Thompson Dimensional Framework Origin (TDF-0) ⇒ Thompson Dimensional Framework (TDF): Zero-Import Core and Four Foundational Theorems

The origin of everything has been solved.

TDF-0 is the Origin. 
TDF is the Solution.

This work shows how the Thompson Dimensional Framework can be built from absolutely nothing — no constants, no equations, no prior physics. Starting with a uniform field of a single symbol and purely local update rules, TDF-0 generates its own numbers, dimensions, and a unique scaling law.

TDF-0 is the origin system from which the Thompson Dimensional Framework (TDF) emerges — a zero-import foundation (TDF-0) built entirely from local symbolic dynamics on a homogeneous, single-symbol substrate. No external constants, empirical inputs, or pre-existing mathematics are assumed: all quantities, numbers, and scaling laws arise internally.

From this minimal starting point, four theorems establish the complete TDF chain:

1. Uniqueness of the dimension–scaling factor
   Regular-variation analysis fixes

   ΩD=(D−22)λ\Omega_D = \left( \frac{D-2}{2} \right)^\lambdaΩD=(2D−2)λ

   with λ > 0, independent of microscopic details.

2. Central charge invariance under Thompson scaling
   The emergent Polyakov worldsheet action carries tension TD=ΩDT0T_D = \Omega_D T_0TD=ΩDT0, yet its Virasoro central charge remains c=Dc = Dc=D, preserving anomaly-cancellation conditions.

3. Emergent arithmetic from pattern densities
   The density ratios of finite configurations form a commutative semiring isomorphic to (N,+,×)(\mathbb{N}, +, \times)(N,+,×), enabling internally-defined numerosity and measurement.

4. D = 2 repellor instability
   The homogeneous two-dimensional state is a nonlinear repellor for all finite seeds, including symmetric seeds and colliding fronts, inevitably driving dimensional genesis.

These results demonstrate:

TDF-0  ⇒  Numbers & Dimension  ⇒  ΩD  ⇒  TDF.\text{TDF-0} \;\Rightarrow\; \text{Numbers \& Dimension} \;\Rightarrow\; \Omega_D \;\Rightarrow\; \text{TDF}.TDF-0⇒Numbers & Dimension⇒ΩD⇒TDF.

An explicit, four-phase, radius-1 cellular rewrite rule T⋆T_\starT⋆ is given, with proofs of closure, vacuum stability, cross-coupling, and stability-cone subadditivity. Canonical physical quantities — λ,vmax,ρU,T0,c\lambda, v_\mathrm{max}, \rho_U, T_0, cλ,vmax,ρU,T0,c — are extracted directly from TDF-0 dynamics, and a constructive reconstruction map carries these microdynamics to smooth continuum fields satisfying the TDF equations.

Where earlier TDF work proved string theory from its scaling law, this paper proves the scaling law — and the entire TDF structure — from first principles alone. The Thompson Dimensional Framework is no longer just inevitable; its origin is now explicit.

TDF is the Theory of Everything