To Lay a Stone with Six Birds: Finite-State Semantics for Packaging, Directionality, and Coarse-Graining

Work on multi-scale organization and audit-based directionality lacks a minimal-substrate theorem: key primitives are often realized only by smuggling in external schedules, ad hoc audits, or declared quotients, making results hard to verify and compare across substrates. We introduce a no-smuggling minimal-substrate semantics for SPT, operationalize it via matched-control audit tests on finite stochastic machines, and demonstrate separations between protocol artifacts, intrinsic drive, and route-dependent packaging defects. Operationally, we define a canonical machine class ( mathcal S_{min} ) of autonomous finite stochastic dynamics with phase-in-state, deterministic lenses, induced packaging endomaps ( E_{tau,f} ), and intrinsic audits (path-reversal KL asymmetry ( Sigma_T ) and ACC affinities), and we implement exact computations together with a reproducible witness suite and a Lean-checked idempotent/closure backbone for packaging equivalence and saturation (de Moura and Ullrich 2021; The mathlib Community 2020). Under calibrated controls, stroboscopic protocols can exhibit ( Sigma_T>0 ) while their autonomous lifts yield ( Sigma_Tapprox 0 ) (protocol trap), whereas turning on internal phase drive produces a clean null-vs-driven separation with ( Sigma_T(Z)>0 ) and a detectable projected macro arrow; moreover deterministic coarse-graining never increases ( Sigma_T ) and can erase it entirely, and packaging along refinement routes exhibits a prototype-driven holonomy defect that is large at moderate timescales while a uniform baseline is near zero and satisfies a gain-plus-mismatch bound. These results supply a machine semantics and separation witnesses for a minimal substrate theorem, reducing false positives and turning SPT from a language into a compilation target up to packaging equivalence. We are explicit that we do not establish a universal physical substrate or continuum limit: our conclusions are semantic and rest on exact finite-state audits and controlled constructions rather than empirical natural-system data.