Separated intervals: CMI with a gap and linearity off criticality

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

Area: Lattice, thermal, massive and gapped geometries (O7, O9, O10) · Card type: result · Knowledge-base id: gap-law

Statement

For the massless Dirac field (lattice Dirac chain), with a gap G between A and B, I(A:C|B) = (1/3) log[(a+g+b)(g+b+c)/((g+b)(a+g+b+c))] exactly (adjacent formula with L_B -> L_B + L_G), so I ~ L_G^-2; -log F decays with the same power and -log F / I -> 0.09. Away from criticality (massive Dirac chain) -log F is LINEAR in the CMI (ratio 0.10-0.17): the quadratic law is a fixed-point-plus-adjacency statement.

How to verify

numerics/lattice outputs; rigor/thermal_rg_gap_summary.md.

Evidence

• numerics/lattice/petz_thermal_rg.py — numerics
• rigor/thermal_rg_gap_summary.md — document

Related claims

None recorded.

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B3.

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.

Documentation map · All claims by area · What is open · Symbols · Definitions