SilverSight/formal/CoreFormalism/HachimojiBase.lean

/-
  HachimojiSubstitution.lean — Greek-symbol re-encoding of the Hachimoji 8 states

  Standalone companion to HachimojiManifoldAxiom.lean.
  Does NOT modify the working axiom file — only adds a Greek-letter variant
  and the bijection between the two encodings.

  The substitution reads the Research Stack's own notation back into the bases:
    Φ (phi)   ←→ A   trivial     — above φ_GCP, fully ordered lattice regime
    Λ (lam)   ←→ T   room        — inside lattice_regime, Barnes-Wall attractor
    Ρ (rho)   ←→ G   tight       — near ρ(J) = 1, STARS spectral radius boundary
    Κ (kap)   ←→ C   marginal    — at BraidBracket.kappa / softplus κ threshold
    Ω (ome)   ←→ B   collision   — Λ = 0 exactly, terminal fixed-point state
    Σ (sig)   ←→ S   symmetric   — σ: entropy/symmetry partner of a known collision
    Π (pi)    ←→ P   potential   — Π: density × area, coverage violation probe
    Ζ (zet)   ←→ Z   zero-region — ζ: near Riemann ζ-zeros; |Λ| ≈ 0, no integer point

  Why this works: the Greek letters are already doing this semantic work in the stack.
  Every occurrence of Κ in BraidBracket, Ρ in BraidEigensolid §9, Φ/Λ in
  ErdosRenyiPipeline, and Ζ in EffectiveBoundDQ maps to the SAME regime in the
  8-state classification. The substitution makes that implicit correspondence explicit.

  The Ζ ↔ Z mapping is the deepest: Riemann ζ non-trivial zeros are exactly the
  canonical "near cancellation with no integer solution" structure — Z state is
  the same phenomenon in the Baker landscape.
-/

import Mathlib.Data.Equiv.Basic
import Mathlib.Tactic
import CoreFormalism.HachimojiManifoldAxiom
import SilverSight.RRCLogogramProjection

-- ============================================================
-- §1  GREEK HACHIMOJI ALPHABET
-- ============================================================

namespace Greek

/-- The 8-state Hachimoji alphabet re-encoded as Greek letters.
    Each letter inherits its semantic meaning from existing Research Stack usage. -/
inductive HachimojiBase where
  | Φ  -- phi:   trivial regime — above φ_GCP density, fully ordered
  | Λ  -- lam:   room regime — inside lattice_regime, Barnes-Wall Λ₁₆ attractor
  | Ρ  -- rho:   tight regime — near spectral radius ρ(J) = 1 stability boundary
  | Κ  -- kap:   marginal — at complementarity threshold κ (BraidBracket.kappa)
  | Ω  -- ome:   collision — Λ(m,n) = 0 exactly, terminal eigensolid state
  | Σ  -- sig:   symmetric partner — σ-symmetry of a known Goormaghtigh solution
  | Π  -- pi:    potential violation — Π density probe below Baker threshold
  | Ζ  -- zet:   zero region — near ζ-zeros; |Λ| ≈ 0 but no integer lattice point
  deriving DecidableEq, Repr, Fintype

theorem HachimojiBase.card_eq : Fintype.card HachimojiBase = 8 := by decide

end Greek

-- ============================================================
-- §2  BIJECTION WITH THE ORIGINAL ENCODING
-- ============================================================

/-- The Greek encoding is in bijection with the Latin HachimojiBase. -/
def hachimojiGreekEquiv : HachimojiBase ≃ Greek.HachimojiBase where
  toFun := fun b => match b with
    | .A => .Φ
    | .T => .Λ
    | .G => .Ρ
    | .C => .Κ
    | .B => .Ω
    | .S => .Σ
    | .P => .Π
    | .Z => .Ζ
  invFun := fun g => match g with
    | .Φ => .A
    | .Λ => .T
    | .Ρ => .G
    | .Κ => .C
    | .Ω => .B
    | .Σ => .S
    | .Π => .P
    | .Ζ => .Z
  left_inv  := by intro b; cases b <;> rfl
  right_inv := by intro g; cases g <;> rfl

