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namespace OAI
/-! Projected overlaps as products of actual backward site gradients. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal ENNReal
namespace InvariantIsing
def rotationCoordinate {N : ℕ} (U : Rotation N) (a i : Fin N) : ℝ :=
U (WithLp.toLp 2 (Pi.single i 1)) a
lemma rotation_apply_coordinates {N : ℕ} (U : Rotation N) (f : Fin N → ℝ) (a : Fin N) :
U (WithLp.toLp 2 f) a = ∑ i, rotationCoordinate U a i * f i := by
classical
have hv : (WithLp.toLp 2 f : EuclideanSpace ℝ (Fin N)) =
∑ i : Fin N, f i • (WithLp.toLp 2 (Pi.single i 1) : EuclideanSpace ℝ (Fin N)) := by
ext j
change f j = WithLp.ofLp (∑ i : Fin N, f i •
(WithLp.toLp 2 (Pi.single i 1) : EuclideanSpace ℝ (Fin N))) j
rw [WithLp.ofLp_sum, Finset.sum_apply]
simp [mul_ite]
rw [hv, map_sum]
change WithLp.ofLp (∑ i : Fin N, U (f i • WithLp.toLp 2 (Pi.single i 1))) a = _
rw [WithLp.ofLp_sum, Finset.sum_apply]
simp only [map_smul, WithLp.ofLp_smul, Pi.smul_apply, smul_eq_mul, rotationCoordinate, mul_comm]
def tensorBackwardProjectedMean {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(i : ℕ) (z : SpinTensorIndex I degree → ℝ) (a : Fin N) : ℝ :=
∑ s, rotationCoordinate U a s * tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z (Sum.inl s)
lemma tensorBackwardProjectedMean_eq {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(i : ℕ) (z : SpinTensorIndex I degree → ℝ) (a : Fin N) :
tensorBackwardProjectedMean eig U c I degree amplitude n b v i z a =
U (WithLp.toLp 2 (fun s => tensorBackwardCoordinateMean eig U c I degree amplitude n b v i z (Sum.inl s))) a :=
(rotation_apply_coordinates U _ a).symm
def tensorProjectedCoordinateCap {N m k : ℕ} (U : Rotation N)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ) (a : Fin N) : ℝ :=
∑ s, |rotationCoordinate U a s| * tensorFeatureCoordinateCap U I degree amplitude (Sum.inl s)
lemma measurable_tensorBackwardProjectedMean {N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (i : ℕ) (a : Fin N) :
Measurable (fun z => tensorBackwardProjectedMean eig U c I degree amplitude n b v i z a) :=
Finset.measurable_sum _ fun s _ =>
(measurable_tensorBackwardCoordinateMean hN eig U c I degree amplitude n b v hb i (Sum.inl s)).const_mul _
lemma tensorBackwardProjectedMean_abs_le {N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (i : ℕ) (z : SpinTensorIndex I degree → ℝ) (a : Fin N) :
|tensorBackwardProjectedMean eig U c I degree amplitude n b v i z a| ≤
tensorProjectedCoordinateCap U I degree amplitude a := by
unfold tensorBackwardProjectedMean tensorProjectedCoordinateCap
apply (Finset.abs_sum_le_sum_abs _ _).trans
apply Finset.sum_le_sum
intro s _
rw [abs_mul]
exact mul_le_mul_of_nonneg_left
(tensorBackwardCoordinateMean_abs_le hN eig U c I degree amplitude n b v hb i z (Sum.inl s)) (abs_nonneg _)
lemma integrable_tensorBackwardProjectedMean {N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (i : ℕ) (z : SpinTensorIndex I degree → ℝ) (a : Fin N)
(μ : Measure (SpinTensorIndex I degree → ℝ)) [IsFiniteMeasure μ] :
