The Horizon as Bisimulation Boundary: Surface Gravity as Relational Divergence Rate¶

Jean-Paul Niko · RTSG Working Paper · March 2026

Status: Formalized

Full LaTeX paper compiled (11 pages, 0 errors). Contains definitions, propositions, theorems with proofs, and clearly separated conjectures. Available as PDF.

Summary of Results¶

Result	Status	Section
λ_bis = κ (Schwarzschild)	Theorem 3.3 — Proven	§3
λ_bis = κ^Kerr	Proposition 5.1 — Proven	§5
Horizon = exact bisimulation boundary	Corollary 4.3 — Proven	§4
Extremal: λ_bis = 0	Proposition 5.1 — Proven	§5
t_kin = S/λ_bis interpretation	Proposition 7.1 — Heuristic	§7
Information preservation via bisimulation	Conjecture 6.1 — Open	§6

Key Definitions¶

Accessible Pointed Graph (APG): Triple (N, E, p) — nodes, directed edges (membership), root. Under AFA, every APG has a unique decoration into the set-theoretic universe.

Bisimulation: Relation R between two APGs where each transition in one graph can be matched by a transition in the other, and vice versa. Under AFA, two sets are equal iff bisimilar (Aczel's Solution Lemma).

ε-Bisimulation: Metric refinement — bisimulation that tolerates distance ε between matched nodes. The bisimulation distance d_bis is the infimum of ε over all ε-bisimulations.

Bisimulation Divergence Rate:

\[\lambda_{\rm bis} := \lim_{T\to\infty} \frac{1}{T} \ln \frac{d_{\rm bis}(T)}{d_{\rm bis}(0)}\]

Main Theorem¶

Theorem 3.3 (Surface Gravity as Bisimulation Divergence Rate): For the Schwarzschild horizon with standard asymptotic Killing normalization:

\[\lambda_{\rm bis} = \kappa = \frac{1}{4M}\]

Proof sketch: Horizon-straddling null geodesics at displacement δ generate sub-APGs whose bisimulation distance grows as d_bis(T) = δ e^{κT}. This follows from the Rindler approximation: both sides of the horizon share the same local causal structure, differing only by the sign of the proper distance ρ. The Rindler reflection ρ → −ρ provides the initial exact bisimulation; the exponential peeling U ∝ −e^{−κu} provides the divergence rate.

Corollary: Horizon Uniqueness¶

The event horizon is the unique codimension-1 surface where interior and exterior sub-APGs are exactly bisimilar at T = 0 and diverge at rate κ for T > 0. No other surface has the Rindler reflection symmetry that provides the initial exact bisimulation.

Kerr Extension¶

\[\lambda_{\rm bis}^{\rm Kerr} = \kappa^{\rm Kerr} = \frac{\sqrt{1-a_*^2}}{2M(1+\sqrt{1-a_*^2})}\]

At extremality (a* → 1): λ_bis → 0. The interior and exterior remain perpetually bisimilar. This is consistent with the third law (T_H = 0, no Hawking radiation) — the bisimulation never diverges, so there is no thermal particle creation.

Information Conjecture¶

Conjecture 6.1: For any finite-dimensional observable O computable from the exterior APG, there exists a bisimilar observable Õ computable from the interior APG. The Page curve is computable entirely from the boundary APG via the bisimulation relation.

This is a precise mathematical formulation of holographic complementarity. NOT proven — stated as a target.

Connection to the GRF Essay¶

The kinematic clock t_kin = S/κ = S/λ_bis is the timescale at which the bisimulation divergence has "explored" all e^S distinguishable relational states on the horizon. The dressing factor C_Page ≈ 95.2 measures how much longer global information dynamics take compared to local bisimulation resolution.

Full LaTeX source available. Sole author: Jean-Paul Niko.

Connection to Will Field GL Action (2026-03-07)¶

The horizon bisimulation boundary is a special case of the general bisimulation quotient \(PS = QS/\!\sim_{\text{bisim}}\). The surface gravity \(\kappa\) at the horizon corresponds to the GL correlation decay rate. The Will Field energy density \(\rho_W\) evaluated at the horizon gives the Bekenstein-Hawking entropy via the holographic Drive D mechanism. See Will Field Universality.