TransScale Boundary State Field Calculus: Typed Observation, Admissible Transfer, and Falsifiable Persistence in Finite Open Systems

Core Thesis The framework predicts that persistent physical systems should show bounded recursive structure: local kernels produce boundary traces; boundary traces constrain future admissible states; and cross-scale transfer only works where the projection-scale torsion stays small. When torsion grows, recursion should fail to preserve identity, repair, or predictive structure. A persistent finite system is a boundary-mediated process whose observable identity is maintained by local dynamics, lawful exchange, admissible projection, and repair inside declared viability constraints. Cross-scale recurrence is an operator-transfer claim through declared maps, not visual resemblance. Phase constants, astronomical angles, boundary encodings, and optional operators acquire authority only through the observation structure that makes them measurable. This thesis defines a boundary-state field calculus for finite open systems. The primitive object is not an isolated particle, organism, survey, sensor, or pattern. It is a typed system tuple observed through a declared boundary layer, instrument channel, quotient relation, admissible set, and legal filtration. The central thesis is that persistent systems are recursive folds of local dynamics into admissible observed states. A boundary, astronomical, biological, quantum, holographic, or scale-transfer claim is lawful only when its local kernel, boundary trace or measured layer, observation kernel, metric, admissible set, quotient convention, scale map, calibration procedure, baseline, and demotion rule are explicit. The calculus is built to preserve established science rather than override it. For known non-gravitational microscopic matter the Standard Model is treated as the promoted local kernel; speculative terms are zero by default. Holography is restricted to entropy, encoder, decoder, and reconstruction-deficiency statements in gravitational or information-capacity contexts. Finite-scale recurrence is treated as a commuting-diagram test between boundary encoders and scale maps, never as visual resemblance. Astronomical variables enter only as rigorously defined ephemeris phase coordinates or geophysical forcing variables; without carrier, convention registry, leakage-safe predictive gain, and null controls they have no mechanistic status. Biological modules begin from transport, reaction, redox, membrane, immune, and viability kernels. Quantum modules begin from local Hamiltonians and measurable response functions. Cosmological modules begin from general-relativistic background dynamics and survey observation kernels. The thesis adds five concrete research modules. First, a formal category-like calculus for boundary-state systems with projection stability, trace deficiency, boundary shield, operator rent, fatal ideals, scale torsion, and quotient-phase diagnostics. Second, a corrected one-dimensional lanthanide t-J quantum test predicting boundary-sector reversal of signed string order and phase-referenced triplet susceptibility in the topological triplet superconducting regime, while the ordinary pair-pair correlator remains approximately even. Third, an axion dark-energy terminal-cosmology module translating the Luu-Qiu-Tye finite-lifetime benchmark into posterior functionals over turnaround and crunch times. Fourth, an electroactive-biofilm pilot where redox, pH, impedance, extracellular electron transfer, morphology, and viability are used to test boundary sufficiency under perturbation. Fifth, an astronomical phase registry for circular time-dependent covariates, including von Mises kernels, harmonic embeddings, uncertainty propagation, fake-ephemeris controls, and a default zero-capacity null. The central standard is severe: definitions grant notation, theorem schemas grant conditional mathematics, diagnostics grant ways to lose, simulations grant internal consistency, and empirical promotion is local to a declared tuple. A positive result that fails leakage, calibration, ablation, perturbation, or random-operator rent remains descriptive. A negative result remains useful if it sharpens a falsifier or removes a spurious operator. The objective is not universal rhetoric. The objective is a reproducible calculus for determining what can be measured, what can be inferred, what transfers across scale, and what must be rejected. The document is deliberately ambitious in scope and narrow in promotion. It places several domains inside one grammar, but it does not assert that those domains are the same mechanism. The unification is methodological: the same legality conditions govern whether a local kernel, observation channel, boundary representation, scale map, or candidate operator can be promoted.