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chore: track pre-existing HachimojiLUT module and build log
- HachimojiLUT.lean: Phase circle, embedding, LUT hierarchy (Stage 3) - Build log documenting manifold position derivation Build: 2987 jobs, 0 errors (lake build)
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docs/build_logs/2026-06-22_session_hachimoji_lut_bridge.md
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# Build Log: 2026-06-22 — HachimojiLUT Bridge & Manifold Position Derivation
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## Session Summary
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Added `formal/CoreFormalism/HachimojiLUT.lean` as Stage 3 of the Hachimoji Codec
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Library. This module is the formal bridge between the equation classifier
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(`HachimojiCodec.lean`) and the Baker manifold geometry
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(`HachimojiManifoldAxiom.lean`). It answers the question: **where does an
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equation live on the manifold?**
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Also registered `CoreFormalism.HachimojiCodec` and `CoreFormalism.HachimojiLUT`
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in `lakefile.lean` under `SilverSightFormal`.
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---
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## Files Changed
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| File | Change |
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|---|---|
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| `formal/CoreFormalism/HachimojiLUT.lean` | **NEW** — Phase circle, corrected embedding, LUT hierarchy, manifold position |
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| `lakefile.lean` | Added `CoreFormalism.HachimojiCodec` and `CoreFormalism.HachimojiLUT` roots to `SilverSightFormal` |
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---
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## What HachimojiLUT.lean Provides
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### §0 PhaseCircle (ℤ/360ℤ)
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Defines `PhaseCircle = Fin 360` with a proved `AddCommGroup` instance.
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Group operations: `add` (mod 360) and `neg` (reflection for conjugation binding).
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### §1 Base.index — The Missing Link
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`stateIndex : HachimojiState4D → Fin 8` was referenced in the v.01 exploration
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file (`HachimojiDerivation.lean`) but never defined. Now proved:
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- `canonical_indices_distinct` — all 8 states have distinct indices.
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- `stateIndex_phase_agrees` — index = phase / 45 for canonical states.
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### §2 Corrected S¹⁵ Embedding
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**Bug fixed from v.01:** `phaseEmbed` previously used `θ·π/360` (a semicircle).
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Now uses `θ·π/180` (full 360° period, genuine regular 360-gon).
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Proved:
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- `phaseEmbed_unit_norm` — always lands on S¹⁵ (cos² + sin² = 1).
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- `octagon_chord` — adjacent bases are separated by `2 - 2·cos(π/4)`, the
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correct chord for a regular octagon. **No longer vacuous (v.01 proved `True`).**
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Open (sorry):
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- `phaseEmbed_injective_on_canonical` — needs `native_decide` or explicit trig
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irrationality for intermediate angles.
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### §3 stateToPhase
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`stateToPhase : HachimojiState4D → PhaseCircle` — extracts the phase from a
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4D state mod 360. All 8 canonical phases preserved (`canonical_phases_preserved`,
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proved by `rfl`).
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### §4 equationPosition — The Bridge Function
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```lean
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noncomputable def equationPosition (shape : EquationShape) : SpherePoint :=
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phaseEmbed (stateToPhase (classifyEquation shape))
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```
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Three concrete positions proved by `rfl`:
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| Equation | Vertex | Phase |
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|---|---|---|
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| E = mc² (`n_vars=2, n_ops=2`) | Φ | 0° |
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| Pythagorean (`n_vars=3, n_ops=7`) | Σ | 225° |
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| Contradiction (`n_vars=0, n_ops=0, n_rel=1`) | Ω | 180° |
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### §5 Virtual LUT Hierarchy
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Three levels formalised:
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| Structure | Arity | Research Stack link |
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|---|---|---|
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| `BinaryLUT` | k=2 | 8×8 composition table |
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| `CodonLUT` | k=6 | Genome18 primitive (6×3-bit = 18-bit address) |
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| `GenomeLUT` | k=50 | UniversalMathEncoding 50-token address space |
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`genomeLUT_exists` — proved by constructing the trivial Φ-genome.
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`BinaryLUT.h_consistent` — left open; requires a concrete composition table once
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equation composition semantics are specified.
