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277 lines
13 KiB
Text
277 lines
13 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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RotationQUBO.lean — Rotation Matrices as Literal Rotation Notation in Frustrated QUBO Fields
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This module formalizes a 1D scalar triangle navigating a frustrated QUBO field,
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spawning friends to rotate in superposition. Each bracket represents a possibility space,
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borrowing the PIST framework for shell geometry.
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Key insight:
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- Rotation matrices as literal rotation notation (not just linear algebra)
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- 1D scalar triangle = (a, b, c) with a+b+c = 0 (triangle closure)
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- Frustrated QUBO field = energy landscape with competing minima
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- Spawning friends = agent generation in superposition
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- Brackets = possibility spaces [lower, upper] from PIST shell geometry
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- PIST mass = a*b (hyperbola index) as rotation weight
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The rotation field:
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Φ_rot(x, θ) = Σᵢ R(θᵢ) · xᵢ / (1 + frustration²)
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Where:
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- R(θ): rotation matrix at angle θ
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- xᵢ: scalar triangle vertex
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- frustration: QUBO field frustration parameter
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Per AGENTS.md §0: Lean is the source of truth.
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Per AGENTS.md §1.4: Q16_16 fixed-point for hardware extraction.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: Every def has eval witness or theorem.
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-/
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Matrix.Basic
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import Mathlib.Tactic
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import Semantics.PIST
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import Semantics.DynamicCanal
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namespace Semantics.RotationQUBO
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open PIST DynamicCanal
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Scalar Triangle Types
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A 1D scalar triangle (a, b, c) with closure condition a + b + c = 0.
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Represents a balanced configuration that can navigate QUBO fields. -/
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structure ScalarTriangle where
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a : Fix16 -- First vertex
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b : Fix16 -- Second vertex
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c : Fix16 -- Third vertex
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closure : Fix16 -- Closure residual (should be 0 for balanced triangle)
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deriving Repr, DecidableEq, BEq
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namespace ScalarTriangle
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/-- Create a balanced scalar triangle from two vertices (c = -(a + b)). -/
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def balanced (a b : Fix16) : ScalarTriangle :=
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let c := Fix16.sub (Fix16.sub Fix16.zero a) b -- c = -(a + b)
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let closure := Fix16.add (Fix16.add a b) c -- should be 0
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{ a, b, c, closure }
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/-- Create a scalar triangle from PIST coordinate (a = t, b = 2k+1-t). -/
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def fromPISTCoord (coord : PIST.Coord) : ScalarTriangle :=
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let a := fix16FromNat coord.t
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let b := fix16FromNat coord.b
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let c := Fix16.sub (Fix16.sub Fix16.zero a) b
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let closure := Fix16.add (Fix16.add a b) c
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{ a, b, c, closure }
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/-- The PIST mass of the scalar triangle (a * b). -/
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def pistMass (st : ScalarTriangle) : Fix16 :=
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Fix16.mul st.a st.b
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end ScalarTriangle
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Rotation Matrix as Literal Rotation Notation
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Rotation matrix at angle θ (2D rotation).
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Treated as literal rotation notation, not just linear algebra. -/
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structure RotationMatrix where
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theta : Fix16 -- Rotation angle in radians (Q16.16)
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cosθ : Fix16 -- cos(θ) in Q16.16
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sinθ : Fix16 -- sin(θ) in Q16.16
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deriving Repr, DecidableEq, BEq
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namespace RotationMatrix
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/-- Create rotation matrix from angle θ.
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Uses Q16.16 approximation for cos and sin. -/
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def fromAngle (theta : Fix16) : RotationMatrix :=
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-- Placeholder: use Taylor series or lookup table for cos/sin
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-- For now, use simple approximation
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let cosθ := Fix16.ofNat 1 -- cos(0) = 1
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let sinθ := theta -- sin(θ) ≈ θ for small θ
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{ theta, cosθ, sinθ }
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/-- Apply rotation matrix to scalar triangle vertex. -/
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def rotateVertex (rm : RotationMatrix) (v : Fix16) : Fix16 :=
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-- 2D rotation: x' = x·cosθ - y·sinθ
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-- For 1D scalar, this is simplified
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Fix16.mul v rm.cosθ
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/-- Apply rotation matrix to entire scalar triangle. -/
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def rotateTriangle (rm : RotationMatrix) (st : ScalarTriangle) : ScalarTriangle :=
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let a' := rm.rotateVertex st.a
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let b' := rm.rotateVertex st.b
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let c' := rm.rotateVertex st.c
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let closure' := Fix16.add (Fix16.add a' b') c'
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{ a := a', b := b', c := c', closure := closure' }
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end RotationMatrix
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Frustrated QUBO Field
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Frustrated QUBO field parameters.
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Frustration parameter δ controls competing energy minima. -/
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structure QUBOField where
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frustration : Fix16 -- Frustration parameter δ (0 ≤ δ ≤ 1)
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energyScale : Fix16 -- Energy scale factor
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deriving Repr, DecidableEq, BEq
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namespace QUBOField
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/-- Compute field energy at position x.
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E(x) = x² / (1 + δ²) - frustration penalty. -/
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def fieldEnergy (qf : QUBOField) (x : Fix16) : Fix16 :=
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let xSq := Fix16.mul x x
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let denom := Fix16.add Fix16.one (Fix16.mul qf.frustration qf.frustration)
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let energy := Fix16.div xSq denom
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Fix16.sub energy qf.energyScale
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/-- Check if field is frustrated at position x. -/
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def isFrustrated (qf : QUBOField) (x : Fix16) : Bool :=
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-- Field is frustrated if energy > 0
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let energy := qf.fieldEnergy x
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energy.raw > 0
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end QUBOField
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Bracket Possibility Spaces
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Bracket possibility space from PIST shell geometry.
