Retained Relational Geometry as a Fading-Memory History Functional in ACRF-4: A Discrete-to-Continuum Theorem and Exact Equal-Area Path Dependence

This paper develops a discrete-to-continuum retained-history construction for the ACRF-4 adaptive registration model and proves that, under exposure-normalized refinement of a prescribed relational path, the retained coupling sector converges to a Volterra-type fading-memory law.

Starting from the discrete retained update

Kn+1=(1−ρm)Kn+ρmK(xn+1),

with exposure-normalized retention

ρm=1−(1−ρ)1/m,

the continuum limit is shown to satisfy

K˙=γ[K(x(t))−K],γ=−ln(1−ρ).

The corresponding solution is an exponentially weighted history functional,

K(T)=e−γTK(0)+γ∫0Te−γ(T−t)K(x(t))dt.

For ACRF-4 near the symmetric state, the retained target admits the exact leading expansion

Kij(x)=1+2340(xixj+12∥x∥2)+O(∥x∥4),

with the cubic term vanishing exactly. This yields an explicit quadratic fading-memory kernel for the retained relational state.

The paper then proves an exact equal-area continuum path-dependence result. Closed rectangular paths in a fixed orthonormal plane of the rank-three relational sector are constructed with identical starting point, endpoint, orientation, enclosed area, number of edges, and total traversal duration. Despite these controls, their retained quadratic history responses differ exactly. In particular, the unit square and a $2:1/2$ rectangle produce unequal retained responses, establishing that the continuum history functional is not reducible to endpoint, oriented area, and total duration alone.

The result is therefore best interpreted as a time-parameterized fading-memory structure that is strictly richer than an area two-form. It does not identify the response with ordinary geometric holonomy, a connection, Riemann curvature, or physical spacetime curvature.

The paper also clarifies the state-space interpretation of the model: dynamics are non-Markovian when viewed only through the reduced relational variable $x$, but Markovian on the enlarged state $(x,K)$, where $K$ carries the retained relational history.

The contribution is intended both as a restricted development within Registration Theory and as a mathematically self-contained result concerning adaptive network dynamics, discrete-to-continuum limits, hereditary systems, and Volterra-type fading-memory functionals.