Y-Theory triadic torus coordinate laboratory

The geometry is calculated from two normalized triadic phase circles. Seven delayed depth-relations instantiate the current source architecture in the default view; diagnostic N controls test implementation sensitivity. This is a partial projection, not the complete Mind engine.

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Proof and fidelity boundary. Exact here means only this: if I+O=1, (O−I)²+ℛ²=1, A>a>0, and the displayed non-negative weights sum to one, then the two phase coordinates map reversibly to a ring torus and 0≤C≤1. The simulator consumes the source-defined seven-depth Mind architecture; it is not being asked to re-prove seven or the twelve-address recursive cell. The delayed phase/residue equations are a bounded prototype, not established canonical Mind dynamics. Each depth is presently compressed to one phase, so the runtime does not yet instantiate its full 3+9 cell, dimensional breadth, same-depth resonance, or whole-field return. A 4D reference-frame projection is a calculated view transform, not observerhood or consciousness. The two-phase address is not the complete delayed history-state.

Model controls

Preset

Projection

Construction stage 4 Motion speed 1.00× Trajectory memory 900

Show 7 depth identities

Show equation-generated mesh

Map closure coherence into solid-torus radius

Geometry

Whole radius A

Local radius a

Diagnostic depth N (source-faithful value = 7; other values are counterfactual controls)

ω local

ω whole

Depth weights

Delay, coupling, residue

τ depth

τ whole→L1

τ L7→whole

κ depth

κ down

κ up

Residue decay γ

Residue gain η

4D reference-frame projection

Rotation q₁–q₃ 0° Rotation q₂–q₄ 0° Rotation q₁–q₄ 0°

Exports

Original derivation Repaired Python model Original validation report Candidate boundaries

depth identities L1…L7collective closurewhole returnrelation/residue

Exact reverse-engineering

Local triad: I=(1−cos θ)/2, O=(1+cos θ)/2, ℛ=sin θ

Whole triad: I_H=(1−cos φ)/2, O_H=(1+cos φ)/2, ℛ_H=sin φ

T(X_L,X_H)=((A+a(O_L−I_L))(O_H−I_H), (A+a(O_L−I_L))ℛ_H, aℛ_L)

Implicit check: (√(x²+y²)−A)²+z²=a²

Under the displayed chart assumptions, each ring-torus address is exactly decodable into a local normalized triadic coordinate and a whole-level normalized triadic coordinate. The selected N layers are delayed instances of the candidate local operator; their weighted circular closure supplies Θ, while φ carries the relation of that closure to the whole.

The torus is a many-to-one two-phase projection of a larger state containing N phases, N+1 residue states, parameter values, and delayed system history. A return to the same torus coordinates does not imply return to the same complete model-state.

Y-Theory triadic torus coordinate laboratory

Exact reverse-engineering

Live numerical substitution

Simulation validation