Horizons, Capacity Bounds, and Entropy in Modal Triplet Theory: A Conditional Interface with Black-Hole Thermodynamics

Abstract

This paper determines exactly what coherence-capacity data can and cannot establish about horizons and entropy in Modal Triplet Theory (MTT). A zero of a declared capacity certificate is a failure boundary for that certificate. It is not, by itself, an event horizon, a trapped surface, an entropy, or an arrow of time. Those notions require additional Lorentzian, state, algebraic, and dynamical data. The positive result is a typed conditional theorem. Let Sigma be a cross-section of a separately specified causal horizon. Suppose a selected model supplies an entropy-transfer density s_Sigma, an integrated capacity-transfer density q_Sigma, and nonnegative constants kappa and sigma_max such that s_Sigma is at most kappa times q_Sigma and q_Sigma is at most sigma_max almost everywhere. Then Delta S(Sigma) is at most kappa sigma_max Area(Sigma). Equality holds precisely when both inequalities saturate almost everywhere, up to null sets. Thus bounded capacity transport can support an area upper bound, but an area equality requires a saturation theorem. Reproducing black-hole entropy further requires an independent source for kappa sigma_max = 1/(4 G hbar) in units c = k_B = 1. Capacity terminology alone does not derive that coefficient. Recovery is also formulated correctly. For an exterior channel E_ext, recovery means that a channel R reconstructs a declared code or observable algebra, with R after E_ext equal to the identity on that domain. It is not a partial right inverse of a noninjective projection, and noninjectivity alone neither proves nor forbids code-relative recovery. Hawking radiation, generalized entropy, island formulas, and Page curves remain imported results of their established regimes. MTT contributes a common typed language and a finite source contract for connecting those results to an upper admissibility model.