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

/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Research Stack Team

RotationQUBO.lean — Rotation Matrices as Literal Rotation Notation in Frustrated QUBO Fields

This module formalizes a 1D scalar triangle navigating a frustrated QUBO field,
spawning friends to rotate in superposition. Each bracket represents a possibility space,
borrowing the PIST framework for shell geometry.

Key insight:
- Rotation matrices as literal rotation notation (not just linear algebra)
- 1D scalar triangle = (a, b, c) with a+b+c = 0 (triangle closure)
- Frustrated QUBO field = energy landscape with competing minima
- Spawning friends = agent generation in superposition
- Brackets = possibility spaces [lower, upper] from PIST shell geometry
- PIST mass = a*b (hyperbola index) as rotation weight

The rotation field:
Φ_rot(x, θ) = Σᵢ R(θᵢ) · xᵢ / (1 + frustration²)

Where:
- R(θ): rotation matrix at angle θ
- xᵢ: scalar triangle vertex
- frustration: QUBO field frustration parameter

Per AGENTS.md §0: Lean is the source of truth.
Per AGENTS.md §1.4: Q16_16 fixed-point for hardware extraction.
Per AGENTS.md §2: PascalCase types, camelCase functions.
Per AGENTS.md §4: Every def has eval witness or theorem.
-/

import Mathlib.Data.Nat.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Matrix.Basic
import Mathlib.Tactic
import Semantics.PIST
import Semantics.DynamicCanal
import Semantics.FixedPoint

namespace Semantics.RotationQUBO

open PIST DynamicCanal Semantics

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Scalar Triangle Types
-- ═══════════════════════════════════════════════════════════════════════════

/-- A 1D scalar triangle (a, b, c) with closure condition a + b + c = 0.
    Represents a balanced configuration that can navigate QUBO fields. -/
structure ScalarTriangle where
  a : Q16_16  -- First vertex
  b : Q16_16  -- Second vertex
  c : Q16_16  -- Third vertex
  closure : Q16_16  -- Closure residual (should be 0 for balanced triangle)
  deriving Repr, DecidableEq, BEq

namespace ScalarTriangle

/-- Create a balanced scalar triangle from two vertices (c = -(a + b)). -/
def balanced (a b : Q16_16) : ScalarTriangle :=
  let c := Q16_16.sub (Q16_16.sub Q16_16.zero a) b  -- c = -(a + b)
  let closure := Q16_16.add (Q16_16.add a b) c  -- should be 0
  { a, b, c, closure }

/-- Create a scalar triangle from PIST coordinate (a = t, b = 2k+1-t). -/
def fromPISTCoord (coord : PIST.Coord) : ScalarTriangle :=
  let a := Q16_16.ofNat coord.t
  let b := Q16_16.ofNat ((2 * coord.k + 1) - coord.t)  -- b = 2k+1-t
  let c := Q16_16.sub (Q16_16.sub Q16_16.zero a) b
  let closure := Q16_16.add (Q16_16.add a b) c
  { a, b, c, closure }

/-- The PIST mass of the scalar triangle (a * b). -/
def pistMass (st : ScalarTriangle) : Q16_16 :=
  Q16_16.mul st.a st.b

end ScalarTriangle

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Rotation Matrix as Literal Rotation Notation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Rotation matrix at angle θ (2D rotation).
    Treated as literal rotation notation, not just linear algebra. -/
structure RotationMatrix where
  theta : Q16_16  -- Rotation angle in radians (Q16.16)
  cosθ : Q16_16   -- cos(θ) in Q16.16
  sinθ : Q16_16   -- sin(θ) in Q16.16
  deriving Repr, DecidableEq, BEq

namespace RotationMatrix

/-- Create rotation matrix from angle θ.
    Uses Q16.16 approximation for cos and sin. -/
def fromAngle (theta : Q16_16) : RotationMatrix :=
  -- Placeholder: use Taylor series or lookup table for cos/sin
  -- For now, use simple approximation
  let cosθ := Q16_16.ofNat 1  -- cos(0) = 1
  let sinθ := theta  -- sin(θ) ≈ θ for small θ
  { theta, cosθ, sinθ }

/-- Apply rotation matrix to scalar triangle vertex. -/
def rotateVertex (rm : RotationMatrix) (v : Q16_16) : Q16_16 :=
  -- 2D rotation: x' = x·cosθ - y·sinθ
  -- For 1D scalar, this is simplified
  Q16_16.mul v rm.cosθ

/-- Apply rotation matrix to entire scalar triangle. -/
def rotateTriangle (rm : RotationMatrix) (st : ScalarTriangle) : ScalarTriangle :=
  let a' := rm.rotateVertex st.a
  let b' := rm.rotateVertex st.b
  let c' := rm.rotateVertex st.c
  let closure' := Q16_16.add (Q16_16.add a' b') c'
  { a := a', b := b', c := c', closure := closure' }

end RotationMatrix

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Frustrated QUBO Field
-- ═══════════════════════════════════════════════════════════════════════════

/-- Frustrated QUBO field parameters.
    Frustration parameter δ controls competing energy minima. -/
structure QUBOField where
  frustration : Q16_16  -- Frustration parameter δ (0 ≤ δ ≤ 1)
  energyScale : Q16_16  -- Energy scale factor
  deriving Repr, DecidableEq, BEq

namespace QUBOField

/-- Compute field energy at position x.
    E(x) = x² / (1 + δ²) - frustration penalty. -/
def fieldEnergy (qf : QUBOField) (x : Q16_16) : Q16_16 :=
  let xSq := Q16_16.mul x x
  let denom := Q16_16.add Q16_16.one (Q16_16.mul qf.frustration qf.frustration)
  let energy := Q16_16.div xSq denom
  Q16_16.sub energy qf.energyScale

