K=1 Chronogeometrodynamics

Background. Time has invariably entered physics as a primitive — an underived input.The present paper proposes a structural framework in which the algebraic type of thelocal cost form is constrained by realizability axioms, with Lorentzian signatureemerging as a consequence rather than an assumption. Framework. Here the cost of change is modelled through an information timedt_info := dPhi/H. The object constrained is not dt_info itself, but the signedleading quadratic form G associated with the local second-order expansion ofdt_info^2. Modelling this form as satisfying the realizability axioms(Axioms R, E, T; see Appendix) yields, under nondegeneracy, Sig(G) = (1,1),equivalently det G < 0; the unit-cost locus is K(x) = x^T G x = 1. Dynamics and thermodynamics. Within the symplectic-dissipative classG A = alpha J - G D, the evolution law takes the uniquely reduced formdx/dt = (J_G - D) grad V. At critical damping the Law III dynamics degeneratesinto a rank-one null flow with a conserved linear combination, foliating the planeinto invariant leaves on each of which K = 1 is reached by exact exponential decay,with a recovery time scaling as c⁻² in the conserved charge c; attraction failsonly on a single measure-zero freeze leaf. Near K = 1, the framework supplies theingredients for a local Clausius relation, thereby providing the thermodynamic inputto the Jacobson argument once the two external identifications T_eff = T_tol andS propto A are supplied. Gravitational content. The thermodynamic perspective above conditionally recoversthe Einstein equations Rmu nu - (1/2) R gmu nu = 8 pi Tmu nu via theJacobson argument. Independently, in the spherical sector with f > 0, thedifferential conditions K_1 = K_2 = 1 for the cost-structural functionalsK_i := sigma_2^2 Box ln sigma_i are shown to be algebraically equivalent to thevacuum Einstein equations Rmu nu = 0 (Theorem). The conditions K_i = 1 areintroduced at the field level as a candidate ansatz, motivated by but not derivedfrom the point-level condition K = 1 of Law I; the Theorem is therefore an algebraicreformulation of vacuum Einstein in cost-structural language, not a derivation fromthe framework axioms. Lorentzian signature is supplied internally at the point levelthroughout.