Foundation Primitive Recognition Calculus Grow Ratio Orbit Lt Trichotomy Lt Q Tr

Ordering the growth ratios

In the Recognition Science framework, growth along a recognition cycle is tracked by a growth ratio, a number that records how a quantity changes from one tick of the cycle to the next. The framework's machine-checked library of formal theorems proves a trichotomy law for these ratios: for any two growth ratios p and q, exactly one of three relations holds: p is strictly less than q, p equals q in the framework's cross-equality sense, or q is strictly less than p. The declaration ltQ_trichotomy states this as a theorem, and the proof in the library shows it follows from the totality of the underlying order and the decidability of equality.

The classical analogue is the ordering of real numbers: for any two real numbers, one is less than the other, or they are equal. The framework's theorem is the same shape, but the equality relation is not ordinary numeric equality. It is cross-equality, a relation defined on the growth-ratio orbits that identifies ratios which yield the same growth behavior across the cycle. The trichotomy therefore says the ordering is total in the framework's own sense: no two ratios are left incomparable, and no ratio is strictly less than itself, a fact the library proves separately as ltQ_irrefl.

What the theorem does not claim is that the growth ratios form a complete ordered field, or that the ordering matches any particular numeric ordering on the real line. It establishes a structural property of the framework's own objects, not a statement about conventional numbers. The trichotomy is a tool for later results: it lets the framework reason about comparisons without case splits, and it underpins the monotonicity and extremal arguments that appear in the forcing chain. In plain terms, it guarantees that the growth model's ledger of ratios is fully ordered, so every pair of entries has a definite rank.

This is a theorem about the framework's definitions, not an empirical claim. It does not say which growth ratios occur in nature, nor does it assign numerical values to them. It only says that the framework's ordering relation is total and irreflexive, a logical scaffold on which the rest of the growth calculus rests.

theorem ltQ_trichotomy (p q : RatioOrbit) :
    ltQ p q ∨ RatioOrbit.crossEq p q ∨ ltQ q p := by
  cases' leQ_total p q with hpq hqp
  · by_cases heq : RatioOrbit.crossEq p q
    · exact Or.inr (Or.inl heq)
    · exact Or.inl ⟨hpq, heq⟩
  · by_cases heq : RatioOrbit.crossEq q p
    · exact Or.inr (Or.inl (RatioOrbit.crossEq_symm heq))
    · exact Or.inr (Or.inr ⟨hqp, heq⟩)

theorem ltQ_irrefl (p : RatioOrbit) : ¬ ltQ p p := by
  intro h
  exact h.2 (RatioOrbit.crossEq_refl p)

What this page does not claim

The theorem does not assign numerical values to growth ratios. The theorem does not claim the growth ratios form a complete ordered field. The theorem does not say which growth ratios occur in nature.

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

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