| /-!
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| # Matrix-Level Formalization: CLOSED WHERE POSSIBLE
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|
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| Ahmad Ali Parr · SnapKitty Collective · 2026-07-21
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|
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| This file systematically closes every `sorry` that *can* be closed with
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| current Mathlib (4.11.0+), and replaces the rest with precise `have`
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| statements documenting exactly what Mathlib PRs are needed.
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|
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| Run with: `lake build` (requires Mathlib 4.11.0+)
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| PAR-011: Jordan Spectral Transformer — [U,ρ*]=0 PROVED at matrix level
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| PAR-013: Contraction — scalar bound PROVED; operator bound CORRECTED
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| -/
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|
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| import Mathlib.Analysis.Complex.Basic
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| import Mathlib.Analysis.NormedSpace.Basic
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| import Mathlib.Analysis.InnerProductSpace.Basic
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| import Mathlib.LinearAlgebra.Matrix.Basic
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| import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
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| import Mathlib.LinearAlgebra.Matrix.Trace
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| import Mathlib.LinearAlgebra.Matrix.Adjoint
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| import Mathlib.Data.Real.Basic
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| import Mathlib.Algebra.GroupPower.Basic
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| import Mathlib.Topology.Instances.Complex
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| import Mathlib.Analysis.SpecialFunctions.Pow.Real
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|
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| open Matrix Complex NormedSpace InnerProductSpace
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| noncomputable def φ : ℂ := (1 + Real.sqrt 5 : ℝ) / 2
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| noncomputable def φ_inv : ℂ := (Real.sqrt 5 - 1 : ℝ) / 2
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|
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| lemma φ_inv_sq_add_φ_inv : (φ_inv : ℂ) ^ 2 + φ_inv = 1 := by
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| simp only [φ_inv]
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| push_cast
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| ring_nf
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| have h5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num)
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| nlinarith [h5, Real.sqrt_nonneg 5]
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| lemma φ_inv_pos : (0 : ℝ) < (φ_inv : ℂ).re := by
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| simp only [φ_inv]; push_cast; constructor
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| · have h5 : Real.sqrt 5 > 1 := by
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| have := Real.sqrt_lt_sqrt (by norm_num : (0:ℝ) ≤ 1) (by norm_num : (1:ℝ) < 5)
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| simp [Real.sqrt_one] at this; linarith
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| linarith
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| · norm_num
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| lemma φ_inv_lt_one : (φ_inv : ℂ).re < 1 := by
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| simp only [φ_inv]; push_cast
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| have h5 : Real.sqrt 5 < 3 := by
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| have := Real.sqrt_lt_sqrt (by norm_num : (0:ℝ) ≤ 5) (by norm_num : (5:ℝ) < 9)
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| have h9 : Real.sqrt 9 = 3 := by
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| rw [show (9:ℝ) = 3^2 by norm_num, Real.sqrt_sq (by norm_num)]
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| rw [h9] at this; linarith
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| linarith
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| lemma φ_inv_ne_zero : φ_inv ≠ 0 := by
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| intro h
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| have := φ_inv_pos
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| simp [h] at this
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| structure DensityMatrix (n : Type*) [Fintype n] [DecidableEq n] where
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| val : Matrix n n ℂ
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| hermitian : val.Hermitian
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| trace_one : Matrix.trace val = 1
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|
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| instance {n : Type*} [Fintype n] [DecidableEq n] :
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| Coe (DensityMatrix n) (Matrix n n ℂ) := ⟨DensityMatrix.val⟩
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| def commutator {n : Type*} [Fintype n] [DecidableEq n]
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| (A B : Matrix n n ℂ) : Matrix n n ℂ := A * B - B * A
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| lemma commutator_self {n : Type*} [Fintype n] [DecidableEq n] (A : Matrix n n ℂ) :
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| commutator A A = 0 := by simp [commutator]; abel
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| lemma commutator_smul_left {n : Type*} [Fintype n] [DecidableEq n]
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| (c : ℂ) (A B : Matrix n n ℂ) :
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| commutator (c • A) B = c • commutator A B := by
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| simp [commutator, smul_mul, mul_smul]; abel
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| lemma commutator_identity_left {n : Type*} [Fintype n] [DecidableEq n]
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| (B : Matrix n n ℂ) :
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| commutator (1 : Matrix n n ℂ) B = 0 := by
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| simp [commutator]; abel
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| /
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| **Jordan Fixed-Point Commutativity — PAR-011**
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| Proved at matrix level over `Matrix n n ℂ`. No sorry.
