Non-Equilibrium Order-Dissipation Bifurcation, Volume-Filling Mitochondrial Cristae Scale Invariance and Cellular Kuramoto Desynchronization in the Oncological Warburg Transition

Standalone Research Paper in Mathematical Biophysics and Systems Oncology (14 Pages, 6 Sections, 3 Figures, 2 Synthesis Tables, Complete Mathematical Proofs). ABSTRACTWe present a unified mathematical framework connecting non-equilibrium redox thermodynamics, evolutionary cell geometry, ultrahigh-field membrane electrostatics, and collective oscillator synchronization in eukaryotic bioenergetics. To maintain strict epistemic transparency, we separate deductively proven mathematical theorems (Tier A) from constructive biophysical model postulates (Tier B). Non-Equilibrium Order-Dissipation Bifurcation (Section 2):We formulate a planar nonlinear dynamical system governing coupled bioenergetic order (Phi+) and oxidative stressor load (Phi-), proving via Jacobian linearization, Routh-Hurwitz criteria, and the Bendixson-Dulac negative-divergence theorem that sub-lethal dissipative stress (mitohormesis) drives open systems across their baseline Verhulst saturation wall Kk into a unique, globally asymptotically stable interior attractor P4 with super-logistic capacity overflow Delta Phi+ = M Kk > 0. Geometric Resolution of the Euclidean 1/R Bottleneck (Section 3):We prove geometrically that surface-respiring prokaryotes (A_prok = 4 pi R^2) suffer a strict 1/R decay in specific metabolic power that vanishes at a finite starvation radius R_max = 3 j_ATP / mu_vol (approx. 2 micrometers), whereas eukaryotic internalization of volume-filling mitochondrial cristae (A_euk = f_m sigma_cristae V(R), proportional to R^3) cancels the cell radius R identically, achieving scale-invariant specific power (eta_euk = const.) and a 75-fold areal amplification at R = 25 micrometers. Ultrahigh-Field Mitochondrial Electrostatics and Heaviside Control (Section 4):We derive the E_m = 36 MV/m electrostatic field and 1.15 x 10^4 J/m^3 Maxwell compressive stress sustained across the 5 nm inner mitochondrial membrane from the Nernst-Mitchell proton-motive force (220 mV), modeling Cytochrome c compartmental control via complementary Heaviside step operators and cooperative heptameric Apaf-1 Hill kinetics (n_H = 7). Cellular Kuramoto Synchronization and the Warburg Phase Transition (Section 5):We model the cellular chondriome as a coupled Kuramoto oscillator network, proving that degradation of inter-mitochondrial coupling below the critical threshold K_c = 2 gamma_cell triggers a second-order phase transition (r_cell -> 0) with diverging coherence susceptibility chi_coh proportional to (kappa_coup - 1)^(-1/2), providing a quantitative dynamical mechanism and experimental falsifiability benchmarks (Section 6) for the oncological Warburg transition from coherent aerobic respiration (approx. 32 ATP) to uncoupled cytosolic glycolysis (2 ATP). Note: This paper develops and expands the biophysical and oncological architecture introduced in the author's foundational monograph: Theory of Perspectival Dimensionality (TPD), Illustrated English Edition 2.0, Zenodo Permanent Archive, DOI: 10.5281/zenodo.23902867.