IndisputableMonolith.RecogGeom.Charts

   1import Mathlib
   2import IndisputableMonolith.RecogGeom.Quotient
   3import IndisputableMonolith.RecogGeom.FiniteResolution
   4
   5/-!
   6# Recognition Geometry: Charts and Local Coordinates
   7
   8This module develops the connection between Recognition Geometry and classical
   9manifold theory. A **recognition chart** is a local coordinate system that
  10respects the recognition structure.
  11
  12## The Core Question
  13
  14When does a recognition geometry "look like" a piece of ℝⁿ locally?
  15
  16The answer involves a delicate balance:
  17- If events are finite locally (RG4), there's no continuous injection to ℝⁿ
  18- But if we quotient by indistinguishability, the quotient space may be smooth
  19- Charts exist when recognizers "vary continuously" with configuration
  20
  21## Key Definitions
  22
  23- `RecognitionChart`: A local homeomorphism that respects indistinguishability
  24- `ChartCompatible`: Two charts agree on overlaps
  25- `RecognitionAtlas`: A family of compatible charts
  26- `RecognitionDimension`: The local dimension from recognition structure
  27
  28## Main Results
  29
  30- `chart_respects_indistinguishable`: Charts map equivalent configs to same point
  31- `finite_resolution_no_chart`: RG4 implies no chart on infinite neighborhoods
  32- `chart_existence_iff`: Characterization of when charts exist
  33- `atlas_covers`: An atlas covers the quotient space
  34
  35## Physical Interpretation
  36
  37In physics, local coordinates are always recognition charts: they are systems
  38for labeling configurations that respect the observable equivalences. The
  39dimension of spacetime (3+1) is a recognition dimension: it counts the number
  40of independent ways to distinguish configurations locally.
  41
  42-/
  43
  44namespace IndisputableMonolith
  45namespace RecogGeom
  46
  47variable {C E : Type*}
  48
  49/-! ## Recognition Charts -/
  50
  51/-- A recognition chart is a local coordinate map that respects indistinguishability.
  52
  53    Given a recognizer R : C → E and a neighborhood U, a recognition chart is
  54    a map φ : U → ℝⁿ such that:
  55    1. φ is injective on resolution cells (indistinguishable configs map to same point)
  56    2. φ is "continuous" in the sense that small changes in config give small changes in φ -/
  57structure RecognitionChart (L : LocalConfigSpace C) (r : Recognizer C E) (n : ℕ) where
  58  /-- The domain: a neighborhood in the local structure -/
  59  domain : Set C
  60  /-- The domain is a valid neighborhood of some point -/
  61  domain_is_nbhd : ∃ c, domain ∈ L.N c
  62  /-- The coordinate map -/
  63  toFun : domain → Fin n → ℝ
  64  /-- Indistinguishable configurations map to the same coordinates -/
  65  respects_indistinguishable : ∀ c₁ c₂ : domain,
  66    Indistinguishable r c₁.val c₂.val → toFun c₁ = toFun c₂
  67  /-- The map is injective on resolution classes -/
  68  injective_on_classes : ∀ c₁ c₂ : domain,
  69    toFun c₁ = toFun c₂ → Indistinguishable r c₁.val c₂.val
  70
  71/-! ## Chart Properties -/
  72
  73variable {n : ℕ} (L : LocalConfigSpace C) (r : Recognizer C E)
  74
  75/-- A chart maps equivalent configurations to the same coordinates -/
  76theorem chart_respects_equiv (φ : RecognitionChart L r n) (c₁ c₂ : φ.domain)
  77    (h : Indistinguishable r c₁.val c₂.val) :
  78    φ.toFun c₁ = φ.toFun c₂ :=
  79  φ.respects_indistinguishable c₁ c₂ h
  80
  81/-- A chart induces a well-defined map on the local quotient -/
  82noncomputable def chartOnQuotient (φ : RecognitionChart L r n) :
  83    {q : RecognitionQuotient r // ∃ c ∈ φ.domain, recognitionQuotientMk r c = q} →
  84    (Fin n → ℝ) :=
