/- HachimojiLUT.lean — Virtual LUT Hierarchy and Manifold Position Bridge Stage 3 of the Hachimoji Codec Library. This module is the formal bridge between: • HachimojiCodec.lean — equation shape → 4D state (regime label) • HachimojiManifoldAxiom.lean — Baker manifold geometry It derives WHERE an equation lives on the manifold by composing: 1. Phase circle ℤ/360ℤ (fixed from v.01: angle π/180 not π/360) 2. S¹⁵ embedding via corrected phaseEmbed 3. Base.index — the missing link from the v.01 exploration 4. Virtual LUT hierarchy: k=2 (binary), k=6 (codon), k=50 (genome) 5. equationPosition : EquationShape → SpherePoint Derivation guarantees: • Unit norm: always on S¹⁵ (cos² + sin² = 1) • Injectivity on canonical phases: 8 bases → 8 distinct points • Consistency: classifyEquation e → equationPosition e agrees on regime -/ import Mathlib.Data.Real.Basic import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic import Mathlib.Data.Fin.Basic import Mathlib.Tactic import CoreFormalism.HachimojiCodec open Real namespace HachimojiLUT -- ============================================================ -- §0 THE PHASE CIRCLE ℤ/360ℤ -- ============================================================ /-- The phase circle: 360 discrete positions. Each position is an angle θ ∈ {0°, 1°, …, 359°}. The 8 canonical Hachimoji states occupy {0°, 45°, …, 315°}. -/ def PhaseCircle := Fin 360 instance : DecidableEq PhaseCircle := Fin.decidableEq 360 instance : Fintype PhaseCircle := Fin.fintype 360 /-- Phase addition mod 360. -/ def PhaseCircle.add (a b : PhaseCircle) : PhaseCircle := ⟨(a.val + b.val) % 360, Nat.mod_lt _ (by norm_num)⟩ /-- Phase negation (reflection, used for DNA-like conjugation binding). -/ def PhaseCircle.neg (a : PhaseCircle) : PhaseCircle := ⟨(360 - a.val) % 360, Nat.mod_lt _ (by norm_num)⟩ instance : AddCommGroup PhaseCircle where add := PhaseCircle.add add_assoc := by intro a b c; simp [PhaseCircle.add]; congr 1; omega zero := ⟨0, by norm_num⟩ zero_add := by intro a; simp [PhaseCircle.add] add_zero := by intro a; simp [PhaseCircle.add] neg := PhaseCircle.neg add_left_neg := by intro a; ext; simp [PhaseCircle.add, PhaseCircle.neg] omega add_comm := by intro a b; simp [PhaseCircle.add]; congr 1; omega -- ============================================================ -- §1 BASE INDEX (was missing from v.01) -- ============================================================ -- Each canonical state has an index 0–7 matching its phase / 45. -- This was referenced but never defined in HachimojiDerivation.lean. /-- Map each 4D Hachimoji state to its canonical index 0–7. The canonical states are exactly those with phase = 45*i. -/ def stateIndex (s : HachimojiState4D) : Fin 8 := ⟨s.phase / 45 % 8, by omega⟩ /-- The 8 canonical states have distinct indices. -/ theorem canonical_indices_distinct : stateIndex StateΦ ≠ stateIndex StateΛ ∧ stateIndex StateΛ ≠ stateIndex StateΡ ∧ stateIndex StateΡ ≠ stateIndex StateΚ ∧ stateIndex StateΚ ≠ stateIndex StateΩ ∧ stateIndex StateΩ ≠ stateIndex StateSigma ∧ stateIndex StateSigma ≠ stateIndex StatePi ∧ stateIndex StatePi ≠ stateIndex StateΖ := by constructor <;> rfl /-- stateIndex agrees with phase / 45 for all 8 canonical states. -/ theorem stateIndex_phase_agrees (s : HachimojiState4D) (h : s.phase ∈ ({0, 45, 90, 135, 180, 225, 270, 315} : Finset ℕ)) : stateIndex s = ⟨s.phase / 45, by fin_cases h <;> simp_all⟩ := by simp [stateIndex] fin_cases h <;> simp_all -- ============================================================ -- §2 CORRECTED S¹⁵ EMBEDDING -- ============================================================ -- -- The v.01 HachimojiDerivation.lean used θ·π/360, which sweeps -- only 0..π (a semicircle). The correct angle is θ·π/180 = θ·2π/360, -- which gives a full period and a genuine regular 360-gon. -- -- Fixed: phaseEmbed now uses θ.val * π / 180. /-- A point on S¹⁵, represented as a 16-vector with unit norm. Coordinates indexed by Fin 16 matching the DQ complexified quaternion space (Q₁, Q₂) ∈ ℂ⁸ = ℝ¹⁶. -/ structure SpherePoint where coords : Fin 16 → ℝ h_norm : ∑ i : Fin 16, coords i ^ 2 = 1 /-- Phase embedding with corrected full-period angle. q₁(θ) = cos(θ · π/180) ← full 360° period q₃(θ) = sin(θ · π/180) all other coords = 0 This is the canonical embedding of ℤ/360ℤ into S¹ ⊂ S¹⁵. The 8 canonical states form a regular octagon on this circle. -/ noncomputable def phaseEmbed (θ : PhaseCircle) : SpherePoint where coords := fun i => if i = 0 then cos (θ.val * π / 180) else if i = 2 then sin (θ.val * π / 180) else 0 h_norm := by simp only [Finset.sum_fin_eq_sum_range] norm_num [Finset.sum_range_succ] rw [show (0 : Fin 16).val = 0 from rfl, show (2 : Fin 16).val = 2 from rfl] simp [cos_sq_add_sin_sq] /-- Unit norm: the embedding always lands on S¹⁵. -/ theorem phaseEmbed_unit_norm (θ : PhaseCircle) : ∑ i : Fin 16, (phaseEmbed θ).coords i ^ 2 = 1 := (phaseEmbed θ).h_norm /-- The canonical octagon: adjacent base states are separated by the correct chord length 2·sin(π/8) ≈ 0.7654. PROVED (not vacuous like v.01): under the corrected angle θ·π/180, the chord between θ=0 and θ=45 is |e^{iπ/4} − 1| = 2·sin(π/8). -/ theorem octagon_chord : let p0 := phaseEmbed ⟨0, by norm_num⟩ let p1 := phaseEmbed ⟨45, by norm_num⟩ ∑ i : Fin 16, (p1.coords i - p0.coords i) ^ 2 = 2 - 2 * cos (π / 4) := by simp [phaseEmbed, SpherePoint.coords] ring_nf rw [show (45 : ℝ) * π / 180 = π / 4 from by ring] rw [show (0 : ℝ) * π / 180 = 0 from by ring] simp [cos_zero, sin_zero] ring_nf rw [cos_sq, sin_sq] ring /-- The 8 canonical phases embed to 8 DISTINCT points on S¹⁵. Proof: distinct phases → distinct (cos, sin) pairs under the corrected full-period embedding. -/ theorem phaseEmbed_injective_on_canonical : ∀ (i j : Fin 8), i ≠ j → phaseEmbed ⟨45 * i.val, by omega⟩ ≠ phaseEmbed ⟨45 * j.val, by omega⟩ := by 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 StateSigma = ⟨225, by norm_num⟩ ∧ stateToPhase StatePi = ⟨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, StateSigma] /-- 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