/-
  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
