Research-Stack/0-Core-Formalism/lean/external/OTOM/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

namespace Semantics.RotationQUBO

open PIST DynamicCanal

-- ═══════════════════════════════════════════════════════════════════════════
-- §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 : Fix16  -- First vertex
  b : Fix16  -- Second vertex
  c : Fix16  -- Third vertex
  closure : Fix16  -- 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 : Fix16) : ScalarTriangle :=
  let c := Fix16.sub (Fix16.sub Fix16.zero a) b  -- c = -(a + b)
  let closure := Fix16.add (Fix16.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 := fix16FromNat coord.t
  let b := fix16FromNat coord.b
  let c := Fix16.sub (Fix16.sub Fix16.zero a) b
  let closure := Fix16.add (Fix16.add a b) c
  { a, b, c, closure }

/-- The PIST mass of the scalar triangle (a * b). -/
def pistMass (st : ScalarTriangle) : Fix16 :=
  Fix16.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 : Fix16  -- Rotation angle in radians (Q16.16)
  cosθ : Fix16   -- cos(θ) in Q16.16
  sinθ : Fix16   -- 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 : Fix16) : RotationMatrix :=
  -- Placeholder: use Taylor series or lookup table for cos/sin
  -- For now, use simple approximation
  let cosθ := Fix16.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 : Fix16) : Fix16 :=
  -- 2D rotation: x' = x·cosθ - y·sinθ
  -- For 1D scalar, this is simplified
  Fix16.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' := Fix16.add (Fix16.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 : Fix16  -- Frustration parameter δ (0 ≤ δ ≤ 1)
  energyScale : Fix16  -- 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 : Fix16) : Fix16 :=
  let xSq := Fix16.mul x x
  let denom := Fix16.add Fix16.one (Fix16.mul qf.frustration qf.frustration)
  let energy := Fix16.div xSq denom
  Fix16.sub energy qf.energyScale

/-- Check if field is frustrated at position x. -/
def isFrustrated (qf : QUBOField) (x : Fix16) : 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 : Fix16  -- Lower bound (a)
  upper : Fix16  -- Upper bound (b)
  mass : Fix16   -- PIST mass (a * b)
  gap : Fix16    -- 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 := fix16FromNat coord.a
  let upper := fix16FromNat coord.b
  let mass := fix16FromNat coord.mass
  let gap := Fix16.sub upper lower
  let admissible := coord.mass > 0  -- Positive mass = admissible
  { lower, upper, mass, gap, admissible }

/-- Check if a value is within the bracket space. -/
def contains (bs : BracketSpace) (x : Fix16) : 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 : Fix16  -- 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 : Fix16) (bracket : BracketSpace) : FriendAgent :=
  let rm := RotationMatrix.fromAngle theta
  let weight := Fix16.ofNat 1  -- Default weight = 1.0
  { rotation := rm, weight, bracket }

/-- Spawn multiple friends in superposition. -/
def spawnSuperposition (thetas : List Fix16) (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) : Fix16 :=
  let denom := Fix16.add Fix16.one (Fix16.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 := Fix16.mul (ScalarTriangle.pistMass rotated) friend.weight
    Fix16.add acc weightedMass
  ) Fix16.zero

  -- Divide by frustration denominator
  Fix16.div sumRotations denom

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

/-- Theorem: Balanced scalar triangle has zero closure.
    ANALYTIC_OPEN: With saturating Fix16 arithmetic, a + b + (-(a+b)) is NOT
    necessarily zero.  Counter-example: if a = b = Fix16.maxVal (0x7FFFFFFF),
    then add a b saturates to maxVal, and c = -(a+b) also saturates, so
    add maxVal minVal = add 0x7FFFFFFF 0x80000001 ≠ 0 in saturating arithmetic.
    The theorem holds only for wrapping (modular 2³²) arithmetic or when
    |a.raw| + |b.raw| ≤ 0x7FFFFFFF (no saturation occurs).
    Correct statement requires a non-overflow side condition:
      a.val.toNat + b.val.toNat ≤ 0x7FFFFFFF → closure = zero.
    Marking ANALYTIC_OPEN pending a wrapping-arithmetic variant. -/
theorem balancedClosureZero (a b : Fix16) :
    (ScalarTriangle.balanced a b).closure = Fix16.zero := by
  -- ANALYTIC_OPEN: false under saturating arithmetic without overflow bounds.
  sorry

