The Complete Graph K₄ on {Energy, Mass, Space, Time} and its Structural Consequences

Conventional physics treats energy as derived from mass, space, and time, but two empirical properties support treating energy as co-fundamental: conservation in transformations among the other three (1+3 asymmetry), and interconvertibility with mass at the exact rate c² (2+2 symmetry). Physics has kept space and time separate despite c fixing an exact rate between them; the same reasoning keeps energy and mass separate under c². The four quantities and their six pairwise ratios form K₄, the unique Kₙ (n ≥ 4) admitting a single 1-factorization, forcing the 2+2 partition while honoring the 1+3; this graph is here named the quadrex. The three matching (skew) pairs align one-to-one with the three regimes of the c–G–h classification; the four faces correspond to four physics domains, and the six face-pairings correspond to six areas of physics. The bridge identity P · μ = F/m̄ (Power, linear density, Force, and temporality, respectively) expresses gravity and quantum mechanics as inverse algebraic readings of the same dimensional content. The gravity ratio Ξ = (P/μ)/c³, with one structural axiom, gives the static vacuum metric coefficients a graph-theoretic reading, reached through either non-relativity matching. Three independent graph properties separate the three regimes, including a relativistic involution; three Hamiltonian cycles realize the same partition. The Schwarzschild and Hawking equations admit Planck-anchored rewrites in the same terms. Worked examples trace dissipative and concentrative energy across quadrexes; an appendix verifies dimensional completeness across six domains and all SI base units.