Vacuum rigidity: the pulled-back vacuum is the vacuum iff the map is Moebius; no protocol recovers exactly

Status: proved — a complete structured proof.

Area: Lattice, thermal, massive and gapped geometries (O7, O9, O10) · Card type: theorem · Knowledge-base id: vacuum-rigidity-exact-recovery

Statement

(a) ELEMENTARY RIGIDITY (Lem 7.1): if Phi: J -> R is C^1 with Phi' > 0 and sqrt(Phi'(x)Phi'(y))/(Phi(x)-Phi(y)) = 1/(x-y) for all x != y in J, then Phi is the restriction of a global Moebius map. Proof: the relation with y = y_1 fixed is the separable ODE g' = g^2/(c (x-y_1)^2) for g = Phi - Phi(y_1), c = Phi'(y_1), whose exact integral (1/g)' = (1/(c(x-y_1)))' gives Phi = Phi(y_1) + c(x-y_1)/(1 + A c (x-y_1)) on each side of y_1; varying the base point and using that two Moebius maps agreeing on an open interval are equal shows the two sides agree. (b) STATE RIGIDITY (Thm 7.3): for the chiral free fermion with vacuum kernel Q(x,y) = (1/2 pi i)(x-y-i0)^{-1}, the state omega o beta_{V_Phi} on F(I) equals the vacuum iff Phi is Moebius iff S(Phi) = 0; the 'only if' follows by testing the two-point function with disjointly supported functions (which avoids the diagonal) and applying (a), the 'if' from [V_g, P] = 0 for g in Moeb (the Moebius group preserves the Hardy space). (c) NO EXACT RECOVERY (Thm 7.3(2,3)): for EVERY non-empty one-sided protocol whose last step keeps a non-empty part, S(Phi) is a non-zero negative measure -- all masses in the general rule -2 sum_m sigma_m G_m'(x_m) delta_{x_m} have the same sign, so they cannot cancel, and the last step's mass is never displaced or swallowed (G_N = id). Hence omega o beta_P != omega and Phi_fid = -log F > 0, using that F = 1 forces equality of the states (Lem 7.2, proved from Alberti's inequality P_tr <= phi(a) psi(a^{-1}) with a = 1 + tb). This needs neither hypothesis (H) nor the second-order law. Degenerate exception: a step conditioning on the ENTIRE block has its corner at an endpoint of I, where beta_I = 0 and zeta = 0; such a step is a pure Moebius compression (Theorem A at a = 0) and is not a recovery in the Petz sense.

How to verify

rigor/g1a_calculus.py section 3 (the swallowed corner and Phi = Moebius o k_2 to 1e-13); the negative-sign argument is algebraic.

Evidence

• rigor/network_corner_calculus.tex — document, at lem:2pt
• rigor/network_corner_calculus.tex — document, at thm:rigidity
• rigor/g1a_calculus.py — numerics
• rigor/ref_p6_3_calculus.py — numerics

Related claims

• Corner calculus: sequential one-sided recovery = Schwarzian point masses at the junctions — proved
• Theorem A: lim Phi/zeta^2 = c/(12 pi^2) — refereed
• Second-order network law: Phi_tot = sum zeta_j zeta_k G(y_j - y_k), G = c Ghat universal — conjectural

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B2.

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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