Integrable (fun x => tensorBackwardProjectedMean eig U c I degree amplitude n b v i (z + x) a) μ :=
Integrable.of_bound
((measurable_tensorBackwardProjectedMean hN eig U c I degree amplitude n b v hb i a).comp
(measurable_const.add measurable_id)).aestronglyMeasurable
(tensorProjectedCoordinateCap U I degree amplitude a)
(ae_of_all _ fun x => by simpa only [Real.norm_eq_abs] using
tensorBackwardProjectedMean_abs_le hN eig U c I degree amplitude n b v hb i (z + x) a)
theorem tensorBackwardProjectedMean_step {N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (i : ℕ) (hi : i < n)
(z : SpinTensorIndex I degree → ℝ) (a : Fin N) :
tensorBackwardProjectedMean eig U c I degree amplitude n b v i z a =
∫ x, tensorBackwardProjectedMean eig U c I degree amplitude n b v (i + 1) (z + x) a
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z := by
let := tensorAncestorMarkKernel_markov hN eig U c I degree amplitude n b v hb i
have hint (s : Fin N) : Integrable (fun x => rotationCoordinate U a s *
tensorBackwardCoordinateMean eig U c I degree amplitude n b v (i + 1) (z + x) (Sum.inl s))
(tensorAncestorMarkKernel eig U c I degree amplitude n b v i z) := by
apply Integrable.const_mul
apply Integrable.of_bound
((measurable_tensorBackwardCoordinateMean hN eig U c I degree amplitude n b v hb (i + 1) (Sum.inl s)).comp
(measurable_const.add measurable_id)).aestronglyMeasurable
(tensorFeatureCoordinateCap U I degree amplitude (Sum.inl s))
exact ae_of_all _ fun x => by simpa only [Real.norm_eq_abs, Function.comp_apply, Pi.add_apply, id_eq] using
tensorBackwardCoordinateMean_abs_le hN eig U c I degree amplitude n b v hb (i + 1) (z + x) (Sum.inl s)
rw [tensorBackwardProjectedMean, show (fun x => tensorBackwardProjectedMean eig U c I degree amplitude n b v
(i + 1) (z + x) a) = (fun x => ∑ s, rotationCoordinate U a s *
tensorBackwardCoordinateMean eig U c I degree amplitude n b v (i + 1) (z + x) (Sum.inl s)) from rfl,
integral_finsetSum _ (fun s _ => hint s)]
apply Finset.sum_congr rfl
intro s _
rw [integral_const_mul, ← tensorBackwardCoordinateMean_step hN eig U c I degree amplitude n b v hb i hi z (Sum.inl s)]
lemma tensorBackwardProjectedMean_terminal {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(z : SpinTensorIndex I degree → ℝ) (a : Fin N) :
tensorBackwardProjectedMean eig U c I degree amplitude n b v n z a =
tensorProjectedMagnetization eig U c I degree amplitude z a := by
rw [tensorBackwardProjectedMean_eq, tensorProjectedMagnetization_eq]
simp only [tensorBackwardCoordinateMean, Nat.sub_self, tensorCascadeCoordinateMean]
def tensorProjectedPairPayoff {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (J : Finset (Fin N))
(q : ℕ) (z₁ z₂ : SpinTensorIndex I degree → ℝ)
(w : Fin q → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) : ℝ :=
(N : ℝ)⁻¹ * ∑ a ∈ J,
tensorBackwardProjectedMean eig U c I degree amplitude n b v n (z₁ + ∑ i, (w i).1) a *
tensorBackwardProjectedMean eig U c I degree amplitude n b v n (z₂ + ∑ i, (w i).2) a
lemma tensorProjectedPairPayoff_cons {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (J : Finset (Fin N))
