NS-circle model: Ext = N^perp exactly, with the dimension identity (L-|I|)^2
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: e2-ext-equals-nperp-circle
Statement
In S10's spectral NS-circle model (exact Hardy projection P of rank k = L/2, I = A u D an arc of m = |I| sites, m > L/2) both spanning maps are DIAGONALISED exactly: with Q = E_I P E_I (eigenvalues q, basis U) one has E_I(P^perp l P + P l P^perp)E_I = (1-Q) l Q + Q l (1-Q), so ||P^perp l P||_2^2 = (1/2) sum_ij den_ij |l_ij|^2 with den_ij = q_i(1-q_j)+q_j(1-q_i); likewise for Z anti-Hermitian on I^c with Qc = E_{I^c} P E_{I^c}. Hence dim_R N = m^2 - 2 z^2 with z = m - L/2 (rank Q = rank(1-Q) = L/2 forces z exact zero and z exact unit eigenvalues, and den vanishes exactly on the (0,0) and (1,1) pairs), dim_R S_2(PH,P^perp H) = L^2/2, and dim_R N^perp = L^2/2 - m^2 + 2z^2 = (L/2 - z)^2 = (L-m)^2 = dim_R Ext. Together with L1 (Ext subset N^perp) this gives Ext = N^perp EXACTLY at every L. Brute-force verification with explicit real bases at L = 32, 48: ranks 316/196 and 711/441 match the formula exactly and all principal angles between Ext and N^perp are 0 (cos = 1.0000000000 to 10 digits); at L = 64 double precision loses 22 of 1319 dimensions of N to roundoff but all principal angles are still 0.
How to verify
run numerics/optimality_all/e2_subspaces.py; part (A) prints 'cos(angle) min=1.0000000000' and ranks equal to the predicted m^2-2z^2 and (L-m)^2
Evidence
- numerics/optimality_all/e2_subspaces.py — numerics
- numerics/optimality_all/e2_common.py — numerics
- rigor/exterior_extension_numerics.tex — document, at
sec:dims
Related claims
- Finite chains: exterior directions span exactly the invisible space (Ext = N^perp) —
numerical 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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