| import OAI.Analysis.CircleDomains.Transfer.RetainedTerminalGeometry |
|
|
| namespace OAI |
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| /-! |
| # Complete geometric assembly for a selected real-coordinate grid |
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| The bands themselves provide the terminal oscillation budgets. After |
| selecting physical terminals off the core, the descending grid construction |
| fills the remaining dust and retains its actual one-active-coordinate |
| boundary condition. No disk family or containment conclusion is assumed. |
| -/ |
|
|
| open Set Function Filter |
| open scoped Topology Manifold ContDiff |
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| noncomputable section |
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| namespace CircleDomainRigidity.FiniteTransfer |
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| attribute [local instance] sphereMetric |
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| theorem boundary_oscillation_le_of_band |
| {U : Set Sphere} {f : Sphere → ℝ} {M a b : ℝ} {p : Sphere} |
| (hM : ∀ x ∈ U, |f x| ≤ M) (hp : p ∈ closure U) |
| (hband : {t : ℝ | (p, t) ∈ intervalFiberHull (clusterGraph U f)} = Icc a b) : |
| limsup f (𝓝[U] p) - liminf f (𝓝[U] p) ≤ b - a := by |
| let : NeBot (𝓝[U] p) := mem_closure_iff_nhdsWithin_neBot.mp hp |
| have hb : ∀ᶠ x in 𝓝[U] p, -M ≤ f x ∧ f x ≤ M := by |
| filter_upwards [self_mem_nhdsWithin] with x hx |
| exact abs_le.mp (hM x hx) |
| have hlo := isBoundedUnder_of_eventually_ge (hb.mono (fun _ hx => hx.1)) |
| have hhi := isBoundedUnder_of_eventually_le (hb.mono (fun _ hx => hx.2)) |
| have hle := liminf_le_limsup hhi hlo |
| have heq := intervalFiberHull_clusterGraph_eq_boundary_band hM hp |
| have ha : liminf f (𝓝[U] p) ∈ Icc a b := by |
| rw [← hband, heq] |
| exact ⟨le_rfl, hle⟩ |
| have hb' : limsup f (𝓝[U] p) ∈ Icc a b := by |
| rw [← hband, heq] |
| exact ⟨hle, le_rfl⟩ |
| linarith [ha.1, hb'.2] |
|
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| theorem exists_geometric_grid_assembly |
| (hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies) |
| (G : Set Sphere) (hG : IsOpen G) (π : Sphere → Sphere) |
| (hπ : Continuous π) (hclosed : IsClosedMap π) (hsurj : Surjective π) |
| (hr : ∀ x y, π x = π y ↔ |
| x = y ∨ (x ∈ Gᶜ ∧ y ∈ connectedComponentIn Gᶜ x)) |
| (hTD : IsTotallyDisconnected (π '' Gᶜ)) |
| (C : Set Sphere) (hC : IsCompact C) (hCG : C ⊆ G) (hinfty : OnePoint.infty ∈ C) |
| {n N : ℕ} {S : Type*} |
| (U : Set Sphere) (f : Fin n → Sphere → ℝ) (M : Fin n → ℝ) |
| (hbounded : ∀ l x, x ∈ U → |f l x| ≤ M l) |
| (hphysical : ∀ z : ℂ, (z : Sphere) ∈ G → π (z : Sphere) ∈ U) |
| (σ : S → Sphere) (hσ : Injective σ) (hσB : ∀ s, σ s ∈ π '' Gᶜ) |
| (hσU : ∀ s, σ s ∈ closure U) |
| (a b : Fin n → S → ℝ) (θ : Fin n → ℝ) (F : Set S) |
| (hband : ∀ l s, {t : ℝ | (σ s, t) ∈ intervalFiberHull (clusterGraph U (f l))} = |
| Icc (a l s) (b l s)) |
| (hfin : (gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ).Finite) |