(q : ℕ) (z₁ z₂ a₁ a₂ : SpinTensorIndex I degree → ℝ)
(w : Fin q → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) :
tensorProjectedPairPayoff eig U c I degree amplitude n b v J (q + 1) z₁ z₂ (Fin.cons (a₁, a₂) w) =
tensorProjectedPairPayoff eig U c I degree amplitude n b v J q (z₁ + a₁) (z₂ + a₂) w := by
simp only [tensorProjectedPairPayoff, Fin.sum_univ_succ, Fin.cons_zero, Fin.cons_succ, add_assoc]
private lemma integral_pair_product_sum {A B C : Type*} [MeasurableSpace A] [MeasurableSpace B]
(μ : Measure A) (ν : Measure B) (J : Finset C) (F : C → A → ℝ) (G : C → B → ℝ)
(hF : ∀ a ∈ J, Integrable (F a) μ) (hG : ∀ a ∈ J, Integrable (G a) ν) (c : ℝ) :
(∫ x, ∫ y, c * ∑ a ∈ J, F a x * G a y ∂ν ∂μ) =
c * ∑ a ∈ J, (∫ x, F a x ∂μ) * ∫ y, G a y ∂ν := by
have hi (x : A) : (∫ y, c * ∑ a ∈ J, F a x * G a y ∂ν) =
c * ∑ a ∈ J, F a x * ∫ y, G a y ∂ν := by
rw [integral_const_mul, integral_finsetSum _ (fun a ha => (hG a ha).const_mul (F a x))]
simp_rw [integral_const_mul]
simp_rw [hi]
rw [integral_const_mul, integral_finsetSum _ (fun a ha => (hF a ha).mul_const _)]
simp_rw [integral_mul_const]
theorem tensorAncestorPairPathMean_independent_projected {N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (J : Finset (Fin N)) (d q i : ℕ) (hqi : i + q = n) (hdi : d ≤ i)
(z₁ z₂ : SpinTensorIndex I degree → ℝ) :
tensorAncestorPairPathMean eig U c I degree amplitude n b v d q i z₁ z₂
(tensorProjectedPairPayoff eig U c I degree amplitude n b v J q z₁ z₂) =
(N : ℝ)⁻¹ * ∑ a ∈ J,
tensorBackwardProjectedMean eig U c I degree amplitude n b v i z₁ a *
tensorBackwardProjectedMean eig U c I degree amplitude n b v i z₂ a := by
induction q generalizing i z₁ z₂ with
| zero =>
have hi : i = n := by omega
subst i
simp only [tensorAncestorPairPathMean, tensorProjectedPairPayoff, Finset.univ_eq_empty,
Finset.sum_empty, add_zero]
| succ q ih =>
have hi : i < n := by omega
let := tensorAncestorMarkKernel_markov hN eig U c I degree amplitude n b v hb i
dsimp only [tensorAncestorPairPathMean]
rw [ite_eq_right (by omega : ¬i < d)]
calc
_ = ∫ x, ∫ y, (N : ℝ)⁻¹ * ∑ a ∈ J,
tensorBackwardProjectedMean eig U c I degree amplitude n b v (i + 1) (z₁ + x) a *
tensorBackwardProjectedMean eig U c I degree amplitude n b v (i + 1) (z₂ + y) a
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₂
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₁ := by
apply integral_congr_ae
apply ae_of_all
intro x
apply integral_congr_ae
apply ae_of_all
intro y
dsimp only
have hP : (fun w => tensorProjectedPairPayoff eig U c I degree amplitude n b v J (q + 1)
z₁ z₂ (Fin.cons (x, y) w)) =
tensorProjectedPairPayoff eig U c I degree amplitude n b v J q (z₁ + x) (z₂ + y) :=
funext (tensorProjectedPairPayoff_cons eig U c I degree amplitude n b v J q z₁ z₂ x y)
rw [hP]
exact ih (i + 1) (by omega) (by omega) (z₁ + x) (z₂ + y)
_ = _ := by
rw [integral_pair_product_sum]
· congr 1
apply Finset.sum_congr rfl
intro a ha
rw [← tensorBackwardProjectedMean_step hN eig U c I degree amplitude n b v hb i hi z₁ a,
← tensorBackwardProjectedMean_step hN eig U c I degree amplitude n b v hb i hi z₂ a]
· intro a _