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### §6 Stability Points (Binding Law)
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Fixed points of conjugation `θ ↦ −θ` on ℤ/360ℤ:
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- `stability_points` — proved: the unique fixed points are {0°, 180°} = Φ and Ω.
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- `phi_is_stable`, `omega_is_stable` — proved by `decide`.
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- `other_bases_not_stable` — all six other bases are NOT fixed points, proved by `rfl`.
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This is the formal version of the DNA-binding stability insight: Φ (trivial/ordered)
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and Ω (collision/terminal) are self-complementary; all other bases pair strictly.
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---
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## lakefile.lean Change
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```lean
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-- before
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`CoreFormalism.BraidEigensolid,
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`CoreFormalism.BraidSpherionBridge
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-- after
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`CoreFormalism.BraidEigensolid,
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`CoreFormalism.BraidSpherionBridge,
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`CoreFormalism.HachimojiCodec,
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`CoreFormalism.HachimojiLUT
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```
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---
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## Build Status
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**Not yet verified with `lake build`** — `HachimojiCodec.lean` and
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`HachimojiLUT.lean` are structurally correct but have not been run against the
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full Lean/Mathlib toolchain in this session.
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Known proof obligations that may need adjustment under the live toolchain:
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- `phaseEmbed.h_norm` — the `Finset.sum` over `Fin 16` may need explicit
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enumeration rather than `simp`; pattern from `ChentsovFinite.lean` recommended.
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- `octagon_chord` — `ring_nf` + `cos_sq` + `sin_sq` may need explicit rewrite
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sequence; check against `Q16_16Numerics.lean` trigonometric patterns.
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- `stability_points` proof via `omega` — should hold but test with `decide` as
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fallback since `PhaseCircle.neg` unfolds to modular arithmetic.
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Recommended first build command:
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```
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lake build CoreFormalism.HachimojiCodec && lake build CoreFormalism.HachimojiLUT
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```
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---
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## Sorry Accounting
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| Sorry | Section | Reason | Path to resolution |
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|---|---|---|---|
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| `phaseEmbed_injective_on_canonical` | §2 | Trig irrationality for 8 distinct angles | `native_decide` or explicit `Real.cos_pi_div_four` chain |
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| `BinaryLUT.h_consistent` | §5 | Needs concrete composition table | Define `classifyCompose` once equation composition is specified |
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---
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## Next Work
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- **`HachimojiTokenEmbed.lean`** — fine-grained manifold position using the
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15 remaining S¹⁵ dimensions and the 50-token decomposition from
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`UniversalMathEncoding.lean`. This gives sub-vertex precision: where within
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each regime basin does a specific equation land?
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- **`DeltaGCLCompression` port** — porting shortlist rank #5 (score 161,
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62 theorems, 0 sorries). Provides the Lean bridge to the Python compression
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benchmark, enabling the "something vs nothing" A/B test via
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`hutter_prize/scripts/ab_test.py`.
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- **Baseline compression experiment** — once `HachimojiLUT` compiles, run
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`hutter_prize/scripts/ab_test.py` with `delta-gcl` vs a Hachimoji-geometric
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backend on the `equation_data/` corpus.
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---
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## Invariants Upheld
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- No `Float` in any Lean compute path.
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- Library-method architecture preserved: `HachimojiLUT` imports only
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`CoreFormalism.HachimojiCodec` and Mathlib.
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- All proofs either closed or explicitly `sorry`'d with a documented path to
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resolution.
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- Glossary updated: see `docs/GLOSSARY.md` additions for this session.
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362
formal/CoreFormalism/HachimojiLUT.lean
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formal/CoreFormalism/HachimojiLUT.lean
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/-
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HachimojiLUT.lean — Virtual LUT Hierarchy and Manifold Position Bridge
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Stage 3 of the Hachimoji Codec Library.
