ∀ m : ℕ, s ≠ -(2 * (m:ℂ)))? Let me check. In LiMoments line 188: `differentiableAt_Gammaℝ_of_not_neg_nat hhalf` with hhalf : ∀ m : ℕ, s / 2 ≠ -(m : ℂ) — hmm it took hhalf about s/2. Let me check the actual signature in LiCircleBound.\n\nLet me look at how LiArchimedean handles the ContDiffAt for the Gammaℝ logDeriv shift — fastest to read the relevant part of LiArchimedean.lean lines 180-230.","signature":""},{"type":"text","text":"All affine-comp lemmas exist unconditionally. Checking how LiArchimedean builds the Gammaℝ ContDiffAt (needed for the add-split)."},{"type":"tool_use","id":"Read_98","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiArchimedean.lean","offset":150,"limit":90},"extra_content":null}]