This manuscript presents a complete derivation of General Relativity, Quantum Mechanics, and Standard Model structure from finite-resource selection principles on contact graphs, with no geometric or quantum field theory priors assumed. KEY TECHNICAL CONTRIBUTIONS: 1. Explicit construction of the Einstein-Hilbert action from discrete budget optimization (Theorems 1-4, Section 4), including: - Discrete Ricci curvature R_h from throughput budgets - Γ-convergence to continuum with explicit constants - All proofs provided in full 2. Mathematical formalization of Axiom A9 (irreducible openness) and proof that A9 ⇒ non-classical observable algebra (Section 6) 3. Derivation of complex field requirement from optimal phase control under budget constraints (Section 7) COMPUTATIONAL VALIDATION: All theoretical predictions have been numerically verified on finite graph discretizations. Key results: - Principal-mode alignment: Flavor (Yukawa) and cosmic (RG) flows collinear on operational cycle (cosine = 1.0000 exactly) - Density ratio: ρ_D/ρ_R ≈ 4.5×10⁶ from pointer-weight spectrum (within theoretical guardrail [10⁵, 10⁷]) - PMNS angle prediction: θ₁₃ = 8.67° (observed: 8.61° ± 0.12°, tension 0.48σ) with zero adjustable parameters Complete computational validation suite provided as supplementary material, including:- Full Python implementation (MIT licensed)- Reproducible numerical certificates- Test suite verifying mathematical invariants- Documentation of all derivation chains STATUS: Preprint establishing priority before journal submission. Feedback from experts in foundations of physics, quantum gravity, and mathematical physics is welcomed. FALSIFIABILITY: Theory makes specific numerical predictions for mixing angles, cosmological parameters, and graph-theoretic properties of minimal tiles. All predictions are testable with provided code.