Compression Synthesis

Compression Synthesis states a single criterion for when a compressed description of a physical system counts as a valid effective theory of an underlying substrate, and uses the criterion to recover classical mechanics, statistical mechanics, quantum mechanics, and equilibrium thermodynamics as different regimes of one structure.

The criterion is closure with substrate-faithfulness: propagating the tracked variables through the substrate dynamics and reading them through the compressed probes must reproduce the propagated variables, within a stated tolerance, with the maximum-entropy reconstruction close to the substrate's reduced state on the cut. Cut-invariance — two such descriptions of one substrate state agreeing on shared events within twice the prediction tolerance — is a derivable corollary, not an additional postulate.

Each compressed description carries four primitive objects: a probe algebra 𝒜, a state m, an entropy functional S, and a propagation rule G. Three pieces locate the contribution. The probe algebra is dynamical: an algebra-evolution map Φ updates the algebra under closure pressure, formalising events such as the water-to-ice and Newton-to-relativity redesigns of physical theory. The cut between system and environment is derived from a cost functional rather than imposed externally. Self-consistency replaces correspondence as the criterion of theory-correctness, with the empirical question (which closed theory matches the realised substrate) separated from the structural question (is the candidate theory closed at all).

The C*-algebra structure of the probe algebra is forced by the structural requirements on probability; classical and quantum probability arise as the commutative and non-commutative regimes of one mathematical structure. The four laws of thermodynamics fall out as derivations from joint maximum entropy at the cut, conservation of ⟨H⟩, and CPTP monotonicity (Lindblad-Uhlmann) of relative entropy under partial trace. Five classes of claim are ruled out by the criterion: theories that postulate the cut from outside, derivations of thermodynamics from microscopic dynamics alone, dynamical-collapse measurement theories, "merely-coarse-graining" emergence claims, and unique-correct-cut universe claims.

Written for readers who have seen quantum mechanics at a standard upper-undergraduate or first-year-graduate level, basic measure-theoretic probability, and elementary operator theory, and who want the structural criterion for theory-correctness made precise. Familiarity with Mori-Zwanzig non-equilibrium statistical mechanics, CPTP channels, and Lindblad-Uhlmann monotonicity accelerates several chapters but is not assumed. The book runs in finite dimensions throughout; infinite-dimensional and field-theoretic extensions are deferred.