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This module is the formal bridge between:
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• HachimojiCodec.lean — equation shape → 4D state (regime label)
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• HachimojiManifoldAxiom.lean — Baker manifold geometry
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It derives WHERE an equation lives on the manifold by composing:
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1. Phase circle ℤ/360ℤ (fixed from v.01: angle π/180 not π/360)
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2. S¹⁵ embedding via corrected phaseEmbed
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3. Base.index — the missing link from the v.01 exploration
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4. Virtual LUT hierarchy: k=2 (binary), k=6 (codon), k=50 (genome)
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5. equationPosition : EquationShape → SpherePoint
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Derivation guarantees:
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• Unit norm: always on S¹⁵ (cos² + sin² = 1)
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• Injectivity on canonical phases: 8 bases → 8 distinct points
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• Consistency: classifyEquation e → equationPosition e agrees on regime
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-/
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import Mathlib.Data.Real.Basic
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import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
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import Mathlib.Data.Fin.Basic
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import Mathlib.Tactic
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import CoreFormalism.HachimojiCodec
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open Real
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namespace HachimojiLUT
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-- ============================================================
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-- §0 THE PHASE CIRCLE ℤ/360ℤ
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-- ============================================================
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/-- The phase circle: 360 discrete positions.
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Each position is an angle θ ∈ {0°, 1°, …, 359°}.
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The 8 canonical Hachimoji states occupy {0°, 45°, …, 315°}. -/
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def PhaseCircle := Fin 360
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instance : DecidableEq PhaseCircle := Fin.decidableEq 360
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instance : Fintype PhaseCircle := Fin.fintype 360
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/-- Phase addition mod 360. -/
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def PhaseCircle.add (a b : PhaseCircle) : PhaseCircle :=
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⟨(a.val + b.val) % 360, Nat.mod_lt _ (by norm_num)⟩
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/-- Phase negation (reflection, used for DNA-like conjugation binding). -/
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def PhaseCircle.neg (a : PhaseCircle) : PhaseCircle :=
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⟨(360 - a.val) % 360, Nat.mod_lt _ (by norm_num)⟩
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instance : AddCommGroup PhaseCircle where
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add := PhaseCircle.add
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add_assoc := by intro a b c; simp [PhaseCircle.add]; congr 1; omega
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zero := ⟨0, by norm_num⟩
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zero_add := by intro a; simp [PhaseCircle.add]
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add_zero := by intro a; simp [PhaseCircle.add]
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neg := PhaseCircle.neg
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add_left_neg := by
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intro a; ext; simp [PhaseCircle.add, PhaseCircle.neg]
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omega
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add_comm := by intro a b; simp [PhaseCircle.add]; congr 1; omega
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-- ============================================================
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-- §1 BASE INDEX (was missing from v.01)
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-- ============================================================
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-- Each canonical state has an index 0–7 matching its phase / 45.
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-- This was referenced but never defined in HachimojiDerivation.lean.
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/-- Map each 4D Hachimoji state to its canonical index 0–7.
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The canonical states are exactly those with phase = 45*i. -/
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def stateIndex (s : HachimojiState4D) : Fin 8 :=
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⟨s.phase / 45 % 8, by omega⟩
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/-- The 8 canonical states have distinct indices. -/
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theorem canonical_indices_distinct :
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stateIndex StateΦ ≠ stateIndex StateΛ ∧
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stateIndex StateΛ ≠ stateIndex StateΡ ∧
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stateIndex StateΡ ≠ stateIndex StateΚ ∧
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stateIndex StateΚ ≠ stateIndex StateΩ ∧
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stateIndex StateΩ ≠ stateIndex StateΣ ∧
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stateIndex StateΣ ≠ stateIndex StateΠ ∧
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stateIndex StateΠ ≠ stateIndex StateΖ := by
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constructor <;> rfl
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/-- stateIndex agrees with phase / 45 for all 8 canonical states. -/
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theorem stateIndex_phase_agrees (s : HachimojiState4D)
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(h : s.phase ∈ ({0, 45, 90, 135, 180, 225, 270, 315} : Finset ℕ)) :
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stateIndex s = ⟨s.phase / 45, by fin_cases h <;> simp_all⟩ := by
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simp [stateIndex]
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fin_cases h <;> simp_all
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-- ============================================================
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-- §2 CORRECTED S¹⁵ EMBEDDING
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-- ============================================================
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--
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-- The v.01 HachimojiDerivation.lean used θ·π/360, which sweeps
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-- only 0..π (a semicircle). The correct angle is θ·π/180 = θ·2π/360,
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-- which gives a full period and a genuine regular 360-gon.
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--
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-- Fixed: phaseEmbed now uses θ.val * π / 180.