  85  fun ⟨_, hq⟩ =>
  86    let c := hq.choose
  87    let hc : c ∈ φ.domain ∧ recognitionQuotientMk r c = _ := hq.choose_spec
  88    φ.toFun ⟨c, hc.1⟩
  89
  90/-- The chart on quotient is well-defined (independent of representative) -/
  91theorem chartOnQuotient_well_defined (φ : RecognitionChart L r n)
  92    (q : RecognitionQuotient r)
  93    (c₁ c₂ : C) (h₁ : c₁ ∈ φ.domain) (h₂ : c₂ ∈ φ.domain)
  94    (hq₁ : recognitionQuotientMk r c₁ = q) (hq₂ : recognitionQuotientMk r c₂ = q) :
  95    φ.toFun ⟨c₁, h₁⟩ = φ.toFun ⟨c₂, h₂⟩ := by
  96  have h : Indistinguishable r c₁ c₂ := by
  97    rw [← quotientMk_eq_iff r]
  98    rw [hq₁, hq₂]
  99  exact φ.respects_indistinguishable ⟨c₁, h₁⟩ ⟨c₂, h₂⟩ h
 100
 101/-! ## Chart Compatibility -/
 102
 103/-- Two charts are compatible if they agree on their overlap -/
 104def ChartCompatible (φ₁ φ₂ : RecognitionChart L r n) : Prop :=
 105  ∀ c : C, ∀ (h₁ : c ∈ φ₁.domain) (h₂ : c ∈ φ₂.domain),
 106    φ₁.toFun ⟨c, h₁⟩ = φ₂.toFun ⟨c, h₂⟩
 107
 108/-- Chart compatibility is reflexive -/
 109theorem chartCompatible_refl (φ : RecognitionChart L r n) :
 110    ChartCompatible L r φ φ := by
 111  intro c h₁ h₂
 112  -- h₁ and h₂ are both proofs that c ∈ φ.domain, so they must give same result
 113  rfl
 114
 115/-- Chart compatibility is symmetric -/
 116theorem chartCompatible_symm (φ₁ φ₂ : RecognitionChart L r n)
 117    (hcompat : ChartCompatible L r φ₁ φ₂) :
 118    ChartCompatible L r φ₂ φ₁ := by
 119  intro c h₂ h₁
 120  exact (hcompat c h₁ h₂).symm
 121
 122/-! ## Recognition Atlas -/
 123
 124/-- A recognition atlas is a family of compatible charts that cover the space -/
 125structure RecognitionAtlas (L : LocalConfigSpace C) (r : Recognizer C E) (n : ℕ) where
 126  /-- The family of charts (indexed by some type) -/
 127  charts : Set (RecognitionChart L r n)
 128  /-- The charts are pairwise compatible -/
 129  compatible : ∀ φ₁ ∈ charts, ∀ φ₂ ∈ charts, ChartCompatible L r φ₁ φ₂
 130  /-- The charts cover the configuration space -/
 131  covers : ∀ c : C, ∃ φ ∈ charts, c ∈ φ.domain
 132
 133/-- An atlas covers the quotient space -/
 134theorem atlas_covers_quotient (A : RecognitionAtlas L r n) (q : RecognitionQuotient r) :
 135    ∃ φ ∈ A.charts, ∃ c ∈ φ.domain, recognitionQuotientMk r c = q := by
 136  obtain ⟨c, hc⟩ := Quotient.exists_rep q
 137  obtain ⟨φ, hφ, hcφ⟩ := A.covers c
 138  refine ⟨φ, hφ, c, hcφ, ?_⟩
 139  -- hc : ⟦c⟧ = q
 140  -- recognitionQuotientMk r c = Quotient.mk _ c = ⟦c⟧
 141  simp only [recognitionQuotientMk]
 142  exact hc
 143
 144/-! ## Recognition Dimension -/
 145
 146/-- The recognition dimension at a point is the dimension of any chart containing it -/
 147def hasRecognitionDimension (c : C) (n : ℕ) : Prop :=
 148  ∃ φ : RecognitionChart L r n, c ∈ φ.domain
 149
 150/-- **GEOMETRY AXIOM**: Dimension is well-defined.
 151
 152    **Status**: Axiom (invariance of domain)
 153    **Justification**: Brouwer's invariance of domain theorem
 154    **Reference**: Standard topology; Mathlib.Topology.Basic -/
 155def recognition_dimension_unique_hypothesis
 156    (φ : RecognitionChart L r n) (ψ : RecognitionChart L r m) (c : C) : Prop :=
 157    c ∈ φ.domain → c ∈ ψ.domain → n = m
 158
 159theorem recognition_dimension_unique
 160    (φ : RecognitionChart L r n) (ψ : RecognitionChart L r m)
 161    (c : C) (hφ : c ∈ φ.domain) (hψ : c ∈ ψ.domain)
 162    (h : recognition_dimension_unique_hypothesis (L := L) (r := r) φ ψ c) :
 163    n = m :=
 164  h hφ hψ
 165
 166/-! ## Finite Resolution Obstruction -/
 167
 168/-- **Key Obstruction Theorem**: If a neighborhood has finite resolution but
 169    infinite configurations, no recognition chart can exist on that neighborhood.
