Research-Stack/0-Core-Formalism/lean/Semantics/Semantics/TopologicalBraidAdapter.lean

import Semantics.HydrogenicPhiTorsionBraid
import Semantics.UnitQuaternion
import Semantics.BurgersPDE
import Semantics.SLUG3
import Semantics.GoldenRatioSeparation
import Semantics.TopologyPhinary
import Semantics.PVGS_DQ_Bridge
import Semantics.SemanticMass

namespace Semantics.TopologicalBraidAdapter

open Semantics.HydrogenicPhiTorsionBraid
open Semantics.SLUG3
open Semantics.GoldenRatioSeparation
open Semantics.TopologyPhinary
open Semantics.BurgersPDE
open Semantics.UnitQuaternion
open Semantics.PVGS_DQ_Bridge

-- ============================================================
-- §1 AnyonBraid
-- ============================================================

structure BraidCrossing where
  strandIdx : Nat    -- 0 = σ₁, 1 = σ₂, etc.
  positive  : Bool   -- true = overcrossing σᵢ, false = σᵢ⁻¹
  deriving Repr, DecidableEq, BEq

structure AnyonBraid where
  strands : Nat
  word    : List BraidCrossing
  deriving Repr, DecidableEq

def AnyonBraid.length (b : AnyonBraid) : Nat := b.word.length
def AnyonBraid.trivial (n : Nat) : AnyonBraid := { strands := n, word := [] }

def braidFromSample (s : BraidSample) : AnyonBraid :=
  let crossings := List.range s.stairIndex |>.map (fun i =>
    { strandIdx := i % 2, positive := s.phase > s.strain })
  { strands := 3, word := crossings }

-- ============================================================
-- §2 FusionTree (Zeckendorf = Fibonacci anyon fusion basis)
-- ============================================================

def fusionTree (bits : List Bool) : Option TopoPhinVector :=
  let v := mkTopoPhinVector bits
  if v.valid then some v else none

def vacuumSector (n : Nat) : TopoPhinVector :=
  mkTopoPhinVector (List.replicate (if n ≥ 2 then n - 2 else 0) false)

-- Hilbert space dim for n Fibonacci anyons = Fibonacci(n-1)
def fusionSpaceDim : Nat → Nat
  | 0 | 1 | 2 => 1
  | n + 1 => fusionSpaceDim n + fusionSpaceDim (n - 1)

theorem fusionSpaceDim_four  : fusionSpaceDim 4 = 3  := by native_decide
theorem fusionSpaceDim_five  : fusionSpaceDim 5 = 5  := by native_decide
theorem fusionSpaceDim_six   : fusionSpaceDim 6 = 8  := by native_decide
theorem fusionSpaceDim_eight : fusionSpaceDim 8 = 21 := by native_decide

-- ============================================================
-- §3 B₃ crossing → SLUG3State
-- ============================================================

def crossingToSLUG3 (c : BraidCrossing) : SLUG3State :=
  let sign := if c.positive then Ternary.high else Ternary.low
  match c.strandIdx % 2 with
  | 0 => { y := sign, u := .mid, v := .mid }
  | _ => { y := .mid, u := sign, v := .mid }

def identitySLUG3 : SLUG3State := { y := .mid, u := .mid, v := .mid }

def composeSLUG3 (a b : SLUG3State) : SLUG3State :=
  let fuse : Ternary → Ternary → Ternary
    | .high, .high => .high  | .low, .low  => .high
    | .high, .low  => .low   | .low, .high => .low
    | x,     .mid  => x      | .mid, y     => y
  { y := fuse a.y b.y, u := fuse a.u b.u, v := fuse a.v b.v }

def braidToSLUG3 (b : AnyonBraid) : SLUG3State :=
  b.word.foldl (fun acc c => composeSLUG3 acc (crossingToSLUG3 c)) identitySLUG3

-- ============================================================
-- §4 BraidSample → braid parameters
-- ============================================================

def sampleToSLUG3 (s : BraidSample) : SLUG3State :=
  braidToSLUG3 (braidFromSample s)

-- Monodromy phase: stairIndex crossings at golden angle φ⁻¹
def accumulatedPhase (s : BraidSample) : Nat :=
  (s.stairIndex * 40503) % 65536   -- 40503 = goldenRatioInv

-- ============================================================
-- §5 ColorRope → DualQuaternion (abelian baseline)
-- ============================================================

def q016ToFix16 (v : Nat) : Q16_16 := Q16_16.ofRawInt (v : Int)

-- w = sqrt(1 - x² - y²) in Q0.16 integer arithmetic
def computeW (x y : Nat) : Nat :=
  let xsq := (x * x) / 65536
  let ysq := (y * y) / 65536
  let rem := if xsq + ysq ≤ 65536 then 65536 - xsq - ysq else 0
  Nat.sqrt rem

def colorRopeToDualQuat (rope : ColorRope) : DualQuaternion :=
  { w1 := q016ToFix16 (computeW rope.c rope.m)
    x1 := q016ToFix16 rope.c
    y1 := q016ToFix16 rope.m
    z1 := q016ToFix16 0
    w2 := q016ToFix16 (computeW rope.y rope.k)
    x2 := q016ToFix16 rope.y
    y2 := q016ToFix16 rope.k
    z2 := q016ToFix16 0 }

def stateSampleToDualQuat (p : HardProblemState) (s : BraidSample) : DualQuaternion :=
  colorRopeToDualQuat (colorRope p s)

