[{"type":"thinking","thinking":"Session 031: ii′-5e-v chunk 2 — the tail bound + the main series circle bound.\n\nPer the design in journal-030:\n\n1. `zero_tail_inv_sq_le` (X ≥ some threshold): Σ_{|γ|>X} m/(1+γ²) ≤ 16·A0count·log X / X — via ENNReal.tsum_fiberwise (free in ℝ≥0∞) with the dyadic key j(ρ) := Nat.log 2 ⌊|γ|/X⌋₊... hmm wait — let me think about the fiberwise decomposition more concretely.\n\nActually — let me reconsider the tail bound approach once more. The ℝ≥0∞ fiberwise route requires:\n1. Convert the real sum to ℝ≥0∞ (ofReal, nonneg terms).\n2. Fiberwise over j(ρ) := the dyadic class of |γ|/X.\n3. Bound each fiber sum by (count)·(term bound).\n4. Sum the dyadic series Σ_j count_j·termbound_j ≤ explicit.\n5. Convert back to ℝ.\n\nThe ℝ≥0∞ conversion dance is itself significant plumbing. Alternative pure-ℝ route: since all terms are nonneg and I have an explicit summable majorant on the INDEX side... Actually — here's a cleaner ℝ-route: use `Summable.tsum_le_tsum_of_inj`? no. Or `tsum_le_tsum_of_inj`? no.\n\nHmm — actually the cleanest pure-ℝ fiberwise: `Summable.tsum_eq_tsum_fiberwise`? Let me check if a real-valued fiberwise exists in this Mathlib. There might be `Summable.tsum_fiberwise` in Topology/Algebra/InfiniteSum/Constructions or NatInt. Let me check.\n\nActually — alternative that avoids fiberwise ENTIRELY: bound the tail by comparison with the full sum's tail property... no.\n\nOr: bound the tail directly via the LAYER CAKE: Σ_{|γ|>X} m/(1+γ²) = Σ m·∫... no.\n\nOr the SIMPLEST structural trick: the tail of the CONVERGENT series Σ m/(1+γ²) — bound the tail at X by (full sum) − (partial up to X)... that gives existence of decay but no rate. The rate needs the count.\n\nOK here's another thought — a MUCH simpler sufficient bound via a DIRECT comparison on the zero subtype without any fiberwise: for ρ with |γ| > X (X ≥ 2):\n\n1/(1+γ²) ≤ 2/(X·|γ|) (since 1+γ² ≥ X·|γ|/... 1+γ² ≥ 2|γ| ≥ X|γ|/... 1+γ² ≥ γ² ≥ X·|γ| for |γ| ≥ X ✓ so 1/(1+γ²) ≤ 1/(X|γ|) ✓ (for |γ| ≥ X)). So Σ_{|γ|>X} m/(1+γ²) ≤ (1/X)·Σ_{|γ|>X} m/|γ| — now I need Σ_{|γ|>X} m/|γ| — and THAT's ≤ Σ m/(|γ|(1+|γ|))·(1+|γ|)/|γ|·... no wait — Σ m/|γ| over |γ| > X DIVERGES?? Σ m/|γ| ~ Σ log γ/γ = ∞ — NO GOOD. So 1/|γ| is not summable — the 1/(1+γ²) tail genuinely needs the quadratic decay's dyadic analysis. Hmm.\n\nWait — actually: Σ_{|γ|>X} m/(1+γ²) ≤ (1/X)·Σ_{|γ|>X} m·|γ|/(1+γ²) — and Σ m·|γ|/(1+γ²) ≤ Σ m/(2|γ|)·... no: |γ|/(1+γ²) ≤ 1/(2√...)·... |γ|/(1+γ²) ≤ 1/|γ|^{1/2}·... — |γ|/(1+γ²) ≤ C/|γ|^{1/2}: γ²+1 ≥ 2|γ| ⟹ |γ|/(1+γ²) ≤ 1/2 — useless. |γ|/(1+γ²) ≤ |γ|/γ² = 1/|γ| — divergent again.\n\nOK so the dyadic fiberwise is unavoidable for the tail. Let me check for a real-valued tsum_fiberwise first.","signature":""},{"type":"text","text":"Session 031 starting: **ii′-5e-v chunk 2** — the dyadic tail bound. First checking for a real-valued fiberwise tsum lemma (to avoid the ℝ≥0∞ dance if possible)."},{"type":"tool_use","id":"Bash_348","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"tsum_fiberwise\" Mathlib/Topology/Algebra/InfiniteSum/*.lean | grep -v ENNReal | head -5; grep -rn \"theorem Summable.tsum_sigma\\|tsum_sigma_of_summable\" Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean Mathlib/Topology/Algebra/InfiniteSum/Group.lean 2>/dev/null | head -4","description":"Real-valued fiberwise/sigma tsum"},"extra_content":null}]