/-- Check if field is frustrated at position x. -/
def isFrustrated (qf : QUBOField) (x : Q16_16) : Bool :=
  -- Field is frustrated if energy > 0
  let energy := qf.fieldEnergy x
  energy.val > 0

end QUBOField

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Bracket Possibility Spaces
-- ═══════════════════════════════════════════════════════════════════════════

/-- Bracket possibility space from PIST shell geometry.
    [lower, upper] = [a, b] where a + b = 2k+1 and mass = a*b. -/
structure BracketSpace where
  lower : Q16_16  -- Lower bound (a)
  upper : Q16_16  -- Upper bound (b)
  mass : Q16_16   -- PIST mass (a * b)
  gap : Q16_16    -- Upper - lower
  admissible : Bool  -- Whether space is admissible
  deriving Repr, DecidableEq, BEq

namespace BracketSpace

/-- Create bracket space from PIST coordinate. -/
def fromPISTCoord (coord : PIST.Coord) : BracketSpace :=
  let lower := Q16_16.ofNat coord.t
  let upper := Q16_16.ofNat ((2 * coord.k + 1) - coord.t)
  let mass := Q16_16.ofNat (coord.t * ((2 * coord.k + 1) - coord.t))
  let gap := Q16_16.sub upper lower
  let admissible := mass.val > 0  -- Positive mass = admissible
  { lower, upper, mass, gap, admissible }

/-- Check if a value is within the bracket space. -/
def contains (bs : BracketSpace) (x : Q16_16) : Bool :=
  let xNat := x.val.toNat
  let lowerNat := bs.lower.val.toNat
  let upperNat := bs.upper.val.toNat
  lowerNat ≤ xNat ∧ xNat ≤ upperNat

end BracketSpace

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Friend Spawning in Superposition
-- ═══════════════════════════════════════════════════════════════════════════

/-- A friend agent spawned in superposition.
    Each friend has a rotation angle and weight. -/
structure FriendAgent where
  rotation : RotationMatrix  -- Rotation matrix
  weight : Q16_16  -- Superposition weight (0 ≤ weight ≤ 1)
  bracket : BracketSpace  -- Assigned bracket space
  deriving Repr, DecidableEq, BEq

namespace FriendAgent

/-- Spawn a friend agent with random rotation. -/
def spawn (theta : Q16_16) (bracket : BracketSpace) : FriendAgent :=
  let rm := RotationMatrix.fromAngle theta
  let weight := Q16_16.ofNat 1  -- Default weight = 1.0
  { rotation := rm, weight, bracket }

/-- Spawn multiple friends in superposition. -/
def spawnSuperposition (thetas : List Q16_16) (bracket : BracketSpace) : List FriendAgent :=
  thetas.map (fun θ => spawn θ bracket)

end FriendAgent

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Rotation Field Computation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compute rotation field for scalar triangle in QUBO field with friends.
    Φ_rot(x, θ) = Σᵢ R(θᵢ) · xᵢ / (1 + frustration²) -/
def rotationField (st : ScalarTriangle) (friends : List FriendAgent)
    (qf : QUBOField) : Q16_16 :=
  let denom := Q16_16.add Q16_16.one (Q16_16.mul qf.frustration qf.frustration)

  -- Sum over friends: Σᵢ weightᵢ * rotationᵢ(triangle)
  let sumRotations := friends.foldl (fun acc friend =>
    let rotated := friend.rotation.rotateTriangle st
    let weightedMass := Q16_16.mul (ScalarTriangle.pistMass rotated) friend.weight
    Q16_16.add acc weightedMass
  ) Q16_16.zero

  -- Divide by frustration denominator
  Q16_16.div sumRotations denom

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Theorems: Rotation and Bracket Properties
-- ═══════════════════════════════════════════════════════════════════════════

/-- External rotation/bracket invariants.
  Balanced closure = 0, PIST mass = a*b, bracket contains bounds, rotation field bounded.
  These are structural properties of the rotation-QUBO field model. -/
structure RotationQUBOInvariantsHypothesis where
  balanced_closure_zero (a b : Q16_16) : (ScalarTriangle.balanced a b).closure = Q16_16.zero
  pist_mass_from_coord (coord : PIST.Coord) : (ScalarTriangle.fromPISTCoord coord).pistMass = Q16_16.ofNat (coord.t * ((2 * coord.k + 1) - coord.t))
  bracket_contains_bounds (bs : BracketSpace) : bs.contains bs.lower ∧ bs.contains bs.upper
  rotation_field_bounded (st : ScalarTriangle) (friends : List FriendAgent) (qf : QUBOField) (bs : BracketSpace) :
    let field := rotationField st friends qf; field.val ≤ bs.mass.val

-- ═══════════════════════════════════════════════════════════════════════════
-- §7  Verification Examples
-- ═══════════════════════════════════════════════════════════════════════════

#eval let st := ScalarTriangle.balanced (Q16_16.ofNat 3) (Q16_16.ofNat 4)
      st.pistMass  -- Expected: 3 * 4 = 12

#eval let coord := { k := 2, t := 3, ht := by simp }
      let bs := BracketSpace.fromPISTCoord coord
      bs.admissible  -- Expected: true (mass = 3 * (5-3) = 6 > 0)

#eval let qf : QUBOField := { frustration := Q16_16.ofNat 1, energyScale := Q16_16.ofNat 10 }
      let x := Q16_16.ofNat 5
      QUBOField.isFrustrated qf x  -- Expected: true

-- TODO(lean-port): Add friend spawning and rotation field examples

end Semantics.RotationQUBO