[Paper Review] Presto! Digitization, Part I: From NKS Number Theory to "XORbitant" Semantics, by way of Cayley-Dickson Process and Zero-Divisor-based "Representations"

de Marrais, P. C. Robert|arXiv (Cornell University)|Mar 13, 2006

Cognitive Computing and Networks3 citations

TL;DR

This paper introduces a novel number-theoretic framework linking Nonlinear Dynamics (CT) and Network Theory (NT) through digitized binary representations in higher-dimensional algebras. By applying the Cayley-Dickson Process to generate 2^N-ions and analyzing their zero-divisors via the Box-Kite formalism, it models complex systems as semantically rich nodules, revealing fractal-like patterns and enabling a unified semantics for chaos, myth, and representation through XOR-based operations and nullity partitions.

ABSTRACT

The objects of the great Nonlinear Revolutions - Catastrophes and Chaos in the 1960s-70s (henceforth, CT); and, small-world and scale-free Network Theory (NT), emerging quite recently - will be spliced together by a New Kind of Number Theory, focused on digitizations (i.e., binary strings). NT nodes then become feature-rich (nodules in a rhizosphere) of CT contents. The Box-Kite formalism of zero-divisors (ZD's) - first showing in the 16-D Sedenions, then in all higher 2^N-ions derived from Imaginaries by Cayley-Dickson Process (CDP) - can model such enriched nodules. Its (bit-string XOR-ing vs. matrix-multiplying) operations unfold into representations of the objects linked in CT with partitions of Nullity: Singularities. The route from here to fractals and Chaos, via CDP extensions to 2^N-ions for growing N, will involve us in graphics of higher-dimensional carry-bit overflow, as manifest in the mandala-like patterns of emanation tables (the rough equivalent, for ZD's, of group theorists' Cayley Tables). I'll lead into this with a quote about Hjelmslev's Net (which I'll claim is the CDP manque') from a famous postmodern text, Deleuze and Guattari's A Thousand Plateaus (where rhizosphere imagery arose). With strong assists from the CT-based structuralism of Jean Petitot, Algirdas Greimas's Semiotic Square will show us how to explicitly link CT to semiotic foundations via ZD representations, while the infinite-dimensional ZD meta-fractal or Sky where Box-Kites fly - first appearing in the 32-D Pathions and incorporating the higher 2^N-ions - will provide sufficient lebensraum for Levi-Strauss's Canonical Formula of mythwork to unfurl in. (These results serve to extend my NKS 2004 paper, available at the Wolfram Science website.)

Motivation & Objective

• To unify Catastrophe Theory (CT) and Network Theory (NT) through a digitized, binary-string-based number system rooted in Nonlinear Science.
• To model CT phenomena as feature-rich nodules in a rhizospheric network by embedding them in zero-divisor structures of higher-dimensional algebras.
• To extend the Box-Kite formalism—originally from 16-D Sedenions—into higher 2^N-ions to represent complex dynamics and singularities via nullity partitions.
• To establish a semantic foundation for chaos and myth by linking zero-divisor representations to semiotic structures like Greimas’s Square and Levi-Strauss’s Canonical Formula.
• To demonstrate that higher-dimensional carry-bit overflow patterns in 2^N-ions generate mandala-like emanation tables, revealing fractal behavior

Proposed method

• Applying the Cayley-Dickson Process (CDP) to iteratively construct 2^N-ions from the complex numbers, starting from quaternions and extending to 32-D Pathions and beyond.
• Using the Box-Kite formalism to identify and analyze zero-divisors (ZDs) in 16-D Sedenions and higher 2^N-ions, treating them as structural units for modeling nodules in a networked system.
• Representing CT dynamics as binary strings (digitizations) and mapping them to ZD structures via bit-string XOR operations, replacing traditional matrix multiplication.
• Constructing 'emanation tables'—analogous to Cayley tables for groups—that visualize ZD relationships and reveal mandala-like, fractal-like patterns in higher-dimensional carry-bit overflow.
• Integrating semiotic frameworks such as Greimas’s Semiotic Square and Levi-Strauss’s Canonical Formula of mythwork into the ZD representation space to ground abstract dynamics in semantic structure.
• Leveraging Deleuze and Guattari’s rhizosphere metaphor and Petitot’s structuralist approach to interpret the resulting algebraic structures as dynamic, self-organizing systems

Experimental results

Research questions

• RQ1 How can zero-divisors in 2^N-ions (generated via the Cayley-Dickson Process) serve as structural units for modeling complex systems in Catastrophe Theory and Network Theory?
• RQ2 What is the role of bit-string XOR operations in representing CT dynamics and singularities, compared to traditional matrix multiplication?
• RQ3 How do emanation tables—derived from ZD relationships in 2^N-ions—generate fractal-like, mandala-like patterns indicative of higher-dimensional carry-bit overflow?
• RQ4 In what way can the Box-Kite formalism in 32-D Pathions and beyond provide sufficient 'lebensraum' for the unfolding of Levi-Strauss’s Canonical Formula of mythwork?
• RQ5 How can semiotic structures like Greimas’s Square be explicitly embedded within zero-divisor representations to link chaos theory with foundational semantics?

Key findings

• The Box-Kite formalism, initially observed in 16-D Sedenions, extends to all higher 2^N-ions generated via the Cayley-Dickson Process, enabling systematic analysis of zero-divisors.
• Bit-string XOR operations on digitized binary representations of CT phenomena provide a computationally efficient and structurally revealing alternative to matrix multiplication in modeling nodules.
• Emanation tables—representing ZD relationships in 2^N-ions—exhibit mandala-like, fractal-like patterns that emerge from higher-dimensional carry-bit overflow during algebraic operations.
• The 32-D Pathions mark the first appearance of the infinite-dimensional ZD meta-fractal or 'Sky,' where Box-Kites can fly, providing a topological space for complex symbolic systems.
• Semiotic frameworks such as Greimas’s Square and Levi-Strauss’s Canonical Formula can be explicitly realized within the ZD representation space, linking chaos theory to mythic and linguistic semantics.
• The framework extends the author’s 2004 NKS paper by embedding nonlinear dynamics and network structures into a unified algebraic and semantic formalism grounded in digitized number theory