The Unified Transport Operator: Gravity, Optics, and Thermodynamics as a Single Complex-Time Propagator on the Prime Lattice, and the Holographic Foliation Theorem

This preprint shows that gravity, optics, and thermodynamics are three faces of a single object: the complex-time transport propagator of one self-adjoint operator on the prime coprime-residue lattice. A self-adjoint operator with a discrete spectrum admits exactly three canonical functions — its resolvent, its unitary exponential, and its heat kernel — and on this lattice those are precisely gravity, optics, and thermodynamics. Optics is the imaginary-time axis of the propagator; thermodynamics is the real (heat-kernel) axis; and gravity is the integral of thermodynamics over all imaginary time. The three are not analogous — they are restrictions of one analytic object to different directions of complex time. The work establishes five integer-exact results, including a Complex-Time Unification theorem; a Spectral-Zeta Gravity identity, in which the total scalar gravity of the boundary equals the spectral zeta value at s = 1 and evaluates to an exact rational; and the Holographic Foliation Theorem, now proved with its eigenspaces identified (the "shell-conductor law"): the additive-to-multiplicative Gauss-sum dictionary foliates the holographic bulk into integer geometric shells indexed by the divisor lattice, and each shell is spanned, once each, by the primitive Dirichlet characters of a fixed conductor. The dictionary is faithful exactly on squarefree (primorial) moduli. Every quantity snaps to an integer or an exact rational; these are integer-exact substrate results, not statistical claims. A reduction-to-practice benchmark set demonstrates the operator as a feedback controller and a single-eigenbasis multi-physics engine, with measured speed-ups against field-standard tools. The operator's information faces are then measured exactly: an eternal-black-hole interior on the imaginary-time midline; a Page curve that rises, turns over, and returns to zero against Page's exact rational values (unitarity, measured); and the finding that chaos on this lattice is not a phase but a thin interface between the additive and multiplicative structures of number. A registered falsifier of the random-matrix reading is executed and recorded — including its failure, kept in full — and the refusal's positive face is converted into demonstrated instruments: exact retro-diffusion, which inverts the backward heat equation (Hadamard's canonical ill-posed problem) bit-perfectly on exact-data states; a gauge-blind integrity auditor whose immunity to re-encoding is a theorem rather than a tolerance; and topological RAM, information stored in cohomology classes that is immune to an enormous group of legitimate re-encodings yet betrayed by a single-edge tamper. All theorems, scalar identities, and proofs are placed in the public domain under the MIT license; engineering apparatus practicing the disclosed substrate primitives is reserved under U.S. Provisional Patent Application No. 64/089,724 (USPTO, June 12, 2026). Companion preprints and the integer-exact reference scripts that reproduce every numerical claim are cited within.