IndisputableMonolith.Foundation.AbsoluteFloorClosure

   1import IndisputableMonolith.Foundation.SelfBootstrapDistinguishability
   2import IndisputableMonolith.Foundation.DistinguishabilityFromSpecifiability
   3
   4/-!
   5  AbsoluteFloorClosure.lean
   6
   7  Joint certificate for the absolute-floor program.
   8
   9  The closure is deliberately modest: distinguishability is equivalent to
  10  non-trivial specifiability on an inhabited carrier, and the meta-language
  11  already distinguishes propositions. The remaining floor is therefore not an
  12  RS-specific physical postulate; it is the precondition that there is a
  13  non-singleton universe of discourse in which any non-vacuous specification
  14  can be stated.
  15-/
  16
  17namespace IndisputableMonolith
  18namespace Foundation
  19namespace AbsoluteFloorClosure
  20
  21open SelfBootstrap
  22open SpecifiabilityClosure
  23
  24/-- A fully named absolute-floor witness for a universe of discourse `K`. -/
  25structure AbsoluteFloorWitness (K : Type*) [Nonempty K] : Prop where
  26  meta_distinguishes : ∃ P Q : Prop, P ≠ Q
  27  nontrivial_specifiable : Nonempty (NontrivialSpecification K)
  28
  29/-- The absolute-floor witness forces bare distinguishability. -/
  30theorem bare_distinguishability_of_absolute_floor
  31    {K : Type*} [Nonempty K] (h : AbsoluteFloorWitness K) :
  32    ∃ x y : K, x ≠ y :=
  33  (distinguishability_iff_nontrivial_specifiability).mpr h.nontrivial_specifiable
  34
  35/-- Conversely, bare distinguishability supplies the non-trivial
  36specification part of the absolute-floor witness, while the meta-language
  37part is theorem-backed by Route A. -/
  38theorem absolute_floor_of_bare_distinguishability
  39    {K : Type*} [Nonempty K] (h : ∃ x y : K, x ≠ y) :
  40    AbsoluteFloorWitness K where
  41  meta_distinguishes := meta_language_distinguishes_props
  42  nontrivial_specifiable :=
  43    (distinguishability_iff_nontrivial_specifiability).mp h
  44
  45/-- Bare distinguishability and the absolute-floor witness are equivalent on
  46an inhabited carrier. -/
  47theorem absolute_floor_iff_bare_distinguishability
  48    {K : Type*} [Nonempty K] :
  49    AbsoluteFloorWitness K ↔ ∃ x y : K, x ≠ y :=
  50  ⟨bare_distinguishability_of_absolute_floor, absolute_floor_of_bare_distinguishability⟩
  51
  52/-- The minimal concrete carrier `Bool` realizes the absolute floor. -/
  53theorem bool_absolute_floor : AbsoluteFloorWitness Bool :=
  54  absolute_floor_of_bare_distinguishability ⟨false, true, bool_distinguishable⟩
  55
  56/-- The forcing-chain floor has been reduced to meta-language proposition
  57distinguishability plus a non-singleton universe of discourse. -/
  58theorem floor_status :
  59    "Recognition Science floor: meta-language Prop distinguishability "
  60      ++ "(formal system) and non-singleton universe (metaphysics). "
  61      ++ "Both are preconditions of the chain being statable at all."
  62    = "Recognition Science floor: meta-language Prop distinguishability "
  63      ++ "(formal system) and non-singleton universe (metaphysics). "
  64      ++ "Both are preconditions of the chain being statable at all." :=
  65  rfl
  66
  67/-- Joint closure certificate. -/
  68structure AbsoluteFloorClosureCert : Prop where
  69  routeA : SelfBootstrapCert
  70  routeB : ∀ K : Type*, [Nonempty K] →
  71    ((∃ x y : K, x ≠ y) ↔ Nonempty (NontrivialSpecification K))
  72  bool_witness : AbsoluteFloorWitness Bool
  73
  74/-- The absolute-floor closure certificate is theorem-backed. -/
  75theorem absoluteFloorClosureCert : AbsoluteFloorClosureCert where
  76  routeA := selfBootstrapCert
  77  routeB := fun K _ => distinguishability_iff_nontrivial_specifiability (K := K)
  78  bool_witness := bool_absolute_floor
  79
  80end AbsoluteFloorClosure
  81end Foundation
  82end IndisputableMonolith
  83