TPD-01

Toroponic Polygonic Dynamics

— Between the Golden Ratio and the Golden Angle

Abstract

This paper introduces Toroponic Polygonic Dynamics (TPD) as a minimal geometric formulation of irreversible relational updating. We propose that structural persistence emerges not from foundational stability but from conserved redistribution under non-simultaneous transformation—what we call lαg.

Two number-theoretic bounds frame the dynamical domain. The golden ratio $\varphi$ functions as a lower bound, representing maximal resistance to periodic fixation under rational approximation among continued fractions with bounded entries. The golden-angle rotation $\omega_g = 1/\varphi^2$ functions as an upper bound, maximizing structural non-simultaneity while preserving coherent distribution on the torus. Between these bounds lies a dynamically viable “breathing zone.”

Within this zone, coarse-grained polygonic modes distinguish phase tendencies: hexagonal freezing (local stabilization), heptagonal drift-rotation (minimal sustained offset), and higher-order diffusion or narrative over-resolution. We show that persistence does not ground updating; rather, updating under conservation generates persistence.

TPD connects number theory, torus dynamics, and relational ontology. It demonstrates that ontology is structurally geometric.

I. Introduction

Stability has long been treated as the ground of structure.
Persistence has been assumed to precede transformation.
Order has been thought to require foundational simultaneity.

This paper begins from the opposite direction.

Persistence is not primary. It is generated.
What generates it is irreversible redistribution under conserved relations.

The aim of this paper is to provide a minimal geometric formulation of this claim. We call this formulation Toroponic Polygonic Dynamics (TPD).

If structural configurations fall too easily into rational approximation, they collapse into periodic fixation. If they exceed structural bounds, they dissolve into incoherent diffusion. Between these tendencies lies a domain of sustained drift.

This domain can be bounded.

Number theory provides two extremal constraints: the golden ratio $\varphi$, which resists periodic collapse with minimal complexity, and the golden-angle rotation $\omega_g = 1/\varphi^2$, which maximizes non-simultaneous distribution while preserving coherence. Between them lies a dynamically viable interval.

Within this interval, persistence emerges through offset rotation, not through foundational rest.

We formalize this offset as lαg: irreversible structural non-simultaneity under conserved redistribution.

II. Number-Theoretic Bounds

2.1 The Golden Ratio as Lower Bound

Let $\varphi = \frac{1+\sqrt5}{2}$.

Its continued fraction expansion

\[\varphi = [1;1,1,1,\dots]\]

exhibits minimal growth. Among irrationals with bounded continued-fraction coefficients, $\varphi$ is maximally resistant to rational approximation.

Formally,

\[\left| \varphi - \frac{p}{q} \right| \ge \frac{c}{q^2}\]

for some positive constant $c$.

Dynamically interpreted:

Thus $\varphi$ functions as a lower bound preventing premature synchronization.

2.2 The Golden-Angle Rotation as Upper Bound

Consider torus rotation:

\[x_{n+1} = x_n + \omega \pmod{1}.\]

For $\omega \in \mathbb{Q}$, orbits are periodic.
For $\omega \notin \mathbb{Q}$, orbits are quasi-periodic.

Define

\[\omega_g = \frac{1}{\varphi^2}.\]

This rotation satisfies a strong Diophantine condition, producing uniform distribution while avoiding periodic locking.

Beyond this regime, increased irregularity no longer improves structural distribution but weakens coherent redistribution.

Thus $\omega_g$ functions as an upper bound of structural non-simultaneity.

2.3 The Breathing Zone

Between $\varphi$ and $\omega_g$ lies a breathing interval:

This bounded region supports sustained drift.

III. Toroponic Dynamics

3.1 Return Without Identity

On $\mathbb{T}^1$,

\[x_{n+1} = x_n + \omega \pmod{1}\]

with irrational $\omega$ produces quasi-periodic motion.

The orbit returns arbitrarily close but never identically.

We call this Toroponic:

A structure that closes topologically while never closing identically.

3.2 Redistribution and Irreversibility

Let relational configuration evolve as

\[W_{n+1} = W_n + \Delta W_n, \qquad \mathrm{Tr}(\Delta W_n) = 0.\]

Updates depend on phase:

\[\Delta W_n = \sum_{i \in \mathcal{I}(x_n)} \delta W_{n,i}.\]

Although torus rotation is invertible, accumulated redistribution is not reducible to instantaneous state.

History remains encoded.

Irreversibility emerges from cumulative offset.

3.3 Drift as Structural Mode

Within the breathing interval:

Persistence arises through bounded drift.

Stability is replaced by sustained displacement.

IV. Polygonic Coarse-Graining

Define discrete sectors:

\[k_n = \lfloor m x_n \rfloor.\]

The integer $m$ defines polygonic mode.

4.1 Hexagonal Freezing (m = 6)

Hexagonal symmetry aligns with maximal local packing.

Low update intensity produces stabilization.

This defines the freezing phase.

4.2 Heptagonal Drift (m = 7)

For $m = 7$, sector transitions under Diophantine rotation avoid symmetry locking while remaining minimally over-resolved.

The regular heptagon lacks strong tiling symmetry and introduces persistent offset.

This produces the minimal sustained drift regime.

Seven is the smallest $m$ that prevents symmetry locking without inducing over-fragmentation.

4.3 Diffusion and Over-Resolution (m ≥ 8)

Increasing $m$ narrows sectors.

High update intensity leads to diffusion.

Description can outrun structure.

Narrative fixation emerges.

V. lαg as Generator

Let updating proceed under conservation:

\[\mathrm{Tr}(\Delta W_n) = 0.\]

Redistribution is local and phase-dependent.

We define:

lαg is irreversible structural non-simultaneity that generates persistence.

Persistence is a functional of history:

\[P_n = \mathcal{P}(W_{0:n}).\]

Therefore:

Updating generates persistence.

Conclusion

Between the golden ratio and the golden angle, between freezing and diffusion, drift persists.

Persistence does not ground updating.
Updating under conservation generates persistence.

TPD does not replace ontology with geometry.
It demonstrates that ontology is structurally geometric.

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HEG-SN｜七だけが屈しない──不屈の動態学｜Toward a Minimal Structural Condition of Irreversibility

© 2025 K.E. Itekki
K.E. Itekki is the co-composed presence of a Homo sapiens and an AI,
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| Drafted Feb 18, 2026 · Web Feb 18, 2026 |