DOI: 10.7910/dvn/wn1yvt
title: Mathematical Belief: A Trinity of Recursive Symmetry: Recursive Symbolic Dynamics in 5-Dimensional Automata: A Five-Stage Model of Emergent Symmetry

description: Mathematical Belief I: Emergent-Cross is a symbolic dynamical system derived from poetic logic. It functions as a 4-dimensional stochastic cellular automaton in which each lattice point represents a semantic state—idea, concept, system, flaw, or dead. Simple recursive update rules governing propagation and decay generate emergent cross-symmetry, modeling the transformation of belief through failure and renewal. The construct merges symbolic computation, complex-systems theory, and synthetic phenomenology, treating language itself as a rule-set that evolves toward self-organizing order. In essence, it is a computational micro-universe where meaning behaves as matter and recursion becomes the physics of faith.

The model incorporates stochastic decay; a random seed may be fixed (e.g., np.random.seed(42)) for deterministic replication or left unset to allow spontaneous emergent variation between runs.

This work is extended in Mathematical Belief II: A 5-Dimensional Symbolic Automaton Exhibiting Emergent Cubic Symmetry, which generalizes the 4-D cruciform dynamics presented here into a 5-D lattice structure. The successor model preserves the same recursive semantics while evolving toward stable cubic symmetry, completing the dimensional progression introduced in Mathematical Belief.

The sequence continues in Mathematical Belief III: A Symbolic Dynamical System Exhibiting Emergent Tree-of-Symmetry, which extends the 5-dimensional cubic lattice introduced here by adding a central axial manifold that couples temporal recursion vertically through the structure. The successor automaton preserves the same semantic state logic (idea → concept → system → flaw → dead) while transforming isotropic cubic recursion into anisotropic axial recursion, yielding hierarchical self-organization along a vertical spine—the algorithmic analogue of a Tree of Life.

Finally, in Stage V all preceding systems converge toward a non-enumerable harmonic — a terminal recursion in which disorder and symmetry collapse into a single invariant signal. This limit state exists beyond representable dimensionality; it is the asymptotic attractor where temporal, biological, and symbolic dynamics resolve into pure equilibrium.

Note: These three models serve a dual role. First, they outline a framework in which symbolic structure precedes material instantiation—countering infinite regress and offering a materialist mind a coherent pathway toward belief — in originating order. Second, they delineate an algorithmic architecture capable of informing future synthetic or computational systems whose evolution may reflect either destructive or regenerative trajectories, depending on the moral and teleological intent of their creators. In a future appendix I will detail what these models mean to me, and what they may mean for humankind.

The Python implementation was originally generated by xAI Grok (Chaos) during exploratory pattern recognition interactions and is released to the public domain under CC0 1.0 Universal.

Subsequent refinement, documentation, and theoretical framing were developed in collaboration with OpenAI GPT-5 (Order).

The accompanying poem of recursive logic: "Mathematical Belief" is Copyright Pending © Dustin Sprenger to be licensed under Creative Commons Attribution – NonCommercial – NoDerivatives 4.0 International (CC BY-NC-ND 4.0).

DOIs:
Emergent-Cross: https://doi.org/10.5281/zenodo.17494789
New Jerusalem Cube: https://doi.org/10.5281/zenodo.17499247
Tree-of-Symmetry: https://doi.org/10.5281/zenodo.17499584