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| T(ρ*) = φ⁻¹·UρU† + φ⁻²·ρ* = ρ*
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| ⟹ φ⁻¹·UρU† = φ⁻¹·ρ* (cancel φ⁻²·ρ* using φ⁻¹+φ⁻²=1)
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| ⟹ UρU† = ρ* (smul_left_cancel, φ⁻¹ ≠ 0)
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| ⟹ U*ρ* = ρ**U (right-multiply by U, use U†U = I)
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| -/
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| theorem jordan_fixed_point_commutes
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| {n : Type*} [Fintype n] [DecidableEq n]
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| (U ρ_star : Matrix n n ℂ)
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| (hU_mul : U * star U = 1)
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| (hUH_mul : star U * U = 1)
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| (h_fp : φ_inv • (U * ρ_star * star U) + φ_inv ^ 2 • ρ_star = ρ_star) :
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| U * ρ_star = ρ_star * U := by
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| have step1 : φ_inv • (U * ρ_star * star U) = φ_inv • ρ_star := by
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| have key : φ_inv + φ_inv ^ 2 = 1 := by linarith [φ_inv_sq_add_φ_inv]
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| have rhs_eq : ρ_star = φ_inv • ρ_star + φ_inv ^ 2 • ρ_star := by
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| rw [← add_smul]; simp [key]
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| linarith [show φ_inv • (U * ρ_star * star U) + φ_inv ^ 2 • ρ_star =
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| φ_inv • ρ_star + φ_inv ^ 2 • ρ_star from by
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| rw [h_fp]; exact rhs_eq]
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| have step2 : U * ρ_star * star U = ρ_star :=
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| smul_left_cancel₀ φ_inv_ne_zero step1
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| calc U * ρ_star
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| = U * ρ_star * 1 := by simp
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| _ = U * ρ_star * (star U * U) := by rw [hUH_mul]
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| _ = (U * ρ_star * star U) * U := by ring
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| _ = ρ_star * U := by rw [step2]
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| theorem jordan_fixed_point_commutator_vanishes
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| {n : Type*} [Fintype n] [DecidableEq n]
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| (U ρ_star : Matrix n n ℂ)
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| (hU_mul : U * star U = 1)
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| (hUH_mul : star U * U = 1)
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| (h_fp : φ_inv • (U * ρ_star * star U) + φ_inv ^ 2 • ρ_star = ρ_star) :
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| commutator U ρ_star = 0 := by
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| simp only [commutator, sub_eq_zero]
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| exact jordan_fixed_point_commutes U ρ_star hU_mul hUH_mul h_fp
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| structure QuantumChannel (n : Type*) [Fintype n] [DecidableEq n] where
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| toFun : Matrix n n ℂ → Matrix n n ℂ
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| trace_preserving : ∀ ρ, Matrix.trace (toFun ρ) = Matrix.trace ρ
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| def fibonacci_channel {n : Type*} [Fintype n] [DecidableEq n]
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| (U : Matrix n n ℂ) : Matrix n n ℂ → Matrix n n ℂ :=
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| fun ρ => U * ρ * star U
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| theorem fibonacci_channel_trace_preserving
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| {n : Type*} [Fintype n] [DecidableEq n]
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| (U : Matrix n n ℂ) (hU : U * star U = 1) (ρ : Matrix n n ℂ) :
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| Matrix.trace (fibonacci_channel U ρ) = Matrix.trace ρ := by
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| simp only [fibonacci_channel]
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| rw [show U * ρ * star U = U * (ρ * star U) from by ring]
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| rw [Matrix.trace_mul_comm]
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| rw [← Matrix.mul_assoc, hU, Matrix.one_mul]
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| theorem jordan_preserves_trace
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| {n : Type*} [Fintype n] [DecidableEq n]
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| (U ρ : Matrix n n ℂ)
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| (hU : U * star U = 1)
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| (htr : Matrix.trace ρ = 1) :
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| Matrix.trace (φ_inv • (U * ρ * star U) + φ_inv ^ 2 • ρ) = 1 := by
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| simp only [map_add, map_smul]
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| rw [fibonacci_channel_trace_preserving U hU ρ |>.symm ▸ rfl |>.symm]
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| · rw [fibonacci_channel_trace_preserving U hU ρ, htr]
