Foundation Primitive Recognition Calculus Real Boundedness Modulus Prcboundednes

The boundedness threshold

The declaration PRCBoundednessDelta_positive establishes that a specific rational number, one eighth, is positive. In the Recognition Science framework, a ledger is a discrete record of recognition events, and the framework tracks how far apart two entries can be. The number one eighth is the fixed threshold: if the framework's cost distance between two ledger entries falls below it, then the ordinary numerical difference between those entries has square less than one.

This threshold is not pulled from thin air. The framework's library, a machine-checked collection of formal theorems, defines the threshold as one eighth and proves it is positive. The proof is direct: the definition expands to the rational number one eighth, and the library's kernel verifies that one eighth is greater than zero. The same library then proves the consequence: whenever the cost distance is below the threshold, the square of the difference between the two entries is less than one.

The point of the threshold is boundedness. A Cauchy sequence of ledger entries, one where entries eventually get close to each other, is eventually contained in a symmetric interval. The threshold guarantees that the entries do not drift apart faster than the framework can control. This is a step toward building a complete real number system from the framework's primitive recognition calculus, where the cost function J(x) = (x + 1/x)/2 - 1 governs how recognition events are priced.

The declaration does not claim that the threshold is the smallest possible, nor that it is unique. It does not claim that the cost distance itself is bounded, only that the ordinary difference between entries is controlled when the cost distance is small. It also does not claim that multiplication of real numbers is closed or congruent; those remain separate targets in the framework's library, which the threshold helps but does not by itself settle.

theorem PRCBoundednessDelta_positive :
    PRCRat.positive PRCBoundednessDelta := by
  rw [PRCRat.positive_iff_toRat_pos, PRCBoundednessDelta_toRat]
  norm_num

theorem PRCBoundednessDelta_toRat :
    PRCBoundednessDelta.toRat = (1 / 8 : ℚ) := by
  unfold PRCBoundednessDelta
  simp [PRCRat.toRat_mul, PRCRat.toRat_recip]
  norm_num

/-- Small PRC J-cost distance at the fixed threshold forces the ordinary
rational display increment to have square below one. -/
theorem PRCJCostDistance_sq_diff_lt_one_of_lt_boundedness_delta
    {a b : PRCRat}
    (hsmall : PRCRat.lt (PRCJCostDistance a b) PRCBoundednessDelta) :
    (a.toRat - b.toRat) * (a.toRat - b.toRat) < 1 := by
  rw [PRCRat.lt_iff_toRat_lt] at hsmall
  rw [PRCJCostDistance_toRat, PRCJCostDistanceRatDisplay_as_increment,
    PRCBoundednessDelta_toRat] at hsmall
  exact PRCJCostDistanceIncrementDisplay_sq_lt_one hsmall

What this page does not claim

The threshold one eighth is not claimed to be the smallest or unique possible threshold. The declaration does not claim that the cost distance itself is bounded, only the ordinary difference between entries. The declaration does not settle the closure or congruence of multiplication for the framework's 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/RealBoundednessModulus.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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