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760f90e | 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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import Mathlib.FieldTheory.Finiteness
import OAI.Computability.UniqueGames.Inverse.KMSAffineRestrictionComplementLemmas
import OAI.Computability.UniqueGames.Inverse.KMSAnalyticBlockFrequenciesLemmas
import OAI.Computability.UniqueGames.Inverse.KMSFourthMomentMixedBaseLemmas
namespace OAI
/-! The full fixed-character induction for the actual KMS small component.
The only local hypothesis bounds squared norms of homogeneous restrictions;
all Fourier recursions, multiplicities and coordinate changes are proved.
-/
namespace UniqueGamesTheorem.Inverse.KMSAnalytic
noncomputable section
open scoped BigOperators Classical
open UniqueGamesTheorem.Integration.BinaryLinear (F2)
open UniqueGamesTheorem.Inverse.KMSBasisInvariant
universe u v w x
local instance mapFintype {V : Type*} {W : Type*}
[AddCommGroup V] [Module F2 V] [AddCommGroup W] [Module F2 W]
[Fintype V] [Fintype W] : Fintype (V →ₗ[F2] W) :=
Fintype.ofInjective (fun L : V →ₗ[F2] W => (L : V → W)) DFunLike.coe_injective
variable {E : Type u} [AddCommGroup E] [Module F2 E]
[FiniteDimensional F2 E] [Fintype E]
private theorem fixedFrequencyEnergy_bound_induction :
∀ (n : ℕ) (F : Type v) [AddCommGroup F] [Module F2 F]
[FiniteDimensional F2 F] [Fintype F]
(I : Type w) [AddCommGroup I] [Module F2 I]
[FiniteDimensional F2 I] [Fintype I]
(J : Type x) [AddCommGroup J] [Module F2 J]
[FiniteDimensional F2 J] [Fintype J],
Module.finrank F2 J = n →
∀ (ι : I →ₗ[F2] E), Function.Injective ι →
∀ (f : (E →ₗ[F2] F) → ℝ), IsBasisInvariant f →
∀ (ε : ℝ), HomogeneousRestrictionBound (Module.finrank F2 I) ε f →
∀ (π : I →ₗ[F2] J), Function.Surjective π → ∀ A : F →ₗ[F2] J,
(2 : ℝ) ^ ((Module.finrank F2 I + Module.finrank F2 J) * Module.finrank F2 E) *
fixedFrequencyEnergy ι f π A ≤
(2 : ℝ) ^ (4 * Module.finrank F2 I * Module.finrank F2 I) * ε := by
intro n
induction n using Nat.strong_induction_on with
| h n ih =>
intro F _ _ _ _ I _ _ _ _ J _ _ _ _ hJn ι hι f hf ε hg π hπ A
let : Finite (KMSKernelFiberCard.KernelClass F I) :=
Finite.of_surjective (KMSKernelFiberCard.fullKernel (F := F) (I := I))
(KMSKernelFiberCard.fullKernel_surjective (F := F) (I := I))
let : Fintype (KMSKernelFiberCard.KernelClass F I) := Fintype.ofFinite _
have hε : 0 ≤ ε := (Finset.expect_nonneg (fun _ _ => sq_nonneg _)).trans
(HomogeneousRestrictionBound.energy_le _ ε f hg)
by_cases hn : n = 0
· exact fixedFrequencyEnergy_zero_dim_bound ι hι f hf ε hg π A (hJn.trans hn)
by_cases hA : Function.Surjective A
· have hp : 0 < Module.finrank F2 J := by omega
let s := KMSAnalyticCompatibleSplitting.chooseSplit π A hπ hA hp
let : Fintype (s.B →ₗ[F2] s.J) := mapFintype
let ι' : (s.J × F2) →ₗ[F2] E := ι.comp s.leftEquiv.symm.toLinearMap
let f' : (E →ₗ[F2] (s.B × F2)) → ℝ := codomainTransport s.rightEquiv f
have hι' : Function.Injective ι' := hι.comp s.leftEquiv.symm.injective
have hf' : IsBasisInvariant f' := basisInvariant_codomainTransport s.rightEquiv f hf
have hg' : HomogeneousRestrictionBound (Module.finrank F2 s.J + 1) ε f' := by
