Part II — Dynamics on P₀

TDGL-like Memory Kernel, Inertial Response, and Present-Surface Dynamics
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Abstract

This second paper in the Dual–Flux (DF) series develops the minimal dynamics supported by the present surface P0 introduced in Part I. Starting from the structural ingredients of the DF ontology—dual informational fluxes, the stabilising mode W0, the quaternionic coheron sector, and the temporal clocks—we ask which evolution law for coheron fields on P0 is compatible with informational conservation, strict Volterra causality, and the internal symmetries identified in Part I.

We show that, within the class of symmetry-preserving dissipative gradient flows with hereditary (Volterra) dissipation, these requirements select a time-dependent Ginzburg–Landau (TDGL) evolution equipped with a retarded memory kernel. The resulting equation is local on P0 but nonlocal in time, encoding the finite causal retention of W0. Linearisation around a stationary coheron yields a spectral decomposition into informational channels, separating short (fast) and long (slow, memory-dominated) sectors. Nonlinear stationary solutions realise the coherons of Part I and exhibit strong suppression of high spatial harmonics, providing dynamical localisation without ad hoc cutoffs. The same stationary problem admits finite-energy defect solutions when the far-field phase is restricted to an embedded subgroup U(1)u ⊆ SU(2), yielding integer winding and a quaternionic internal orientation u ∈ S2.

In linear response, projecting onto slow collective coordinates produces a deterministic hereditary (generalized Langevin-type) equation. In the DF slow-variation regime, an effective inertial scale emerges as a causal memory cost controlled by causal moments of the Volterra kernel, multiplied by a geometric translation metric fixed by the coheron profile. In the underdamped long-channel regime, slow modulations admit an envelope description written intrinsically as J ∂t(·) = … , where the DF clock structure supplies an internal complex structure J with J2 = −𝕀 (a coarse-grained remnant of the ℤ3t cycle). In a representation J ≃ i, this yields Schrödinger-type dynamics for envelopes on P0, while the quaternionic internal algebra realises the Pauli structure and allows a controlled Dirac-type first-order factorisation at the effective level. Part III builds on the same long-channel sector to derive DF’s effective metric response and an operator formulation of equivalence in the long-memory regime.

Equivalence Principle at a glance
Figure 1 — Emergent Dynamics at a glance

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