Entanglement Equilibrium on a Solvable Gaussian Network: The Contact Term Derived, Geometry Reconstructed, Causal Cones Emergent, and Equilibrium Explained
Year: 2026
Type: preprint
Abstract: Jacobson's entanglement-equilibrium argument derives the Einstein equation from (i) an area law with finite Newton constant, (ii) a first law for the matter modular Hamiltonian and (iii) stationarity of the vacuum entropy of small geodesic balls at fixed volume. I test these ingredients on a solvable family of network states: a pair sector whose entanglement encodes a Riemannian metric, combined with the Gaussian ground state of a scalar field on the same weighted graph, for which entropies and modular Hamiltonians of arbitrary regions are computed exactly. Every assumption is isolated as a hypothesis and classified as proved, derived, computed or imposed by construction, and five structural questions (derivation or fit, recovered or imposed geometry, causal structure, equilibrium mechanism, convergence off the regular grids) are answered with the proof status of every step made explicit. (1) Derivation (conditional). The wave-packet quadratic form of the exact massless modular Hamiltonian of a ball, computed by Longo and Morsella, is identical to the bulk form K_bulk = 2π ∫ κ T_min + (2π ξ_c n/ℓ) ∫ φ² exhibited by the lattice data; the contact-term coefficient χ = 2π ξ_c and the response R = 1 + ξ_c n(n+2)/(ωℓ/c)² are therefore identified, not fitted. The step from the continuum identity to the lattice is a modular-convergence statement (Conjecture C1) which I do not prove: a proved proposition on the non-analyticity of the entropy function at the band edge, together with a heuristic accumulation argument, shows why the spectral convergence of the matter sector (H2) does not imply it. The lattice data support C1: q_∞ = 1.02 ± 0.02 in n = 2 (up to 1.09 in n = 3 at the accessible radii ℓ ≤ 8). (2) Geometry (recovered, not emergent). The metric is imposed on the pair sector by construction; what is tested is the inverse map. An identifiability lemma shows the entropic distance matrix determines the manifold up to isometry; on S² the blind pipeline recovers the dimension (c₂ = 3.13 ≈ π), detects curvature through the negative tail of the MDS spectrum, and measures the curvature radius to 4%. H1 becomes a tested statement, not an input. No bound on matter contamination of the reconstruction is proved. (3) Causal structure (bounded, then computed). The classical Lieb–Robinson bound is proved only for bounded interaction terms (the pair sector); it does not apply to the unbounded oscillator Hamiltonian. For the oscillator sector I prove a field-commutator cone bound with explicit constants: the commutator kernel is a c-number, bounded by a path-counting sinh series with cone velocity v_φ = min_μ √(M_μ)/μ = 3.05 = 3.1 c_lat, valid on arbitrary point sets. The exact field commutator shows that the cone the physics uses is tighter: the front propagates at c_lat, with exponential tails outside of e-folding 1.25 ξ, and Lorentz violation controlled by β = Ren(6,m)/16 Ren(4,m) = 0.384 as (kξ)². (4) Equilibrium. H5 is proved equivalent to the reconstructed metric having constant scalar curvature; the further identification with the Einstein condition holds in n = 2. Passivity supplies an exact constrained second-variation identity: at fixed geometry and fixed modular energy the vacuum is a local maximum of the generalised entropy along every constrained matter direction; without the modular-energy constraint no extremum statement holds. The entropic flow whose fixed points are the equilibrium geometries is defined, not derived; in n = 2 it coincides with the normalised Ricci flow, whose convergence is Hamilton's theorem. (5) Spectral convergence (leading order, black box). For deterministic quasi-uniform sequences on curved manifolds the form-consistency step is proved explicitly, with the discrete Hilbert space made precise, and eigenvalue convergence is obtained only conditionally, by citing the graph discretisation of the Laplace–Beltrami operator of Burago–Ivanov–Kurylev as a black box. The curvature renormalisation of the speed of sound, c_eff² = c²(1 − R_s Ren(n+4,m) ξ² / 6n Ren(n+2,m)), is the leading term (verified on S² to 1%). For non-uniform random point sets the raw operator converges to a drift operator (σ/σ̄)(Δ + 2 ∇ln σ · ∇) (multi-seed mean eigenvalue deficits growing from ≈1% to ≈20% with wavenumber), and a blind density correction computed from entropic ball counts repairs it, recovering the manifold spectrum to within ≈0.2–2.3% plus a counting bias of order 1/N̄_rb on quasi-uniform sets. All numbers are generated by the programs distributed with this work.
Author: Weinmann, T.
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225423041