Foundation Logic Realization Positive Ratio Interpret Injective

The injective embedding

In mathematics, an injective function is one that never sends two different inputs to the same output; each output has at most one preimage. The declaration positiveRatio_interpret_injective establishes that a specific mapping, from a discrete arithmetic structure into the positive real numbers, is injective. This mapping is built from a recognition process, a discrete record of events where each step multiplies a running value by a fixed positive factor. The theorem proves that two distinct steps in this record always yield two distinct positive real values.

The proof is short and rests on a prior result about the embedding of arithmetic from logic. The mapping in question sends the identity element to the number 1 and each subsequent step to the previous value multiplied by a chosen generator. The theorem positiveRatio_interpret_injective shows this map cannot collapse: if two records produce the same real number, then the records themselves must be identical. This is the property called faithfulness in the framework's terminology, and the declaration positiveRatio_faithful packages it together with a second condition, that the zero step never equals a later step.

In Recognition Science, this result matters because it guarantees that the arithmetic forced by the laws of logic can be studied inside the real numbers without losing information. The framework models a comparison operator that satisfies five plain conditions, and from those conditions it derives a cost function and a chain of structural consequences. The injectivity result is a technical foundation: it says the internal orbit of the recognition process, though generated by a single step, does not wrap around or repeat when viewed in the positive reals. This is what allows the framework to treat the real-valued interpretation as a faithful mirror of the discrete arithmetic.

The theorem does not claim that every recognition process has this property. The framework explicitly notes that periodic realizations, such as modular carriers, need not satisfy faithfulness; their internal orbit can remain free while the carrier interpretation is periodic. The result also does not identify the generator or compute the cost function; it only concerns the injectivity of the orbit map. It does not assert that the positive reals are the only possible carrier, nor that the embedding is surjective. The declaration is a statement about one specific construction, not about all possible models.

/-- 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 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 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)

/-- A realization whose internal forced arithmetic embeds faithfully into its
ambient carrier. Periodic realizations, such as modular carriers, need not
satisfy this; their internal orbit is still free while the carrier
interpretation is periodic. -/
structure FaithfulArithmeticInterpretation (R : LogicRealization) : Prop where
  injective : Function.Injective R.interpret
  zero_step_noncollapse : ∀ n : R.Orbit, R.interpret R.orbitZero ≠ R.interpret (R.orbitStep n)

What this page does not claim

The theorem does not identify the generator or compute the cost function. The result does not assert that the embedding is surjective onto the positive reals. The declaration does not apply to periodic realizations such as modular carriers.

Verify this page

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

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