Foundation Primitive Recognition Calculus Real Order Congruence Rat Sq Lt Sq Bou

The square bound lemma

The declaration rat_sq_lt_sq_bounds is a lemma about rational numbers. It says: if gamma is a positive rational, and x is a rational whose square is less than gamma squared, then x lies strictly between -gamma and gamma. In symbols, from x*x < gamma*gamma and 0 < gamma, it proves -gamma < x and x < gamma. This is the familiar fact that the square function is increasing on the positive half of the number line, stated for rationals and checked mechanically.

The lemma matters because it is the engine behind a larger result in the Recognition Science framework. The framework's ledger, a discrete record of recognition events, builds real numbers as equivalence classes of Cauchy sequences of rationals. To compare two such real numbers, the framework needs to know that the ordering on the underlying rational sequences is well-defined, meaning that replacing a sequence by an equivalent one does not change which is eventually larger. The square bound lemma is what makes that proof go through: it shows that if the cost distance between two sequences is small, then the difference of their terms is small in absolute value, which is exactly the step needed to transfer the rational order to the real order.

In Recognition Science, the framework models the real numbers this way so that its cost function, the forced function J(x) = (x + 1/x)/2 - 1, can be evaluated on a continuum. The lemma itself is tagged as a theorem in the machine-checked library of formal theorems, meaning it is proved from the ambient type theory's standard axioms with no framework-specific assumptions. It does not by itself establish any property of the cost function; it is a technical bridge about rational arithmetic.

What the lemma does not claim is broader. It does not say that the square function is increasing on all rationals, only on the positive side. It does not imply that the real order is total, nor that the Cauchy construction is the only way to build the reals. It says nothing about the cost function's uniqueness or about the golden ratio. The lemma is a tool, not a conclusion; its payoff is that the framework's real numbers behave like the ordinary real line, which is a prerequisite for the cost function's theorems to mean anything.

private theorem rat_sq_lt_sq_bounds {x gamma : ℚ}
    (hgamma : 0 < gamma) (hsq : x * x < gamma * gamma) :
    -gamma < x ∧ x < gamma := by
  constructor
  · by_contra hnot
    have hxle : x ≤ -gamma := by linarith
    have hnonneg : 0 ≤ -x - gamma := by linarith
    have hprod : 0 ≤ (-x - gamma) * (-x + gamma) := by
      have hright : 0 ≤ -x + gamma := by linarith
      exact mul_nonneg hnonneg hright
    nlinarith
  · by_contra hnot
    have hxge : gamma ≤ x := by linarith
    have hnonneg : 0 ≤ x - gamma := by linarith
    have hprod : 0 ≤ (x - gamma) * (x + gamma) := by
      have hright : 0 ≤ x + gamma := by linarith
      exact mul_nonneg hnonneg hright
    nlinarith

theorem PRCRealOrderCongruenceTarget_proved :
    PRCRealOrderCongruenceTarget := by
  intro u u' v v' huu hvv
  constructor
  · intro hle
    exact PRCRawEventuallyLe_of_null_equiv huu hvv hle
  · intro hle
    exact PRCRawEventuallyLe_of_null_equiv
      (PRCNullEquivalent.symm huu) (PRCNullEquivalent.symm hvv) hle

What this page does not claim

The lemma does not prove the cost function is unique or that the golden ratio is forced. It does not establish that the square function is increasing on all rationals, only on the positive side. It does not show that the Cauchy construction is the only way to build the real numbers.

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

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