Recursive Dimensionality Theory VI: A Geometric Completion with Gravitational Wave Predictions

This paper presents a geometric and dynamical completion of Recursive Dimensionality Theory (RDT), embedding the phenomenological framework developed in Papers I–V within a rigorous scalar-tensor theory with conformal matter coupling. The effective spatial dimensionality d_eff = 3 + rb'/b emerges naturally from a radial warp factor in the spatial metric, with dynamics governed by a chameleon-type scalar field. In dense regions where the scalar field gradient is negative, this yields d_eff < 3, recovering the phenomenology established in earlier papers. The formulation automatically satisfies the GW170817 constraint (gravitational wave speed equals c exactly), provides natural screening mechanisms via thin-shell effects, and generates falsifiable predictions for gravitational wave observations. A central result is the identification of a fundamental tradeoff between two observationally distinct parameter regimes: Stellar-matched regime (scalar mass m ≳ 10⁻⁶ eV): Reproduces Papers I–III phenomenology for solar neutrinos and stellar structure, but gravitational wave dipole radiation is Yukawa-suppressed below detectability. GW-loud regime (scalar mass m ≲ 10⁻¹² eV): Produces observable dipole radiation in neutron star–black hole binaries, but with different density-crossover behavior than the stellar-matched case. The differential signature—NS-BH systems showing RDT effects while BH-BH systems match general relativity exactly (due to the scalar no-hair theorem)—provides a robust observational discriminant. We present constraints from binary pulsar timing, Solar System tests (Cassini), and current gravitational wave observations, mapping the allowed parameter space and outlining detection strategies for next-generation detectors (Einstein Telescope, Cosmic Explorer). This work completes the theoretical foundation of RDT by providing an action principle, field equations, stability proofs, and quantitative observational predictions, while honestly confronting the theory with current data and identifying the tradeoff between stellar-structure fits and gravitational wave detectability as a central scientific result.