Foundation Absolute Floor Closure Bool Absolute Floor

The minimal carrier

The declaration bool_absolute_floor establishes, in the framework's machine-checked library of formal theorems, that the two-element Boolean universe Bool (whose only members are false and true) realizes the absolute floor. The absolute floor is the framework's name for the minimal condition that a universe of discourse must satisfy before any recognition event can be stated: it must contain at least two distinguishable objects. The theorem proves that Bool meets this condition, because false and true are distinct.

The deeper content is the equivalence the theorem relies on. The framework proves, for any inhabited carrier, that the existence of two distinguishable elements is equivalent to the existence of a non-trivial specification: a predicate that is neither always true nor always false. This equivalence is the real mathematical receipt. The bool_absolute_floor theorem is then a concrete instance: since false and true differ, Bool admits a non-trivial specification, and therefore satisfies the absolute floor.

The framework's own documentation is explicit about what this does not claim. The absolute floor is not an RS-specific physical postulate. It reduces to two ingredients: the meta-language already distinguishes propositions, and there is a non-singleton universe of discourse in which any non-vacuous specification can be stated. The theorem adds no new physical content; it is a joint certificate that the framework's foundational route is consistent with the minimal precondition of any language that can talk about distinct things.

In plain terms, the declaration says: if you can tell two things apart, you already have enough structure to begin the framework's program. The floor is not a discovery about the physical world; it is a precondition of the framework's own vocabulary. What the theorem changes is that the framework's foundational chain can be audited down to this minimal condition, and Bool serves as the smallest concrete witness that the condition is satisfiable.

/-- The minimal concrete carrier `Bool` realizes the absolute floor. -/
theorem bool_absolute_floor : AbsoluteFloorWitness Bool :=
  absolute_floor_of_bare_distinguishability ⟨false, true, bool_distinguishable⟩

/-- Bare distinguishability and the absolute-floor witness are equivalent on
an inhabited carrier. -/
theorem absolute_floor_iff_bare_distinguishability
    {K : Type*} [Nonempty K] :
    AbsoluteFloorWitness K ↔ ∃ x y : K, x ≠ y :=
  ⟨bare_distinguishability_of_absolute_floor, absolute_floor_of_bare_distinguishability⟩

/-- Joint closure certificate. -/
structure AbsoluteFloorClosureCert : Prop where
  routeA : SelfBootstrapCert
  routeB : ∀ K : Type*, [Nonempty K] →
    ((∃ x y : K, x ≠ y) ↔ Nonempty (NontrivialSpecification K))
  bool_witness : AbsoluteFloorWitness Bool

What this page does not claim

This answer does not claim that bool_absolute_floor derives any physical constant or law. This answer does not claim that the absolute floor is a physical discovery about the universe. This answer does not claim that the framework's full forcing chain is established by this single declaration.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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