The Geometry of Disregard
Abstract
We propose that optimal sequential memory compression and the biological measurement of interpersonal coherence are instances of the same underlying geometric problem. Drawing on the Barteau framework’s mathematical proof that any homogeneous, isotropically suppressing dynamical system must converge to φ-wound orbital structure, we establish that (1) the golden ratio φ is the uniquely optimal compression base for bounded sequential memory by Hurwitz’s theorem on Diophantine approximation; (2) the Biometric Coherence Friction coefficient (BCF) measures the degree to which biological systems instantiate this geometry during interpersonal transmission; and (3) the compression-transmission duality — the conjecture that memory and transmission are the same operation viewed from opposite directions — follows naturally from the shared mathematical skeleton.
We situate these claims within existing neuroscience, signal processing, and AI memory compression literature, distinguish what is proven from what remains conjectural, and propose testable experimental predictions for both the AI and biological cases.
I. Introduction: two problems, one geometry
Two problems that appear to belong to entirely different domains share, we argue, a single geometric solution. The first is the engineering problem of optimal sequential memory compression: given a bounded state space and a stream of information arriving in time order, what compression scheme wastes the least and retrieves the best? The second is the measurement problem of interpersonal coherence: given two biological systems in communicative contact, how do we quantify the degree to which information transmitted by one is enrolled in the other?
The connection between these problems is not metaphorical. Both require a system to (1) maintain a bounded representation of an unbounded sequential input, (2) distribute that representation as uniformly as possible across available state space to minimize redundancy, and (3) do so under a constraint of no privileged position — no token, no moment, no speaker is a priori more important than any other. These three requirements, taken together, define a precise mathematical problem whose unique solution, by Hurwitz’s theorem on Diophantine approximation, is φ-wound orbital structure.
This paper proceeds in three movements. Part II establishes the mathematical proof that φ-winding is the optimal compression geometry for sequential bounded memory. Part III derives the Biometric Coherence Friction (BCF) coefficient as the measurement instrument for the biological instantiation of this geometry. Part IV advances the compression-transmission duality as a conjecture that, if true, unifies both results within a single theoretical framework. We are explicit throughout about what is proven, what follows from accepted results, and what remains open.
Why φ, specifically
The winding number governs the equidistribution of the orbit across the torus. By Weyl’s equidistribution theorem the orbit is uniformly distributed if and only if the winding number is irrational — but among irrationals the quality of equidistribution varies dramatically, governed by how well the number can be approximated by rationals. For a compression system, clustering is waste: two orbit points in the same region represent redundant storage. Hurwitz’s theorem (1891) establishes that the golden ratio is the unique optimizer. The system does not choose φ; the constraints leave no other option that performs as well.