Elementary Phase-Transition Gravity: From the Kramers Visibility Threshold to a Bulk–Boundary (Casimir) Cross-Scale Closure

## OverviewThis record contains the manuscript **“Elementary Phase-Transition Gravity: From the Kramers Visibility Threshold to a Bulk–Boundary (Casimir) Cross-Scale Closure” (Paper I)**. We propose a minimal, computable, and falsifiable framework in which:- **macroscopic gravity** is described as a bulk (medium) free-energy rearrangement that yields a unified effective potential, and- **Casimir-like forces** arise as a boundary free-energy response induced by spectral truncation under geometric constraints. ## Core state variables (coarse-grained)The observable state is represented by:- ρ : effective density- u : flow velocity- Θ : fluctuation/noise energy scale- χ : order parameter for nucleation and phase-transition boundaries ## Threshold criterion (rare-event visibility)Phase-transition triggering is modeled as rare events crossing a barrier ΔW.We define a dimensionless visibility cost: - chi_vis = ΔW / Θ A minimal “one e-fold” turning point is adopted: - exp(−ΔW/Θ) = e^(−1) ≈ 0.37 This is used as an operational normalization convention for when exponential suppression becomes non-negligible. ## Macroscopic gravity module (bulk response)A unified effective potential Φ_eff[ρ, Θ, χ] is introduced to control both:- orbital dynamics (rotation curves / Jeans analysis), and- gravitational lensing (line-of-sight projection of Φ_eff). In the static, spherically symmetric, source-exterior limit, the Laplace structure recovers Φ_eff ~ 1/r and the inverse-square law as the universal bulk response. ## Two falsification chains1) **Lens → dynamics parameter migration test** Parameters inferred from lensing must, without adding independent free functions, migrate to predict rotation curves or velocity dispersion. 2) **Spectral inversion test (S(k, ω))** Linearized closure yields propagation/relaxation branches and an experimentally invertible route using the dynamical structure factor S(k, ω), allowing independent constraints on: - a low-energy propagation speed limit c (from the long-wavelength regime), - relaxation time scales, and - correlation length xi_chi = sqrt(kappa / m_chi^2). ## Bulk–Boundary closure (Casimir as boundary free-energy response)Boundary constraints alter the available mode spectrum and induce a free-energy difference ΔF_bdry(a).The corresponding force is: - F_Cas(a) = − d/da [ΔF_bdry(a)] A key claim is that the same geometric scales (xi_chi, ell) can enter both:- astronomical-scale screening / environment dependence, and- tabletop boundary-force turning behavior,forming a cross-scale closure point. ## Relativistic consistency (minimal requirements)Paper I focuses on consistency-level statements needed for viability:- no measurable “ether wind” in the vacuum-like isotropic steady state,- gravitational-wave speed consistency away from criticality: c_g = c,while noting that recovering GR’s full TT polarization structure likely requires additional tensor/shear degrees of freedom (planned for extended models).