EXTERNAL-DIMENSION MECHANICS (EDM) The Single Field Reconstruction of Time, Geometry, and Gravitation

External‑Dimension Mechanics (EDM) is a full‑scale reconstruction of spacetime physics that does something almost no alternative‑gravity framework even attempts: it replaces the ontological foundation of General Relativity rather than modifying its equations. The manuscript develops a single‑field, Riemann–Cartan geometric theory in which time, curvature, torsion, causal structure, and gravitational dynamics all emerge from the monotonic evolution of one scalar field along an external dimension. EDM is not a phenomenological patch or a parameterized deviation — it is a mathematically complete, axiomatic, and self‑consistent alternative to the spacetime postulate itself. Across these pages of derivations, the manuscript delivers: • A unique induced metric derived from first principles No arbitrary functions, no free tensors. The metric, torsion, and connection are fixed by locality, covariance, regularity, and the Newtonian limit. The result is a Riemann–Cartan geometry with a totally antisymmetric torsion sector sourced algebraically by scalar gradients. • A horizonless, singularity‑free strong‑field sector Static and rotating solutions exhibit finite curvature invariants, a regular core, and a high‑redshift surface instead of an event horizon — all proven analytically and numerically. The rotating sector introduces a helical Killing symmetry that resolves the classic “scalar fields cannot rotate” no‑go theorem. • A cosmology that resolves current tensions without dark-sector epicycles The theory naturally produces: – a local Hubble‑tension resolution via geometric lapse gradients, – a scale‑dependent effective Newton constant that suppresses S₈, – accelerated early structure formation via an elevated effective G, – an exact mapping from a static 3‑manifold to an apparent FRW expansion, – preservation of the CMB blackbody spectrum without a hot Big Bang. • A full perturbation and gravitational‑wave sector EDM predicts two tensor modes plus a massive scalar breathing mode, with torsion‑modified polar potentials and a split QNM spectrum. The 2.5PN radiation‑reaction terms include monopole and dipole channels absent in GR, producing distinctive inspiral phasing signatures for LISA. • A quantum sector emerging from medium dynamics and SU(2) holonomy Spin‑½ behavior arises geometrically from torsion‑induced SU(2) holonomy in the extended manifold. The Schrödinger and Dirac equations emerge from the statistical dynamics of microscopic fluctuations of the single scalar field. • A complete observational test suite The manuscript provides falsifiable predictions for: – ngEHT photon‑ring asymmetries and torsion‑induced birefringence, – LIGO/LISA scalar‑mode radiation and QNM splitting, – Euclid/Rubin scale‑dependent growth and gravitational slip, – CMB spectral preservation and Sachs–Wolfe structure, – MeV/X‑ray polarimetric torsion signatures. • A fully reproducible numerical pipeline Including ODE/PDE solvers, spectral methods, QNM finders, ray‑tracing engines, and Bayesian inference frameworks — all validated with convergence tests and HPC scaling benchmarks.