Chapter F Time, Energy Conservation, and Irreversibility as Lattice Properties

Four statements about time and energy, traditionally treated as separate postulates or empirical facts, are shown to be theorems of the Fano spacetime lattice and the Huniverse = Hlattice identity (Paper XCVII). (1) Conservation of energy = shape memory. Noether's theorem applied to the Fano lattice gives dHlattice/dt = 0 ⇔ ∂L/∂t = 0 ⇔the Fano lattice topology (G2 symmetry, 7 nodes, Poisson ratio ν = −1) is time-invariant. This time-invariance is the shape memory: the Weissenberg number Wi →∞ensures the lattice never relaxes, so G2 symmetry is permanent, so Hlattice is conserved. Energy conservation and shape memory are the same statement in dierent languages. (2) Time is a phenomenon of brane expansion. The arrow of time is the direction of increasing H+ real-unit advance, which is identical to the direction of increasing Hubble parameter H(t) > 0 (brane expansion). The rate of time at any point equals the local lattice propagation speed divided by c: dτ/dt = clocal(r)/c = p gtt(r). (3) Time reversal is topologically prohibited. T-reversal requires H(t) →−H(t), i.e., a0 →ia0 (imaginary sump tilt). This crosses from the H+ sector to the H−sector of O = H+ ⊕H− a topological sector change, not a statistical uctuation. The H−sector is the CPT-conjugate universe (Boyle & Turok 2018), permanently separated from H+ after the nucleation event. T is violated by topology, not by any asymmetric microphysics. (4) Lattice dimension change = local time dilation. A change in the local Fano lattice metric by factor f(r) = p 1 −rs/r (Schwarzschild) produces an identical change in local clock rate for external observers. Spatial lattice compression ⇒temporal lattice stretching ⇒slower clocks: gravitational time dilation is a lattice metric identity. Four new predictions are made (P56P59). Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.