openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Character /Existence.lean
| 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 | |