Foundation Logic Realization Positive Ratio Faithful

Faithful arithmetic

The positive real numbers are the familiar line of numbers greater than zero, where multiplication and division behave well. The declaration positiveRatio_faithful establishes that a certain structure built from these numbers, called a realization, interprets its internal arithmetic without any two distinct elements collapsing to the same value. In plainer terms: if the framework's laws force two different counting steps, then those steps correspond to two different positive real numbers, never to one.

The proof works through a chain of definitions. A realization is a carrier set with a comparison cost, an identity element, and a step action. The positive-ratio realization takes the positive reals as its carrier and a chosen non-trivial generator as its step. The theorem positiveRatio_interpret_injective shows that the map from the abstract counting type into the positive reals is injective, meaning no two distinct counting positions land on the same real number. A second condition, positiveRatio_hasIdentityStep, rules out the specific failure where the identity element and some step produce the same real value. Together these two conditions form the FaithfulArithmeticInterpretation structure, and positiveRatio_faithful assembles them into one theorem.

The point of this result is to guarantee that the arithmetic the framework extracts from its laws is not an illusion created by a periodic carrier. A modular carrier, for instance, wraps around and identifies values that are distinct in the counting structure. The positive reals do not wrap, and the theorem certifies that the framework's internal arithmetic embeds into them without such collisions. This is a structural guarantee, not a numerical one: it says the embedding exists and is faithful, not what particular real numbers the steps land on.

In Recognition Science, this theorem is part of the bridge between the abstract laws of logic and the concrete arithmetic of positive ratios. It does not prove that the laws themselves hold, nor that the chosen generator is the golden ratio. It only certifies that, given a comparison operator satisfying the laws, the arithmetic forced by those laws can be read faithfully in the positive reals. The theorem is machine-checked in the framework's library of formal theorems, so the reasoning is verified step by step.

What the declaration does not claim is worth stating plainly. It does not establish which comparison operator satisfies the laws, nor does it derive the value of the generator. It does not prove that every realization is faithful; periodic realizations are explicitly noted as needing no such guarantee. The theorem is a conditional bridge: if the laws hold, then the arithmetic they force embeds faithfully into the positive reals. That is the whole content, and it is a precise and useful one.

/-- The continuous positive-ratio realization interprets its forced arithmetic
faithfully into the positive real carrier. -/
theorem positiveRatio_faithful
    (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
    FaithfulArithmeticInterpretation (ofPositiveRatioComparison C h) where
  injective := by
    intro a b hab
    exact positiveRatio_interpret_injective C h hab
  zero_step_noncollapse := by
    intro n hcollapse
    exact LogicNat.zero_ne_succ n (positiveRatio_interpret_injective C h hcollapse)

/-- The continuous positive-ratio orbit interpretation is injective. -/
theorem positiveRatio_interpret_injective
    (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
    Function.Injective (positiveRatioOrbitInterpret C h) := by
  intro a b hab
  have hval := congrArg Subtype.val hab
  rw [positiveRatioOrbitInterpret_val, positiveRatioOrbitInterpret_val] at hval
  exact ArithmeticFromLogic.embed_injective
    (ArithmeticFromLogic.generatorOfLawsOfLogic h) hval

/-- The continuous positive-ratio realization satisfies the abstract
identity-step predicate. -/
theorem positiveRatio_hasIdentityStep
    (C : ComparisonOperator) (h : SatisfiesLawsOfLogic C) :
    (ofPositiveRatioComparison C h).hasIdentityStep :=
  hasIdentityStep_of_nontrivial _

What this page does not claim

The theorem does not prove that any particular comparison operator satisfies the laws of logic. The theorem does not derive the numerical value of the non-trivial generator. The theorem does not assert that every realization is faithful, only the positive-ratio one.

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/LogicRealization.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: