One of the core ideas in the VERSF framework is that physical events leave traces — not metaphorically, but literally, through a field called the κ-field. When an irreversible physical event occurs, it sources a perturbation in this field, and that perturbation propagates and persists. The result is that the present state of a physical system is influenced not just by current conditions but by the accumulated record of past events.

A key question is: what exactly is the mathematical shape of that accumulated memory? Previous work in the programme established that the answer involves an oscillating, algebraically decaying function — specifically cos(mτ)/τ, where m is the κ-field mass and τ is the time elapsed since a past event. This paper provides a more complete derivation of that result and extends it in two new directions.

Deriving the memory kernel. The cos(mτ)/τ form had been established using stationary-phase arguments. This paper derives it more directly: by integrating the full spatial propagator of the κ-field over a narrow coherence tube using a Gaussian weighting function, then evaluating the integral at late times. A single algebraic step — the appearance of a factor of i in the late-time limit — converts the sine function you would naively expect into a cosine. This makes the mechanism transparent in a way the earlier derivation did not.

Unifying two pictures. Two separate mathematical descriptions of commitment-modulated decay had been developed in earlier papers: a local description using an ordinary differential equation for a geometric field g(t), and a nonlocal description using a Volterra integral equation for the observable population N(t). These were known to be related but the precise relationship had not been proved. This paper proves it as a theorem: eliminating the geometric field from the local description yields an exact equation from which the Volterra description follows by two controlled approximations. The two pictures are not different theories — they are different levels of the same theory.

Gravitational consequences. Since spacetime geometry in VERSF is sourced by the accumulated record of committed events — and that record now includes the derived memory kernel — curvature becomes history-dependent. The extended gravitational equations derived here predict that metric perturbations oscillate as cos(mt)/t: a slowly decaying oscillatory signal distinct from the predictions of standard general relativity.

One open problem remains: proving that the width of the coherence tube equals the VERSF coherence scale ξ from first principles. This is named explicitly and is the subject of ongoing work.