A First-Principles Axiomatic System for the Cosmic Continuum—The Component Model Based on Scale Topos and Braided Tensor Categories

Fundamental physics faces three profound difficulties: ontological incompatibility ($G,c,\hbar$ presuppose different ontological categories), the chasm between continuity and discreteness (classical continuum vs. quantum discreteness), and divergent mathematical languages (differential geometry, Hilbert spaces, renormalization groups). This paper proposes a first-principles axiomatic system for the Cosmic Continuum. A0 asserts: All state changes in the universe arise from energy redistribution; there is no state change without energy change. This axiom replaces Newton's "inertia" as the most fundamental physical assumption—inertia is merely the special case $\Delta E = 0$. Built upon this first principle, the axiomatic system is structured through ontological axioms (A1–A3) as the cornerstone, structural axioms (A4–A7) as the skeleton, dynamical axioms (A8–A12) as the realization, and meta-axioms (A13–A14) as verification. The framework clearly distinguishes mass, energy, and dark mass beings and their corresponding space, time, and dark space dimensions, unified via the New Equivalence Principle (A3). The component (A4) serves as the skeleton, unifying particles and gauge fields into an inseparable tensor product. The scale topos and the $U_{q}(\mathfrak{e}_{8})$ modular tensor category provide the mathematical realization, rigorously capturing the relative continuum (A6) and embedding the Standard Model gauge group. From these axioms we derive: the mirror 2-morphism $M$ is equivalent to CPT; the singularity is a phase boundary from ordinary spacetime to dark space; the singularity flux is quantized as $dN/dt = \mathrm{sgn}(-t)/t_{P}$; dark space entropy $S_{\mathrm{dark}} = k_{B}\ln 2\cdot N$ resolves the black hole information paradox; the mirror cyclic universe predicts $w_{a} > 0$ (dark energy weakens over time), consistent with current DESI/Planck data at $1.3\sigma$; wavefunction collapse is interpreted as a natural projection in the fibred category, with coherent information entering dark space. The framework is fundamentally deterministic (causal): the Born rule probabilities arise from limited access to information stored in dark space, not from intrinsic randomness. Testable predictions include Planck-scale CMB oscillations ($\alpha \approx 0.032$), a gravitational wave background peak at $f_{\mathrm{peak}}\approx 0.2\,\mathrm{Hz}$, and quantum corrections to black hole shadows ($\gamma \approx 0.3$). The core dynamical part is falsifiable: exclusion of $w_{a} > 0$ at $>5\sigma$ etc. would falsify the framework.