[{"type":"thinking","thinking":"`zero_sum_inv_sq` exists: `Summable (fun ρ : (zetaZeros hs).carrier => (zeroMult ρ : ℝ) / (1 + normSq (gammaOf ρ)))` where `gammaOf ρ` — need to check: gammaOf ρ = (ρ − 1/2)/i presumably (the \"gamma\" coordinate), so 1+‖γ_ρ‖² ~ |ρ|²-ish? ‖gammaOf ρ‖² = |ρ.im|² + (1/2−ρ.re)². So the summable family is mult/(1 + γ_im² + (1/2−re)²) — comparable to mult/(1+|ρ|²) only if |ρ.re| is controlled... |γ_ρ|² = ρ.im² + (1/2 − ρ.re)², while |ρ|² = ρ.im² + ρ.re². For zeros, ρ.re ∈ (0,1) bounded, so |γ_ρ|² ≥ ρ.im² and 1+|γ_ρ|² ≥ ρ.im² ≥ ... hmm, need Σ mult/|ρ|²: |ρ|² = ρ.re² + ρ.im² ≤ 1 + ρ.im² ≤ 1 + ‖γ_ρ‖². So mult/|ρ|² ≥ mult/(1+‖γ_ρ‖²) — direction wrong for comparison (we need Σ mult/|ρ|² < ∞, and mult/(1+‖γ‖²) ≤ mult/(1+ρ.im²)... and |ρ|² ≤ (1 + ρ.im²)·2-ish for re ≤ 1: |ρ|² = re²+im² ≤ 1+im². And 1+‖γ_ρ‖² = 1 + im² + (1/2−re)² ≤ 2 + im². So mult/|ρ|² ≤ mult/(re²+im²) and mult/(1+‖γ‖²) ≥ mult/(2+im²) ≥ mult/(2(1+im²)) ≥ mult/(2·2|ρ|²)... for |ρ| ≥ 1: 1+im² ≤ |ρ|² + 1 ≤ 2|ρ|². So mult/(1+‖γ‖²) ≥ mult/(4|ρ|²). Summability of mult/(1+‖γ‖²) ⟹ summability of mult/(4|ρ|²) restricted to |ρ| ≥ 1 ⟹ Σ mult/|ρ|² converges ✓. Good — the needed Σ mult·|ρ|⁻² summability follows from zero_sum_inv_sq by a comparison. That's the L7 engine.\n\nNote `(zetaZeros hs).carrier` is the subtype over IsNontrivialZero — SAME subtype as liCoeff uses ({ρ // IsNontrivialZero ρ} vs (zetaZeros hs).carrier — zetaZeros hs is a ZeroConfig with carrier = {ρ | IsNontrivialZero ρ}; the subtype coerces). And the theorem needs `hs : ZetaSeam` — the seam obligations (discharged by `zetaSeam` per SeamClosed.lean). So we can instantiate at zetaZeroConfig... `zero_sum_inv_sq zetaSeam : Summable ...` over (zetaZeros zetaSeam).carrier = zetaZeroConfig.carrier. Need to check carrier of zetaZeroConfig is definitionally {ρ | IsNontrivialZero ρ} and the liCoeff subtype matches. Fine — engineering detail for the skeleton.\n\nNow the axiom check: are zero_sum_inv_sq / zeta_local_zero_count / riemannVonMangoldt sorry-free? D-061 said zeta-23-lean full build green with sorries only in trusted challenge stubs, and the comparator PrintAxioms matched AUDIT.md. The audit checked \"27 main + 16 amendment statements\". Is zero_sum_inv_sq among them? Let me just run a quick PrintAxioms on the ones we care about. That's a lean env command — short ones allowed in foreground per handoff notes. Let me write a tiny scratch file and run `lake env lean` on it from the NguyenRH1 dir. Also check RectangleIntegral'_eq_sumResiduesIn and zeta_local_zero_count axioms. And gammaOf def.\n\nActually, let me keep this quick and focused: one scratch file printing axioms of the 4-5 key dependencies. Compile time should be seconds (imports already built).","signature":""},{"type":"text","text":"`zero_sum_inv_sq` is exactly the L7 engine we need. Now the critical axiom check on the Zeta23 dependencies we plan to reuse, plus the `gammaOf` definition."},{"type":"tool_use","id":"Bash_35","name":"Bash","input":{"command":"grep -rn \"def gammaOf\" /d/Python/Quantum/zeta-23-lean/Zeta23/ | head -3","description":"Find gammaOf definition"},"extra_content":null},{"type":"tool_use","id":"Write_36","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean","content":"/- Session 014 scratch: axiom check of planned tier-ii′ dependencies. -/\nimport Mathlib\nimport Zeta23.WeilEF.ZeroSummability\nimport Zeta23.RvM.LocalCount\nimport Zeta23.RvM.Statement\nimport Zeta23.FromPNTPlus.ResidueCalcOnRectangles\nimport Zeta23.ZetaReflect\n\nopen Zeta23\n\n#print axioms WeilEF.zero_sum_inv_sq\n#print axioms RvM.zeta_local_zero_count\n#print axioms zeta_reflect_zero\n#print axioms zeta_mult_reflect\n#print axioms RectangleIntegral'_eq_sumResiduesIn\n#print axioms ResidueMult\n#print axioms logDerivResidue\n#print axioms riemannZetaLogDerivResidue\n"},"extra_content":null}]