[{"type":"thinking","thinking":"Now I have the complete picture. Let me summarize precisely for the user.\n\nThe user's proposed four-part stack:\n1. Platt–Trudgian RH verification to 3×10¹² ✓ (this is the `RHUpTo` hypothesis — declared perimeter, NOT proved by us; it's a published computational certificate)\n2. Rigorously certified finite Li verification up to splice point ✓ (`LiVerifiedUpTo 5×10⁵` — declared perimeter: our E-011/E-015 to 3000 + Misguich–Voros to 5×10⁵)\n3. **Lean-verified contour bound** — ✗ NOT done. This is exactly `SfBoundProp`, which is ASSUMED as a named hypothesis, not proved.\n4. Lean-verified archimedean bound, convexity, assembly ✓ (tier (i): sInfty_lower_bound, marginTilde_convex, chain_rule_sqrt_shift all PROVED; tier (ii) steps 1+2: assembly + base points PROVED)\n\nSo the answer to the user's question \"Please check if the research do this?\" is: NO — three of the four components are in place, but the third one (Lean-verified contour bound) is NOT. The current theorem `lambda_pos_finite_height_verified` has SfBoundProp as a hypothesis. D-062 §4 says tier ii′ (the contour layers L0–L3 inside SfBoundProp) is POST-PUBLICATION.\n\nWhat IS done on the contour side:\n- The full D-037/D-056 contour WAS calculated analytically (pen-and-paper with all constants traced): D-036 (contour constants), D-037 §4 (assembly of B(n)), D-056 (reference-point R-bound repair, A = 10.5, B₀ = 48, proved honestly with the C₀ log log term), E-028 (26× slack verification), E-029 (convexity). The analytic chain is complete and survived the adversarial review (D-054 found the holes; 4.5a–4.5d repaired them).\n- But the Lean formalization of that contour argument is exactly what's isolated in SfBoundProp. D-062 §3 triages its eventual proof into layers L0–L7: L0 (rectangle contour identity — the main lift, Mathlib meromorphic-residue machinery is thin), L1–L3 (Mathlib-tractable modulo L4), L4 (Trudgian N(T)/S(T) — decided PERIMETER), L5 (bridge — perimeter via RHUpTo), L6 (M₀ — computational perimeter after E-023b), L7 (summability).\n\nSo the user's four-part decomposition is EXACTLY the structure the research has — but with component 3 still open (it's the declared hypothesis, honestly tagged in the paper). The paper itself says this explicitly: paperA.tex line 347ff \"machine-proved from exactly three named hypotheses\", line 351 SfBoundProp = \"the traced contour bound\", line 370-371 \"The remaining formalization target inside this paper is (ii′) the contour estimate inside SfBoundProp\".\n\nOne more check — the user's item 1 says \"Platt–Trudgian RH verification\" and item 2 \"certified finite Li verification\" — in the current Lean theorem these are also hypotheses (RHUpTo, LiVerifiedUpTo), i.e., trusted perimeter, not Lean-proved. The user's list presents them as ingredients to trust, which matches the perimeter decision. But note the user's framing says \"plus rigorously certified finite Li-coefficient verification\" — that's the perimeter too, not Lean-internal. So in the user's proposed stack, items 1 and 2 are external certificates (trusted), and 3, 4 are Lean-verified. Current state: 1 ✓ (as hypothesis), 2 ✓ (as hypothesis), 4 ✓ (proved), 3 ✗ (hypothesis, not proved).\n\nAlso worth noting: the range. SfBoundProp covers n ∈ [10⁵, N₁]; LiVerifiedUpTo covers n ≤ 5×10⁵ (overlap 5× at the splice 10⁵); the assembly covers everything. The final conclusion ∀ n, 1 ≤ n → n ≤ (3×10¹²−1)² → 0 < liCoeff n.re is exactly the user's stated yield.\n\nSo the answer: No — the research does NOT currently achieve that full stack. It achieves it modulo SfBoundProp. The contour bound |S_f(n)| ≤ B(n) over [10⁵, N₁] is the single remaining analytic hypothesis. It was triaged (D-062) and deferred to post-publication (tier ii′) per the mathematician's review guidance. Everything else in the user's stack is in place.\n\nAlso relevant: the paper is HONEST about this — it claims exactly \"machine-verified modulo one named analytic hypothesis (the traced contour bound) + the declared computational perimeter.\" So no overclaim. If the user wants the full stack as they describe, the remaining work is tier ii′: formalizing L0–L3 (with L4/L6 as declared perimeter). D-062 estimates L0 (rectangle contour identity with residues) is the main lift — Mathlib's meromorphic residue machinery on rectangles is thin.\n\nLet me write a concise, precise answer with file:line references.","signature":""},{"type":"text","text":"Clear answer: **No — the research does not yet do this.** Three of your four components are in place, but the third — the Lean-verified contour bound — is exactly the one remaining open hypothesis. It is assumed, not proved, and the paper says so explicitly.\n\nComponent-by-component against your stack:\n\n| Your component | Status |\n|---|---|\n| Platt–Trudgian RH to 3×10¹² | In Lean as named hypothesis `RHUpTo (3*10^12)` — trusted perimeter, not re-proved (matches your framing) |\n| Certified Li verification to splice | In Lean as `LiVerifiedUpTo (5*10^5)` — trusted perimeter (E-011/E-015 to 3000 + Misguich–Voros to 5×10⁵) |\n| **Lean-verified contour bound \\|S_f(n)\\| ≤ B(n)** | **NOT DONE.** This is precisely `SfBoundProp` (LiSkeleton.lean:2131-2133), assumed as a named hypothesis over n ∈ [10⁵, N₁] |\n| Lean-verified archimedean bound, convexity, assembly | DONE — `sInfty_lower_bound`, `marginTilde_convex` (tier i), assembly + certified base points (tier ii steps 1–2), all sorry-free |\n\nWhat exists now is `lambda_pos_finite_height_verified` (LiBasePoints.lean:631-637): λₙ > 0 for 1 ≤ n ≤ (3×10¹²−1)² proved from exactly three hypotheses — `RHUpTo`, `LiVerifiedUpTo`, **`hSf : SfBoundProp`**. PrintAxioms = 3 standard axioms, no sorryAx.\n\n**On the contour itself:** the full D-037/D-056 contour *was* calculated analytically — all constants traced (A = 10.5, B₀ = 48 with the honest C₀ log log term, D-056; B(n) assembly D-037 §4; E-028 verified with 26× slack), and it survived the four-instance adversarial review plus repairs (D-054 → D-056–D-059). But that is pen-and-paper mathematics, not Lean. The Lean formalization of the contour argument is what `SfBoundProp` stands in for.\n\nD-062 §3 triages what closing it takes: L0 (rectangle contour identity with residues — the main lift; Mathlib's meromorphic-residue machinery on rectangles is thin), L1–L3 (Mathlib-tractable), L4 (Trudgian N(T)/S(T) — decided perimeter), L6 (M₀ — perimeter after E-023b). This is \"tier ii′\", explicitly deferred to **post-publication** per the session-012 mathematician review, and Paper A declares it as the remaining formalization target (paperA.tex:370-371).\n\nSo: the theorem statement and the paper's claim are honest — \"machine-verified modulo one named analytic hypothesis\" — but the stack you describe, with the contour bound Lean-proved, is the tier ii′ project, not the current state. Do you want to take up tier ii′ now (starting with L0, the contour identity), or proceed with submission with `SfBoundProp` declared?"}]