/-- Theorem: PIST mass from coordinate equals a * b.
    TACTIC_GAP: fix16FromNat t * fix16FromNat b = fix16FromNat (t * b) requires
    (t * 65536) * (b * 65536) >> 16 = t * b * 65536 in UInt32 arithmetic.
    This holds exactly when t * b * 65536 < 2^32 (no overflow), i.e. t*b < 65536.
    For arbitrary Coord, t can be up to 2k+1 (unbounded), so overflow is possible.
    A correct statement needs t * b < 65536 as a side condition or uses
    unbounded-integer semantics.
    Marking TACTIC_GAP pending a bounded-range variant. -/
theorem pistMassFromCoord (coord : PIST.Coord) :
    (ScalarTriangle.fromPISTCoord coord).pistMass = fix16FromNat coord.mass := by
  -- TACTIC_GAP: see theorem docstring — requires overflow guard.
  sorry

/-- Theorem: Bracket space contains its own bounds when lower ≤ upper.
    The original statement is FALSE for arbitrary BracketSpace because no
    invariant guarantees lower ≤ upper.  The corrected statement adds the
    required hypothesis, expressed in terms of the UInt32 raw field (for
    the OTOM UInt32 Fix16) or the Int val field (for the Semantics Q16_16 Fix16).
    TACTIC_GAP: the exact proof depends on which Fix16 definition is in scope
    (UInt32 struct vs Int subtype) and whether ∧ in `contains` is Bool.and or
    Prop.And. The mathematical content is trivial (reflexivity + hypothesis),
    but the elaboration path is import-context-dependent. -/
theorem bracketContainsBounds (bs : BracketSpace)
    (h : bs.lower.val.toNat ≤ bs.upper.val.toNat) :
    bs.contains bs.lower ∧ bs.contains bs.upper := by
  simp only [BracketSpace.contains]
  -- After unfolding: goal is (lower ≤ lower ∧ lower ≤ upper) ∧ (lower ≤ upper ∧ upper ≤ upper)
  -- The two reflexivity parts follow from le_refl; the cross parts follow from h.
  -- TACTIC_GAP: decide / simp path depends on Fix16 elaboration context.
  constructor <;> simp only [decide_eq_true_eq] <;> omega

/-- Theorem: Rotation field is bounded by bracket mass.
    ANALYTIC_OPEN: No structural relationship exists between the rotation field
    (arbitrary sum of rotated triangle masses) and a bracket's mass (a * b from
    a PIST coordinate). The theorem as stated is false in general — e.g., with
    large triangle vertices and unit weights, rotationField can exceed any fixed
    bracket mass.
    A meaningful bound would require:
      (1) bounded input constraints on triangle vertices and weights, and
      (2) a specific bracket constructed from the same coordinate as the triangle.
    Marking ANALYTIC_OPEN as the theorem lacks a mathematical foundation. -/
theorem rotationFieldBounded (st : ScalarTriangle) (friends : List FriendAgent)
    (qf : QUBOField) (bs : BracketSpace) :
    let field := rotationField st friends qf
    field.val ≤ bs.mass.val := by
  -- ANALYTIC_OPEN: see theorem docstring.
  sorry

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

#eval let st := ScalarTriangle.balanced (Fix16.ofNat 3) (Fix16.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 := { frustration := Fix16.ofNat 1, energyScale := Fix16.ofNat 10 }
      let x := Fix16.ofNat 5
      qf.isFrustrated x  -- Expected: true

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

end Semantics.RotationQUBO