Finite chains: exterior directions span exactly the invisible space (Ext = N^perp)

Status: numerical — measured, with an error bar and a convergence order; not a theorem.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: result · Knowledge-base id: finite-dim-exterior-density

Statement

On the half-filled hopping chain with P the Fermi projection and I a contiguous block (n = 8..16, |I| = 3..7), the real span N of the visible directions P^perp l P (l Hermitian on I) has dimension |I|^2, the span Ext of the exterior directions P^perp Z P (Z anti-Hermitian on I^c) is orthogonal to N and has dimension 2k(n-k) - |I|^2, and N + Ext is all of S_2(PH, P^perp H): Ext = N^perp exactly in every case. Algebraically, B = P^perp X P is orthogonal to N iff E_I(B + B^*)E_I = 0 and to Ext iff E_{I^c}(B - B^*)E_{I^c} = 0; with H = B + B^* (Hermitian, P-off-diagonal) and S = 1 - 2P one has B - B^* = S H, so (Q) is the two-projection statement: a P-off-diagonal Hermitian H with E_I H E_I = 0 and E_{I^c} S H E_{I^c} = 0 vanishes. This is the finite-dimensional shadow of question (Q) of the Phase 4 brief; the continuum statement (density, Hardy projection, interval) is track E1's target.

How to verify

run numerics/optimality_all/e0_finite_density.py (numpy only): every line ends with 'Ext = N^perp: True'.

Evidence

• numerics/optimality_all/e0_finite_density.py — numerics
• rigor/phase4_brief.md — document

Related claims

• phase4-plan — done, no page of its own
• Is the compression beaten in EXACT fidelity? — numerical

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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