Step-down and main theorem from exactly the remaining contracts

A8Remaining β C₀ collects the still-unproduced inputs of step-down (both conjuncts) — each a contract in producer form — and nothing else:

• Big-bound estimate (inside step-down part (i)): Normie3CenteredSourceContract at the datum amplitude;
• Step-down part (i) at the datum amplitude: HmSupContract (feeds both InitialLayerContract, via initialLayer_from_HmSup, and SectionFourAnsatzContract, via section_four_ansatz_bound) and SectionFourIterateContract;
• Step-down part (ii) at the relative amplitude S, under the gate: the seven relative fields of A8ContractProducers (gradient, material, jets, relative_initial, terms, leading, temperature). All of them reduce to the S-amplitude θ profile, but the composition is currently blocked by constant-order mismatches between the θ-profile producer and its consumers, so they stay hypotheses. Every other big-bound/step-down input is discharged here by its datum-amplitude producer.

theorem AVenhance.Infra.Section5.Contracts.a7Producers_of_remaining (β C₀ : ℝ) (hNormie3 : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA7Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.Normie3CenteredSourceContract I hΦ m (I.kappaSeq κ M m) T C √(l2NormSq θ₀)) : A7ContractProducers β C₀

The big-bound estimate producers from the single remaining big-bound estimate contract (refactor of bigbound_of_remaining).

structure AVenhance.Infra.Section5.Contracts.A8Remaining (β C₀ : ℝ) : Prop

Exactly the still-unproduced inputs of step-down (and hence of main theorem, with classical well-posedness), each in producer form. The relative fields are those of A8ContractProducers (amplitude S, gate).

• normie3 : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA7Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.Normie3CenteredSourceContract I hΦ m (I.kappaSeq κ M m) T C √(l2NormSq θ₀)

  Big-bound estimate slot 8 at the datum amplitude.

• hmSup : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA7Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.HmSupContract I hΦ m (I.kappaSeq κ M m) θ₀ T C

  e.Hm.Linfty at the datum amplitude.

• iterate : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (_hΦ : IsStreamSeq I _Φ) (_κ : ℝ) (_M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.SectionFourIterateContract I m θ₀ θprev T C

  e.Tm.thetam at the datum amplitude.

• gradient : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (_hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.TGradientContract β (I.kappaSeq κ M (m - 1)) T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))

  Relative TGradientContract at S.

• material : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (_hΦ : IsStreamSeq I Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.TMaterialContract I Φ m (I.kappaSeq κ M (m - 1)) T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))

  Relative TMaterialContract at S.

• jets : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (_hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.TPositiveJetsContract I m T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))

  Relative TPositiveJetsContract at S.

• relative_initial : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.InitialLayerContract I hΦ m (I.kappaSeq κ M m) θm T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))

  Relative InitialLayerContract at S.

• terms : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.RelativeTermsContract I hΦ m (I.kappaSeq κ M m) (I.kappaSeq κ M (m - 1)) T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))

  Relative RelativeTermsContract at S (includes slot 8 = normie3 at S).

• leading : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.LeadingErrorContract I hΦ m (I.kappaSeq κ M m) T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))

  Relative LeadingErrorContract at S.

• temperature : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (_hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (R : ℝ) (_θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θm θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => 6 / 5 ≤ β → epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2 - delta β) ≤ R → Integration.TemperatureErrorContract I m (I.kappaSeq κ M (m - 1)) θprev T C (√(I.kappaSeq κ M (m - 1)) * √(spaceTimeGradNormSq fun (s : ℝ) (x : Homogenization.Vec 2) => spaceGrad (θprev s) x))

  Relative TemperatureErrorContract at S.

theorem AVenhance.Infra.Section5.Contracts.initialLayer_datum_of_hmSup (β C₀ : ℝ) (hS : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA7Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.HmSupContract I hΦ m (I.kappaSeq κ M m) θ₀ T C) : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA8Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (θm _θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.InitialLayerContract I hΦ m (I.kappaSeq κ M m) θm T C √(l2NormSq θ₀)

The datum InitialLayerContract producer from the HTilde sup contract (initialLayer_from_HmSup).

theorem AVenhance.Infra.Section5.Contracts.sectionFourAnsatz_datum_of_hmSup (β C₀ : ℝ) (hS : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA7Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.HmSupContract I hΦ m (I.kappaSeq κ M m) θ₀ T C) : ∃ (C : ℝ) (C₁ : ℝ), Integration.OnA7Instances β C₀ C₁ fun (I : Ingredients β) (_Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I _Φ) (κ : ℝ) (M : ℕ) (_R : ℝ) (θ₀ : Homogenization.Vec 2 → ℝ) (m : ℕ) (_θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ) => Integration.SectionFourAnsatzContract I hΦ m (I.kappaSeq κ M m) θ₀ T C

The datum SectionFourAnsatzContract producer from the HTilde sup contract and the first-order slice jet.

theorem AVenhance.Infra.Section5.Contracts.a8Producers_of_remaining (β C₀ : ℝ) (h : A8Remaining β C₀) : A8ContractProducers β C₀

The step-down estimate producers from exactly the remaining contracts.

theorem AVenhance.Infra.Section5.Contracts.indystepdown_of_remaining (β C₀ : ℝ) (h : A8Remaining β C₀) : Integration.IndyStepDownStatement β C₀

Step-down from the remaining contracts: the literal step-down statement (both conjuncts).