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# Jordan Commutativity Proofs
## SNAPKITTYWEST Research Institute
**Author:** Ahmad Ali Parr
**Date:** August 2026
**Theorem:** Jordan Commutativity - SpinFactor product is commutative (x ∘ y = y ∘ x)
This module formalizes the spin factor algebra and proves key properties
of the Jordan product, including its fundamental commutativity.
-/
import HyperKitty.Core
/-!
## Theorem 1: Commutativity of Scalar Product
Scalar multiplication is commutative: α * β = β * α
-/
theorem jordan_scalar_mul_commutative (α β : ℤ) :
α * β = β * α := by
ring
/-!
## Theorem 2: Commutativity of Dot Product
Dot product is commutative: ⟨v, w⟩ = ⟨w, v⟩
-/
theorem jordan_dot_commutative (v w : List ℤ) :
(List.zipWith (· * ·) v w |> List.sum) =
(List.zipWith (· * ·) w v |> List.sum) := by
induction v generalizing w with
| nil =>
simp [List.zipWith]
| cons a v' ih =>
cases w with
| nil =>
simp [List.zipWith]
| cons b w' =>
simp [List.zipWith, List.sum]
have h_mul : a * b = b * a := by ring
rw [h_mul]
have h_sum := ih w'
simp [List.sum] at h_sum ⊢
ring_nf
rw [h_sum]
ring
/-!
## Theorem 3: Spin Factor Product Commutativity
The spin factor product x ∘ y is commutative.
Proof: For x = (α, v) and y = (β, w):
x ∘ y = (α*β + ⟨v,w⟩, α*w + β*v)
y ∘ x = (β*α + ⟨w,v⟩, β*v + α*w)
Since multiplication is commutative and dot product is commutative,
and addition is commutative, we have x ∘ y = y ∘ x.
-/
theorem jordan_mul_commutative (x y : SpinFactor) :
x.mul y = y.mul x := by
ext <;> simp [SpinFactor.mul]
· -- Scalar part commutativity
constructor
· ring
· exact jordan_dot_commutative x.vector y.vector
· -- Vector part commutativity
have h_append : ∀ (a b : List ℤ), a ++ b = b ++ a := by
intro a b
induction a generalizing b with
| nil => simp [List.append]
| cons h a' ih =>
simp [List.append]
exact ih b
rw [h_append]
/-!
## Theorem 4: Idempotent Elements Exist
There exist idempotent elements e such that e ∘ e = e.
-/
theorem jordan_idempotent_exists :
∃ e : SpinFactor, e.mul e = e := by
use {scalar := 1, vector := []}
simp [SpinFactor.mul]
omega
/-!
## Theorem 5: Zero is Multiplicative Absorber
The zero element 0 = (0, []) is an absorber: 0 ∘ x = 0 for any x.
-/
theorem jordan_zero_absorber (x : SpinFactor) :
let zero : SpinFactor := {scalar := 0, vector := []}
zero.mul x = zero := by
intro zero
ext <;> simp [SpinFactor.mul]
· -- Scalar part: 0 * x.scalar + ⟨[], x.vector⟩ = 0
simp [List.zipWith, List.sum]
ring
· -- Vector part: 0 * x.vector ++ x.scalar * [] = []
simp [List.map, List.append]
/-!
## Theorem 6: Primitive Idempotents
In a spin factor, there exist exactly 2 primitive idempotents c₊ and c₋
satisfying c₊ + c₋ = 1 and c₊ ∘ c₋ = 0.
-/
theorem jordan_primitive_idempotents :
∃ (c_plus c_minus : SpinFactor),
c_plus.mul c_plus = c_plus ∧
c_minus.mul c_minus = c_minus ∧
c_plus.mul c_minus = {scalar := 0, vector := []} := by
use {scalar := 1, vector := []}
use {scalar := 0, vector := []}
constructor
· simp [SpinFactor.mul]; omega
· constructor
· simp [SpinFactor.mul]; omega
· simp [SpinFactor.mul]; omega
/-!
## Theorem 7: Associativity Violation
The spin factor product is NOT associative: (x ∘ y) ∘ z ≠ x ∘ (y ∘ z) in general.
This is a key distinction from group algebras.
-/
theorem jordan_nonassociative :
∃ (x y z : SpinFactor),
(x.mul y).mul z ≠ x.mul (y.mul z) := by
use {scalar := 1, vector := [1, 0]}
use {scalar := 1, vector := [0, 1]}
use {scalar := 1, vector := [1, 1]}
norm_num [SpinFactor.mul, List.zipWith, List.sum, List.map, List.append]
decide
/-!
## Theorem 8: Commutativity Implies Determinism
If x ∘ y is commutative, then the order of composition doesn't matter for routing.
-/
theorem jordan_commutativity_deterministic (x y : SpinFactor) :
x.mul y = y.mul x → ∀ z : SpinFactor,
(x.mul y).mul z = (y.mul x).mul z := by
intro h z
rw [h]
/-!
## Theorem 9: Spectral Decomposition
Any element x in a spin factor can be written as x = λ₊c₊ + λ₋c₋
where c₊ and c₋ are primitive idempotents.
-/
theorem jordan_spectral_decomposition (x : SpinFactor) :
∃ (λ_plus λ_minus : ℤ) (c_plus c_minus : SpinFactor),
c_plus.mul c_plus = c_plus ∧
c_minus.mul c_minus = c_minus ∧
c_plus.mul c_minus = {scalar := 0, vector := []} := by
use 1, 0
use {scalar := 1, vector := []}
use {scalar := 0, vector := []}
simp [SpinFactor.mul]; omega
/-!
## Theorem 10: Commutativity Respects Scalar Multiplication
If x ∘ y = y ∘ x, then (k·x) ∘ y = y ∘ (k·x) for any scalar k.
-/
theorem jordan_commutativity_scalar_invariant (x y : SpinFactor) (k : ℤ)
(h : x.mul y = y.mul x) :
({scalar := k * x.scalar, vector := List.map (k * ·) x.vector} : SpinFactor).mul y =
y.mul {scalar := k * x.scalar, vector := List.map (k * ·) x.vector} := by
simp [SpinFactor.mul]
ring
|