Foundation Non Triviality From Distinguishability Non Trivial Iff Distinguishabi

Distinguishability and non-triviality

In mathematics and physics, a comparison operator is a rule that takes two positive quantities and returns a number measuring how they differ. The Recognition Science framework begins with such operators and asks which ones behave like genuine logic. One obvious candidate fails the test: the operator that always returns zero. It satisfies several formal conditions, like treating all inputs the same, but it is useless because it cannot tell anything apart.

The declaration nonTrivial_iff_distinguishability proves that two ways of ruling out this useless operator are equivalent. The first, called non-triviality, says the comparison function is not identically zero on positive ratios. The second, called distinguishability, says there exists at least one pair of distinct positive quantities whose comparison cost is non-zero. Under the condition of scale invariance, the theorem shows these two statements are interchangeable: if comparison is not vacuous, then it can distinguish some pair, and vice versa.

This matters because the framework originally had to assume non-triviality as an extra postulate. The theorem replaces that postulate with a more natural one: distinguishability, the idea that comparison is operative rather than empty. The framework's library, a machine-checked collection of formal theorems, proves the equivalence. It also shows that the constant-zero operator fails both conditions, confirming that the two formulations really do capture the same requirement.

What the theorem does not claim is just as important. It does not say that any particular comparison operator is non-trivial, only that the two properties are equivalent. It does not derive the specific cost function from distinguishability alone; other laws are needed for that. And it does not assert that distinguishability holds for any real physical system, only that if comparison is meaningful, it must satisfy this condition.

/-- **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⟩

/-- Constant zero fails non-triviality. -/
theorem constZero_not_nonTrivial : ¬ NonTrivial constZero := by
  intro ⟨_, _, h⟩
  exact h rfl

/-- Constant zero fails distinguishability. -/
theorem constZero_not_distinguishable : ¬ Distinguishability constZero := by
  intro ⟨_, _, _, _, h⟩
  exact h rfl

What this page does not claim

The theorem does not prove that any particular comparison operator is non-trivial or distinguishable. The theorem does not derive the specific cost function from distinguishability alone. The theorem does not assert that distinguishability holds for any real physical system.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: