| import Mathlib |
| import OAI.Probability.SKGap.Localization.LiteralMargin |
| import OAI.Probability.SKGap.Stability.LiteralStableEvent |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Matrix |
| namespace SKGapCutoff.Recipe |
| open SKGap Primary Static SKGap.ObservationBridge SKGap.Stein |
| variable {n : ℕ} |
|
|
| lemma vector_column_opNorm (M : Interaction n) (k : Fin n) : |
| vectorNorm (fun i=>M i k)≤SKGap.opNorm M := by |
| apply nonneg_le_nonneg_of_sq_le_sq (norm_nonneg _) |
| simp only [←pow_two] |
| rw [vectorNorm_sq] |
| exact SKGap.SourceError.matrix_column_square_le M k |
|
|
| lemma vector_flip_opNorm (F : VectorFields n) (x : Spin n) (k : Fin n) : |
| vectorNorm (F (flip x k)-F x)≤2*SKGap.opNorm (derivativeMatrix F x) := by |
| have H:=congrArg norm (halfDiff_column F x k) |
| rw [norm_smul,Real.norm_eq_abs,abs_div,abs_spin_eq_one,abs_of_pos (by norm_num : (0:ℝ)<2),norm_sub_rev] at H |
| have hcol:=vector_column_opNorm (derivativeMatrix F x) k |
| change (1/2:ℝ)*vectorNorm (F (flip x k)-F x)=vectorNorm (fun i=>derivativeMatrix F x i k) at H |
| linarith |
|
|
| lemma vector_flip_buffer (F : VectorFields n) (x : Spin n) {B ρ : ℝ} |
| (hD : SKGap.opNorm (derivativeMatrix F x)≤B) |
| (hsmall : 2*B≤ρ*Real.sqrt (n:ℝ)) |
| (hF : vectorNorm (F x)≤ρ*Real.sqrt (n:ℝ)) (k : Fin n) : |
| vectorNorm (F (flip x k))≤(2*ρ)*Real.sqrt (n:ℝ) := by |
| have H : vectorNorm (F (flip x k))≤vectorNorm (F x)+vectorNorm (F (flip x k)-F x) := by |
| calc |
| _ = vectorNorm (F x+(F (flip x k)-F x)) := by congr 1;ext i;simp |
| _ ≤ _ := SKGap.vectorNorm_add_le _ _ |
| exact H.trans ((add_le_add hF ((vector_flip_opNorm F x k).trans |
| ((mul_le_mul_of_nonneg_left hD (by norm_num)).trans hsmall))).trans_eq (by ring)) |
|
|
| lemma residualCutoff_flip_buffer (hn : 0<n) {j ρ B : ℝ} (hρ : 0<ρ) |
| (J : Interaction n) (h : Fin n→ℝ) (l : ℕ) (x : Spin n) |
| (hcut : residualCutoff ρ j J h l x≠0) |
| (hD₀ : SKGap.opNorm (derivativeMatrix (Primary.residual j J h l) x)≤B) |
| (hD₁ : SKGap.opNorm (derivativeMatrix (Primary.residual j J h (l+1)) x)≤B) |
| (hsmall : 2*B≤ρ*Real.sqrt (n:ℝ)) : |
| ∀k,vectorNorm (Primary.residual j J h l (flip x k))≤(2*ρ)*Real.sqrt (n:ℝ) ∧ |
| vectorNorm (Primary.residual j J h (l+1) (flip x k))≤(2*ρ)*Real.sqrt (n:ℝ) := by |
| obtain ⟨h0,h1⟩:=residualCutoff_support hn hρ J h l x hcut |
| intro k |
| exact ⟨vector_flip_buffer _ x hD₀ hsmall h0 k,vector_flip_buffer _ x hD₁ hsmall h1 k⟩ |
|
|
| end SKGapCutoff.Recipe |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Matrix.Norms.Frobenius |
| namespace SKGapCutoff.Recipe |
| open Primary SKGap.Stein |
| variable {n : ℕ} |
|
|
| lemma auxiliary_derivative (j q : ℝ) (b : Spin n→ℝ) (x : Spin n) : |
| derivativeVector (fun y=>j*(1-b y-q)) x= -j • derivativeVector b x := by |
| ext i |
| simp only [derivativeVector,WithLp.ofLp_toLp,PiLp.smul_apply,halfDiff] |
| ring |
|
|
| lemma primary_literal_derivative_diagnostics (hn : 0<n) {j K B : ℝ} |
| (hK : 0≤K) (hB : 0≤B) (J : Interaction n) (h : Fin n→ℝ) |
| (hop : SKGap.opNorm J≤K) (k : ℕ) |
| (hformal : ∀x l,l<k+3→ShapeBound (formalField j J h x l) B) |
| (r : Fin n→ℝ) (x : Spin n) : |
| let C:=localPrimaryBudget j K B (k+3) |
| let T:=(1+|j|)*C |
| let a:=fun v=>j*(1-Primary.onsager j J h (k+1) v-SKGap.overlap r) |
| SKGap.opNorm (derivativeMatrix (fld j J h (k+1)) x)+‖derivativeVector a x‖≤T ∧ |
| SKGap.opNorm (derivativeMatrix (mag j J h (k+1)) x)≤T ∧ |
| ‖derivativeMatrix (fld j J h (k+1)) x-(primaryTree j J h x (k+1)).fieldMatrix j J‖≤C ∧ |
| ‖derivativeMatrix (mag j J h (k+1)) x-(primaryTree j J h x (k+1)).sourceMatrix j J‖≤C ∧ |
| (∑i,∑v,(derivativeMatrix (fld j J h (k+1)) x i v)^4)≤(2*C)^4 ∧ |
| (∑i,(derivativeMatrix (fld j J h (k+1)) x i i)^2)≤T^2 := by |
| dsimp only |
| let C:=localPrimaryBudget j K B (k+3) |
| have hC : 0≤C:=localPrimaryBudget_nonneg hK hB _ |
| have hT : C≤(1+|j|)*C := by nlinarith [abs_nonneg j] |
| have hh:=primary_differentiation J h x hn hK hB hop (k+1) |
| (fun l hl=>hformal x l (by omega)) |
| have hb:=(localPrimaryBudget_bounds (j:=j) hK hB (k+3) ⟨k+1,by omega⟩).1 |
| have hz : ShapeBound (derivativeMatrix (fld j J h (k+1)) x) C := hh.2.2.2.mono hb |
| have ha := onsager_derivative_bound (j:=j) J h x hn hK hB hop (k+3) |
| (hformal x) (show k+1≤k+3 by omega) |
| refine ⟨?_,(hh.2.2.1.trans hb).trans hT,?_,?_,?_,?_⟩ |
| · rw [auxiliary_derivative,norm_smul,Real.norm_eq_abs,abs_neg] |
| exact (add_le_add hz.1 (mul_le_mul_of_nonneg_left ha (abs_nonneg j))).trans_eq (by dsimp [C];ring) |
| · rw [(primaryTree_matrices j J h x (k+1)).2] |
| exact hh.1.trans hb |
| · rw [(primaryTree_matrices j J h x (k+1)).1] |
| exact hh.2.1.trans hb |
| · exact (fourth_sum_le hC hz).trans (by nlinarith [pow_nonneg hC 4]) |
| · have H:=pow_le_pow_left₀ (norm_nonneg _) (hz.2.1.trans hT) 2 |
| change (SKGap.diagonalSeminorm (derivativeMatrix (fld j J h (k+1)) x))^2≤((1+|j|)*C)^2 at H |
| simpa only [SKGap.diagonalSeminorm_sq] using H |
|
|
| end SKGapCutoff.Recipe |
|
|
| noncomputable section |
| open scoped BigOperators Matrix.Norms.Frobenius |
| namespace SKGapCutoff.Recipe |
| open Primary SKGap.Stein |
|
|
| theorem stable_selected_initial_diagnostics {j K B A ε c r₀ R ρ : ℝ} |
| (hj : 0≤j) (hK : 0≤K) (hB : 0≤B) (hA : 1≤A) (hc : 0<c) |
| (hr₀ : 0<r₀) (hρ : 0<ρ) (hε : 0≤ε) (hAR : Real.exp (R/2)≤A) |
| (hbuffer : (K+4*j)*(2*ρ)<r₀) |
| (hR : (1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ)≤R) |
| (hsmall : (3*Real.exp (R/2)+2)*((1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ))≤ε) |
| (himplicit : (1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ)≤ |
| c/(2*(|j| *Real.exp (R/2)*(3*Real.exp (R/2)+16)+1))) |
| (k : ℕ) (f : KernelExpr) : |
| ∃V≥0,∃S≥0,∃F≥0,∀n : ℕ,0<n→∀(J : Interaction n) (h : Fin n→ℝ), |
| J.IsSymm→SKGap.opNorm J≤K→¬SKGap.rootBad j A ε c r₀ J h→ |
| (∀x l,l<k+3→ShapeBound (formalField j J h x l) B)→ |
| 2*(residualDerivativeBudget j K B k+residualDerivativeBudget j K B (k+1))≤ρ*Real.sqrt (n:ℝ)→ |