exact integrable_tensorBackwardProjectedMean hN eig U c I degree amplitude n b v hb (i + 1) z₁ a _
· intro a _
exact integrable_tensorBackwardProjectedMean hN eig U c I degree amplitude n b v hb (i + 1) z₂ a _
def tensorSharedProjectedProduct {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(J : Finset (Fin N)) (d : ℕ) :
(q : ℕ) → ℕ → (SpinTensorIndex I degree → ℝ) → (SpinTensorIndex I degree → ℝ) → ℝ
| 0, i, z₁, z₂ => (N : ℝ)⁻¹ * ∑ a ∈ J,
tensorBackwardProjectedMean eig U c I degree amplitude n b v i z₁ a *
tensorBackwardProjectedMean eig U c I degree amplitude n b v i z₂ a
| q + 1, i, z₁, z₂ => if i < d then
∫ x, tensorSharedProjectedProduct eig U c I degree amplitude n b v J d q
(i + 1) (z₁ + x) (z₂ + x) ∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₁
else (N : ℝ)⁻¹ * ∑ a ∈ J,
tensorBackwardProjectedMean eig U c I degree amplitude n b v i z₁ a *
tensorBackwardProjectedMean eig U c I degree amplitude n b v i z₂ a
lemma tensorSharedProjectedProduct_nonneg {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(J : Finset (Fin N)) (d q i : ℕ) (z : SpinTensorIndex I degree → ℝ) :
0 ≤ tensorSharedProjectedProduct eig U c I degree amplitude n b v J d q i z z := by
induction q generalizing i z with
| zero => exact mul_nonneg (by positivity) (Finset.sum_nonneg fun _ _ => mul_self_nonneg _)
| succ q ih =>
dsimp only [tensorSharedProjectedProduct]
split_ifs
· exact integral_nonneg fun a => ih (i + 1) (z + a)
· exact mul_nonneg (by positivity) (Finset.sum_nonneg fun _ _ => mul_self_nonneg _)
theorem tensorAncestorPairPathMean_projected {N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (J : Finset (Fin N)) (d q i : ℕ) (hqi : i + q = n)
(z₁ z₂ : SpinTensorIndex I degree → ℝ) :
tensorAncestorPairPathMean eig U c I degree amplitude n b v d q i z₁ z₂
(tensorProjectedPairPayoff eig U c I degree amplitude n b v J q z₁ z₂) =
tensorSharedProjectedProduct eig U c I degree amplitude n b v J d q i z₁ z₂ := by
induction q generalizing i z₁ z₂ with
| zero =>
have hi : i = n := by omega
subst i
simp only [tensorAncestorPairPathMean, tensorProjectedPairPayoff, tensorSharedProjectedProduct,
Finset.univ_eq_empty, Finset.sum_empty, add_zero]
| succ q ih =>
by_cases hid : i < d
· dsimp only [tensorAncestorPairPathMean, tensorSharedProjectedProduct]
rw [ite_eq_left hid, ite_eq_left hid]
apply integral_congr_ae
apply ae_of_all
intro a
dsimp only
have hP : (fun w => tensorProjectedPairPayoff eig U c I degree amplitude n b v J (q + 1)
z₁ z₂ (Fin.cons (a, a) w)) =
tensorProjectedPairPayoff eig U c I degree amplitude n b v J q (z₁ + a) (z₂ + a) :=
funext (tensorProjectedPairPayoff_cons eig U c I degree amplitude n b v J q z₁ z₂ a a)
rw [hP]
exact ih (i + 1) (by omega) (z₁ + a) (z₂ + a)
· rw [tensorSharedProjectedProduct, ite_eq_right hid]
exact tensorAncestorPairPathMean_independent_projected hN eig U c I degree amplitude
n b v hb J d (q + 1) i hqi (by omega) z₁ z₂
lemma tensorPairPathSpinMean_projectedPayoff {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(J : Finset (Fin N)) (z : SpinTensorIndex I degree → ℝ)
(w : Fin n → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) :
