A Paradigm to Replace General Relativity: Pressure-Element Medium Thermodynamics with Microscopic Phase Transition

ABSTRACT. We propose a candidate replacement for general relativity. In the pressure-stress ontology (PES) the gravitational field is not fundamental: what exists is a pre-stressed pressure-element medium, whose complete stress state — pressure in the broad sense, the full stress tensor — generates the optical geometry, and the metric is the macroscopic appearance of that state rather than its cause. The break with general relativity (GR) is therefore ontological before it is quantitative: the two frameworks do not agree on what is real. This paper does not argue that point; it makes the quantitative comparison complete, explicit and archived. The framework is not free. In the light sector it carries one new dimensionless number, the areal mass misfit $\delta$ between the time-time and radial masses ($\eta=1+\delta,\ \gamma_{PES}=1/\eta$), together with a pressure-response shape that is structural rather than adjustable. On that basis it reproduces every light-sector prediction that GR has passed — the solar-grazing deflection at the $10^{-6}$ level of the first PPN coefficient, the photon-sphere critical curve, and the logarithmic divergence as $b\to b_c$ — while its departures, computable at every order, lie to five orders of magnitude below present and committed sensitivities. Within the errors of every experiment considered here, the two frameworks are indistinguishable. What it predicts from below is different in kind. The transition width $\sigma$ and the misfit $\delta$ are not fitted: they are outputs of microscopic configuration counting, $\sigma=(1+\tfrac12\ln g)/n_T$ with $g=\Omega_c/\Omega_u$ an integer. With the two postulates $\(g=2\)$ and $n_T=1$ the framework gives $\sigma=1.1733$ and $\delta=-2.6549\times10^{-6}$. The postulates are hidden: without them what remains is the discrete spectrum $\sigma\in\{1,\,1.1733,\,1.2747,\dots\}$, the band $1<\sigma<1.6084$ is in the thermal reading, and the unconditional sign $\delta<0$. The light field is settled to the following eight statements. (i) Every weak-field coefficient is rescaled by a single shape-independent factor, $\alpha_k=\alpha_k^{GR}\left(1+\zeta_k\delta+O(\delta^2)\right)$ with $\zeta_k=-k(2k-1)/(3k-1)$; because $|\zeta_k|$ grows like $\(2k/3\)$, the signature of $\delta$ is a uniform recalibration of the series rather than a higher-order tail. (ii) The photon-sphere impact parameter shifts by $-2\delta/3$ in Kepler normalisation and by $+\delta/3$ in the (area) book normalisation — two readings of one and the same spacetime. (iii) The near-critical deflection is $\alpha=-c_1\ln(b/b_c-1)+C$ with $c_1=\Omega_c\sqrt{2/\pi}$, and $C$ differs from its GR value by $O(\delta)$. (iv) The deflection integral is hyperelliptic of genus $g=\deg\Omega+1$; for the quadratic $\Omega$ of the standard derivation $\(g=3\)$, the $3\times6$ period matrix satisfies the Riemann bilinear relations to $4\times10^{-23}$ (relative), and the Picard-Fuchs operator has order $2g$ with regular singularities exactly at the critical values. (v) The transition width is not a bulk-viscosity width: both probes sit on the frozen branch ($\omega\tau=3.4\times10^{29}$ and $5.7\times10^{16}$), the identification $\Delta=|D|$ is self-consistent in a factor 15, and five statements of the v1.0 closure can be withdrawn (erratum, §7.1). (vi) The width is the Landau width of a two-state cell, $\sigma=(k_BT+J/4)/J$, which equals $1+T_c/T=1/(4h'\chi)$ at $n_T=1$ and contains no $H$; both transfers of the double-transit profile share it, and only $T$ has a channel-independent logarithmic rate ($\rho$- and $H$-driven biases would split the width by 29.4%). (vii) Entropic coupling with integer counting gives the discrete spectrum above, reduced to $\(g=2\)$ by minimality. (viii) The black-hole shadow offsets are two to five orders of magnitude below present and planned sensitivities, so the strong-field channel cannot test $\delta$: the binding constraint remains the weak-field PPN bound, $|\delta|\leq4\times10^{-5}$. Every claim carries one of five grades — A (analytic identity, verified symbolically and numerically), B (derived and numerically verified inside a stated domain), C (unconditional sign, bound or order of magnitude), D (discrete candidate set), D' (single value under stated postulates). Every falsifier is catalogued in a master table that separates structural from observational paths. Nothing is asserted beyond its grade. The scripts and logs behind every number are indexed in Appendix D. This preprint is a declaration, not a verdict. It states a complete, internally closed first stage — derivation, microscopic basis, falsifiable prediction — and marks exactly where the comparison with general relativity must be. The observational chain that would decide it is the next stage, and is not claimed here. Keywords: alternative gravitation; pressure-element medium; medium thermodynamics; light deflection; post-Newtonian expansion; strong-field