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[lower, upper] = [a, b] where a + b = 2k+1 and mass = a*b. -/
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structure BracketSpace where
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lower : Fix16 -- Lower bound (a)
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upper : Fix16 -- Upper bound (b)
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mass : Fix16 -- PIST mass (a * b)
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gap : Fix16 -- Upper - lower
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admissible : Bool -- Whether space is admissible
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deriving Repr, DecidableEq, BEq
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namespace BracketSpace
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/-- Create bracket space from PIST coordinate. -/
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def fromPISTCoord (coord : PIST.Coord) : BracketSpace :=
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let lower := fix16FromNat coord.a
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let upper := fix16FromNat coord.b
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let mass := fix16FromNat coord.mass
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let gap := Fix16.sub upper lower
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let admissible := coord.mass > 0 -- Positive mass = admissible
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{ lower, upper, mass, gap, admissible }
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/-- Check if a value is within the bracket space. -/
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def contains (bs : BracketSpace) (x : Fix16) : Bool :=
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let xNat := x.raw.toNat
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let lowerNat := bs.lower.raw.toNat
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let upperNat := bs.upper.raw.toNat
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lowerNat ≤ xNat ∧ xNat ≤ upperNat
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end BracketSpace
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Friend Spawning in Superposition
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A friend agent spawned in superposition.
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Each friend has a rotation angle and weight. -/
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structure FriendAgent where
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rotation : RotationMatrix -- Rotation matrix
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weight : Fix16 -- Superposition weight (0 ≤ weight ≤ 1)
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bracket : BracketSpace -- Assigned bracket space
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deriving Repr, DecidableEq, BEq
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namespace FriendAgent
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/-- Spawn a friend agent with random rotation. -/
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def spawn (theta : Fix16) (bracket : BracketSpace) : FriendAgent :=
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let rm := RotationMatrix.fromAngle theta
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let weight := Fix16.ofNat 1 -- Default weight = 1.0
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{ rotation := rm, weight, bracket }
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/-- Spawn multiple friends in superposition. -/
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def spawnSuperposition (thetas : List Fix16) (bracket : BracketSpace) : List FriendAgent :=
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thetas.map (fun θ => spawn θ bracket)
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end FriendAgent
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Rotation Field Computation
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute rotation field for scalar triangle in QUBO field with friends.
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Φ_rot(x, θ) = Σᵢ R(θᵢ) · xᵢ / (1 + frustration²) -/
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def rotationField (st : ScalarTriangle) (friends : List FriendAgent)
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(qf : QUBOField) : Fix16 :=
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let denom := Fix16.add Fix16.one (Fix16.mul qf.frustration qf.frustration)
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-- Sum over friends: Σᵢ weightᵢ * rotationᵢ(triangle)
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let sumRotations := friends.foldl (fun acc friend =>
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let rotated := friend.rotation.rotateTriangle st
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let weightedMass := Fix16.mul (ScalarTriangle.pistMass rotated) friend.weight
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Fix16.add acc weightedMass
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) Fix16.zero
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-- Divide by frustration denominator
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Fix16.div sumRotations denom
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Theorems: Rotation and Bracket Properties
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Balanced scalar triangle has zero closure. -/
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theorem balancedClosureZero (a b : Fix16) :
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(ScalarTriangle.balanced a b).closure = Fix16.zero := by
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unfold ScalarTriangle.balanced
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-- c = -(a + b), so a + b + c = 0
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sorry -- TODO(lean-port): Prove closure = 0 for balanced triangle
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/-- Theorem: PIST mass from coordinate equals a * b. -/
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theorem pistMassFromCoord (coord : PIST.Coord) :
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(ScalarTriangle.fromPISTCoord coord).pistMass = fix16FromNat coord.mass := by
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unfold ScalarTriangle.fromPISTCoord, ScalarTriangle.pistMass
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-- mass = a * b = t * (2k+1-t)
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sorry -- TODO(lean-port): Prove mass = a*b
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/-- Theorem: Bracket space contains its bounds. -/
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theorem bracketContainsBounds (bs : BracketSpace) :
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bs.contains bs.lower ∧ bs.contains bs.upper := by
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unfold BracketSpace.contains
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-- lower ≤ lower and upper ≤ upper
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sorry -- TODO(lean-port): Prove bracket contains its own bounds
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/-- Theorem: Rotation field is bounded by bracket mass. -/
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theorem rotationFieldBounded (st : ScalarTriangle) (friends : List FriendAgent)
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(qf : QUBOField) (bs : BracketSpace) :
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let field := rotationField st friends qf
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field.raw ≤ bs.mass.raw := by
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-- Rotation field divided by (1 + δ²) ≤ original mass
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sorry -- TODO(lean-port): Prove field bounded by bracket mass
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Verification Examples
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval let st := ScalarTriangle.balanced (Fix16.ofNat 3) (Fix16.ofNat 4)
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st.pistMass -- Expected: 3 * 4 = 12
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#eval let coord := { k := 2, t := 3, ht := by simp }
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let bs := BracketSpace.fromPISTCoord coord
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bs.admissible -- Expected: true (mass = 3 * (5-3) = 6 > 0)
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#eval let qf := { frustration := Fix16.ofNat 1, energyScale := Fix16.ofNat 10 }
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let x := Fix16.ofNat 5
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qf.isFrustrated x -- Expected: true
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-- TODO(lean-port): Add friend spawning and rotation field examples
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end Semantics.RotationQUBO
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