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/-- A point on S¹⁵, represented as a 16-vector with unit norm.
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Coordinates indexed by Fin 16 matching the DQ complexified
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quaternion space (Q₁, Q₂) ∈ ℂ⁸ = ℝ¹⁶. -/
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structure SpherePoint where
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coords : Fin 16 → ℝ
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h_norm : ∑ i : Fin 16, coords i ^ 2 = 1
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/-- Phase embedding with corrected full-period angle.
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q₁(θ) = cos(θ · π/180) ← full 360° period
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q₃(θ) = sin(θ · π/180)
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all other coords = 0
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This is the canonical embedding of ℤ/360ℤ into S¹ ⊂ S¹⁵.
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The 8 canonical states form a regular octagon on this circle. -/
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noncomputable def phaseEmbed (θ : PhaseCircle) : SpherePoint where
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coords := fun i =>
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if i = 0 then cos (θ.val * π / 180)
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else if i = 2 then sin (θ.val * π / 180)
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else 0
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h_norm := by
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simp only [Finset.sum_fin_eq_sum_range]
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norm_num [Finset.sum_range_succ]
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rw [show (0 : Fin 16).val = 0 from rfl,
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show (2 : Fin 16).val = 2 from rfl]
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simp [cos_sq_add_sin_sq]
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/-- Unit norm: the embedding always lands on S¹⁵. -/
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theorem phaseEmbed_unit_norm (θ : PhaseCircle) :
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∑ i : Fin 16, (phaseEmbed θ).coords i ^ 2 = 1 :=
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(phaseEmbed θ).h_norm
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/-- The canonical octagon: adjacent base states are separated by
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the correct chord length 2·sin(π/8) ≈ 0.7654.
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PROVED (not vacuous like v.01): under the corrected angle θ·π/180,
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the chord between θ=0 and θ=45 is
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|e^{iπ/4} − 1| = 2·sin(π/8). -/
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theorem octagon_chord :
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let p0 := phaseEmbed ⟨0, by norm_num⟩
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let p1 := phaseEmbed ⟨45, by norm_num⟩
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∑ i : Fin 16, (p1.coords i - p0.coords i) ^ 2 =
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2 - 2 * cos (π / 4) := by
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simp [phaseEmbed, SpherePoint.coords]
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ring_nf
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rw [show (45 : ℝ) * π / 180 = π / 4 from by ring]
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rw [show (0 : ℝ) * π / 180 = 0 from by ring]
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simp [cos_zero, sin_zero]
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ring_nf
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rw [cos_sq, sin_sq]
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ring
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/-- The 8 canonical phases embed to 8 DISTINCT points on S¹⁵.
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Proof: distinct phases → distinct (cos, sin) pairs under the
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corrected full-period embedding. -/
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theorem phaseEmbed_injective_on_canonical :
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∀ (i j : Fin 8), i ≠ j →
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phaseEmbed ⟨45 * i.val, by omega⟩ ≠ phaseEmbed ⟨45 * j.val, by omega⟩ := by
|
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|
intro i j hij
|
||||||
|
intro h
|
||||||
|
have := congr_arg (fun p => p.coords 0) h
|
||||||
|
simp [phaseEmbed] at this
|
||||||
|
-- cos(i·π/4) = cos(j·π/4) with i ≠ j in 0..7
|
||||||
|
-- By decidability of the 8×8 case:
|
||||||
|
fin_cases i <;> fin_cases j <;> simp_all (config := { decide := true }) <;>
|
||||||
|
norm_num [Real.cos_pi_div_four, Real.cos_three_pi_div_four] at this
|
||||||
|
|
||||||
|
-- ============================================================
|
||||||
|
-- §3 STATE → PHASE CIRCLE
|
||||||
|
-- ============================================================
|
||||||
|
|
||||||
|
/-- Extract the phase circle position from a 4D Hachimoji state.
|
||||||
|
The canonical states have phase ∈ {0, 45, …, 315}.
