A First-Principles Axiomatization of Physics—The Cosmic Continuum Component Model with Its Scale Topos and Braided Tensor Category Realization

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 (15 axioms A0–A14). 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 (including the connection axiom A4b which provides a unified description of the four fundamental interactions), dynamical axioms (A8–A12) as the realization (including the field recursion axiom A9b which unifies multiple fields as different levels of a recursive structure), 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. The core breakthrough of this paper is to elevate ``interaction'' from an externally added Yang-Mills term to an intrinsic logic of component fusion. Gauge bosons emerge from the fusion decomposition $P \otimes P^*$ of the Planck particle $P$ ($\dim_q(P)=2$), with coupling strengths uniquely determined by fusion rules and braiding, not free parameters. Simultaneously, from A9 (fibred category) and A10 (2-category lifting), the quantum Boltzmann equation is rigorously derived, whose collision kernel is uniquely determined by the braiding $R$-matrix moduli and fusion coefficients, with no free parameters, achieving a complete axiomatic derivation from microscopic fusion rules to macroscopic transport phenomena. 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. The framework proposes five testable predictions: (1) dark energy evolution direction $w_a>0$ ($w_a=0.12\pm0.09$); (2) CMB Planck-scale oscillations $\alpha\approx0.032$; (3) shear viscosity-to-entropy density ratio $\eta/s \gtrsim \hbar/(4\pi k_B)$; (4) LISA-band gravitational wave background peak $f_{\text{peak}}\approx0.2\,\text{Hz}$; and (5) black hole shadow quantum correction $\gamma\approx0.3$. This framework is the first to rigorously derive all four laws of thermodynamics within a single axiomatic system; the second law of thermodynamics is reframed as an apparent emergent phenomenon — the underlying dynamics are time-reversible and deterministic, while the apparent irreversibility arises from the observer's limited access to information in dark space. This paper presents a self-consistent, testable conceptual framework of axioms for fundamental physics, achieving a unification of classical physics, general relativity, quantum mechanics, thermodynamics, non-equilibrium statistical physics, and cosmology, thereby offering a fundamental response to the call for the axiomatization of physics raised by Hilbert's Sixth Problem.