-- ============================================================
-- §3  GREEK CLASSIFIER AND FIELD
-- ============================================================

/-- Classify a lattice point using the Greek-symbol encoding. -/
noncomputable def hachimojiClassifyGreek (Λ_val B_threshold : ℝ) : Greek.HachimojiBase :=
  hachimojiGreekEquiv (hachimojiClassify Λ_val B_threshold)

/-- Hachimoji state at (m,n) in Greek encoding. -/
noncomputable def hachimojiBakerFieldGreek (x y C : ℕ) (m n : ℕ) : Greek.HachimojiBase :=
  hachimojiGreekEquiv (hachimojiBakerField x y C m n)

/-- The Greek and Latin classifiers agree up to the bijection. -/
theorem greek_latin_agree (Λ_val B_threshold : ℝ) :
    hachimojiClassifyGreek Λ_val B_threshold =
    hachimojiGreekEquiv (hachimojiClassify Λ_val B_threshold) := rfl

-- ============================================================
-- §4  SEMANTIC CROSS-REFERENCE (DOCUMENTATION)
-- ============================================================

/-
  STACK CROSS-REFERENCE

  Κ (kappa / marginal):
    · BraidBracket.kappa — per-strand complementarity residual
    · softplusRetraction κ — IPM complementarity parameter (BraidEigensolid §10)
    · IsTopologicallyTrivial: kappa ≤ Q16_16.ofRawInt 16384 (= 0.25)
    · The marginal Baker regime is where b_κ(v)·b_κ(−v) = κ becomes tight

  Ρ (rho / tight):
    · BraidEigensolid §9: strandResidue proxy for ρ²(J) (STARS JSRR loss)
    · IsEigensolid ↔ ρ(J★) < 1 (crossStep = Φ_θ, BraidState = h^(t))
    · The tight Baker regime is where ρ(J) ≈ 1 — loop stability boundary

  Φ (phi / trivial):
    · ErdosRenyiPipeline §9: φ_LT, φ_RCP, φ_GCP — RCP phase boundaries
    · SpherionTwinPrime §13: φ_LT = 127/200, φ_RCP = 16/25, φ_GCP = 13/20
    · Above φ_GCP: BW16 lattice_regime, Barnes-Wall attractor — trivial Baker

  Λ (lambda / room):
    · ErdosRenyiPipeline: lattice_regime = Set.Icc φ_RCP φ_GCP
    · BraidEigensolid: kissingNumberBW16 = 4320 (vs. E8×E8 = 480); 9× basin advantage
    · The room Baker regime corresponds to density inside the ordered lattice phase

  Ζ (zeta / zero-region):
    · Riemann ζ non-trivial zeros: canonical near-cancellation without integer solutions
    · Baker landscape Z-state: |Λ| ≈ 0 but no (m,n) integer point exists
    · The connection: both are "apparent zeros" that resist a Sidon-type proof

  Ω (omega / collision):
    · GoormaghtighEnumeration: only two Ω-states exist: (2,5,5,3) and (2,13,90,3)
    · BraidEigensolid §8: ZeroGenusLayer = eigensolid ∧ topologically trivial
    · Ω is the terminal state — the eigensolid fixed point in the braid dynamics
-/

-- ============================================================
-- §5  CHIRALITY AND PHASE (OMINDIRECTION)
-- ============================================================
-- Each Greek state has a phase in ℤ/360ℤ (45° per state).
-- Chirality is derived from phase per Omindirection Principle 3:
--   ambidextrous = phase 0 or 180
--   left         = phase 1..179
--   right        = phase 181..359
-- Direction:
--   forward  = phases 0..179  (Φ Λ Ρ Κ — normal Baker regime)
--   reverse  = phases 180..359 (Ω Σ Π Ζ — quarantine/tearing regime)

/-- Chirality class per Omindirection principle 3. -/
inductive Chirality where
  | ambidextrous
  | left
  | right
  deriving DecidableEq, Repr