| ∃r:Fin n→ℝ,SKGap.tapField j J h r=0 ∧ (∀v,SKGap.tapField j J h v=0→v=r) ∧ |
| ∀e:Fin n→ℝ,vectorNorm e≤1→∃w y : VectorFields n,∃c₀:Spin n→ℝ, |
| ∀x:Spin n,residualCutoff ρ j J h k x≠0→ |
| let a:=fun v=>j*(1-Primary.onsager j J h (k+1) v-SKGap.overlap r) |
| LiteralEquations J j f (fld j J h (k+1)) (mag j J h (k+1)) w y a c₀ r e x ∧ |
| LiteralInitialDiagnostics |
| (literalInitial J j (Real.sqrt (n:ℝ)) f (fld j J h (k+1)) (mag j J h (k+1)) a c₀ r e) |
| w y (primaryTree j J h x (k+1)) x V S F (Real.exp (R/2)) := by |
| let η:=(1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ) |
| let C:=localPrimaryBudget j K B (k+3) |
| let T:=(1+|j|)*C |
| have hη : 0≤η:=by dsimp [η];positivity |
| have hC : 0≤C:=localPrimaryBudget_nonneg hK hB _ |
| have hT : 0≤T:=by dsimp [T];positivity |
| obtain ⟨V,hV,S,hS,F,hF,hdiag⟩:=literal_initial_diagnostics j R η c T K (2*C) C f hj hc hη hT |
| (mul_nonneg (by norm_num) hC) himplicit |
| refine ⟨V,hV,S,hS,F,hF,?_⟩ |
| intro n hn J h hJ hop hbad hformal hdim |
| have hact : ∀v,vectorNorm (J.mulVec v)≤K*vectorNorm v := fun v=> |
| (vectorNorm_matrix_mul J v).trans (mul_le_mul_of_nonneg_right hop (vectorNorm_nonneg v)) |
| obtain ⟨r,hr,hu,hstable⟩:=stable_primary_buffer hn hj hK hA hc hr₀ |
| (mul_nonneg (by norm_num) hρ.le) hε hAR hbuffer hR hsmall J hJ h hact hbad |
| refine ⟨r,hr,hu,?_⟩ |
| intro e he |
| let a:Spin n→ℝ:=fun v=>j*(1-Primary.onsager j J h (k+1) v-SKGap.overlap r) |
| obtain ⟨w,y,c₀,heq,hd⟩:=hdiag n hn J (fld j J h (k+1)) (mag j J h (k+1)) a r e hJ hop he |
| refine ⟨w,y,c₀,?_⟩ |
| intro x hcut |
| have hD₀ : SKGap.opNorm (derivativeMatrix (Primary.residual j J h k) x)≤ |
| residualDerivativeBudget j K B k+residualDerivativeBudget j K B (k+1) := |
| (residual_derivative_bound hn hK hB J h hop k |
| (fun v l hl=>hformal v l (by omega)) x).trans |
| (le_add_of_nonneg_right (residualDerivativeBudget_nonneg hK hB _)) |
| have hD₁ : SKGap.opNorm (derivativeMatrix (Primary.residual j J h (k+1)) x)≤ |
| residualDerivativeBudget j K B k+residualDerivativeBudget j K B (k+1) := |
| (residual_derivative_bound hn hK hB J h hop (k+1) |
| (fun v l hl=>hformal v l (by omega)) x).trans |
| (le_add_of_nonneg_left (residualDerivativeBudget_nonneg hK hB _)) |
| have hf:=residualCutoff_flip_buffer hn hρ J h k x hcut hD₀ hD₁ hdim |
| obtain ⟨h0,h1⟩:=residualCutoff_support hn hρ J h k x hcut |
| have hρ2 : ρ*Real.sqrt (n:ℝ)≤(2*ρ)*Real.sqrt (n:ℝ) := |
| mul_le_mul_of_nonneg_right (by linarith) (Real.sqrt_nonneg _) |
| have H:=hstable k x (h0.trans hρ2) (h1.trans hρ2) |
| have Hf (v:Fin n):=hstable k (flip x v) (hf v).1 (hf v).2 |
| have hb:=primary_literal_derivative_diagnostics hn hK hB J h hop k hformal r x |
| change LiteralEquations J j f (fld j J h (k+1)) (mag j J h (k+1)) w y a c₀ r e x ∧ _ |
| exact ⟨heq x H,hd x (primaryTree j J h x (k+1)) H Hf hb.1 hb.2.1 |
| (primaryState_mag_bounded j J h (k+1) x) hb.2.2.1 hb.2.2.2.1 hb.2.2.2.2.1 hb.2.2.2.2.2⟩ |
|
|
| end SKGapCutoff.Recipe |
|
|
| end |
| end |
| end |
|
|
| end OAI |
|
|