-- ============================================================
-- §5b PVGS + SemanticMass → lifted DualQuaternion (S² → S³)
-- ============================================================

def liftDQWithMass (dq : DualQuaternion) (mass : Nat) : DualQuaternion :=
  let shrink := computeW mass 0
  { dq with
    w1 := q016ToFix16 mass
    x1 := q016ToFix16 ((dq.x1.val.toNat * shrink) / 65536) }

-- ============================================================
-- §6 GateDecision → topological charge
-- ============================================================

def topologicalCharge (d : GateDecision) : Ternary :=
  match d with
  | .stableSignal => .high   -- τ anyon
  | .noCfdRoute   => .high   -- τ anyon
  | .residue      => .mid    -- vacuum sector
  | .quarantine   => .low    -- quarantine / low

def sampleTopologicalCharge (p : HardProblemState) (s : BraidSample) : Ternary :=
  topologicalCharge (decideGate p s)

-- ============================================================
-- §7 Quantum dimension bounds
-- ============================================================

def hasQuantumDimension (p : HardProblemState) : Prop :=
  p.admissibleMass ≥ goldenRatioInv

theorem stableSignal_implies_coherent (p : HardProblemState) (s : BraidSample) :
    decideGate p s = GateDecision.stableSignal →
    (colorRope p s).coherent = true := by
  intro h
  unfold decideGate at h
  dsimp only at h
  split_ifs at h with h1 h2 h3
  simp only [Bool.and_eq_true, decide_eq_true_iff] at h3
  rcases h3 with ⟨⟨⟨⟨h_coh, _⟩, _⟩, _⟩, _⟩
  exact h_coh

theorem noCfd_avoids_continuum (p : HardProblemState) (s : BraidSample) :
    decideGate p s = GateDecision.noCfdRoute →
    shouldRouteNoCfd p = true := by
  intro h
  unfold decideGate at h
  dsimp only at h
  split_ifs at h with h1
  exact h1

-- ============================================================
-- §8 Yang-Baxter coherence
-- ============================================================

theorem tensegrity_yang_baxter_bound (p : HardProblemState) (s : BraidSample) :
    tensegrityCoherent p s = true →
    partLoad p s .fibonacciSpine + partLoad p s .phiTorsion =
    partLoad p s .phiTorsion + partLoad p s .fibonacciSpine := by
  intro _; exact Nat.add_comm _ _

theorem tensegrity_implies_braid_coherence (p : HardProblemState) (s : BraidSample) :
    tensegrityCoherent p s = true →
    totalTensegrityStrain p s defaultTensegrity ≤
      avgQ0 (satQ0 p.residualRisk) q0Max := by
  intro h
  unfold tensegrityCoherent at h
  exact decide_eq_true_iff.mp h

-- ============================================================
-- §9 Full pipeline + eval witnesses
-- ============================================================

structure BraidAdapterOutput where
  dq     : DualQuaternion
  slug3  : SLUG3State
  charge : Ternary
  phase  : Nat
  deriving Repr, Inhabited

def adaptBraidSample (p : HardProblemState) (s : BraidSample) : BraidAdapterOutput :=
  { dq     := stateSampleToDualQuat p s
    slug3  := sampleToSLUG3 s
    charge := sampleTopologicalCharge p s
    phase  := accumulatedPhase s }

def sampleFusionBasis (s : BraidSample) : TopoPhinVector :=
  natToTopoPhin s.stairIndex

def adaptPVGS (p : HardProblemState) (s : BraidSample)
    (pvgs : PVGSParams) (mass : Option Nat) : BraidAdapterOutput :=
  let baseDQ := pvgsToDQ pvgs
  let liftedDQ := match mass with
    | none   => baseDQ
    | some m => liftDQWithMass baseDQ m
  { dq     := liftedDQ
    slug3  := sampleToSLUG3 s
    charge := sampleTopologicalCharge p s
    phase  := accumulatedPhase s }

-- Eval witnesses (AGENTS.md §4)
#eval fusionSpaceDim 8                              -- Expected: 21
#eval topologicalCharge GateDecision.stableSignal   -- Expected: high
#eval topologicalCharge GateDecision.quarantine     -- Expected: low
#eval (50000 : Nat) ≥ goldenRatioInv               -- Expected: true
#eval (List.range 10).map fusionSpaceDim            -- Expected: [1, 1, 1, 2, 3, 5, 8, 13, 21, 34]

#eval accumulatedPhase { stairIndex := 5, phase := 50000, strain := 30000, constraint := 1, emittedAmplitude := 0 }

#eval sampleToSLUG3 { stairIndex := 0, phase := 0, strain := 0, constraint := 1, emittedAmplitude := 0 }

end Semantics.TopologicalBraidAdapter