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

Abstract

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 full 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. d ≠ 4 ruled out 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 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 on current data it is insufficient by ~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), TdeSitter_entropy, Thorizon_reciprocity, 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 Codebase backing. Every theorem traces to a named check function in the canonical apf/ codebase (v7.5; 454 bank-registered theorems, 471 verify_all checks, 39 modules + standalone). Paper 6's Einstein equations, cosmological constant T11, dark-matter T12/T12E, and black-hole thermodynamics live in apf/gravity.py, apf/cosmology.py, and apf/spacetime.py. About the APF series. The Admissibility Physics Framework derives Standard Model and cosmological structure from a single primitive — finite enforcement capacity — through a chain of papers plus a machine-verifiable engine. Each paper has a main text and, where applicable, a Technical Supplement, deposited on Zenodo and collected in the admissibility_physics community; the canonical engine (APF-Engine v24.3.427, 3,918 bank-registered checks) backs every quantitative claim. The full series index is at admissibilityphysics.com. DOIs below are concept DOIs (they always resolve to the latest version). Paper 0 — What Physics Permits: A Constraint-First Framework for Physics — 10.5281/zenodo.18439523 Paper 1 — The Enforceability of Distinction — 10.5281/zenodo.18439200 · supplement Paper 2 — Finite Admissibility and the Failure of Global Description — 10.5281/zenodo.18439274 · supplement Paper 3 — Entropy, Time, and Accumulated Cost — 10.5281/zenodo.18439363 · supplement Paper 4 — Admissibility Constraints and Structural Saturation — 10.5281/zenodo.18439397 · supplement Paper 5 — Quantum Structure from Finite Enforceability — 10.5281/zenodo.18439433 · supplement Paper 6 — Dynamics and Geometry as Optimal Admissible Reallocation — 10.5281/zenodo.18439445 · supplement Paper 7 — A Minimal Quantum of Action from Finite Admissibility — 10.5281/zenodo.18439513 · supplement Paper 8 — The Admissibility-Capacity Ledger — 10.5281/zenodo.19721384 · supplement Paper 9 — The Geometric Substrate as Cost Structure of Comparison Continuations — 10.5281/zenodo.20041675 · supplement Paper 10 — The Calculus of Finite Continuability — 10.5281/zenodo.20041680 · supplement Paper 11 — Forced Universality from Capacity-Bounded Admissibility — 10.5281/zenodo.20684198 · supplement Paper 12 — The Gauge Sector of Admissibility Physics — 10.5281/zenodo.21212521 · supplement Paper 13 — The Minimal Admissibility Core — 10.5281/zenodo.18361446 Paper 14 — The Quantum Story — 10.5281/zenodo.21212188 · supplement Paper 16 — Markov Breakdown and the Hard Problems — 10.5281/zenodo.20684207 Paper 18 — The Electroweak Sector as a Capacity Equilibrium — 10.5281/zenodo.20684209 Paper 19 — Large-Angle CMB Suppression as a Finite-Continuability Constraint — 10.5281/zenodo.21199218 Paper 20 — The Enforcement Crystal — 10.5281/zenodo.18531732 · supplement Engine (Paper 21) — Admissibility Physics: An Executable Constraint Framework and Constants Map — 10.5281/zenodo.18529115 Paper 23 — Magnetically Dark Winding-Class Quantum Processors — 10.5281/zenodo.21199220 Paper 24 — The Recruitment-Radius Extension — Foundations — 10.5281/zenodo.20684211 · supplement Paper 25 — The Recruitment-Radius Extension: Applications — 10.5281/zenodo.21199222 · supplement Paper 28 — Absolute Mass Scales from Electroweak Capacity Saturation — 10.5281/zenodo.20684215 Paper 29 — Plaquette Representation Dominance and Confinement — 10.5281/zenodo.20684218 Paper 30 — A Tube Mechanism for the Lattice Mass Gap — 10.5281/zenodo.20684220 Paper 31 — Osterwalder–Schrader Structure of Lattice Yang–Mills — 10.5281/zenodo.20684222 Paper 32 — Quark Anchors as APF Trace Masses — 10.5281/zenodo.21199226 Paper 33 — Trace-to-Scheme Export Architecture — 10.5281/zenodo.20684224 · supplement Paper 35 — The Dark Sector as a Two-Role Capacity Decomposition — 10.5281/zenodo.20684228 · supplement Paper 36 — The Missing Floor — 10.5281/zenodo.21199228 Paper 37 — Collapse as Realignment — 10.5281/zenodo.21199231 · supplement Paper 38 — Unification as Ledger — 10.5281/zenodo.21199233 · supplement Paper 39 — Coherent Phase Regimes in the APF Interface Engine — 10.5281/zenodo.21199235 · supplement Paper 40 — The Thermodynamics of Finite Distinction (Between Symmetry and the Void) — 10.5281/zenodo.20684235 · supplement Paper 41 — The Horizon as a Continuation Ledger — 10.5281/zenodo.20684241 · supplement Paper 42 — The Weak Mixing Angle Is Not Free — 10.5281/zenodo.20684245 Paper 43 — The Neutrino Sector — 10.5281/zenodo.21194191 Paper 44 — What a Finite Theory Owes the Continuum — 10.5281/zenodo.21195874 · supplement Paper 45 — What the Vacuum Can Keep — 10.5281/zenodo.21194195 · supplement Paper 46 — The Strong CP Angle Is Not Free — 10.5281/zenodo.21194197 · supplement Paper 47 — The Confinement Antecedent Is Not Forced — 10.5281/zenodo.21194199 · supplement Paper 48 — The Light-Bending Weight Is Not Free — 10.5281/zenodo.21199627 · supplement Technical Supplements cross-link to their main paper via isSupplementTo and to the companion GitHub repository (github.com/Ethan-Brooke) 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