The Braid Field Papers: DIAT, AVMR, and Quasi-Periodic Integer Geometry

This work introduces a computational and geometric framework derived from integer structure, composed of three primary components:

(1) DIAT (Dual-Interval Algebraic Transform), which encodes each integer n in terms of its distances to adjacent perfect squares, producing a minimal algebraic representation (a, b, ab, a-b) that captures geometric position, polarity, and interaction potential.

(2) AVMR (Algebraic Vector Mountain Range), a hierarchical append-only structure that lifts DIAT into a vector-valued computational system. Each leaf encodes local braid-field state, including tip coordinates, standing-wave field residues, interaction scores, timing parameters (slot, phase), parity, jitter tolerance, and codon window structure. Internal nodes merge vector states using an algebraic rule incorporating additive composition, parity propagation, and cross-boundary resonance detection.

(3) A quasi-periodic integer geometry perspective inspired by Erdős-style combinatorial reasoning, in which integers are partitioned into quadratic shells and mapped into wave-like structures. Linear traversals through this space reveal structured intersections corresponding to factor-like relationships, suggesting a geometric sieve interpretation.

Together, these components define a system in which integers are transformed into interacting signals, signals into structured timing-coded events, and events into hierarchical vector summaries. The resulting framework supports interpretations as a signal-processing substrate, a geometry-native instruction system, and an append-only proof structure with embedded computation.

This work was developed by Brandon Schneider (No One Everywhere LLC) as part of an independent research effort exploring geometry-native computation and signal-structured number systems.

AI Assistance Disclosure:
Portions of the conceptual structuring, formalization, and code scaffolding in this work were developed with the assistance of OpenAI’s ChatGPT (GPT-5.3). The model was used as a tool to help organize, refine, and express the underlying ideas; all core concepts, system design, and research direction originate from the author.