/-- Flow direction per Omindirection principle 2. -/
inductive FlowDirection where
  | forward   -- LTR, normal projection lane
  | reverse   -- RTL, quarantine projection lane
  deriving DecidableEq, Repr

/-- Phase angle in ℤ/360ℤ for each Greek state (45° steps). -/
def Greek.HachimojiBase.phase : Greek.HachimojiBase → ℕ
  | .Φ => 0
  | .Λ => 45
  | .Ρ => 90
  | .Κ => 135
  | .Ω => 180
  | .Σ => 225
  | .Π => 270
  | .Ζ => 315

/-- Chirality derived from phase per Omindirection Principle 3. -/
def Greek.HachimojiBase.chirality (g : Greek.HachimojiBase) : Chirality :=
  match g.phase with
  | 0   => .ambidextrous   -- Φ: phase 0, perfect symmetry
  | 45  => .left           -- Λ: left-leaning lattice
  | 90  => .ambidextrous   -- Ρ: spectral boundary, balanced
  | 135 => .left           -- Κ: near-left marginal
  | 180 => .ambidextrous   -- Ω: perfect inversion, balanced
  | 225 => .right          -- Σ: symmetric partner, right-handed
  | 270 => .right          -- Π: violation probe, right (quarantine)
  | _   => .right          -- Ζ: 315°, right-handed near-reverse

/-- Flow direction: forward for phases 0-135° (Φ Λ Ρ Κ),
    reverse for phases 180-315° (Ω Σ Π Ζ). -/
def Greek.HachimojiBase.direction (g : Greek.HachimojiBase) : FlowDirection :=
  if g.phase < 180 then .forward else .reverse

/-- The four forward states are the "normal Baker regime" (non-quarantine). -/
theorem forward_states_are_normal (g : Greek.HachimojiBase)
    (h : g.direction = .forward) :
    g = .Φ ∨ g = .Λ ∨ g = .Ρ ∨ g = .Κ := by
  cases g <;> simp [Greek.HachimojiBase.direction, Greek.HachimojiBase.phase] at h ⊢ <;>
  first | exact Or.inl rfl | exact Or.inr (Or.inl rfl) |
           exact Or.inr (Or.inr (Or.inl rfl)) | exact Or.inr (Or.inr (Or.inr rfl)) |
           simp at h

-- ============================================================
-- §6  BIDIRECTIONAL QAOA DECODER → LOGOGRAM RECEIPT
-- ============================================================
-- The QAOA circuit produces an 8-qubit measurement bitstring.
-- Each bit selects whether its Greek-state strand is "active".
-- The dominant active state (lowest phase among active bits)
-- determines the LogogramReceipt fields.
--
-- Bit → Greek state mapping (matches braid_receipt_to_qubo variable order):
--   bit 0 → Φ   bit 1 → Λ   bit 2 → Ρ   bit 3 → Κ
--   bit 4 → Ω   bit 5 → Σ   bit 6 → Π   bit 7 → Ζ

open Semantics.RRCLogogramProjection

/-- Decode a single bit index to its Greek state. -/
def bitToGreek (i : Fin 8) : Greek.HachimojiBase :=
  match i.val with
  | 0 => .Φ | 1 => .Λ | 2 => .Ρ | 3 => .Κ
  | 4 => .Ω | 5 => .Σ | 6 => .Π | _ => .Ζ

/-- Decode a Greek state to the LogogramReceipt Bool fields it controls.
    Returns (payloadBound, contradictionWitness, tearBoundary, detachedMass, residualLane). -/
def greekToReceiptBits (g : Greek.HachimojiBase) :
    Bool × Bool × Bool × Bool × Bool :=
  match g with
  | .Φ => (true,  false, false, false, false)  -- payloadBound only
  | .Λ => (true,  false, false, false, false)  -- lattice = also bounded
  | .Ρ => (false, false, false, false, true)   -- residualLane active
  | .Κ => (false, false, false, true,  false)  -- detachedMass (marginal)
  | .Ω => (true,  true,  true,  true,  true)   -- full tear repair witness
  | .Σ => (false, false, true,  false, false)  -- tearBoundary (symmetric)
  | .Π => (false, false, false, false, false)  -- no Bool fields; regime=horrible
  | .Ζ => (false, false, false, true,  false)  -- detachedMass (near-zero)

