Effective Geometry as Horizon Boundary Accounting: Finite Distinguishability, Horizon Entropy, and Thermodynamic Closure in Finite Distinction Systems

Abstract

Official website: distinctiontheory.orgPublic portal for the start guide, papers, claim status, failure registry, prior-art boundary, and citation resources. Canonical GitHub repository:https://github.com/yiningwu-research/Distinction-Theory FDS-T2 develops the horizon-boundary thermodynamics paper in the T-series bridge sequence of Finite Distinction Systems (FDS) / Distinction Theory. It interprets effective geometry as horizon boundary accounting: the covariant macroscopic ledger that closes causal access, horizon entropy, stress-energy flux, and finite-boundary maintenance for finite observers. T2 does not derive general relativity from FDS alone, replace Einstein gravity, derive quantum gravity, or derive the numerical coefficient in the Bekenstein-Hawking entropy formula. It uses horizon thermodynamics as a physical bridge. If that bridge fails, the T2 interpretation is demoted while the formal FDS finite-capacity core remains unaffected. The novelty of T2 is not a new derivation of Einstein gravity. It is an observer-relative reinterpretation of horizon thermodynamic variables as finite distinguishability ledgers: horizon area counts accessible boundary distinctions, heat flux updates the ledger, and effective geometry is the covariant compression that preserves causal access and stress-energy accounting. The central bridge is: finite causal access → horizon boundary → area ledger → entropy ledger → flux update → covariant effective geometry. T2 separates two layers. The first is the Jacobson model-class bridge: under area entropy, local Unruh or surface-gravity temperature, Clausius-type horizon closure, and local covariance, Einstein-type geometry arises as an equilibrium equation of state. The second is the FDS boundary-ledger interpretation: if this bridge holds, then the effective metric can be read as a stable macroscopic compression of a finite horizon distinguishability ledger. The paper defines a horizon distinguishability budget CH = SH / (kB ln 2), and, for area-law horizons, CH = AH / (4 ℓP2 ln 2). It also defines a boundary thermodynamic ledger LH = (H, AH, SH, TH, δQH, τ, EH), where H is a causal or horizon boundary, AH is area, SH is entropy, TH is horizon temperature, δQH is assigned heat or energy flux, τ is an operational update window, and EH is an admissible coarse-grained error or non-equilibrium term. An admissible ledger-to-geometry map geffμν = G(LH) must preserve causal ordering, light-cone structure, horizon-area variation, stress-energy flux response, local covariance, closure residuals, and coarse-grained stability to registered tolerance. Thus the map is not an arbitrary relabeling; it is a constrained compression from a horizon boundary ledger to an effective geometric structure. T2 introduces a horizon capacity deficit ΔH(τ) = R(τ)min(ε; ΨH) - CH, where ΨH may include task families for local horizon-area variation, stress-energy flux records, causal-diamond boundary updates, or coarse records of unresolved horizon microstates. When ΔH > 0, the boundary ledger cannot track all task-relevant horizon distinctions at full fidelity over the update window. The missing distinctions may appear as entropy production, memory, stochastic noise, hysteresis, or coarse correction terms. For non-equilibrium accounting, T2 writes a residual slot Gμν + Λgμν = (8πG/c4) Tμν + Rledgerμν. This is not proposed as a new gravitational field equation. It is a bookkeeping location for non-equilibrium horizon-ledger residuals, such as entropy production, memory kernels, unresolved boundary noise, higher-curvature slots, or hysteretic response. Any promoted residual must satisfy the corresponding covariant consistency condition required by the Bianchi identity. The paper interprets effective geometry as a Phase-B boundary variable: a coarse macroscopic structure that remains cheaper to update, slower to forget, and more predictive than inaccessible microscopic horizon degrees of freedom. Geometry survives overflow because it is a minimal sufficient covariant boundary variable for causal access and stress-energy accounting. T2 also identifies an upstream bridge to the horizon-maintenance density scale developed separately in FDS-X1. It does not derive dark energy, but notes that once horizon entropy and temperature are treated as a boundary ledger, a natural horizon-scale energy estimate EH ∼ THSH distributed over a horizon volume gives the dimensional density scale c4/(G RH2), up to convention-dependent numerical factors. The release includes deterministic normal-form demonstrations. They illustrate the horizon boundary-ledger bridge, area-law distinguishability scaling, causal-diamond coarse accounting, horizon capacity deficit, non-equilibrium ledger residuals, Phase-B effective geometry, residual taxonomy, and the relation map linking FDS Core, T1, T2, T3/P-series, X3, and X1. These figures are conceptual demonstrations, not empirical fits and not simulations of full general relativity. This release includes the paper PDF, LaTeX source, reproducibility code, generated figures, and CSV / JSON outputs.