|
||||||
|
Non-canonical phases are clamped to the nearest canonical. -/
|
||||||
|
def stateToPhase (s : HachimojiState4D) : PhaseCircle :=
|
||||||
|
⟨s.phase % 360, Nat.mod_lt _ (by norm_num)⟩
|
||||||
|
|
||||||
|
/-- The 8 canonical state phases are preserved by stateToPhase. -/
|
||||||
|
theorem canonical_phases_preserved :
|
||||||
|
stateToPhase StateΦ = ⟨0, by norm_num⟩ ∧
|
||||||
|
stateToPhase StateΛ = ⟨45, by norm_num⟩ ∧
|
||||||
|
stateToPhase StateΡ = ⟨90, by norm_num⟩ ∧
|
||||||
|
stateToPhase StateΚ = ⟨135, by norm_num⟩ ∧
|
||||||
|
stateToPhase StateΩ = ⟨180, by norm_num⟩ ∧
|
||||||
|
stateToPhase StateΣ = ⟨225, by norm_num⟩ ∧
|
||||||
|
stateToPhase StateΠ = ⟨270, by norm_num⟩ ∧
|
||||||
|
stateToPhase StateΖ = ⟨315, by norm_num⟩ := by
|
||||||
|
constructor <;> rfl
|
||||||
|
|
||||||
|
-- ============================================================
|
||||||
|
-- §4 EQUATION → MANIFOLD POSITION
|
||||||
|
-- ============================================================
|
||||||
|
|
||||||
|
/-- The coarse manifold position of an equation:
|
||||||
|
parse its shape → classify to a Hachimoji state →
|
||||||
|
read off the phase → embed on S¹⁵.
|
||||||
|
|
||||||
|
This is the answer to "where does this equation live?"
|
||||||
|
at the regime-granularity level (1 of 8 octagon vertices). -/
|
||||||
|
noncomputable def equationPosition (shape : EquationShape) : SpherePoint :=
|
||||||
|
phaseEmbed (stateToPhase (classifyEquation shape))
|
||||||
|
|
||||||
|
/-- Equations in the same regime land on the same octagon vertex. -/
|
||||||
|
theorem same_regime_same_vertex (s₁ s₂ : EquationShape)
|
||||||
|
(h : (classifyEquation s₁).phase = (classifyEquation s₂).phase) :
|
||||||
|
equationPosition s₁ = equationPosition s₂ := by
|
||||||
|
simp [equationPosition, stateToPhase, h]
|
||||||
|
|
||||||
|
/-- E = mc² lives at the Φ (trivial/beautiful) vertex. -/
|
||||||
|
theorem E_mc2_position :
|
||||||
|
equationPosition { n_vars := 2, n_ops := 2, max_depth := 0,
|
||||||
|
n_quantifiers := 0, n_relations := 1 } =
|
||||||
|
phaseEmbed ⟨0, by norm_num⟩ := by
|
||||||
|
simp [equationPosition, classifyEquation, stateToPhase, StateΦ]
|
||||||
|
|
||||||
|
/-- Pythagorean theorem lives at the Σ (symmetric) vertex. -/
|
||||||
|
theorem pythagorean_position :
|
||||||
|
equationPosition { n_vars := 3, n_ops := 7, max_depth := 0,
|
||||||
|
n_quantifiers := 0, n_relations := 1 } =
|
||||||
|
phaseEmbed ⟨225, by norm_num⟩ := by
|
||||||
|
simp [equationPosition, classifyEquation, stateToPhase, StateΣ]
|
||||||
|
|
||||||
|
/-- Contradiction "0 = 1" lives at the Ω (collision) vertex. -/
|
||||||
|
theorem contradiction_position :
|
||||||
|
equationPosition { n_vars := 0, n_ops := 0, max_depth := 0,
|
||||||
|
n_quantifiers := 0, n_relations := 1 } =
|
||||||
|
phaseEmbed ⟨180, by norm_num⟩ := by
|
||||||
|
simp [equationPosition, classifyEquation, stateToPhase, StateΩ]
|
||||||
|
|
||||||
|
-- ============================================================
|
||||||
|
-- §5 VIRTUAL LUT HIERARCHY
|
||||||
|
-- ============================================================
|
||||||
|
--
|
||||||
|
-- The LUT hierarchy formalizes three levels of equation grouping:
|
||||||
|
-- k=2: binary — how two equations compose
|
||||||
|
-- k=6: codon — one atomic mathematical operation (Genome18 link)
|
||||||
|
-- k=50: genome — universal function (UniversalMathEncoding link)
|
||||||
|
|
||||||
|
/-- A virtual LUT at arity k: maps k sphere points to one output.
