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cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 | import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Spectrum.StateObstruction
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Spectrum.MultiplicativeStates
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Spectrum.Obstruction
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Tensor.Characters
namespace OAI
/-!
# Detecting characters for complex matrix multiplication
This proves Lemma 3.1, including Appendix A. Finite inconsistency would
produce a positive scalar-gain catalyst, ruled out by the exact-rank exponent.
A compact normalized state space then yields a multiplicative state, which
transfers to the concrete coefficient-level character interface.
-/
noncomputable section
namespace MatrixMultiplication.AuxiliarySeparation
open MatrixMultiplication.Foundation
/-- Lemma 3.1: normalized, additive, multiplicative, restriction-monotone
characters detect every integer strictly below the exact-rank growth value. -/
theorem exists_detecting_character {d k : ℕ} (hd : 2 ≤ d)
(hk : (k : ℝ) < (d : ℝ) ^ exactRankExponent) :
∃ χ : Character, (k : ℝ) ≤ χ.value (Tensor.matrixMultiplication d d d) := by
have hnonempty := normalizedStates_nonempty_of_no_catalyst
TensorSemiring.rank (TensorSemiring.matrixClass d) k
TensorSemiring.zero_le TensorSemiring.le_rank
(fun _ _ _ _ h₁ h₂ => TensorSemiring.add_mono h₁ h₂)
(fun D s m hm hcat => no_tensor_catalytic_gain (by omega) hk ⟨D, s, m, hm, hcat⟩)
have hdom (z x : TensorSemiring.TensorClass) :
z * x ≤ (TensorSemiring.rank x : TensorSemiring.TensorClass) * z := by
calc
z * x ≤ z * (TensorSemiring.rank x : TensorSemiring.TensorClass) :=
TensorSemiring.mul_mono le_rfl (TensorSemiring.le_rank x)
_ = (TensorSemiring.rank x : TensorSemiring.TensorClass) * z := mul_comm _ _
obtain ⟨φ, hφ, _, hdetect⟩ := exists_monotone_semiringHom_of_normalizedStates
TensorSemiring.rank (TensorSemiring.matrixClass d) (k : ℝ)
(fun z hz => TensorSemiring.one_le_of_ne_zero hz)
(fun _ _ _ h => TensorSemiring.mul_mono le_rfl h) hdom hnonempty
exact ⟨Character.ofTensorClassHom φ hφ, hdetect⟩
end MatrixMultiplication.AuxiliarySeparation
end
end OAI
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