[{"type":"thinking","thinking":"Constants verified. Two notes:\n1. `log 11 >= 2.3979` is FALSE (log 11 = 2.397895273 < 2.3979) — need log 11 ≥ 2.3978 for the lower bound (0.833/2.3978 = 0.34740 ≤ 0.3475 ✓ still fine).\n2. `log 14 >= 2.6391` is FALSE (log 14 = 2.63905733) — need log 14 ≥ 2.639 (0.1942/2.639 = 0.07359 ≤ 0.0736 ✓).\n\nAlso S₃ exact-term bounds sum: 1.3475495 + 0.25 = 1.5975495?? wait no — \"sum of bounds: 1.3475495168\" — that's already including the +0.25? Let me look: the last print was sum of the four bounds + 0.25 = 1.3475495168. And the S3 bound 1.3476 ✓ (1.3475495 ≤ 1.3476 ✓ margin 5e-5).\n\nPiece3 = 12.1284·C₀ + 7.242013 — so the constant 7.2425 works (7.242013 ≤ 7.2425 ✓).\n\nWait — main constant: 4.5·0.55·S3 = 3.33531 (I had 3.33594 before — because I used S₃ = 1.3476: 4.5·0.55·1.3476 = 3.33531. Yes.) Total: 3.33531+2.73394+1.17277 = 7.24202 ≤ 7.2425 ✓.\n\nLet me fix: log 11 ≥ 2.3978 (certify: e^2.3978 ≤ 11 — e^2.3978 = ? we need the LOWER bound direction: log 11 ≥ 2.3978 ⟺ 11 ≥ e^2.3978. e^2.3978 ≈ 10.9987 ≤ 11 ✓). And log 14 ≥ 2.639 (e^2.639 ≈ 13.9965 ≤ 14 ✓).\n\nUpdated Tail_R: 0.833/log11 with log 11 ≥ 2.3978: 0.833/2.3978 = 0.34740 ≤ 0.3475 ✓ same. Fine.\n\nNow — one more check on the main bucket family: the per-bucket count 2C₀+0.55 where C₀ = C0hat(t+1). And the weight Σ_{k ∈ Ico 1 (⌊t⌋₊+1)} farW k ≤ S₃. \n\nHmm wait — one thing to double check: the MAIN finset's bucket function k(ρ) = ⌊|t − γ|⌋₊ for far zeros (x ≥ 1): k ≥ 1 ✓. Main = {ρ far, k(ρ) ≤ ⌊t⌋₊}. For ρ in main: x < k+1 ≤ ⌊t⌋₊+1 ≤ t+1 ⟹ |γ| ≤ t+x ≤ 2t+1 ⟹ finite ball ✓.\n\nAnd the far/tail split: far = {x ≥ 1} (nearZerosSub complement). Tail = far with k(ρ) ≥ ⌊t⌋₊+1. Tail ⟹ x ≥ ⌊t⌋₊+1 > t ⟹ γ > 2t or γ < 0 ✓.\n\nThe g-function: g ρ = if ρ ∈ nearZerosSub t then 0 else 4.5·m/(x·√(1+x²)). Note: g uses x·√(1+x²) not the bucket form. The per-term bound: farTerm ≤ 4.5·m/(x√(1+x²)) ✓ and then in buckets ≤ 4.5·m·W(k) with W(k) = 1/(k√(1+k²)) since x ≥ k and u√(1+u²) monotone ✓.\n\nHmm — actually, for cleanliness, maybe define g directly with the bucket: g ρ = if near then 0 else 4.5·m·W(k(ρ))? The per-term bound would need x ≥ k(ρ) ⟹ (x√(1+x²))⁻¹ ≤ W(k(ρ))·... — one extra step but then g's bucket structure is manifest. But g-summability needs the comparison with m/(1+γ²): W(k(ρ)) ≤ 1/k(ρ)² and k(ρ) vs γ... k(ρ) ≥ |x|−1 ≥ |γ|−t−1 — the comparison gets messier. Keep g = the x-form, and convert to bucket weight inside the family lemmas (per-element bound: g ρ ≤ 4.5·m·W(k(ρ))).\n\ng-summability: g ρ ≤ (if near 0 else) 4.5·m/(x√(1+x²)). For the dominator: for x ≥ 1 (g ≠ 0 ⟹ far ⟹ x ≥ 1): 4.5/(x√(1+x²)) ≤ 4.5/x². And x = |t−γ|: 1/x² ≤ ?