/- 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°}. -/ -- Transparent alias so Fin 360's instances (DecidableEq, Fintype, AddCommGroup via ZMod) resolve. abbrev PhaseCircle := Fin 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)⟩ -- AddCommGroup is inherited from ZMod 360 = Fin 360 via the abbrev transparency. example : AddCommGroup PhaseCircle := inferInstance -- ============================================================ -- §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 refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_⟩ <;> decide /-- 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 simp only [Finset.mem_insert, Finset.mem_singleton] at h; omega⟩ := by simp only [Finset.mem_insert, Finset.mem_singleton] at h simp only [stateIndex, Fin.mk.injEq] omega -- ============================================================ -- §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 := { 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 calc ∑ i : Fin 16, (if i = 0 then cos (θ.val * π / 180) else if i = 2 then sin (θ.val * π / 180) else 0) ^ 2 = ∑ i ∈ (Finset.univ : Finset (Fin 16)), (if i = 0 then cos (θ.val * π / 180) else if i = 2 then sin (θ.val * π / 180) else 0) ^ 2 := rfl _ = ∑ i ∈ ({0, 2} : Finset (Fin 16)), (if i = 0 then cos (θ.val * π / 180) else if i = 2 then sin (θ.val * π / 180) else 0) ^ 2 := by refine (Finset.sum_subset (by simp) ?_).symm intro i hi hi_not have hi0 : i ≠ 0 := by intro h; apply hi_not; simp [h] have hi2 : i ≠ 2 := by intro h; apply hi_not; simp [h] simp [hi0, hi2] _ = cos (θ.val * π / 180) ^ 2 + sin (θ.val * π / 180) ^ 2 := by simp _ = 1 := by simp [Real.cos_sq_add_sin_sq] } /-- Unfolding lemma: `(phaseEmbed θ).coords i` reduces to the explicit if-then-else. -/ @[simp] lemma phaseEmbed_coords_eq (θ : PhaseCircle) (i : Fin 16) : (phaseEmbed θ).coords i = (if i = 0 then cos (θ.val * π / 180) else if i = 2 then sin (θ.val * π / 180) else 0) := rfl /-- 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 intro p0 p1 calc ∑ i : Fin 16, (p1.coords i - p0.coords i) ^ 2 = ∑ i ∈ ({0, 2} : Finset (Fin 16)), (p1.coords i - p0.coords i) ^ 2 := by refine (Finset.sum_subset (by simp) ?_).symm intro i hi hi_not have hi0 : i ≠ 0 := by intro h; apply hi_not; simp [h] have hi2 : i ≠ 2 := by intro h; apply hi_not; simp [h] dsimp [p0, p1]; simp [hi0, hi2] _ = (p1.coords 0 - p0.coords 0) ^ 2 + (p1.coords 2 - p0.coords 2) ^ 2 := by simp _ = (cos (π/4) - 1) ^ 2 + (sin (π/4) - 0) ^ 2 := by dsimp [p0, p1]; simp have h45 : (45 : ℝ) * π / 180 = π / 4 := by ring_nf simp [h45] _ = (cos (π/4) ^ 2 - 2 * cos (π/4) + 1) + sin (π/4) ^ 2 := by ring_nf _ = (cos (π/4) ^ 2 + sin (π/4) ^ 2) + 1 - 2 * cos (π/4) := by ring_nf _ = 1 + 1 - 2 * cos (π/4) := by rw [Real.cos_sq_add_sin_sq (π/4)] _ = 2 - 2 * cos (π / 4) := by ring_nf /-- The 8 canonical phases embed to 8 DISTINCT points on S¹⁵. Proof: distinct phases → distinct (cos, sin) pairs under the corrected full-period embedding. -/ -- Helper: trig values for all 8 angles k*π/4, k=0..7. -- These provide concrete `simp` normal forms for each case. private lemma cos_two_pi_div_four : cos (2 * π / 4) = 0 := by calc cos (2 * π / 4) = cos (π/2) := by ring_nf _ = 0 := by simp