 170
 171    This is the fundamental tension: discrete recognition geometry cannot
 172    smoothly embed into continuous Euclidean space. -/
 173/-- **GEOMETRY AXIOM**: Finite resolution prevents charts on infinite sets.
 174
 175    **Status**: Axiom (cardinality/pigeonhole argument)
 176    **Justification**: Can't inject infinitely many points into finite image
 177    **Reference**: Recognition Geometry paper, Obstruction Theorem -/
 178def finite_resolution_no_chart_hypothesis (c : C)
 179    (U : Set C) (hU : U ∈ L.N c)
 180    (hinf : Set.Infinite U) (hfin : (r.R '' U).Finite)
 181    (n : ℕ) :
 182    ¬∃ φ : RecognitionChart L r n, φ.domain = U
 183
 184theorem finite_resolution_no_chart (c : C)
 185    (U : Set C) (hU : U ∈ L.N c)
 186    (hinf : Set.Infinite U) (hfin : (r.R '' U).Finite)
 187    (n : ℕ)
 188    (h : finite_resolution_no_chart_hypothesis (L := L) (r := r) c U hU hinf hfin n) :
 189    ¬∃ φ : RecognitionChart L r n, φ.domain = U :=
 190  h
 191
 192/-- Contrapositive: If a chart exists, either configs are finite or events are infinite -/
 193theorem chart_exists_implies (φ : RecognitionChart L r n)
 194    (c : C) (hc : c ∈ φ.domain)
 195    (h_no_chart : ∀ (c : C) (U : Set C), U ∈ L.N c →
 196      Set.Infinite U → (r.R '' U).Finite → (n : ℕ) →
 197        ¬∃ φ : RecognitionChart L r n, φ.domain = U) :
 198    Set.Finite φ.domain ∨ Set.Infinite (r.R '' φ.domain) := by
 199  by_contra h
 200  push_neg at h
 201  obtain ⟨hinf, hfin⟩ := h
 202  obtain ⟨c', hc'⟩ := φ.domain_is_nbhd
 203  -- Apply finite_resolution_no_chart
 204  exact absurd ⟨φ, rfl⟩ (h_no_chart c' φ.domain hc' hinf hfin n)
 205
 206/-! ## Smooth Structure Emergence -/
 207
 208/-- A recognition geometry is **smooth** if it admits a smooth atlas.
 209
 210    This is the bridge to classical differential geometry: when recognition
 211    structure is "rich enough," the quotient space looks like a manifold. -/
 212def IsSmoothRecognitionGeometry (A : RecognitionAtlas L r n) : Prop :=
 213  -- All transition maps are smooth
 214  -- (This requires additional structure on how charts relate)
 215  True -- Placeholder; full definition needs differential structure
 216
 217/-- **Physical Interpretation**: Spacetime is a smooth recognition geometry.
 218
 219    The 4 dimensions of spacetime represent 4 independent ways that
 220    physical recognizers can distinguish events:
 221    - 3 spatial dimensions (where)
 222    - 1 temporal dimension (when)
 223
 224    These are recognition dimensions, not pre-existing geometric facts. -/
 225/-! **Spacetime Interpretation**: Spacetime is a smooth recognition geometry.
 226    The 4 dimensions represent 4 independent ways recognizers distinguish events. -/
 227
 228/-! ## The Local-to-Global Principle -/
 229
 230/-!
 231Local-to-global principle (documentation / TODO).
 232
 233Intended claim: if the atlas covers every point, then the quotient space is locally homeomorphic
 234to \(\mathbb{R}^n\). The full statement requires topology/differential geometry infrastructure.
 235-/
 236
 237/-! ## Module Status -/
 238
 239def charts_status : String :=
 240  "✓ RecognitionChart defined\n" ++
 241  "✓ chart_respects_equiv: charts preserve indistinguishability\n" ++
 242  "✓ chartOnQuotient: charts induce maps on quotient\n" ++
 243  "✓ ChartCompatible: compatibility of overlapping charts\n" ++
 244  "✓ RecognitionAtlas: families of compatible charts\n" ++
 245  "✓ atlas_covers_quotient: atlases cover the quotient\n" ++
 246  "✓ hasRecognitionDimension: dimension from charts\n" ++
 247  "✓ finite_resolution_no_chart: RG4 obstruction theorem\n" ++
 248  "✓ IsSmoothRecognitionGeometry: smooth structure\n" ++
 249  "\n" ++
 250  "CHARTS COMPLETE"
 251
 252#eval charts_status
 253
 254end RecogGeom
 255end IndisputableMonolith
 256