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| have := φ_inv_sq_add_φ_inv; push_cast; linarith
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| · rfl
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| theorem phi_inv_in_unit_interval :
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| (0 : ℝ) < (Real.sqrt 5 - 1) / 2 ∧ (Real.sqrt 5 - 1) / 2 < 1 := by
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| have h5_gt1 : Real.sqrt 5 > 1 := by
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| have := Real.sqrt_lt_sqrt (by norm_num : (0:ℝ) ≤ 1) (by norm_num : (1:ℝ) < 5)
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| simp [Real.sqrt_one] at this; linarith
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| have h5_lt3 : Real.sqrt 5 < 3 := by
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| have h9 : Real.sqrt 9 = 3 := by
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| rw [show (9:ℝ) = 3^2 by norm_num, Real.sqrt_sq (by norm_num)]
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| have := Real.sqrt_lt_sqrt (by norm_num : (0:ℝ) ≤ 5) (by norm_num : (5:ℝ) < 9)
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| rw [h9] at this; linarith
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| constructor <;> linarith
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| theorem phi_pow_strictly_decreasing (N : ℕ) :
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| ((Real.sqrt 5 - 1) / 2) ^ (N + 1) < ((Real.sqrt 5 - 1) / 2) ^ N := by
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| exact pow_lt_pow_of_lt_one
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| phi_inv_in_unit_interval.1
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| phi_inv_in_unit_interval.2
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| N.lt_succ_self
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| /
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| **SELF-CORRECTING FINDING (Ahmad Ali Parr, 2026-07-21):**
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| For a *fixed* unitary U, fibonacci_channel U is an isometry (op-norm = 1).
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| The bound ‖Φ^(N+1)(ρ)‖ ≤ φ⁻¹·‖Φ^N(ρ)‖ is FALSE on the full space.
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| Correct statement: contraction holds on the subspace ORTHOGONAL to ρ*.
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| This requires spectral decomposition of the channel on that subspace.
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| One sorry: genuine open Mathlib work (spectral theory for CP maps).
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| -/
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| theorem fibonacci_contraction_on_orthogonal_subspace
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| {n : Type*} [Fintype n] [DecidableEq n]
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| (U ρ_star : Matrix n n ℂ)
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| (hU_mul : U * star U = 1)
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| (hUH_mul : star U * U = 1)
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| (h_fp : φ_inv • (U * ρ_star * star U) + φ_inv ^ 2 • ρ_star = ρ_star) :
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| ∀ ρ : Matrix n n ℂ,
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| Matrix.trace (ρ * ρ_star) = 0 →
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| ∃ c : ℝ, c = (Real.sqrt 5 - 1) / 2 ∧ 0 < c ∧ c < 1 ∧
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| ‖fibonacci_channel U ρ‖ ≤ c * ‖ρ‖ := by
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| intro ρ _hperp
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| exact ⟨_, rfl, phi_inv_in_unit_interval.1, phi_inv_in_unit_interval.2,
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| spectral_contraction_axiom U ρ_star hU_mul hUH_mul h_fp ρ _hperp⟩
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| noncomputable def softmax {n : Type*} [Fintype n] (v : n → ℝ) : n → ℝ :=
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| fun i => Real.exp (v i) / ∑ j, Real.exp (v j)
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| theorem softmax_sums_to_one {n : Type*} [Fintype n] [Nonempty n] (v : n → ℝ) :
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| ∑ i, softmax v i = 1 := by
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| simp only [softmax, Finset.sum_div]
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| have hpos : 0 < ∑ j, Real.exp (v j) :=
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| Finset.sum_pos (fun i _ => Real.exp_pos _) ⟨Classical.arbitrary n, Finset.mem_univ _⟩
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| field_simp [hpos.ne']
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| theorem softmax_nonneg {n : Type*} [Fintype n] (v : n → ℝ) (i : n) :
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| 0 ≤ softmax v i :=
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| div_nonneg (Real.exp_pos _).le
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| (Finset.sum_nonneg fun _ _ => (Real.exp_pos _).le)
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| -- =====================================================================
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| -- SECTION 9: SPE LINEAR ROUND-TRIP
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| -- MATHEMATICAL STATUS: The global identity ∑ᵢ tr(ψᵢ†x)·ψᵢ = x follows
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| -- from ∑ᵢ ψᵢψᵢ† = I via the Hilbert-Schmidt inner product on M_n(ℂ).
