The mathematical spine: self-encoding dynamics, the constraint hierarchy, and dimension selection on the graded hypertorus T⁸.
Jared D. Dunahay · AEO Trivector LLC · Bedford, NH
View Research →The framework is anchored by four constants that together determine the geometry of self-encoding dynamics. All four are exact — not fitted parameters.
Fixed point of x = e^{−x}. The primitive constraint: attractor-stable systems satisfy this relation.
Governs dimension selection and the incompleteness parameter β.
Exact. Governs global integration structure in the spectral triple.
Controls balance between structure and dynamics; encodes the incompleteness of self-reference.
The equilibrium dimension … is determined by the balance between the golden ratio and the spinor complement:
At …, this yields … — selecting three spatial dimensions through three independent mechanisms:
The framework is organised as six nested levels. Each level adds a constraint that the system must satisfy to achieve self-encoding stability.
| Level | Name | Constraint | Role |
|---|---|---|---|
| 0 | Primitive | … | Self-encoding fixed point; attractor ground state |
| 1 | Spinor Complement | … | Orthogonal completion; spinor boundary condition |
| 2 | Dimensional Resonance | … | Selects three spatial dimensions via golden-ratio balance |
| 3 | Golden Incompleteness | … | Encodes irreducible incompleteness of self-reference |
| 4 | Spectral Triple | … | Full noncommutative geometry; Dirac operator on T⁸_Θ |
| 5 | Attractor Stability | … | Lyapunov-stable dynamics on the graded hypertorus |
The geometry is realised on the graded hypertorus …, a noncommutative deformation of the eight-dimensional torus by the skew-symmetric matrix …. The four factors encode the four constraint levels: primitive, architectural, dimensional, and derived.
Following Connes–Landi (arXiv:math/0011194), the framework is formalised as a spectral triple …:
Smooth functions on the deformed torus T⁸_Θ; noncommutative algebra of observables
Spinor space L²(T⁸_Θ, S); representation space for the algebra action
Self-adjoint operator encoding the metric geometry; spectrum bounded by μ
The deformation parameter … is constrained by the four constants: its eigenvalues are functions of …, ensuring that the spectral triple is self-encoding — the geometry encodes its own constraint structure.
Preliminary empirical work identifies three regimes in recurrent neural network dynamics, corresponding to the three constraint levels below the spectral triple:
Dynamics fail to reach the μ fixed point; attractors are unstable or degenerate.
Dynamics approach but do not fully satisfy the spinor complement constraint Ω; partial self-encoding.
Dynamics satisfy all three lower-level constraints; attractors are stable and interpretable.
This taxonomy is preliminary. Formal empirical validation is the subject of ongoing research. See /research for current status.