/-- Derive SemanticRegime from the dominant Greek state. -/
def greekToRegime (g : Greek.HachimojiBase) : SemanticRegime :=
  match g with
  | .Φ | .Λ => .beautifulTopologicalFolding
  | .Ρ | .Κ => .uglyAsymmetricPruning
  | .Ω | .Σ | .Π | .Ζ => .horribleManifoldTearing

/-- Full bidirectional decoder: QAOA bitstring → LogogramReceipt.
    Uses the Greek state of the LOWEST active bit as the dominant state.
    (Lowest phase = most stable = closest to Φ.) -/
def fromQAOABitstring (bits : Fin 8 → Bool) : LogogramReceipt :=
  -- Find dominant state: lowest active bit index
  let dominant : Greek.HachimojiBase :=
    if bits ⟨0, by omega⟩ then .Φ
    else if bits ⟨1, by omega⟩ then .Λ
    else if bits ⟨2, by omega⟩ then .Ρ
    else if bits ⟨3, by omega⟩ then .Κ
    else if bits ⟨4, by omega⟩ then .Ω
    else if bits ⟨5, by omega⟩ then .Σ
    else if bits ⟨6, by omega⟩ then .Π
    else .Ζ
  -- Accumulate Bool fields from ALL active bits
  let fold8 (init : Bool × Bool × Bool × Bool × Bool)
      (f : Fin 8 → Bool × Bool × Bool × Bool × Bool → Bool × Bool × Bool × Bool × Bool)
      : Bool × Bool × Bool × Bool × Bool :=
    f 7 (f 6 (f 5 (f 4 (f 3 (f 2 (f 1 (f 0 init)))))))
  let acc := fold8 (false, false, false, false, false) (fun i prev =>
    if bits i then
      let (b, cw, tb, dm, rl) := prev
      let (b', cw', tb', dm', rl') := greekToReceiptBits (bitToGreek i)
      (b || b', cw || cw', tb || tb', dm || dm', rl || rl')
    else prev)
  let (payloadBound, contradictionWitness, tearBoundary, detachedMass, residualLane) := acc
  { shape                := .logogramProjection
    status               := if bits ⟨7, by omega⟩ then .hold else .candidate
    regime               := greekToRegime dominant
    payloadBound
    contradictionWitness
    tearBoundary
    detachedMass
    residualLane }

/-- Backward: extract the 8-bit "Greek signature" from a LogogramReceipt.
    This is the RTL direction: receipt → bitstring → QUBO → circuit update. -/
def toQAOABitstring (r : LogogramReceipt) : Fin 8 → Bool
  | ⟨0, _⟩ => r.payloadBound                                           -- Φ
  | ⟨1, _⟩ => r.regime == .beautifulTopologicalFolding                 -- Λ
  | ⟨2, _⟩ => r.residualLane                                           -- Ρ
  | ⟨3, _⟩ => r.detachedMass                                           -- Κ
  | ⟨4, _⟩ => r.contradictionWitness                                   -- Ω
  | ⟨5, _⟩ => r.tearBoundary                                           -- Σ
  | ⟨6, _⟩ => r.regime == .horribleManifoldTearing                    -- Π
  | ⟨7, _⟩ => r.status == .hold                                        -- Ζ
  | ⟨i, _⟩ => false

-- ============================================================
-- §7  COLLISION STATES IN GREEK ENCODING
-- ============================================================

/-- The known Goormaghtigh collisions are exactly the Ω-states. -/
def knownOmegaStates : List (ℕ × ℕ × ℕ × ℕ) :=
  [(2, 5, 5, 3), (5, 3, 2, 5), (2, 13, 90, 3), (90, 3, 2, 13)]

/-- The Ω-state receipt: all quarantine witnesses present, horrible tearing regime. -/
def omegaLogogramReceipt : LogogramReceipt :=
  fromQAOABitstring (fun i => i.val == 4)  -- only bit 4 (Ω) active