IndisputableMonolith.RecogGeom.Core

   1import Mathlib
   2
   3/-!
   4# Recognition Geometry: Core Definitions
   5
   6This module establishes the foundational types for Recognition Geometry—
   7a framework where space is not primitive, but emerges from the structure
   8of recognition maps.
   9
  10## Key Insight
  11
  12Standard geometry starts with a space and puts structure on top.
  13Recognition geometry flips the story:
  14- The primitive objects are **recognizers** and **events**
  15- "Space" is whatever is needed to make sense of what can be recognized
  16
  17## Axiom RG0: Nonempty Configuration Space
  18
  19The first axiom simply asserts that there exists something to recognize.
  20
  21-/
  22
  23namespace IndisputableMonolith
  24namespace RecogGeom
  25
  26/-! ## Configuration Space (RG0) -/
  27
  28/-- A configuration space is a type of possible states of the world.
  29    Configurations are what the world "really" has—recognizers never
  30    get the configuration itself; they get events.
  31
  32    RG0: There exists a nonempty configuration space. -/
  33class ConfigSpace (C : Type*) where
  34  /-- The configuration space is nonempty -/
  35  nonempty : Nonempty C
  36
  37/-- Extract a witness configuration from a ConfigSpace -/
  38noncomputable def ConfigSpace.witness (C : Type*) [cs : ConfigSpace C] : C :=
  39  cs.nonempty.some
  40
  41/-! ## Event Space -/
  42
  43/-- An event space is a type of observable outcomes.
  44    Events are things like "the needle points this direction,"
  45    "the detector clicks," or "the image matches this template." -/
  46class EventSpace (E : Type*) where
  47  /-- The event space has at least two distinct events
  48      (otherwise recognition is trivial) -/
  49  nontrivial : ∃ e₁ e₂ : E, e₁ ≠ e₂
  50
  51/-- An event space with decidable equality -/
  52class DecEventSpace (E : Type*) extends EventSpace E where
  53  /-- Decidable equality on events -/
  54  decEq : DecidableEq E
  55
  56attribute [instance] DecEventSpace.decEq
  57
  58/-! ## Basic Properties -/
  59
  60/-- **THEOREM**: A configuration space has at least one element.
  61    Replaces the vacuous `∃ c : C, True` with a constructive witness. -/
  62theorem config_exists (C : Type*) [cs : ConfigSpace C] : ∃ c : C, c = ConfigSpace.witness C :=
  63  ⟨ConfigSpace.witness C, rfl⟩
  64
  65/-- An event space has at least two distinct elements -/
  66theorem event_nontrivial (E : Type*) [EventSpace E] : ∃ e₁ e₂ : E, e₁ ≠ e₂ :=
  67  EventSpace.nontrivial
  68
  69/-! ## Recognition Triple -/
  70
  71/-- A recognition triple bundles a configuration space, event space,
  72    and the implicit structure connecting them. This is the basic
  73    object of study in recognition geometry. -/
  74structure RecognitionTriple where
  75  /-- The type of configurations -/
  76  Config : Type*
  77  /-- The type of events -/
  78  Event : Type*
  79  /-- Configurations form a valid configuration space -/
  80  configSpace : ConfigSpace Config
  81  /-- Events form a valid event space -/
  82  eventSpace : EventSpace Event
  83
  84/-! ## Module Status -/
  85
  86def core_status : String :=
  87  "✓ ConfigSpace defined (RG0)\n" ++
  88  "✓ EventSpace defined\n" ++
  89  "✓ Nonemptiness axiom established\n" ++
  90  "✓ Nontriviality axiom established\n" ++
  91  "✓ RecognitionTriple bundle defined\n" ++
  92  "\n" ++
  93  "CORE FOUNDATION COMPLETE"
  94
  95#eval core_status
  96
  97end RecogGeom
  98end IndisputableMonolith
  99