Foundation Primitive Recognition Calculus Prcnative Cost Minimality Certificate

The calibration boundary

A uniqueness theorem says that only one object can satisfy a list of rules. The Recognition Science framework's central result is such a theorem: it proves that any recognition cost, a function that assigns a price to each possible comparison between two observations, must take one specific form. The proof works because the rules are tight. The declaration PRCSlimSansTwoCalibrationUniquenessTarget_refuted is a machine-checked demonstration that the rules are exactly as tight as they need to be: remove one of them, and the uniqueness collapses.

The rule in question is the two-point calibration condition, which pins down the cost's value at two specific reference points. The theorem, proved in the framework's machine-checked library of formal theorems, shows that without this condition there exists a different cost function, called the Liouville twist cost, that satisfies every remaining rule but is not the canonical one. The proof works by exhibiting this counterexample explicitly: on the rational number 2, the Liouville twist cost returns -2 instead of 2, while still obeying all the other structural laws. This is a refutation, not a confirmation: it proves the statement "every cost satisfying the remaining rules is canonical" is false.

The method matters as much as the result. The counterexample is built from the Liouville sign, a classical arithmetic function that assigns +1 or -1 to a rational number based on the parity of its total number of prime factors. The twist multiplies each rational by this sign, which flips the sign of 2 but leaves 1 and -1 unchanged. The proof verifies that this twisted function meets all the weakened hypotheses, then shows it fails the canonical equality at 2. This is a constructive counterexample, not an abstract existence argument: the library names the exact function and checks every condition by computation.

Within the framework, this result does not weaken the main theorem. It sharpens it. The full set of hypotheses, including the two-point calibration, is what forces the unique cost function. The refuted target shows that each hypothesis is doing real work: remove one and the conclusion no longer follows. This is the standard way a uniqueness proof earns its strength, by demonstrating that no premise is redundant. The framework's library contains parallel refutations for other dropped conditions, each one a separate witness that the full theorem is not over-stated.

What the declaration does not claim is broader. It does not say the full uniqueness theorem is false, only that the version missing the calibration condition is. It does not identify the canonical cost function, which is proved elsewhere. It does not say anything about whether the Liouville twist cost is physically meaningful; it is a mathematical counterexample, constructed to test the boundary of the theorem. The result is a boundary marker, not a new candidate theory.

/-- **Two-point-anchor necessity.** Slim minus two-calibration admits the
Liouville twist, so the anchor cannot be dropped. -/
theorem PRCSlimSansTwoCalibrationUniquenessTarget_refuted :
    ¬ PRCSlimSansTwoCalibrationUniquenessTarget := by
  intro huniq
  exact liouvilleTwistNativeCost_two_not_canonical
    (huniq liouvilleTwistNativeCost
      liouvilleTwistNativeCost_sans_two_hypotheses two)

theorem liouvilleTwistDisplay_two : liouvilleTwistDisplay 2 = -2 := by
  have h2 : liouvilleSign (2 : ℚ) = -1 := by
    have := liouvilleSign_natCast_prime (p := 2) Nat.prime_two
    simpa using this
  rw [liouvilleTwistDisplay, h2]
  norm_num

/-- The Liouville-type sign of a rational display: `(-1)` to the total number
of prime factors (with multiplicity) of numerator and denominator of the
reduced fraction; Mathlib's `cardFactors` totalizes `Ω 0 = 0`, so the sign is
`+1` at `0`. `-1` on every prime, `±1` everywhere. -/
def liouvilleSign (t : ℚ) : ℚ :=
  (-1) ^ (Ω t.num.natAbs + Ω t.den)

What this page does not claim

The full uniqueness theorem for the canonical cost function is false. The Liouville twist cost is a physically meaningful alternative to the canonical cost. The refutation identifies which specific hypothesis is the weakest link.

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/PRCNativeCostMinimalityCertificate.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

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