Dimensional Deployment Theory (DDT): A Self-Contained Stage-1 Framework

We present the complete, self-contained framework of Dimensional Deployment Theory (DDT), with all derivations from first principles. Observable 3+1-dimensional physics—spacetime geometry, U(1) gauge fields, topologically quantized matter, and quantum-mechanical behavior—emerges from a single dynamical scalar field X₂ on a 2+1-dimensional Lorentzian substrate (M₃, gₐᵦ). The framework rests on one ontological commitment (the dimensionality of space is deployed, not given) and one derived constant (X²ᵥₐ꜀ = ln 2, forced by bidirectional consistency). From these, four sectors of physics emerge: (I) cosmological spacetime with propagating fronts and a universal −8.98% lensing reduction (boundary-regime: R_F⁽ᵇᵒᵘⁿᵈᵃʳʸ⁾ ≈ 0.9102); (II) a U(1) gauge sector with coupling g²ₑբբ = μ∗(ln 2)²; (III) topologically quantized matter with gap mass M∞ = (ln 2)√(λ/μ∗); (IV) effective Schrödinger dynamics with the Born rule P = |ψ|². Every theorem is proved within this document. Every admissibility condition is stated with explicit gates. Every prediction carries a falsification criterion with precommitted thresholds. No external reference is required to verify any claim. The framework is formulated as a pure geometrical deployment: every dynamical quantity is a projective invariant of the deployment field X₂, with no additive decomposition into independent contributions. The parameters (β, μ∗, κ_g) plus extension parameter (λ) appear as Taylor coefficients of the single projective action |∇ ln X₂|² expanded around the vacuum; they are not independently free but constrained by the vacuum ln 2. An appendix of worked examples demonstrates operational reproducibility: a single parameter set produces numerical predictions across all four sectors with cross-sector consistency verified to four significant figures. PACS numbers: 04.60.-m, 98.80.-k, 11.15.-q, 03.65.Ta, 11.25.-w