Foundation Non Triviality From Distinguishability Existing Of Absolute Floor

The absolute floor

In mathematics, a comparison operator is a function that takes two numbers and returns a value measuring their difference. The Recognition Science framework studies such operators under a set of axioms meant to capture the laws of logic. A trivial operator, one that returns zero for every pair, vacuously satisfies many of these axioms. The framework's library of formal theorems proves that adding a single, natural condition rules out this trivial case.

The condition is called an absolute floor: the existence of a smallest positive ratio, a fundamental unit of comparison. The theorem existing_of_absoluteFloor states that if a comparison operator satisfies the laws of logic and detects this floor, meaning it gives a non-zero result for any two distinct positive quantities, then it satisfies the full original laws of logic. In plain terms, a logic that can tell any two things apart is a real logic, not a vacuous one.

This result replaces an older, more algebraic assumption called NonTrivial, which simply asserted the cost function was not identically zero. The newer theorem derives that same fact from the more fundamental, Aristotelian notion of distinguishability: comparison is operative, it actually does something. The framework's library proves that under the other axioms, distinguishability and non-triviality are equivalent, making the new condition the canonical content.

The theorem is a formal result in the framework's machine-checked library. It does not claim that the absolute floor exists in the physical world, nor does it specify what that smallest ratio is. It only shows that if such a floor is assumed, the logic is guaranteed to be non-vacuous. The existence of the floor itself remains a separate assumption, not a consequence of the theorem.

/-- The absolute-floor form induces the existing algebraic form used by
downstream modules. -/
theorem existing_of_absoluteFloor
    (C : ComparisonOperator) (h : SatisfiesLawsOfLogicAbsoluteFloor C) :
    SatisfiesLawsOfLogic C :=
  (canonical_iff_existing C).mp (canonical_of_absoluteFloor C h)

/-- **Equivalence theorem**: under scale invariance, distinguishability
and non-triviality are the same condition. -/
theorem nonTrivial_iff_distinguishability
    (C : ComparisonOperator) (hSI : ScaleInvariant C) :
    NonTrivial C ↔ Distinguishability C :=
  ⟨distinguishability_of_nonTrivial C, nonTrivial_of_distinguishability C hSI⟩

What this page does not claim

The theorem does not prove that an absolute floor exists. The theorem does not identify the value of the smallest positive ratio. The theorem does not claim that the absolute floor is a physical constant.

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

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