cft-cmi

Exterior directions exhaust the invisible space: closure(Ext) = N^perp (question (Q) settled, YES)

Status: proved — a complete structured proof.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: theorem · Knowledge-base id: exterior-exhaust-invisible

Statement

In the real Hilbert space S_2(PH, P^perp H) with inner product Re Tr(X^* Y), let N = {P^perp l P : l = l^* = E_I l E_I} (visible space) and Ext = {P^perp Z_e P : Z_e^* = -Z_e = E_{I^c} Z_e E_{I^c}} (exterior directions). Then closure(Ext) = N^perp = {B : E_I(B + B^*)E_I = 0}, and finite-rank Z_e already suffice. Proof (invariant, no Halmos normal form): B orthogonal to both spaces means, with Xi := B + B^*, Gamma := 2P-1: Gamma Xi + Xi Gamma = 0, E_I Xi E_I = 0, E_{I^c} Gamma Xi E_{I^c} = 0. Compressing Gamma Xi Gamma = -Xi with E_{I^c} and inserting 1 = E_I + E_{I^c} twice gives Gamma' pi Gamma' = pi with pi = E_{I^c} Xi E_{I^c}, Gamma' = E_{I^c}(2P-1)E_{I^c}; since Gamma' is a self-adjoint contraction with no eigenvalue +-1 (generic position) Gamma'^n -> 0 strongly, and pi is compact, so pi = 0; then E_{I^c} P E_I (E_I Xi E_{I^c}) = 0 with E_{I^c} P E_I injective gives Xi = 0. FINITE-RANK exterior generators already suffice (closure(Ext_fr) = closure(Ext) = N^perp). Compactness of the test element is NOT needed (referee R3): for any BOUNDED pi with pi = Gamma'^n pi Gamma'^n one has |<xi, pi eta>| <= ||Gamma'^n xi|| ||pi|| ||Gamma'^n eta|| -> 0, so pi = 0 by two-sided strong convergence alone; the earlier claim that bounded solutions exist (from approximate eigenvectors) is false, since approximate eigenvectors at +-1 tend weakly to 0. Generic position IS essential: one failure in each of the two groups breaks it (f in h_{I^c} cap PH and g in h_I cap P^perp H give B = |g><f| nonzero in N^perp cap Ext^perp). Only a CLOSURE is claimed because Z_e -> P^perp Z_e P has unbounded inverse on N^perp (E2 Sec. 2.3), not because of the fixed-point equation. Second, independent proof: N and Ext are the (1,1)-sector standard subspaces of F(I) and F(I)', so (Q) IS Haag/twisted duality (Tomita: K^perp_R = i K').

How to verify

rigor/exact_fidelity_upper_half.tex Sec. 5 (Lemma 5.1 generic position, Thm 5.3 invariant proof, Thm 5.4 duality (which, for a bounded interval, uses twisted duality for the disconnected complement plus additivity, circle picture) proof); numerically corroborated by E0 card finite-dim-exterior-density.

Evidence

Related claims

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A1.

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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