·1/(1+γ²): |γ| ≤ |t−γ|+t = x+t ≤ x+t·x (x ≥ 1... x+t ≤ x(1+t) ⟺ x+t ≤ x+tx ⟺ t ≤ tx ✓ since x ≥ 1, t ≥ 10). So |γ| ≤ (1+t)·x ⟹ 1+γ² ≤ 1+(1+t)²x² ≤ (1+t)²(1+x²)·... ≤ 2(1+t)²x² (1+x² ≤ 2x² for x ≥ 1). So 1/x² ≤ 2(1+t)²/(1+γ²). g ≤ 9·(1+t)²·m/(1+γ²) — summable via zero_sum_inv_sq (the (1+t)² is a constant factor w.r.t. ρ ✓). Hmm — zero_sum_inv_sq in the form used: `(WeilEF.zero_sum_inv_sq zetaSeam) : Summable (fun ρ => (zeroMult ρ : ℝ)/(1+normSq (gammaOf ρ)))` — note it's gammaOf = (ρ − 1/2)/i-ish — the LiCircleBound tailTerm_summable proof has the comparison normSq(gammaOf) = im² + (1/2−re)² — clone that ✓.\n\nAlso needed: the per-term ‖seriesDiffTerm‖ bound:\n‖seriesDiffTerm s s₀ ρ‖ = ‖(m:ℂ)·(1/(s−ρ) − 1/(s₀−ρ))‖ = m·‖(s₀−s)/((s−ρ)(s₀−ρ))‖ = m·‖s₀−s‖/(‖s−ρ‖·‖s₀−ρ‖).\n- algebra: 1/(s−ρ) − 1/(s₀−ρ) = (s₀−s)/((s−ρ)(s₀−ρ)): needs s ≠ ρ, s₀ ≠ ρ. s₀ = 2+it: re s₀ = 2 ≠ re ρ < 1 ⟹ s₀ ≠ ρ ✓. s ≠ ρ via hζ ✓.\n- ‖s₀−s‖ ≤ 4.5 (hdiff form from one_div_diff_le: s₀−s = (2−s.re:ℝ) — I'll restate inline).\n- ‖s−ρ‖ ≥ x: |im(s−ρ)| = |t−γ| = x ✓ (abs_im_le_norm).\n- ‖s₀−ρ‖ ≥ √(1+x²): ‖s₀−ρ‖² = normSq = (2−β)²+(t−γ)² ≥ 1+x² (|2−β| ≥ 1 since β < 1). Then ‖s₀−ρ‖ ≥ √(1+x²) via Real.le_sqrt-ish: √(1+x²) ≤ ‖s₀−ρ‖ ⟺ 1+x² ≤ ‖s₀−ρ‖² (sqrt_le_iff or Real.sqrt_le_sqrt + sq). \n- combine: m·4.5/(x·√(1+x²)) — division bounds with x ≥ 1 > 0.\n\nOK — now the assembly structure. The final theorem:\n\n```\ntheorem far_series_le (hZC : ZeroCountProp) (hNLZ : NoLowZeros) {t : ℝ} (ht : 10 ≤ t)\n    {s : ℂ} (hσlo : -5/2 ≤ s.re) (hσhi : s.re ≤ 3) (him : s.im = t)\n    (hζ : riemannZeta s ≠ 0) :\n    ‖∑' ρ : {ρ : ℂ // IsNontrivialZero ρ},\n      if ρ ∈ nearZerosSub t then (0 : ℂ) else seriesDiffTerm s (2 + t * Complex.I) ρ‖\n      ≤ 12.1284 * C0hat (t + 1) + 7.2425\n```\n\nWait — I should double-check the main bucket-count uses C0hat(2t+1) ≤ C₀+0.55 — C₀ := C0hat(t+1) and the assembly is in terms of C0hat(t+1) ✓ consistent.\n\nOne more sanity check on the main: Σ_{k ∈ Ico 1 (⌊t⌋₊+1)} W(k)·(2C₀+0.55)·4.5 ≤ (2C₀+0.55)·4.5·1.3476 — yes since the Ico-sum ≤ S₃-tsum ≤ 1.3476 (finite ≤ tsum for nonneg summable — `Summable.sum_le_tsum` ✓).\n\nAlso the LEFT TAIL's bucket: ℓ(ρ) = ⌊|γ|⌋₊ for γ < 0. NoLowZeros gives 14 ≤ |γ| ⟹ ℓ ≥ 14 ✓. Fiber ℓ: |γ| ∈ [ℓ, ℓ+1), γ < 0 ⟹ conj im = |γ| ∈ [ℓ, ℓ+1) ⊆ zeroWindow ℓ [ℓ, ℓ+1] ✓ count ≤ C0hat(ℓ+1) (hZC ℓ, ℓ ≥ 14 ≥ 2 ✓).\n\nHmm wait — the fiber map: fibers are on the SUBTYPE (with the conj structure). The fiber mult-sum ≤ window count: fiber → image under (conj ∘ val)?? zeroMult(conj ρ) = zeroMult(ρ)? Need that! zeroMult = analyticOrderAt ζ ·.toNat — under conjugation: analyticOrderAt ζ (conj ρ) = analyticOrderAt ζ ρ — does this exist? In LiReflect there's `analyticOrderAt_zeta_one_sub` — for conj: `riemannZeta_conj` gives ζ(conj s) = conj(ζ(s)). The order invariance: hmm. Let me check — for the mult-sum over the fiber, I need Σ_fiber m(ρ) ≤ count in zeroWindow(ℓ) where the window counts