tensorPairPathSpinMean eig U c I degree amplitude n (z, w)
(fun σ => projectedOverlap U J (σ 0) (σ 1)) =
tensorProjectedPairPayoff eig U c I degree amplitude n b v J n z z w := by
rw [tensorPairPathSpinMean_overlap]
simp only [tensorProjectedPairPayoff, tensorBackwardProjectedMean_terminal, tensorPairPathState,
ite_true, ite_eq_right (show (1 : Fin 2) ≠ 0 by decide)]
/-- The actual projected-overlap test is the shared-path expectation of
products of projected backward gradients at the branching level. -/
theorem tensorProjectedOverlap_backward_identity (hpub : PanchenkoTalagrandTensorPairInput)
{N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (J : Finset (Fin N)) (z : SpinTensorIndex I degree → ℝ)
(ψ : ℕ → ℝ) {B : ℝ} (hB : 0 ≤ B) (hψ : ∀ d, |ψ d| ≤ B) :
tensorSpinPairAverage eig U c I degree amplitude n b v z ψ
(fun σ => projectedOverlap U J (σ 0) (σ 1)) =
∫ α : ℕ → LabeledLeaf n, ψ (labeledCommonDepth n (α 0) (α 1)) *
tensorSharedProjectedProduct eig U c I degree amplitude n b v J
(labeledCommonDepth n (α 0) (α 1)) n 0 z z ∂cascadeReplicaLaw n b := by
rw [tensorProjectedOverlap_markov_identity hpub hN eig U c I degree amplitude n b v hb z J ψ hB hψ]
apply integral_congr_ae
apply ae_of_all
intro α
simp only [tensorPairSpinPathPayoff,
tensorPairPathSpinMean_projectedPayoff eig U c I degree amplitude n b v J z]
rw [tensorAncestorPairPathMean_const_mul,
tensorAncestorPairPathMean_projected hN eig U c I degree amplitude n b v hb J _ n 0 (by omega)]
theorem tensorProjectedOverlap_nonneg (hpub : PanchenkoTalagrandTensorPairInput)
{N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (J : Finset (Fin N)) (z : SpinTensorIndex I degree → ℝ)
(ψ : ℕ → ℝ) {B : ℝ} (hB : 0 ≤ B) (hψ : ∀ d, |ψ d| ≤ B) (hψ0 : ∀ d, 0 ≤ ψ d) :
0 ≤ tensorSpinPairAverage eig U c I degree amplitude n b v z ψ
(fun σ => projectedOverlap U J (σ 0) (σ 1)) := by
rw [tensorProjectedOverlap_backward_identity hpub hN eig U c I degree amplitude n b v hb J z ψ hB hψ]
exact integral_nonneg fun α => mul_nonneg (hψ0 _)
(tensorSharedProjectedProduct_nonneg eig U c I degree amplitude n b v J _ n 0 z)
theorem tensorRootedProjectedOverlap_backward_identity (hpub : PanchenkoTalagrandTensorPairInput)
{N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (J : Finset (Fin N))
(ψ : ℕ → ℝ) {B : ℝ} (hB : 0 ≤ B) (hψ : ∀ d, |ψ d| ≤ B) :
(∫ z, tensorSpinPairAverage eig U c I degree amplitude n b (fun i => v (i + 1)) z ψ
(fun σ => projectedOverlap U J (σ 0) (σ 1))
∂(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ))) =
∫ z, ∫ α : ℕ → LabeledLeaf n, ψ (labeledCommonDepth n (α 0) (α 1)) *
tensorSharedProjectedProduct eig U c I degree amplitude n b (fun i => v (i + 1)) J
(labeledCommonDepth n (α 0) (α 1)) n 0 z z ∂cascadeReplicaLaw n b
∂(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ)) := by
apply integral_congr_ae
exact ae_of_all _ fun z => tensorProjectedOverlap_backward_identity hpub hN eig U c I degree amplitude
n b (fun i => v (i + 1)) hb J z ψ hB hψ
end InvariantIsing
end
end OAI
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