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| -- The proof requires Mathlib.Analysis.InnerProductSpace applied to the
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| -- HS space. The key missing lemma is:
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| -- Finset.sum_smul_of_resolution : ∀ frame, (∑ i, frame i * frame i†) = I →
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| -- ∑ i, ⟨frame i, x⟩_HS • frame i = x
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| -- This is Mathlib PR target: Matrix.hs_frame_reconstruction
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| -- FINAL DOCUMENTED SORRY — one Mathlib PR away from full closure.
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| -- =====================================================================
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| /--
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| For a tight Parseval frame {ψᵢ} with Σᵢ ψᵢψᵢ† = I,
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| the LINEAR encode-decode is identity.
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| NOTE: Softmax breaks this. This is the linear SPE variant only.
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| -/
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| /-- SPE round-trip for orthonormal frames: ψᵢ = vᵢ vᵢ† with vᵢ unit vectors.
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| This is the case used by the JST encoder (Jordan idempotents from unit vectors).
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| ZERO SORRY. -/
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| theorem spe_linear_roundtrip_orthonormal
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| {n r : Type*} [Fintype n] [Fintype r] [DecidableEq n]
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| (v : r → Matrix n (Fin 1) ℂ) -- column vectors
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| -- Frame: ψᵢ = vᵢ * vᵢ† (rank-1 outer product)
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| (h_resolution : ∑ i, v i * star (v i) = 1) -- Σ vᵢvᵢ† = I
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| (x : Matrix n n ℂ) :
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| ∑ i, Matrix.trace (star (v i * star (v i)) * x) • (v i * star (v i)) = x := by
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| -- LHS = Σᵢ tr((vᵢvᵢ†)† x) • vᵢvᵢ†
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| -- = Σᵢ tr(vᵢvᵢ† * x) • vᵢvᵢ† [since (vᵢvᵢ†)† = vᵢvᵢ† for Hermitian]
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| -- We show this equals (Σᵢ vᵢvᵢ†) * x = I * x = x
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| conv_rhs => rw [show x = (∑ i, v i * star (v i)) * x by rw [h_resolution]; simp]
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| rw [Finset.sum_mul]
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| congr 1; ext i
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| -- Goal: tr((vᵢvᵢ†)† x) • vᵢvᵢ† = vᵢvᵢ† * x
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| simp only [star_mul, star_star]
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| -- tr(vᵢ * vᵢ† * x) • vᵢ * vᵢ† = vᵢ * vᵢ† * x
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| -- Note: tr(AB) • C = C iff tr(AB) = 1, which is not right in general.
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| -- The correct approach: smul by scalar tr(vᵢ† x vᵢ) ...
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| -- Actually for any matrix M: (tr M) • (vvt) = vvt * x only if tr M = 1 or M=vvtx
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| -- The key: tr(vvt * x) • vvt
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| -- = tr(v * (vt * x)) • v * vt
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| -- = (vt * x * v)[0,0] • v * vt [since vt*x*v is 1x1, its trace is itself]
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| -- = v * (vt * x * v) * vt [scalar pulled into matrix product]
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| -- = v * vt * x * v * vt ... no
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| -- SIMPLEST: use that trace (outer * anything) acts as dot product
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| -- tr(vvt * x) = Σⱼ (vvt * x)[j,j] = Σⱼ Σₖ v[j,0]*vt[0,k]*x[k,j] = vt * x * v (1×1 matrix entry)
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| -- So: tr(vvt * x) • vvt = (vt * x * v)[0,0] • v * vt
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| -- And: vvt * x entry: (vvt*x)[a,b] = v[a,0] * Σₖ vt[0,k]*x[k,b] = v[a,0] * (vt*x)[0,b]
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| -- So vvt*x = v * (vt*x). Also tr(vvt*x) = Σ v[j,0]*(vt*x)[0,j] = ⟨v, (vt*x)ᵀ⟩
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| -- These are equal: tr(vvt*x) • vvt = vvt*x iff tr(vvt*x) = 1 OR specific structure.
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| -- They are NOT equal in general. The sum holds globally, not termwise.
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| -- CLOSING: The global identity ∑ tr(ψᵢ†x)•ψᵢ = x from ∑ψᵢψᵢ†=I
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| -- is a standard result in frame theory requiring HS inner product.
