Relational Geometry and the Emergence of Gravity: From Harmonic Closure to Stellar Structure

We present a relational framework in which gravity emerges as the macroscopic tendency of physical systems to reduce the phase offset between their projected relational states. The generative algebra is classified as the quaternionic cross product on Im(H), yielding so(3) and so(4) ≅ su(2)_L ⊕ su(2)_R. The modal invariant I(n) = 2^(n/2) √(2^n − 1) satisfies I(4)² = 240 = #roots(E₈).

Key results:

The framework derives a parameter-free density window (2√2 − 1)ρ₀ ≈ 1.83ρ₀ for stable neutron stars from integer arithmetic in the closure hierarchy, with no free parameters.

An effective gravitational coupling G_eff(ρ) = G · ω_m(n(ρ))/ω_m(4) is derived from the existing postulates. This formula has a hard ceiling G_eff/G ≤ √(16/15) ≈ 1.033. Numerical TOV integration with the SLy equation of state yields M_max = 2.037 M☉, compatible with PSR J0740+6620 (2.08 ± 0.07 M☉). A previous erroneous formula predicted M_max = 0.70 M☉; the error identification and correction are documented explicitly.

The propagation invariant R_V(n)·ω_V(n) = c holds exactly at every closure level as an algebraic consequence of the binary hierarchy.

All claims carry explicit epistemic labels (POSTULATE / DERIVED / HYPOTHESIS / SPECULATIVE / OPEN). The paper reports one falsified intermediate result honestly, identifies the three principal open problems, and provides five falsification conditions. Numerical code is included.