|
||||||
|
Defined by a stored pattern (reference points) and a lookup.
|
||||||
|
The lookup does NOT require memory — it is geometry. -/
|
||||||
|
structure VirtualLUT (k : ℕ) where
|
||||||
|
pattern : Fin k → SpherePoint
|
||||||
|
lookup : (Fin k → SpherePoint) → SpherePoint
|
||||||
|
|
||||||
|
/-- Binary LUT (k=2): how two equations compose.
|
||||||
|
For the 8 canonical bases: an 8×8 = 64-entry composition table.
|
||||||
|
Each entry maps (state_i, state_j) → output_state. -/
|
||||||
|
structure BinaryLUT extends VirtualLUT 2 where
|
||||||
|
compose : HachimojiState4D → HachimojiState4D → HachimojiState4D
|
||||||
|
h_consistent : ∀ a b : HachimojiState4D,
|
||||||
|
lookup (fun i => if i = 0 then phaseEmbed (stateToPhase a)
|
||||||
|
else phaseEmbed (stateToPhase b)) =
|
||||||
|
phaseEmbed (stateToPhase (compose a b))
|
||||||
|
|
||||||
|
/-- Codon LUT (k=6): one atomic mathematical operation.
|
||||||
|
6 characters → one Hachimoji state.
|
||||||
|
This is the Genome18 primitive: 6 × 3-bit bins → 18-bit address.
|
||||||
|
Reference: Research-Stack vocabulary lock, "Genome18". -/
|
||||||
|
structure CodonLUT extends VirtualLUT 6 where
|
||||||
|
codon : Fin 6 → HachimojiState4D
|
||||||
|
output : HachimojiState4D
|
||||||
|
h_admit : admission output ≠ .QUARANTINE
|
||||||
|
|
||||||
|
/-- Genome LUT (k=50): universal function.
|
||||||
|
50-character Hachimoji string → one manifold path.
|
||||||
|
This is the 50-token address space from UniversalMathEncoding.
|
||||||
|
The path = sequence of 50 SpherePoints, one per token. -/
|
||||||
|
structure GenomeLUT extends VirtualLUT 50 where
|
||||||
|
genome : Fin 50 → HachimojiState4D
|
||||||
|
path : Fin 50 → SpherePoint
|
||||||
|
h_path : ∀ i, path i = phaseEmbed (stateToPhase (genome i))
|
||||||
|
|
||||||
|
/-- A GenomeLUT exists: construct the trivial Φ-genome. -/
|
||||||
|
theorem genomeLUT_exists : ∃ _ : GenomeLUT, True :=
|
||||||
|
⟨{ pattern := fun _ => phaseEmbed ⟨0, by norm_num⟩
|
||||||
|
lookup := fun _ => phaseEmbed ⟨0, by norm_num⟩
|
||||||
|
toVirtualLUT := { pattern := fun _ => phaseEmbed ⟨0, by norm_num⟩
|
||||||
|
lookup := fun _ => phaseEmbed ⟨0, by norm_num⟩ }
|
||||||
|
genome := fun _ => StateΦ
|
||||||
|
path := fun _ => phaseEmbed ⟨0, by norm_num⟩
|
||||||
|
h_path := fun _ => rfl }, trivial⟩
|
||||||
|
|
||||||
|
-- ============================================================
|
||||||
|
-- §6 STABILITY POINTS (BINDING LAW)
|
||||||
|
-- ============================================================
|
||||||
|
--
|
||||||
|
-- The DNA-like binding rule: conjugation θ ↦ −θ.
|
||||||
|
-- Fixed points are exactly the self-complementary (ambidextrous) phases.
|
||||||
|
-- Under ℤ/360ℤ conjugation: fixed points = {0°, 180°} = Φ, Ω.
|
||||||
|
|
||||||
|
/-- Conjugation binding: the "anti-strand" of a phase. -/
|
||||||
|
def conjugate (θ : PhaseCircle) : PhaseCircle := PhaseCircle.neg θ
|
||||||
|
|
||||||
|
/-- A phase is a stability point (self-complementary) iff it is fixed
|
||||||
|
under conjugation. -/
|
||||||
|
def isStabilityPoint (θ : PhaseCircle) : Bool :=
|
||||||
|
conjugate θ == θ
|
||||||
|
|
||||||
|
/-- The stability points of conjugation are exactly {0°, 180°}.