| (O : (gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ) → Set Sphere) |
| (hO : ∀ s : gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ, |
| O s ∈ 𝓝 (σ s)) |
| (hfree : ∀ x ∈ π '' Gᶜ, x ∉ range σ → ∀ l, |
| ∃ v : ℝ, {t : ℝ | (x, t) ∈ intervalFiberHull (clusterGraph U (f l))} = {v}) |
| (hcompat : OrderedBandGridCompatibility U f θ N (π '' Gᶜ) σ) : |
| ∃ (D : PhysicalTerminalDisks G π (π '' Gᶜ) |
| (fun s : gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ => σ s) O) |
| (m : ℕ) (H : Fin m → ℂ ≃ₜ ℂ) (γ : Fin m → Circle → ℂ), |
| (∀ s, Disjoint (closedTerminalCap (D.H s)) C) ∧ |
| C ⊆ openCompletedRegion D.H ∧ |
| (∀ s l, ∀ δ η : ℝ, ∀ x ∈ frontier (jordanInterior (D.H s)), |
| ∀ y ∈ frontier (jordanInterior (D.H s)), |
| |shiftedStaircase (reciprocalGridStep N) δ η (f l (π (x : Sphere))) - |
| shiftedStaircase (reciprocalGridStep N) δ η (f l (π (y : Sphere)))| ≤ |
| (b l s - a l s) + 4 * reciprocalGridStep N) ∧ |
| Pairwise (Disjoint on fun j => closure (jordanInterior (H j))) ∧ |
| (∀ s j, Disjoint (closure (jordanInterior (D.H s))) (closure (jordanInterior (H j)))) ∧ |
| (∀ j, Disjoint (closedTerminalCap (H j)) C) ∧ |
| π ⁻¹' (π '' Gᶜ) ⊆ sphericalDiskUnion D.H ∪ sphericalDiskUnion H ∧ |
| C ⊆ completionCore D.H H ∧ completionCore D.H H ⊆ G ∧ |
| (∀ j, ContMDiff (𝓡 1) (𝓘(ℝ, ℂ)) ∞ (γ j) ∧ Topology.IsEmbedding (γ j) ∧ |
| frontier (jordanInterior (H j)) = range (γ j) ∧ |
| (∀ z ∈ frontier (jordanInterior (H j)), (z : Sphere) ∈ G ∧ π (z : Sphere) ∉ π '' Gᶜ) ∧ |
| ∃ k : ℕ, ∀ l : Fin n, l.val ≠ k → |
| Disjoint (frontier (jordanInterior (H j))) |
| ((fun z : ℂ => π (z : Sphere)) ⁻¹' (⋃ r, reciprocalBandLevel U (f l) (θ l) N r))) := by |
| let : Fintype (gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ) := hfin.fintype |
| obtain ⟨D, hD, hCQ, hosc⟩ := exists_retained_physical_terminals_with_staircase_bound |
| hsard hsch hG hC hCG hinfty π hπ hclosed hsurj hr hTD |
| (fun s : gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ => σ s) |
| (hσ.comp Subtype.val_injective) (fun s => hσB s) O hO |
| (fun _ => U) (fun _ l => f l) (fun _ l => M l) (fun _ => hbounded) |
| (fun _ z hz _ => hphysical z hz) (fun _ _ => True) (fun s l => b l s - a l s) |
| (fun s l _ => boundary_oscillation_le_of_band (hbounded l) (hσU s) (hband l s)) |
| (reciprocalGridStep_pos N) |
| obtain ⟨m, H, γ, hHd, hDH, hHC, hcover, hCC, hCG', hcurves⟩ := |
| exists_geometric_completion_for_grid hsard hsch G π hπ hclosed hsurj hr |
| (hG.isClosed_compl.isCompact.image hπ).isClosed rfl hTD hC hCG hinfty |
| U f M hbounded σ hσB a b θ F hband hfin O D hD |
| (fun x hx => hfree x hx.1) (fun x hx => hcompat x hx.1) |
| exact ⟨D, m, H, γ, hD, hCQ, fun s l => hosc s l trivial, |
| hHd, hDH, hHC, hcover, hCC, hCG', hcurves⟩ |
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| end CircleDomainRigidity.FiniteTransfer |
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| end |
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| end OAI |
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