IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Inevitability

   1/-
   2  PrimitiveRecognitionCalculus/Inevitability.lean
   3
   4  Round-trip source:
   5    δ/PRC_Universal_Foundation_Execution_Plan_20260526.html
   6
   7  Spec anchor:
   8    Build Order step 13: prove that every admissible expressive foundation
   9    admits a PRC trace core.
  10
  11  "Admissible" here is deliberately exact: the foundation has been parsed into
  12  the `FormalSystem` interface and supplies endpoint distinction. Parsing
  13  external historical foundations into this interface is a later corpus task,
  14  not an implicit axiom in this theorem.
  15-/
  16
  17import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.FormalSystem
  18
  19namespace IndisputableMonolith
  20namespace Foundation
  21namespace PrimitiveRecognitionCalculus
  22
  23/-- An admissible foundation for the first inevitability theorem is a formal
  24system that is expressive enough to distinguish the two endpoints of δ. -/
  25structure AdmissibleFoundation where
  26  system : FormalSystem
  27  expressive : system.Expressive
  28
  29/-- The exact first inevitability target. -/
  30def PRCInevitabilityTarget : Prop :=
  31  ∀ A : AdmissibleFoundation, Nonempty (PRCEmbeddingInto A.system)
  32
  33/-- Any foundation already parsed into the admissible interface presupposes a
  34PRC trace core. -/
  35theorem any_foundation_presupposes_distinction :
  36    PRCInevitabilityTarget := by
  37  intro A
  38  exact FormalSystemEmbeddingTarget_proved A.system A.expressive
  39
  40/-- PRC itself is an admissible foundation in this interface. -/
  41def PRCAdmissibleFoundation : AdmissibleFoundation where
  42  system := PRCFormalSystem
  43  expressive := PRCFormalSystem_expressive
  44
  45theorem PRCAdmissibleFoundation_embeds :
  46    Nonempty (PRCEmbeddingInto PRCAdmissibleFoundation.system) :=
  47  any_foundation_presupposes_distinction PRCAdmissibleFoundation
  48
  49/-- The exact remaining external parsing target schema: once a corpus of
  50external foundations and a faithful-parse relation are fixed, every external
  51object in that corpus must parse to an expressive formal system before the
  52inevitability theorem applies to it. -/
  53def ExternalFoundationParsingTarget
  54    (ExternalFoundation : Type)
  55    (FaithfulParse : ExternalFoundation → FormalSystem → Prop) : Prop :=
  56  ∀ E : ExternalFoundation,
  57    ∃ F : FormalSystem, FaithfulParse E F ∧ F.Expressive
  58
  59/-- Step 13 certificate. The admissible-interface theorem is closed; the
  60external parsing workload is named separately so it is not hidden inside the
  61theorem. -/
  62structure PRCInevitabilityCertificate : Prop where
  63  admissible_foundation_surface : Nonempty AdmissibleFoundation
  64  prc_admissible : Nonempty AdmissibleFoundation
  65  inevitability_target : PRCInevitabilityTarget
  66  any_foundation_embedding :
  67    ∀ A : AdmissibleFoundation, Nonempty (PRCEmbeddingInto A.system)
  68  prc_embedding : Nonempty (PRCEmbeddingInto PRCAdmissibleFoundation.system)
  69  external_parsing_target_schema :
  70    ∀ (ExternalFoundation : Type)
  71      (FaithfulParse : ExternalFoundation → FormalSystem → Prop),
  72      ExternalFoundationParsingTarget ExternalFoundation FaithfulParse =
  73        ExternalFoundationParsingTarget ExternalFoundation FaithfulParse
  74  strength_tag : StrengthTag.deltaOnly = StrengthTag.deltaOnly
  75
  76theorem prc_inevitability_certificate :
  77    PRCInevitabilityCertificate where
  78  admissible_foundation_surface := ⟨PRCAdmissibleFoundation⟩
  79  prc_admissible := ⟨PRCAdmissibleFoundation⟩
  80  inevitability_target := any_foundation_presupposes_distinction
  81  any_foundation_embedding := any_foundation_presupposes_distinction
  82  prc_embedding := PRCAdmissibleFoundation_embeds
  83  external_parsing_target_schema := by
  84    intro ExternalFoundation FaithfulParse
  85    rfl
  86  strength_tag := rfl
  87
  88end PrimitiveRecognitionCalculus
  89end Foundation
  90end IndisputableMonolith
  91