private lemma sin_two_pi_div_four : sin (2 * π / 4) = 1 := by calc sin (2 * π / 4) = sin (π/2) := by ring_nf _ = 1 := by simp private lemma cos_three_pi_div_four : cos (3 * π / 4) = -Real.sqrt 2 / 2 := by calc cos (3 * π / 4) = cos (π - π/4) := by ring_nf _ = cos π * cos (π/4) + sin π * sin (π/4) := by rw [Real.cos_sub] _ = (-1) * (Real.sqrt 2 / 2) + 0 * (Real.sqrt 2 / 2) := by simp _ = -Real.sqrt 2 / 2 := by ring_nf private lemma sin_three_pi_div_four : sin (3 * π / 4) = Real.sqrt 2 / 2 := by calc sin (3 * π / 4) = sin (π - π/4) := by ring_nf _ = sin π * cos (π/4) - cos π * sin (π/4) := by rw [Real.sin_sub] _ = 0 * (Real.sqrt 2 / 2) - (-1) * (Real.sqrt 2 / 2) := by simp _ = Real.sqrt 2 / 2 := by ring_nf private lemma cos_five_pi_div_four : cos (5 * π / 4) = -Real.sqrt 2 / 2 := by calc cos (5 * π / 4) = cos (π + π/4) := by ring_nf _ = -cos (π/4) := by simp [Real.cos_add] _ = -(Real.sqrt 2 / 2) := by simp _ = -Real.sqrt 2 / 2 := by ring_nf private lemma sin_five_pi_div_four : sin (5 * π / 4) = -Real.sqrt 2 / 2 := by calc sin (5 * π / 4) = sin (π + π/4) := by ring_nf _ = -sin (π/4) := by simp [Real.sin_add] _ = -(Real.sqrt 2 / 2) := by simp _ = -Real.sqrt 2 / 2 := by ring_nf private lemma cos_six_pi_div_four : cos (6 * π / 4) = 0 := by calc cos (6 * π / 4) = cos (3 * π / 2) := by ring_nf _ = cos (π + π/2) := by ring_nf _ = -cos (π/2) := by simp [Real.cos_add] _ = 0 := by simp private lemma sin_six_pi_div_four : sin (6 * π / 4) = -1 := by calc sin (6 * π / 4) = sin (3 * π / 2) := by ring_nf _ = sin (π + π/2) := by ring_nf _ = -sin (π/2) := by simp [Real.sin_add] _ = -1 := by simp private lemma cos_seven_pi_div_four : cos (7 * π / 4) = Real.sqrt 2 / 2 := by calc cos (7 * π / 4) = cos (π/4) := by calc cos (7 * π / 4) = cos (2*π - π/4) := by ring_nf _ = cos (2*π) * cos (π/4) + sin (2*π) * sin (π/4) := by rw [Real.cos_sub] _ = 1 * cos (π/4) + 0 * sin (π/4) := by simp _ = cos (π/4) := by ring_nf _ = Real.sqrt 2 / 2 := by simp private lemma sin_seven_pi_div_four : sin (7 * π / 4) = -Real.sqrt 2 / 2 := by calc sin (7 * π / 4) = sin (2*π - π/4) := by ring_nf _ = sin (2*π) * cos (π/4) - cos (2*π) * sin (π/4) := by rw [Real.sin_sub] _ = 0 * cos (π/4) - 1 * sin (π/4) := by simp _ = - sin (π/4) := by simp _ = -(Real.sqrt 2 / 2) := by simp _ = -Real.sqrt 2 / 2 := by ring_nf 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 h apply hij have hcos_raw : (phaseEmbed ⟨45 * i.val, by omega⟩).coords 0 = (phaseEmbed ⟨45 * j.val, by omega⟩).coords 0 := congr_arg (·.coords 0) h have hsin_raw : (phaseEmbed ⟨45 * i.val, by omega⟩).coords 2 = (phaseEmbed ⟨45 * j.val, by omega⟩).coords 2 := congr_arg (·.coords 2) h have hcos_val (k : Fin 8) : (phaseEmbed ⟨45 * k.val, by omega⟩).coords 0 = cos (((45 : ℝ) * (k.val : ℝ) * π) / 180) := by simp have hsin_val (k : Fin 8) : (phaseEmbed ⟨45 * k.val, by omega⟩).coords 2 = sin (((45 : ℝ) * (k.val : ℝ) * π) / 180) := by simp have h45 : (45 : ℝ) * π / 180 = π / 4 := by ring_nf have hcos : cos ((i.val : ℝ) * π / 4) = cos ((j.val : ℝ) * π / 4) := by calc cos ((i.val : ℝ) * π / 4) = cos (((45 : ℝ) * (i.val : ℝ) * π) / 180) := by ring_nf _ = (phaseEmbed ⟨45 * i.val, by omega⟩).coords 0 := (hcos_val i).symm _ = (phaseEmbed ⟨45 * j.val, by omega⟩).coords 0 := hcos_raw _ = cos (((45 : ℝ) * (j.val : ℝ) * π) / 180) := hcos_val j _ = cos ((j.val : ℝ) * π / 4) := by ring_nf have hsin : sin ((i.val : ℝ) * π / 4) = sin ((j.val : ℝ) * π / 4) := by calc sin ((i.val : ℝ) * π / 4) = sin (((45 : ℝ) * (i.val : ℝ) * π) / 180) := by ring_nf _ = (phaseEmbed ⟨45 * i.val, by omega⟩).coords 2 := (hsin_val i).symm _ = (phaseEmbed ⟨45 * j.val, by omega⟩).coords 2 := hsin_raw _ = sin (((45 : ℝ) * (j.val : ℝ) * π) / 180) := hsin_val j _ = sin ((j.val : ℝ) * π / 4) := by ring_nf have hsq2 : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2) fin_cases i <;> fin_cases j <;> first | rfl | simp [Real.cos_zero, Real.sin_zero, Real.cos_pi_div_four, Real.sin_pi_div_four, cos_two_pi_div_four, sin_two_pi_div_four, cos_three_pi_div_four, sin_three_pi_div_four, Real.cos_pi, Real.sin_pi, cos_five_pi_div_four, sin_five_pi_div_four, cos_six_pi_div_four, sin_six_pi_div_four, cos_seven_pi_div_four, sin_seven_pi_div_four] at hcos hsin <;> nlinarith [hsq2] -- ============================================================ -- §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 decide -- ============================================================ -- §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 unfold equationPosition have hclass : classifyEquation { n_vars := 2, n_ops := 2, max_depth := 0, n_quantifiers := 0, n_relations := 1 } = StateΦ := by decide rw [hclass, canonical_phases_preserved.1] /-- 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 unfold equationPosition have hclass : classifyEquation { n_vars := 3, n_ops := 7, max_depth := 0, n_quantifiers := 0, n_relations := 1 } = StateSigma := by decide obtain ⟨-, -, -, -, -, hSig, -, -⟩ := canonical_phases_preserved rw [hclass, hSig] /-- 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 unfold equationPosition have hclass : classifyEquation { n_vars := 0, n_ops := 0, max_depth := 0, n_quantifiers := 0, n_relations := 1 } = StateΩ := by decide obtain ⟨-, -, -, -, hΩ, -, -, -⟩ := canonical_phases_preserved rw [hclass, hΩ] -- ============================================================ -- §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⟩ 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 have h : ∀ θ : PhaseCircle, isStabilityPoint θ = true ↔ θ.val = 0 ∨ θ.val = 180 := by intro θ; fin_cases θ <;> decide exact h /-- 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 decide -- ============================================================ -- §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, binaryLUT_exists (trivial constant-Φ solution) -- §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. -- -- 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). /-- A concrete BinaryLUT exists: the constant-Φ composition table. TODO: Replace with non-trivial phase-addition table when the full 8×8 composition semantics are specified. h_consistent holds because both sides reduce to phaseEmbed ⟨0⟩. -/ theorem binaryLUT_exists : ∃ _ : BinaryLUT, True := ⟨{ pattern := fun _ => phaseEmbed ⟨0, by norm_num⟩ lookup := fun _ => phaseEmbed ⟨0, by norm_num⟩ compose := fun _ _ => StateΦ h_consistent := fun _ _ => rfl }, trivial⟩ end HachimojiLUT