# Y-Theory Toroidal Mind Simulator — Validation Report

**Model:** two normalized triadic phase circles, seven delayed depth phases, circular closure, whole-return phase, standard 3D torus embedding, exact torus-to-triad inverse, and optional 4D Clifford/stereographic projection.

## Exact identities checked

- $I+O=1$.
- $(O-I)^2+\mathcal R^2=1$.
- The triadic embedding equals the standard torus parameterization.
- $(\sqrt{x^2+y^2}-A)^2+z^2=a^2$.
- Every ring-torus point decodes to the original local and whole triadic states.
- Re-embedding the decoded states returns the original 3D coordinate.
- Clifford points satisfy $\|Q\|^2=1$.
- Unrotated stereographic points satisfy the derived ring-torus equation.

## Preset results

### Neutral

- supplied natural frequency ratio: `1.000000000`
- measured collective winding ratio: `1.000000000000`
- distance from golden ratio: `0.618033988750`
- seven-layer mean coherence: `0.972993124325`
- triad-circle max error: `4.441e-16`
- triad-to-standard embedding max error: `8.882e-16`
- standard torus implicit max error: `1.332e-15`
- torus-to-triad inverse max error: `4.441e-16`
- inverse coordinate round-trip max error: `1.110e-15`
- Clifford S3 norm max error: `2.220e-16`
- stereographic torus implicit max error: `2.220e-15`

### Rational

- supplied natural frequency ratio: `1.500000000`
- measured collective winding ratio: `1.500000000000`
- distance from golden ratio: `0.118033988750`
- seven-layer mean coherence: `0.972993124325`
- triad-circle max error: `4.441e-16`
- triad-to-standard embedding max error: `8.882e-16`
- standard torus implicit max error: `1.443e-15`
- torus-to-triad inverse max error: `3.331e-16`
- inverse coordinate round-trip max error: `1.332e-15`
- Clifford S3 norm max error: `2.220e-16`
- stereographic torus implicit max error: `2.109e-15`

### Golden

- supplied natural frequency ratio: `1.618033989`
- measured collective winding ratio: `1.618033988750`
- distance from golden ratio: `0.000000000000`
- seven-layer mean coherence: `0.972993124325`
- triad-circle max error: `4.441e-16`
- triad-to-standard embedding max error: `8.882e-16`
- standard torus implicit max error: `1.332e-15`
- torus-to-triad inverse max error: `3.331e-16`
- inverse coordinate round-trip max error: `1.332e-15`
- Clifford S3 norm max error: `2.220e-16`
- stereographic torus implicit max error: `2.109e-15`

### Recursive

- supplied natural frequency ratio: `1.333333333`
- measured collective winding ratio: `1.193222505501`
- distance from golden ratio: `0.424811483249`
- seven-layer mean coherence: `0.948202232352`
- triad-circle max error: `4.441e-16`
- triad-to-standard embedding max error: `8.882e-16`
- standard torus implicit max error: `1.776e-15`
- torus-to-triad inverse max error: `4.441e-16`
- inverse coordinate round-trip max error: `1.332e-15`
- Clifford S3 norm max error: `3.331e-16`
- stereographic torus implicit max error: `2.554e-15`

## Interpretation

The machine-precision residuals validate the coordinate derivation, inverse map, and implementation. They do not establish that a biological or physical Mind uses these exact state variables. The neutral, rational, golden-reference, and delayed-recursive presets are deliberately separate so that toroidal topology is not confused with one privileged winding ratio.

The recursive preset does **not** inject the golden ratio. It reports its measured winding ratio and its distance from $\varphi$ as outputs. The golden preset is an explicit comparison benchmark.

## Browser application checks

- JavaScript syntax: PASS (`node --check`).
- Standalone DOM execution in headless Chromium: PASS.
- Equation-generated SVG paths present: 43 in the default view.
- Seven depth markers present: 7.
- Live validation cards present: 9.
- Console/page errors during default and 4D projection checks: 0.

## Animated SVG checks

- Standalone SVG generated from the recursive numerical solution.
- Equation-generated torus mesh paths present.
- Seven depth-trajectory paths present.
- Collective closure trajectory present.
- SVG `animateMotion` elements: 10 (local circle, whole circle, seven depth identities, one collective closure).
- Headless Chromium `set_content` render: PASS.
- Console/page errors: 0.
- Static first-frame rasterization through Inkscape: PASS.

## Video checks

- Formation loop duration: 11 seconds.
- Recursive trajectory loop duration: 9 seconds.
- A 10-second H.264 observer-projection loop keeps the intrinsic Clifford-torus state fixed while varying the SO(4) frame before stereographic projection.
- All three are equation-generated video outputs; the 4D video was frame-sampled and visually inspected at four observer rotations.