APF Paper 6 Technical Supplement — Dynamics and Geometry as Optimal Admissible Reallocation

Technical Supplement to Paper 6 of the Admissibility Physics Framework (APF), Dynamics and Geometry as Optimal Admissible Reallocation. Paper 6 derives Einstein's field equations Gμν + Λgμν = 8πG Tμν, the cosmological constant Λ, dark matter as external capacity Cext, and the complete Planck 2018 cosmological-parameter match — all from the optimal reallocation of admissibility capacity under PLEC. Thirteen formal sections. §1 Assumption inventory — PLEC imports from Papers 1–5, the formal Regime R definition with its five-type regime-exit taxonomy (Types I–V), and four paper-specific hypotheses H1–H4. §2 Least-cost dynamics — the Euler–Lagrange form ∂μ (∂ℒ/∂ (∂μφ) ) − ∂ℒ/∂φ = 0 derived as the admissibility-saturation condition. §3 Five-type regime-exit classification — Types I (runaway cost), II (degenerate floor), III (capacity overflow), IV (non-coherent), V (admissible exit: the only physically realised type). §4 Geodesic geometry from admissibility cost; geodesics as least-cost worldlines. §5 Metric-tensor theorem T7B — the cost kernel induces a symmetric rank-2 tensor satisfying the signature constraints of Lorentzian (−, +, +, +) geometry. §6 Spacetime-dimension theorem T8 — d = 4 from Lovelock uniqueness in the cost-functional's higher-derivative expansion, with explicit exclusion of d ≠ 4 by degree-of-freedom counting. §7 Curvature from capacity gradients. §8 Einstein field equations via the unified A9-closure (T9, grav, T10) plus Lovelock uniqueness. §9 Cosmological-constant theorem T11 — Λ from residual capacity, giving ΩΛ = 42/61 ≈ 0. 6885. §10 Planck 2018 match. The APF capacity partition maps directly onto the ΛCDM density fractions: Ωb, Ωc, Ωm, ΩΛ match Planck 2018 within 1. 2%; APF predicts H0 = 67. 76 km/s/Mpc, consistent with CMB-constrained measurements. The formal H0 tension falsifier is stated in §11. 4 of the main paper: the 7. 09σ tension with H0DN 2026 (H0 = 73. 50 ± 0. 81 km/s/Mpc) is the framework's primary empirical confrontation. Route V (local inhomogeneity) is the only APF-admissible path and is insufficient by approximately a factor of 2. §11 Dark matter as external capacity Cext via T12, T12E — structural existence and properties derived; particle identity is an open problem. §12 Black-hole thermodynamics — TBek (Bekenstein entropy SBH = A/4ℓP2), TdeSitterₑntropy, Thorizonᵣeciprocity, Tgraviton. §13 Neutrino/cosmology confrontation. Appendices: countermodels, red-team, theorem index, dependency diagram, changelog. Readable without prior exposure to the APF series. Code and reproducibility. GitHub repository Colab walkthrough notebook (one-click) Interactive dependency DAG About the APF series. The Admissibility Physics Framework is a ten-paper derivation chain plus core infrastructure, extending a single axiom (finite information capacity) through the Standard Model gauge group, fermion content, quantum formalism, Lorentzian spacetime, Einstein field equations, cosmological constant, and minimum quantum of action. Each paper's main text and Technical Supplement is deposited separately on Zenodo; each paper has a companion GitHub repository with the vendored apf/ codebase (v6. 9, 376 bank-registered theorems across 23 modules, 48 quantitative predictions), a one-click Colab notebook, and an interactive D3. js dependency DAG. Engine — Admissibility Physics Unified Theorem Bank & Verification Engine — DOI 10. 5281/zenodo. 18604548 · GitHub Paper 0 — What Physics Permits: A Constraint-First Framework for Physics — DOI 10. 5281/zenodo. 18605692 · GitHub Paper 1 — The Enforceability of Distinction — DOI 10. 5281/zenodo. 18604678 · GitHub Paper 2 — Finite Admissibility and the Failure of Global Description — DOI 10. 5281/zenodo. 18604839 · GitHub Paper 3 — Entropy, Time, and Accumulated Cost — DOI 10. 5281/zenodo. 18604844 · GitHub Paper 4 — Admissibility Constraints and Structural Saturation — DOI 10. 5281/zenodo. 18604845 · GitHub Paper 5 — Quantum Structure from Finite Enforceability — DOI 10. 5281/zenodo. 18604861 · GitHub Paper 6 — Dynamics and Geometry as Optimal Admissible Reallocation — DOI 10. 5281/zenodo. 18604874 · GitHub Paper 7 — A Minimal Quantum of Action from Finite Admissibility — DOI 10. 5281/zenodo. 18604875 · GitHub Paper 8 — The Admissibility-Capacity Ledger — main paper DOI pending · GitHub Paper 13 — The Minimal Admissibility Core — DOI 10. 5281/zenodo. 18614663 · GitHub Companion derivation: The Weak Mixing Angle as a Capacity Equilibrium — DOI 10. 5281/zenodo. 18603209 Technical Supplement DOIs for Papers 1–8 (this series of deposits) cross-link to each main paper DOI via isSupplementTo and to each companion GitHub repository via isDocumentedBy. Author. Ethan Brooke, Independent Researcher, San Anselmo, California, USA. ORCID: 0009-0001-2261-4682 LinkedIn: linkedin. com/in/ethanbrooke GitHub: github. com/Ethan-Brooke Contact: brooke. ethan@gmail. com