The Dual-Axiom Fixed Point: Deduction of the Minimal Necessary Architectural Class for Physical Execution

1. The Methodological Foundation (Chapter 1): Axiomatizing the absolute boundary of empirical science. By enforcing the Principle of Empirical Primacy and the Axiom of the Embedded Observer, we establish that scientific prediction evaluates strictly as finite, localized physical computation within the Computable Domain (TTG). <br> <br> 2. The Objective Function (Chapter 2): Deriving the Universal Cost Ledger. We formalize the thermodynamic penalty for continuous mathematical bloat. Foundational scientific progress evaluates not as syntactic elegance, but strictly as the algorithmic minimization of static memory allocation and dynamic execution trace. <br> <br> 3. The Architectural Class (Chapter 3): The execution of the deductive trap. We run continuous assumptions through the universal cost compiler, proving they trigger unresolvable hardware halts and phase-space divergence. By algorithmically excluding the uncomputable, we deduce the exact minimal necessary architectural class necessitated to compile the macroscopic Evidence Vector: a local, discrete, Base-72 symplectic state-machine.&nbsp; <br> <br> Note: These lemmas constrain the generative class to a narrow finite-discrete family with zero continuous degrees of freedom. The concrete forcing term f within this class remains to be calibrated against raw observational data D, per Axiom I. <br> <br> 4. The Checksum Protocol (Chapter 4): Dynamically unrolling the architectural class. We verify that the macroscopic phenomenological invariants of the universe—emergent Lorentz invariance, quantum measurement bounds, fractal self-similarity, objective causality, and thermodynamic irreversibility—are structural invariants guaranteed by the architectural class itself. We conclude by isolating the exact, falsifiable dispersion limit where continuous spacetime mathematically fails. <br> <br> Structural validation proceeds sequentially. The reader must verify the inescapable thermodynamic bounds before the collapse of continuous modeling. Execution trace commences in Chapter 1. <br> <br>