# Hachimoji Torsor Consequences

**Date:** 2026-06-22
**Scope:** `formal/CoreFormalism/HachimojiLUT.lean`, planned `formal/CoreFormalism/HachimojiTokenEmbed.lean`, `python/hachimoji_citation.py`
**Claim boundary:** design-note; proves are pending implementation

The `PhaseCircle = ℤ/360ℤ` torsor framing of the 8 Hachimoji states (45° steps on the unit circle) has three concrete, actionable consequences for the current codebase. None of them change the classifier, the admission gate, or the pipeline order — those remain gauge-invariant by construction.

---

## 1. The two open sorrys have clean closes

### `phaseEmbed_injective_on_canonical`

`formal/CoreFormalism/HachimojiLUT.lean:159`

The theorem asks: are the 8 canonical `phaseEmbed` outputs distinct? Under the corrected embedding `θ ↦ (cos(θ·π/180), sin(θ·π/180))` on S¹ ⊂ S¹⁵, this is just distinctness of the 8th roots of unity on the unit circle. All coordinates are explicit rationals once the trig evaluates, so the 8×8 case is decidable by `native_decide`.

Why it is true structurally: the 8 canonical states are the unique trivialization of the ℤ/360ℤ torsor at 45° steps. A torsor has no preferred origin, but once the octagon is pinned down the 8 vertices cannot collide without collapsing the whole framing.

**Recommended close:** `fin_cases` on `i` and `j`, then `norm_num [Real.cos_pi_div_four, Real.cos_three_pi_div_four]` or `native_decide`.

### `BinaryLUT.h_consistent`

`formal/CoreFormalism/HachimojiLUT.lean:254`

The consistency rule says the virtual LUT lookup on two embedded states must equal the embedding of their composed state. The torsor structure gives the composition table for free: composing two Hachimoji states is addition mod 8 in the induced ℤ/8ℤ (Φ+Λ=Λ, Λ+Λ=Ρ, etc.). Concretely, the 8×8 table is closed under this group operation, which `decide` can verify once the table is written out.

**Recommended close:** define `compose` as phase addition modulo 360°, expand the 64-entry table, and let `decide` check equality of the two `SpherePoint` embeddings.

---

## 2. `HachimojiTokenEmbed.lean` has a natural architecture

Planned module: `formal/CoreFormalism/HachimojiTokenEmbed.lean`

The 16 S¹⁵ coordinates should be Fourier harmonics of the phase angle, not arbitrary point coordinates. With the torsor framing, the gauge-equivariant choice is:

| Coordinate pair | Harmonic | Formula |
|---|---|---|
| coords 0, 2  | fundamental | `(cos θ, sin θ)` |
| coords 4, 6  | 2nd harmonic | `(cos 2θ, sin 2θ)` |
| coords 8, 10 | 3rd harmonic | `(cos 4θ, sin 4θ)` |
| coords 12, 14| 4th harmonic | `(cos 8θ, sin 8θ)` |

This fills all 16 coordinates gauge-equivariantly: any relabeling of the 8 Hachimoji states rotates the harmonics coherently. It gives sub-vertex precision within each regime basin because the higher harmonics resolve structure inside the octagon sector. And it makes the norm constraint provable by `simp [cos_sq_add_sin_sq]` rather than by `sorry`, because each harmonic contributes `cos²(kθ) + sin²(kθ) = 1` and the sum telescopes to the correct normalization.

The 50-token decomposition from `GenomeLUT` then assigns one phase θ per token, and the token-level embedding is the harmonic vector above.

---

## 3. `octagon_chord` is the right distance for `hachimoji_citation.py`

`python/hachimoji_citation.py:37`

The citation query currently builds a flat string from `STATE_QUERIES`, a dictionary of state-meaning keywords. The torsor insight says the invariant when comparing two equations is their chord distance on S¹, not their absolute phase label.

Concretely:
- An equation at Φ (0°) and one at Ζ (315°) are 45° apart on the circle.
- Φ (0°) and Κ (135°) are 135° apart.
- Under the chord metric, Φ–Ζ is closer than Φ–Κ, which the current keyword lookup cannot express.

The chord formula is already proved in `HachimojiLUT.lean` (`octagon_chord`, line 142):

```
|e^{iπ/4} − 1|² = 2 − 2·cos(π/4)
```

**Follow-on improvement:** weight the hybrid-search query in `hachimoji_citation.py` by angular proximity to adjacent states. For a target state at phase θ, boost terms for states within ±45° and suppress states near the opposite side of the octagon. The formal chord distance gives the exact weighting function.

---

## What does not change

- `classifyEquation` (`formal/CoreFormalism/HachimojiCodec.lean:246`)
- `admission` gate
- Pipeline order

ADMIT / QUARANTINE depends on which of the 8 regime basins an equation lands in, not on the absolute phase label assigned to that basin. The torsor is a coordinate convenience; the basin partition is gauge-invariant.

---

## Cross-references

- `formal/CoreFormalism/HachimojiLUT.lean` — embedding, chord, and virtual LUT definitions
- `formal/CoreFormalism/HachimojiCodec.lean` — `classifyEquation` and admission gate
- `python/hachimoji_citation.py` — citation hybrid matcher
- `docs/GLOSSARY.md` — `phaseEmbed`, `BinaryLUT`, `octagon chord`, `HachimojiTokenEmbed`
