The theta-problem in the modular frame of the half-line is a parameter-free quadratic Wiener-Hopf problem (F1 Prop. 6.2)

Status: proved — a complete structured proof.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: result · Knowledge-base id: theta-wiener-hopf-frame

Statement

In the modular frame y = -log(1-x) of I = (-inf,1), 1 - theta = inf_G g_Q([Q, d + G]) / g_Q([Q, d]) over bounded anti-self-adjoint G supported in {y > 0}, with d = E_I D_chi E_I and g_Q([Q,X]) = (1/4) int int K(p,p') |X^(p,p')|^2 dp dp', K(p,p') = sinh^2(pi(p-p')) / (cosh(pi(p-p')) [cosh(pi(p+p')) + cosh(pi(p-p'))]), so theta is the value of an explicit quadratic Wiener-Hopf problem independent of the geometry.

How to verify

No verification recipe recorded on this card.

Evidence

• rigor/exact_optimum_tangent_problem.tex — document, at prop:thetaframe

Related claims

None recorded.

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A2.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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