VWZ four-interval protocols: Phi(1):Phi(2):Phi(3) -> 1:2:4; middle-out is the worst order

Status: conjectural — believed, with partial evidence; not proved.

Area: Lattice, thermal, massive and gapped geometries (O7, O9, O10) · Card type: conjecture · Knowledge-base id: vwz-protocol-ordering

Statement

In Vardhan-Wei-Zou's symmetric setup (L_A = L_D = a, L_B = L_C = b, their eta (5.9) = a/(a+2b)), the corner calculus (card corner-calculus, open) gives exactly: single step (Protocol 1(A) = 1(B)) one corner of strength zeta_1 = a/b; L->R Protocol 2 corners zeta_2 = a(a+2b)/(2b(a+b)) at p_2 and zeta_3 = a/b at p_3 with e^{Delta_23} = 1/eta; Protocol 3 one double corner of strength 2a/b at p_3 (all checked by REF-P6-0). Hence, given the corner calculus, -log F^(1) : -log F^(3) = Phi_A(a/b) : Phi_A(2a/b) exactly for all eta (-> 1 : 4 as eta -> 0, Theorem A twice, no hypothesis on Ghat), and, CONDITIONAL on the second-order network law with a remainder uniform in the corner separation and on Ghat(Delta) -> 0 as Delta -> inf (both open, corner-correlation-kernel), -log F^(2) -> 2 x -log F^(1); exactly at second order Phi^(2)/Phi^(1) = (zeta_2^2 + zeta_3^2)/zeta_1^2 + (2 zeta_2 zeta_3/zeta_1^2) Ghat(log 1/eta)/Ghat(0). Middle-out is the worst of the three orders at second order with no sign hypothesis: by Cauchy-Schwarz Phi^(2) <= f2 (zeta_2 + zeta_3)^2 < f2 (2 zeta_1)^2 = Phi^(3) because zeta_2 < zeta_1. VWZ (5.10)-(5.11), Fig. 25 state: "-log F^(2) ~ -log F^(3) ∝ f' c eta^2 with f' < f, confirming that the sequential recovery is less effective than the single-step case" -- their inequality and their prose point opposite ways; the calculus contradicts the inequality (f'_3 = 4f exactly) and, at small eta, the approximate equality of Protocols 2 and 3. Their single-step coefficient (2.13), f2 ~ 0.070 c, matches our ordinary-Petz prediction 8 f2 = 0.0675 c (zeta = 2z). Fig. 25 CANNOT be read as a measurement of Ghat (REF-P6-1, rigor/ref_p6_1_fig25.py): Delta_23 = log(1/eta) is bounded below by their axis (eta^2 <= 0.4 gives Delta_23 >= 0.46), the corner strengths there are O(1) (zeta_1 = 2 eta/(1-eta) >= 0.22 for eta >= 0.1) so the second-order law does not apply, and Phi^(2) = Phi^(3) at second order would need Ghat(Delta_23)/Ghat(0) = 1.1 ... 2.1 > 1, which positive-definiteness forbids; no value or sign of Ghat can be extracted from their figure -- G3 measures Ghat directly. Corner measures occurring for four intervals: {p_2} (1(A) = 1(B)), {p_2, p_3} (L->R), {p_3, p_4} (R->L), {2 x p_3} (middle-out); union conditioning regions can move corners off the junctions (corner-calculus).

How to verify

G2: lattice_chain.py at 4 x 18 sites, exact 2^n validation at n <= 10; Protocol 3 against paper 1's Phi_A table; G1a for the exact statements.

Evidence

• rigor/phase6_brief.md — document, at W4
• refs/VWZ_2307.14434.pdf — document, at Sec5; held privately, not redistributed

Related claims

• Corner calculus: sequential one-sided recovery = Schwarzian point masses at the junctions — proved
• Second-order network law: Phi_tot = sum zeta_j zeta_k G(y_j - y_k), G = c Ghat universal — conjectural
• f2 = c/(12 pi^2) — proved

Review

Review record: passed 2026-09-16 by REF-P6-2.

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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