TriOcta Toy Model v4.0 — A Numerical Laboratory for Geometric Dominance, Metastability, and Structured Regime Transitions

This release (v4.0) presents a consolidated version of the TriOcta Toy Model together with a new diagnostic framework for probing its internal geometric organization.

The model combines a three-node nonlinear phase system, discrete phase scaffolding, and an emergent geometric manifold constructed from both macro-scale corridor geometry and micro-scale chiral phase structure. While previous versions established stability boundaries, metastability, and delayed instability, v4.0 introduces a reproducible method for analyzing how the system organizes internally across these regimes.

A new diagnostic script (diagnose_bottom_lid.py) defines alignment observables between:

• the macro corridor geometry,

• the chiral embedding derived from phase relations, and

• the blended emergent orientation manifold.

From these observables, a geometric dominance order parameter

Δ(t)=⟨Z^vec,Z^macro⟩−⟨Z^vec,Z^chiral⟩\Delta(t)=\langle \hat Z_{\mathrm{vec}},\hat Z_{\mathrm{macro}}\rangle -\langle \hat Z_{\mathrm{vec}},\hat Z_{\mathrm{chiral}}\rangleΔ(t)=⟨Z^vec,Z^macro⟩−⟨Z^vec,Z^chiral⟩

is constructed, which cleanly separates macro-dominant, chiral-dominant, and competitive regimes.

Parameter sweeps near the stability boundary demonstrate that internal organization is an emergent and tunable property of the dynamics. Small changes in the phase synchronization parameter can qualitatively shift dominance between organizing structures, even though the underlying model equations remain unchanged. Importantly, no evidence is found for a simple geometric inversion (e.g. a 180° top–bottom flip), constraining earlier speculative interpretations and strengthening the geometric interpretation of metastability.

This release therefore marks a transition from exploratory behavior to a structured numerical laboratory: the model now supports systematic investigation of regime structure, dominance transitions, and internal geometry.

All results are fully reproducible using the included scripts and documented command-line examples.