have h := HomogeneousRestrictionBound.codomain_equiv _ ε f hg s.rightEquiv
simpa only [s.left_finrank] using h
have hs : Module.finrank F2 s.K0 < n := by
have hdim : Module.finrank F2 J = Module.finrank F2 s.K0 + 1 := s.target_finrank
omega
have hgfirst : HomogeneousRestrictionBound (Module.finrank F2 s.J) ε
(fun X : E →ₗ[F2] s.B => f' ((LinearMap.inl F2 s.B F2).comp X)) :=
HomogeneousRestrictionBound.codomain_pullback _ 1 ε f' hg'
(LinearMap.inl F2 s.B F2) (by intro x y h; exact congrArg Prod.fst h)
(by simp only [Module.finrank_prod, Module.finrank_self]; omega)
have hglower : HomogeneousRestrictionBound (Module.finrank F2 s.J) ε f' :=
HomogeneousRestrictionBound.mono (Nat.le_succ _) ε f' hg'
have hfirst := ih (Module.finrank F2 s.K0) hs s.B s.J s.K0 rfl
(leftEmbedding ι') (leftEmbedding_injective ι' hι')
(fun X : E →ₗ[F2] s.B => f' ((LinearMap.inl F2 s.B F2).comp X))
(basisInvariant_codomain_pullback f' hf' (LinearMap.inl F2 s.B F2))
ε hgfirst s.leftMap s.leftMap_surjective s.rightMap
have hlower (w : s.J) := ih (Module.finrank F2 s.K0) hs (s.B × F2) s.J s.K0 rfl
(leftEmbedding ι') (leftEmbedding_injective ι' hι') f' hf'
ε hglower s.leftMap s.leftMap_surjective (hyperplaneExtend s.rightMap (s.leftMap w))
have hrec := fixedFrequencyEnergy_product_bound ι' hι' f' hf'
s.leftMap s.rightMap ε hε hfirst hlower
have htransport : fixedFrequencyEnergy ι' f'
(s.leftMap.prodMap (LinearMap.id : F2 →ₗ[F2] F2))
(s.rightMap.prodMap (LinearMap.id : F2 →ₗ[F2] F2)) =
fixedFrequencyEnergy ι f π A := by
rw [← s.left_conjugation, ← s.right_conjugation,
fixedFrequencyEnergy_postcomp_injective _ _ _ _ s.targetEquiv.toLinearMap
s.targetEquiv.injective]
change fixedFrequencyEnergy (ι.comp s.leftEquiv.symm.toLinearMap)
(codomainTransport s.rightEquiv f) (π.comp s.leftEquiv.symm.toLinearMap)
(A.comp s.rightEquiv.symm.toLinearMap) = _
rw [fixedFrequencyEnergy_codomainTransport, fixedFrequencyEnergy_transport]
rw [htransport, ← s.leftEquiv.finrank_eq, ← s.targetEquiv.finrank_eq] at hrec
exact hrec
· rw [fixedFrequencyEnergy_eq_zero_of_not_surjective ι f π hπ A hA, mul_zero]
positivity
/-- KMS's fixed-character estimate, with actual homogeneous restriction
control. The injection is the only ambient-dimension assumption. -/
theorem fixedFrequencyEnergy_le_of_homogeneous
{F : Type v} [AddCommGroup F] [Module F2 F]
[FiniteDimensional F2 F] [Fintype F]
{I : Type w} [AddCommGroup I] [Module F2 I]
[FiniteDimensional F2 I] [Fintype I]
{J : Type x} [AddCommGroup J] [Module F2 J]
[FiniteDimensional F2 J] [Fintype J]
(ι : I →ₗ[F2] E) (hι : Function.Injective ι)
(f : (E →ₗ[F2] F) → ℝ) (hf : IsBasisInvariant f)
(ε : ℝ) (hg : HomogeneousRestrictionBound (Module.finrank F2 I) ε f)
(π : I →ₗ[F2] J) (hπ : Function.Surjective π) (A : F →ₗ[F2] J) :
(2 : ℝ) ^ ((Module.finrank F2 I + Module.finrank F2 J) * Module.finrank F2 E) *
fixedFrequencyEnergy ι f π A ≤
(2 : ℝ) ^ (4 * Module.finrank F2 I * Module.finrank F2 I) * ε :=
fixedFrequencyEnergy_bound_induction (Module.finrank F2 J) F I J rfl ι hι f hf ε hg π hπ A
end
end UniqueGamesTheorem.Inverse.KMSAnalytic
end OAI
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