zeros in ℂ (the zeroWindow_finite t toFinset sums over ℂ zeros with mult). The bridge: fiber (subtype) → ℂ via val... but the fiber has γ < 0 and the window is at +ℓ. So map ρ ↦ conj(ρ.1): zeroMult(conj ρ.1) = zeroMult(ρ.1) — NEED this lemma. Does it exist? Let me search for conj + zeroMult/analyticOrderAt. \n\nIn LiReflect (session 025): `sum_nearZeros_neg_reflect` — \"zeroMult invariant via analyticOrderAt_zeta_one_sub\" — that's for ρ ↦ 1−ρ, not conj. Hmm — maybe there's `analyticOrderAt_riemannZeta_conj`? Let me grep.\n\nActually — alternative: avoid conj-mult entirely: map the fiber to zerosIn(−(ℓ+1), −ℓ] (the NEGATIVE window at −ℓ): ρ ↦ ρ.1 directly (no conj!). Then the count: Ncount(−(ℓ+1), −ℓ) — but ZeroCountProp is for POSITIVE windows [a, a+1], a ≥ 2!! The perimeter only gives positive windows. The negative window count needs conj symmetry of the Ncount... Zeta23.Ncount(−(ℓ+1), −ℓ) = Ncount(ℓ, ℓ+1)? Is there a conj symmetry lemma for Ncount or zerosIn? Hmm. Let me grep for zerosIn conj or Ncount conj in Zeta23.\n\nOR — cleanest: bound the negative-window count via the band lemma `sum_zeroMult_le_of_im_band` — NO, that uses A₀ (opaque). \n\nOR: prove zeroMult_conj : zeroMult (conj ρ) = zeroMult ρ — via: analyticOrderAt ζ (conj z) = analyticOrderAt ζ z. Proof sketch: ζ = conj ∘ ζ ∘ conj (riemannZeta_conj: ζ(conj s) = conj(ζ(s))). analyticOrderAt under pre/post composition with analyticity-preserving maps... There might be a Mathlib lemma `analyticOrderAt_comp` for equivalences? Conj is a ℂ→ℂ map, analytic? NO — conj is NOT complex-analytic! But: ζ(conj s) = conj(ζ(s)) means: near conj ρ, write s = conj w: ζ(s) = ζ(conj w) = conj(ζ(w)) — as w ranges near ρ. The map w ↦ conj(ζ(conj w))... the order of vanishing of conj∘ζ∘conj at conj ρ equals order of ζ at ρ because conj is a bijection with... hmm — analyticOrderAt is defined via the Nat-order (the order of the first nonzero Taylor coefficient). conj is real-analytic and swaps Taylor coefficients by conjugation (zero-ness preserved). This could be fiddly. Let me check if Zeta23 or NguyenRH1 already has it.","signature":""},{"type":"text","text":"Constants verified. Two fixes: `log 11 ≥ 2.3978` (not 2.3979) and `log 14 ≥ 2.639` (not 2.6391). Now checking the conj-multiplicity bridge for the left tail."},{"type":"tool_use","id":"Grep_28","name":"Grep","input":{"pattern":"zeroMult.*conj|conj.*zeroMult|analyticOrderAt.*conj|conj.*analyticOrderAt","path":"D:\\Python\\Quantum\\research\\lean","output_mode":"content","-n":true,"head_limit":20},"extra_content":null},{"type":"tool_use","id":"Grep_29","name":"Grep","input":{"pattern":"zerosIn.*conj|Ncount.*conj|conj.*zerosIn","path":"D:\\Python\\Quantum\\research\\lean","output_mode":"content","-n":true,"head_limit":15},"extra_content":null}]