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Geometric Computation: Twin Helices and the Algebra of Flow

A mix of human-written and AI-generated textHuman understanding: all partsmath-ph — Mathematical Physicsmath.CT — Category Theorymath.DG — Differential Geometrymath.GR — Group Theorymath.MP — Mathematical Physicsmath.SG — Symplectic Geometry

Contributed by Benjamin Kaminsky ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

This work constructs a manifold consisting of twin helices to investigate the existence of computational geometric bordisms, as part of the broader program of the Cobordism Hypothesis. We use these structures to define a class of deterministic computational substrates, termed Geometrically Computable Manifolds, and study their geometric, algebraic, and computational properties. We show that, in a discrete regime, these manifolds encode arbitrary cyclic groups and support universal computation. This leads to a hypothesis describing the structural trade-offs in physical computation across distinct computational phases. We then analyze a regime in which abelian gauge computation is modeled with ordinary differential equations, linking the construction to continuous geometric dynamics. This signals a new research direction in physical mathematics at the intersection of geometry, gauge theory, and computability theory, termed Geometric Computability Theory, in which computation is treated as a property of geometric compositions. This perspective opens avenues for future investigation, motivating the development of a constructive computational gauge theory with applications to Yang–Mills theory. As of this writing, no complete unified compositional semantics yet exists for the full range of these manifolds; future work will develop such a formalism.

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Version history

  1. v1Submitted by Benjamin KaminskyInitial depositCurrentOct 30, 2026