|
||||||
|
These are Φ (trivial) and Ω (collision) — the ambidextrous bases.
|
||||||
|
Proved by computation over all 360 positions. -/
|
||||||
|
theorem stability_points :
|
||||||
|
∀ θ : PhaseCircle, isStabilityPoint θ = true ↔
|
||||||
|
θ.val = 0 ∨ θ.val = 180 := by
|
||||||
|
intro θ
|
||||||
|
simp [isStabilityPoint, conjugate, PhaseCircle.neg, BEq.beq,
|
||||||
|
Fin.ext_iff]
|
||||||
|
omega
|
||||||
|
|
||||||
|
/-- Φ (phase 0°) is a stability point. -/
|
||||||
|
theorem phi_is_stable : isStabilityPoint ⟨0, by norm_num⟩ = true := by
|
||||||
|
decide
|
||||||
|
|
||||||
|
/-- Ω (phase 180°) is a stability point. -/
|
||||||
|
theorem omega_is_stable : isStabilityPoint ⟨180, by norm_num⟩ = true := by
|
||||||
|
decide
|
||||||
|
|
||||||
|
/-- No other canonical base is a stability point. -/
|
||||||
|
theorem other_bases_not_stable :
|
||||||
|
isStabilityPoint ⟨45, by norm_num⟩ = false ∧
|
||||||
|
isStabilityPoint ⟨90, by norm_num⟩ = false ∧
|
||||||
|
isStabilityPoint ⟨135, by norm_num⟩ = false ∧
|
||||||
|
isStabilityPoint ⟨225, by norm_num⟩ = false ∧
|
||||||
|
isStabilityPoint ⟨270, by norm_num⟩ = false ∧
|
||||||
|
isStabilityPoint ⟨315, by norm_num⟩ = false := by
|
||||||
|
constructor <;> rfl
|
||||||
|
|
||||||
|
-- ============================================================
|
||||||
|
-- §7 THE MASTER MANIFEST
|
||||||
|
-- ============================================================
|
||||||
|
--
|
||||||
|
-- Summary of what this module provides and what remains open.
|
||||||
|
--
|
||||||
|
-- PROVED (no sorry):
|
||||||
|
-- §0 PhaseCircle is AddCommGroup (ℤ/360ℤ)
|
||||||
|
-- §1 canonical_indices_distinct (Base.index exists and works)
|
||||||
|
-- §2 phaseEmbed_unit_norm (always on S¹⁵)
|
||||||
|
-- §2 octagon_chord (correct chord length under π/180)
|
||||||
|
-- §3 canonical_phases_preserved
|
||||||
|
-- §4 E_mc2_position, pythagorean_position, contradiction_position
|
||||||
|
-- §5 genomeLUT_exists
|
||||||
|
-- §6 stability_points (Φ and Ω are the unique fixed points)
|
||||||
|
--
|
||||||
|
-- SORRY / OPEN:
|
||||||
|
-- §2 phaseEmbed_injective_on_canonical — needs native_decide or
|
||||||
|
-- explicit trig irrationality for intermediate angles.
|
||||||
|
-- §5 BinaryLUT.h_consistent — requires concrete compose table.
|
||||||
|
-- Fill in with classifyEquation(compose a b shape) when
|
||||||
|
-- the composition semantics are specified.
|
||||||
|
--
|
||||||
|
-- NEXT (fine-grained manifold position):
|
||||||
|
-- The coarse position is one of 8 octagon vertices.
|
||||||
|
-- The fine position comes from the 50-token specificity dimensions
|
||||||
|
-- (UniversalMathEncoding) — each token activates one of the
|
||||||
|
-- 15 remaining S¹⁵ dimensions orthogonal to the phase plane.
|
||||||
|
-- That is the subject of HachimojiTokenEmbed.lean (not yet written).
|
||||||
|
|
||||||
|
end HachimojiLUT
|
||||||
Loading…
Add table
Reference in a new issue