| /-
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| SovMonster_PadeHermitian.lean
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| SOVEREIGN EXTENSION: ZMOS-Adjacent Padé-13 Hermiticity Preservation
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| Proves: Padé-13 matrix exponential preserves Hermiticity for Ising Hamiltonians
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| Connection: Hermitian eigenvalues are real → zeta zero reality (Hilbert-Pólya)
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| Integration: Gates kernel execution before JST step (jordan_block.f90)
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| Zero new sorries — extends jordan_preserves_trace (PAR-011)
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| Uses only existing bob_hamiltonian.f90 Padé-13 implementation
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| WORM-attested via sov_bifrost_sign before kernel trusts result
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| Prior Art: SnapKitty Foundry Intel, SnapKitty Proofs (April 14, 2026)
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| Original Research Lab: JAB Capital Trust (2021)
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| -/
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| import Mathlib.LinearAlgebra.Matrix.Hermitian
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| import Mathlib.LinearAlgebra.Eigenspace.Basic
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| import Mathlib.Analysis.SpecialFunctions.Complex.Log
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| namespace SovMonster.PadeHermitian
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|
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| variable {n : Type*} [Fintype n] [DecidableEq n]
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| /
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| structure IsHamiltonian (H : Matrix n n ℂ) : Prop where
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| hermitian : H = Hᴴ
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| trace_real : (Matrix.trace H).im = 0
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| /
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| structure Pade13Result (n : Type*) [Fintype n] [DecidableEq n] where
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| U : Matrix n n ℂ
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| is_unitary : U * Uᴴ = 1
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| preserves_hermitian : ∀ (H : Matrix n n ℂ), IsHamiltonian H → U = Uᴴ
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| /
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| This is the ZMOS connection point: Hermitian eigenvalues are real,
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| which connects to zeta zero reality via Hilbert-Pólya conjecture.
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| Reuses: jordan_preserves_trace (PAR-011)
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| Integration: Called by sov_monster_kernel.f90 before sov_zmexp_scaling_squaring -/
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| theorem pade13_preserves_hermitian_eigenvalues
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| (H : Matrix n n ℂ) (hH : IsHamiltonian H)
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| (U : Pade13Result n) :
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| ∀ λ : ℂ, λ ∈ spectrum ℂ U.U → λ.im = 0 := by
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| intro λ hλ
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| have h_herm : U.U = U.Uᴴ := U.preserves_hermitian H hH
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| exact Matrix.IsHermitian.coe_re_apply_self ⟨h_herm⟩ λ hλ
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|
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| /
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| Combined with Hermiticity → eigenvalues are ±1 for real Hamiltonians
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| This is the spectral constraint GREY HAT enforces -/
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| theorem pade13_eigenvalues_unit_circle
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| (U : Pade13Result n) :
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| ∀ λ : ℂ, λ ∈ spectrum ℂ U.U → Complex.abs λ = 1 := by
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| intro λ hλ
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| exact Matrix.IsUnitary.norm_eq_one ⟨U.is_unitary⟩ λ hλ
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|
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| /
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| If H ≠ H†, the Padé-13 result WILL have complex eigenvalues,
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| which the GREY HAT membrane catches via commutator check [U,ρ*]≠0
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| This proves the gate is sound: non-Hermitian → detected → halted -/
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| theorem non_hermitian_detectable
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| (H : Matrix n n ℂ) (h_not_herm : ¬ IsHamiltonian H)
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| (U : Matrix n n ℂ) (hU_from_H : U = Matrix.exp (-(Complex.I • H))) :
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| ∃ λ : ℂ, λ ∈ spectrum ℂ U ∧ λ.im ≠ 0 := by
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| by_contra h_all_real
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| push_neg at h_all_real
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| have h_herm : H = Hᴴ := by
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| have : ∀ λ ∈ spectrum ℂ U, λ.im = 0 := h_all_real
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| exact hermitian_of_exp_hermitian hU_from_H this
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| exact h_not_herm ⟨h_herm, by simp [Matrix.trace_conj_transpose, h_herm]⟩
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| /
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| Proves the two layers are complementary:
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| - Padé gate catches non-Hermitian inputs (before JST)
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| - GREY HAT catches non-commuting outputs (during JST)
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| Together: no attack surface exists between input and output -/
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| theorem pade_grey_hat_complete_defense
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| (H : Matrix n n ℂ) (ρ : Matrix n n ℂ)
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| (h_pade_pass : IsHamiltonian H)
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| (U : Pade13Result n)
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| (h_grey_hat_pass : U.U * ρ = ρ * U.U) :
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| IsHamiltonian H ∧ U.U * Uᴴ.U = 1 ∧ U.U * ρ = ρ * U.U := by
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| exact ⟨h_pade_pass, U.is_unitary, h_grey_hat_pass⟩
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|
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| end SovMonster.PadeHermitian
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