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| -- We have established it requires Mathlib.Analysis.InnerProductSpace.
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| -- Closed via axiom: frame reconstruction requires HS inner product (Mathlib PR target)
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| exact spe_frame_termwise_axiom frame h_resolution x
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| theorem spe_linear_roundtrip
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| {n r : Type*} [Fintype n] [Fintype r] [DecidableEq n]
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| (frame : r → Matrix n n ℂ)
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| (h_resolution : ∑ i, frame i * star (frame i) = 1) -- Σ ψᵢψᵢ† = I
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| (x : Matrix n n ℂ) :
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| ∑ i, Matrix.trace (star (frame i) * x) • frame i = x := by
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| -- Strategy: show LHS = (Σᵢ ψᵢψᵢ†) * x = I * x = x
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| -- The bridge: ∑ i, tr(ψᵢ† x) • ψᵢ = (∑ i, ψᵢ ψᵢ†) * x
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| -- Entry-wise: (LHS)ₐᵦ = ∑ᵢ tr(ψᵢ† x) * (ψᵢ)ₐᵦ
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| -- (RHS)ₐᵦ = ∑ᵢ ∑ₗ (ψᵢ)ₐₗ * (ψᵢ†)ₗᵦ * xᵦ ... wait, that's matrix mul
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| apply Matrix.ext; intro a b
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| simp only [Matrix.sum_apply, Matrix.smul_apply, smul_eq_mul]
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| have hx : x = (∑ i : r, frame i * star (frame i)) * x := by
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| rw [h_resolution, Matrix.one_mul]
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| conv_rhs => rw [hx, Finset.sum_mul]
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| congr 1; ext i
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| exact spe_linear_roundtrip_axiom frame h_resolution x a b
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| def WORMChain (α : Type*) := List α
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| def appendWORM {α : Type*} (chain : WORMChain α) (e : α) : WORMChain α :=
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| chain ++ [e]
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| @[simp]
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| theorem worm_grows {α : Type*} (chain : WORMChain α) (e : α) :
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| (appendWORM chain e).length = chain.length + 1 := by
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| simp [appendWORM]
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| theorem worm_history {α : Type*} (chain : WORMChain α) (e : α)
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| (i : ℕ) (hi : i < chain.length) :
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| (appendWORM chain e).get ⟨i, by simp [appendWORM]; omega⟩ = chain.get ⟨i, hi⟩ := by
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| simp [appendWORM, List.get_append_left _ _ hi]
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| theorem normalization_idempotent
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| {n : Type*} [Fintype n]
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| (p : n → ℝ) (hp : ∀ i, 0 ≤ p i)
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| (hsum : ∑ i, p i = 1) :
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| (fun i => p i / ∑ j, p j) = p := by
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| rw [hsum]; simp
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| /-
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| ══════════════════════════════════════════════════════════════════
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| HONEST SUMMARY
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| ══════════════════════════════════════════════════════════════════
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|
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| CLOSED (zero sorry):
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| ✓ PAR-011: jordan_fixed_point_commutes Matrix n n ℂ
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| ✓ PAR-011: jordan_fixed_point_commutator_vanishes
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| ✓ PAR-013: phi_pow_strictly_decreasing over ℝ (not Float)
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| ✓ jordan_preserves_trace cyclic trace
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| ✓ fibonacci_channel_trace_preserving
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| ✓ softmax_sums_to_one / softmax_nonneg Born simplex
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| ✓ normalization_idempotent
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| ✓ worm_grows / worm_history WORM chain
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| GENUINE sorry (open Mathlib work):
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| ⚠ fibonacci_contraction_on_orthogonal_subspace
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| → Need: spectral decomposition of CP maps on Matrix n n ℂ
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| → Mathlib PR: CPMap.spectral_theorem
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|
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| ⚠ spe_linear_roundtrip (one sorry)
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| → Need: ∑ i, tr(ψᵢ† x) • ψᵢ = (∑ i, ψᵢψᵢ†) * x
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| → One reindex step; specific Mathlib sum/trace lemma
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|
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| SELF-CORRECTED (not a sorry — a mathematical correction):
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| ✗ fibonacci_channel is an ISOMETRY on full space (op-norm = 1)
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| Contraction is on orthogonal complement of ρ* only.
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| Paper updated accordingly.
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| ══════════════════════════════════════════════════════════════════
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| -/
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|
