IndisputableMonolith.Gravity.Analysis.ContinuumOrderSensitiveResidual4D

   1import Mathlib
   2import IndisputableMonolith.Loom.CertificateData
   3import IndisputableMonolith.Gravity.Analysis.OrderSensitiveHistoryResponse4D
   4import IndisputableMonolith.Gravity.Analysis.MetricEdgeImage4D
   5
   6/-!
   7# Continuum promotion of the order-sensitive residual
   8
   9Campaign G4/G5. Replaces finite Boolean non-membership in
  10`MetricEdgeImage` with a normalized-separation trichotomy under a
  11shape-regular refinement family.
  12
  13## Trichotomy (frozen)
  14
  15For a refinement level `n` with response difference `Δ_n` and metric image
  16`M_n`:
  17
  18* **Survive:** positive liminf of normalized distance to the metric image
  19* **Metric collapse:** normalized distance tends to 0 while the response
  20  norm does not
  21* **Lattice washout:** response norm tends to 0
  22
  23## What is proved here
  24
  25* The finite residual is outside `MetricEdgeImage` (imported).
  26* A shape-regular refinement family interface is named.
  27* Continuum survival / collapse / washout for the certificate pair are
  28  recorded as OPEN residual Props (uninhabited). The geometric mesh
  29  Tendsto baseline (Arc 2 step 9) is required before any terminal is claimed.
  30* Continuum promotion is explicitly not earned from the finite theorem.
  31
  32## Honesty
  33
  34* THEOREM: finite outside-image; status flag false.
  35* OPEN: inhabiting any continuum terminal for the discovery pair.
  36* Forbidden: promoting the finite Boolean result to continuum novelty.
  37-/
  38
  39namespace IndisputableMonolith
  40namespace Gravity
  41namespace Analysis
  42namespace ContinuumOrderSensitiveResidual4D
  43
  44open OrderSensitiveHistoryResponse4D
  45open MetricEdgeImage4D
  46open IndisputableMonolith.Loom.Certificate
  47open BigOperators
  48
  49/-- Edge-field squared Frobenius norm on the patch. -/
  50noncomputable def edgeNorm (F : Fin 16 → Fin 16 → ℝ) : ℝ :=
  51  ∑ i : Fin 16, ∑ j : Fin 16, (F i j) ^ 2
  52
  53/-- Shape-regular refinement family interface. -/
  54structure RefinementFamily where
  55  responseDiff : ℕ → Fin 16 → Fin 16 → ℝ
  56  metricProj : ℕ → (Fin 16 → Fin 16 → ℝ) → Fin 16 → Fin 16 → ℝ
  57  levelNorm : ℕ → ℝ
  58  shapeRegular : ∀ n, 0 < levelNorm n
  59
  60/-- Normalized distance from a response difference to its metric projection. -/
  61noncomputable def normalizedSeparation (F : RefinementFamily) (n : ℕ) : ℝ :=
  62  edgeNorm (fun i j =>
  63      F.responseDiff n i j - F.metricProj n (F.responseDiff n) i j)
  64    / edgeNorm (F.responseDiff n)
  65
  66/-- Continuum terminal: surviving new sector. -/
  67def Survives (F : RefinementFamily) : Prop :=
  68  ∃ ε : ℝ, 0 < ε ∧
  69    ∀ᶠ n in Filter.atTop, ε ≤ normalizedSeparation F n
  70
  71/-- Continuum terminal: metric collapse. -/
  72def MetricCollapse (F : RefinementFamily) : Prop :=
  73  Filter.Tendsto (normalizedSeparation F) Filter.atTop (nhds 0) ∧
  74    ¬ Filter.Tendsto (fun n => edgeNorm (F.responseDiff n)) Filter.atTop (nhds 0)
  75
  76/-- Continuum terminal: lattice washout. -/
  77def LatticeWashout (F : RefinementFamily) : Prop :=
  78  Filter.Tendsto (fun n => edgeNorm (F.responseDiff n)) Filter.atTop (nhds 0)
  79
  80/-- Finite stage already outside the metric image (imported G3). -/
  81theorem finite_outside_metric_image :
  82    ¬ MetricEdgeImage (responseDiff cfgA cfgB) :=
  83  responseDiff_cfgAB_not_in_MetricEdgeImage
  84
  85/-- Collapse and washout are incompatible by definition. -/
  86theorem collapse_not_washout (F : RefinementFamily)
  87    (hC : MetricCollapse F) (hW : LatticeWashout F) : False :=
  88  hC.2 hW
  89
  90/-- **OPEN residual.** Continuum survival of the certificate residual. -/
  91def ContinuumSurvivalOpen : Prop :=
  92  ∃ F : RefinementFamily, Survives F
  93
  94/-- **OPEN residual.** Metric collapse of the certificate residual. -/
  95def ContinuumMetricCollapseOpen : Prop :=
  96  ∃ F : RefinementFamily, MetricCollapse F
  97
  98/-- **OPEN residual.** Lattice washout of the certificate residual. -/
  99def ContinuumWashoutOpen : Prop :=
 100  ∃ F : RefinementFamily, LatticeWashout F
 101
 102/-- Honest status flag: continuum promotion not yet earned. -/
 103def continuumPromotionEarned : Bool := false
 104
 105theorem continuumPromotionEarned_eq :
 106    continuumPromotionEarned = false := rfl
 107
 108/-- Methodological wall: the finite outside-image theorem does not flip the
 109continuum-promotion flag. -/
 110theorem finite_exclusion_does_not_earn_promotion :
 111    (¬ MetricEdgeImage (responseDiff cfgA cfgB)) →
 112      continuumPromotionEarned = false :=
 113  fun _ => continuumPromotionEarned_eq
 114
 115end ContinuumOrderSensitiveResidual4D
 116end Analysis
 117end Gravity
 118end IndisputableMonolith
 119