| import OAI.Probability.InvariantIsing.Cavity.CavityRetainedNorm |
| import OAI.Probability.InvariantIsing.Cavity.CavityConcreteFrameOrbit |
|
|
| namespace OAI |
|
|
| /-! A single physical base eigenframe identifies every retained group |
| overlap with the corresponding projected spin overlap. -/ |
|
|
| noncomputable section |
| open scoped BigOperators Matrix |
|
|
| namespace InvariantIsing |
|
|
| lemma cavity_physical_frame_coordinates {N n : ℕ} {ι : Type*} [Fintype ι] |
| (e : ι ≃ Fin N) (U : SpecialOrthogonal (N + n)) |
| (F : Matrix (Fin (N + n)) ι ℝ) |
| (hF : ∀ i : Fin n, ∀ j, |
| ((U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ).transpose * F) (Fin.natAdd N i) j = 0) |
| (V : Orthogonal N) |
| (hV : ∀ i j, (V : Matrix (Fin N) (Fin N) ℝ) i j = |
| cavityReservoirRows ((U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ).transpose * F) |
| i (e.symm j)) |
| (σ : Spin (N + n)) (j : ι) : |
| matrixRotation V⁻¹ (spinVector (fun i => σ (Fin.castAdd n i))) (e j) = |
| (F.transpose *ᵥ (specialRotation U (spinVector σ)).ofLp) j := by |
| let P := (U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ).transpose * F |
| have he : F.transpose *ᵥ (specialRotation U (spinVector σ)).ofLp = |
| P.transpose *ᵥ (spinVector σ).ofLp := by |
| change F.transpose *ᵥ ((U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ) *ᵥ |
| (spinVector σ).ofLp) = _ |
| simp only [P, Matrix.transpose_mul, Matrix.transpose_transpose, Matrix.mulVec_mulVec] |
| rw [he] |
| change (∑ i : Fin N, (V : Matrix (Fin N) (Fin N) ℝ) i (e j) * |
| spinValue (σ (Fin.castAdd n i))) = ∑ i : Fin (N + n), P i j * spinValue (σ i) |
| rw [Fin.sum_univ_add] |
| simp only [hF, zero_mul, Finset.sum_const_zero, add_zero, hV, |
| Equiv.symm_apply_apply, cavityReservoirRows, P] |
|
|
| def cavityBaseGroup {N m d : ℕ} (k : Fin m → ℕ) |
| (e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N) |
| (a₀ : Fin d → Fin m) (a : Fin m) : Finset (Fin N) := |
| Finset.univ.filter (fun j => Sum.elim (fun v => v.1) a₀ (e.symm j) = a) |
|
|
| theorem cavity_base_group_overlap {N n m d : ℕ} |
| (g : Fin (N + n) → Fin m) (U : SpecialOrthogonal (N + n)) |
| (B : Matrix (Fin (m * n)) (Fin d) ℝ) |
| (k : Fin m → ℕ) (R : (a : Fin m) → Matrix (Fin (N + n)) (Fin (k a)) ℝ) |
| (e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N) |
| (a₀ : Fin d → Fin m) |
| (hF : ∀ i : Fin n, ∀ j, |
| ((U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ).transpose * |
| Matrix.fromCols (cavityRetainedStack k R) |
| (cavityEigenspaceFrame (cavitySpectralImage g |
| (cavityColumns (cavitySpecialOrthogonal U))) * B)) (Fin.natAdd N i) j = 0) |
| (V : Orthogonal N) |
| (hV : ∀ i j, (V : Matrix (Fin N) (Fin N) ℝ) i j = |
| cavityReservoirRows ((U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ).transpose * |
| Matrix.fromCols (cavityRetainedStack k R) |
| (cavityEigenspaceFrame (cavitySpectralImage g |
| (cavityColumns (cavitySpecialOrthogonal U))) * B)) i (e.symm j)) |
| (a : Fin m) (σ τ : Spin (N + n)) : |
| projectedOverlap (matrixRotation V⁻¹) (cavityBaseGroup k e a₀ a) |
| (fun i => σ (Fin.castAdd n i)) (fun i => τ (Fin.castAdd n i)) = |
| cavityRetainedBaseOverlap g U B k R |
| (fun a => Finset.univ.filter (fun j => a₀ j = a)) a σ τ := by |
| let F := Matrix.fromCols (cavityRetainedStack k R) |
| (cavityEigenspaceFrame (cavitySpectralImage g (cavityColumns (cavitySpecialOrthogonal U))) * B) |
| let z := fun σ : Spin (N + n) => F.transpose *ᵥ (specialRotation U (spinVector σ)).ofLp |
| have hc (σ : Spin (N + n)) (j) : |
| matrixRotation V⁻¹ (spinVector (fun i => σ (Fin.castAdd n i))) (e j) = z σ j := |
| cavity_physical_frame_coordinates e U F hF V hV σ j |
| have hr (σ : Spin (N + n)) (b : Fin m) (i : Fin (k b)) : |
| z σ (Sum.inl ⟨b, i⟩) = |
| ((R b).transpose *ᵥ (specialRotation U (spinVector σ)).ofLp) i := rfl |
| have hy (σ : Spin (N + n)) (i : Fin d) : |
| z σ (Sum.inr i) = cavityFullSpecialCoordinates g B U σ i := rfl |
| rw [projectedOverlap, cavityBaseGroup, Finset.sum_filter] |
| rw [← Equiv.sum_comp e] |
| simp only [Equiv.symm_apply_apply, hc, Fintype.sum_sum_type, Fintype.sum_sigma, |
| Sum.elim_inl, Sum.elim_inr, hr, hy] |
| simp only [Finset.sum_ite_irrel] |
| rw [cavityRetainedBaseOverlap, Finset.sum_filter, div_eq_inv_mul] |
| simp only [Finset.sum_const_zero, Finset.sum_ite_eq', Finset.mem_univ, ite_true, dotProduct] |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
