Discrete Wave Mechanics in ROG-H

This paper presents the exact discrete dynamical laws of wave propagation within the quaternionic state space of Relational Operator Geometry (ROG-ℍ). It rejects, on principled rather than merely computational grounds, every continuum limit ordinarily used to justify a partial differential equation as the description of a physical field. In its place, we give the finite-difference update rule that governs a biquaternionic spinor field on the nodes of a discrete relational graph, and we show that the familiar equations of mathematical physics — the classical wave equation, the Schrödinger equation, the Maxwell equations, the Weyl and Dirac equations, the Yang–Mills equation, and linearised gravity — are each recovered as a named special case of one master update rule, not as independent postulates. The quaternionic operator dynamics inherited from the corrected mechanics of ROG-ℍ assign SINK the exclusive role of writing the Now (Δ) into Memory (Φ₁), and PUMP the exclusive role of reading Expectation (Φ₃) into the Now. Because these operators act on disjoint pairs of axes, their commutator is not the symmetric, Cartan-type torsion of the two-dimensional theory; it is a single directed transfer from Expectation into Memory, [Ŝ, P̂](q) = (0, γδΦ₃, 0, 0)ᵀ. This is the Expectation-Dependence of Torsion: temporal torsion cannot arise in a system with no forward-directed disposition. We then isolate the sub-case relevant to ordinary radiative propagation — the Traditional Wave, defined by the exclusion of PUMP — and show that its dynamics reduce, in the Now–Memory plane, to repeated application of a single unipotent shear matrix of unit determinant. Because the determinant is exactly one at every tick, the shear is symplectic: it conserves phase-space area while continuously converting kinematic presence into historical record. Applying this shear N times to an emitted wave yields a closed-form expression for the accumulated ratio of Memory to Now, and hence a closed-form expression for redshift: z = Nγ. Redshift is the count of relational ticks a signal has traversed, multiplied by a single structural constant of the graph. No metric expansion is invoked at any point in the derivation.