Second-order network law: Phi_tot = sum zeta_j zeta_k G(y_j - y_k), G = c Ghat universal

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

Area: Lattice, thermal, massive and gapped geometries (O7, O9, O10) · Card type: conjecture · Knowledge-base id: network-second-order-law

Statement

For corners at modular positions y_k with strengths zeta_k (frame of the whole chain), -log F(omega, omega o beta_Phi) = sum_{j,k} zeta_j zeta_k G(y_j - y_k) + o(|zeta|^2) with G(Delta) = c Ghat(Delta), Ghat(0) = 1/(12 pi^2) (Theorem A), Ghat even and positive-definite as the Fourier transform of the positive spectral density of the first-order corner defect in the modular frame (positive-definiteness does NOT imply Ghat >= 0 pointwise; the sign for Delta > 0 is open, card corner-correlation-kernel). The remainder must be shown UNIFORM in the corner separations for the applications in which zeta -> 0 and Delta -> inf together (vwz-protocol-ordering). Lower bound: data processing + Legendre form, linear in the defect. Upper bound: Uhlmann + C^{1,1} flow extension with finitely many second-derivative jumps; expected global bound Phi_tot <= -(1/2) log(1 - 2 sum zeta zeta G). Separation constraint: admissible protocols ALL of whose corners are weak (zeta_j <= eps for every j) cannot have two of them close at distinct points -- for the L->R Petz chain (e^{Delta_k} - 1) zeta_k^(I) = 2 r_k (1+v_k)/(1+r_k) >= 2 r_k/(1+r_k) exactly, and zeta_j <= eps for all j forces e^{Delta_k} >= 2/eps, PROVED in rigor/network_corner_calculus.tex Thm 11.1 (track G1a; REF-P6-3: stands; sharp as eps -> 0, exact minima 20.0499, 66.6817, 200.0050, 666.6682 at eps = 0.1, 0.03, 0.01, 0.003, rigor/ref_p6_3_separation.out; 1/eps for general admissible s_k >= lambda_k; card corner-separation-constraint); with only the pair weak the separation can stay O(1). Hence, for all-weak L->R chains and given the uniform remainder, at strict second order Phi_tot = f2 sum_points (sum_steps zeta)^2 + O(eps^2 sup_{Delta >= log(2/eps)} |Ghat(Delta)|): coherent addition at a common junction, additivity in zeta^2 across distinct junctions. Universality of Ghat by the H_T argument of Theorem B.

How to verify

Track G1b (rigor/network_second_order.tex); G3 measures Ghat(Delta) in two frames; G2 tests the sum rule on the lattice.

Evidence

• rigor/phase6_brief.md — document, at W2
• rigor/network_corner_calculus.tex — document, at label{thm:sep}
• rigor/ref_p6_3_separation.py — numerics

Related claims

• phase6-plan — active, no page of its own
• Corner calculus: sequential one-sided recovery = Schwarzian point masses at the junctions — proved
• corner-correlation-kernel — open, no page of its own
• Theorem A: lim Phi/zeta^2 = c/(12 pi^2) — refereed
• Theorem B: f2 = c/(12 pi^2) for every diffeomorphism-covariant net — refereed

Review

Review record: passed 2026-09-16 by REF-P6-3b.

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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