Foundation Primitive Recognition Calculus Prcnative Cost Minimality Prcsigned St

A refuted target

In the Recognition Science framework, the central object is the cost, a function that assigns a price to every recognition event, where a recognition event is a discrete record of something being identified. The framework's main theorem proves that any cost satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. That theorem is proved in a machine-checked library of formal theorems, meaning every step is verified by a computer.

The declaration in question, PRCSignedStrengthenedNativeCostSignedAdmissibleCharacterFactorizationTarget_refuted, establishes that one particular proposed route to that theorem is closed. The proposal was to derive the uniqueness of the cost by factoring it through a special kind of character, a structure-preserving map on the ratio orbits that the framework uses to represent possible costs. The declaration proves this factorization target is false: no such derivation can work, because the stronger uniqueness claim it would imply is itself false.

Why is the stronger claim false? The proof exhibits a counterexample, a cost function called the zero-flat cost. This function assigns zero to the identity orbit and satisfies all the signed-strengthened hypotheses, which include the base conditions plus pairs and a signed unit, but it does not have the zero-orbit field that the stronger uniqueness claim requires. The factorization target, if it held, would force the stronger uniqueness claim, but the counterexample shows that stronger claim cannot hold. Therefore the factorization target is refuted.

This refutation does not touch the main theorem. The machine-checked library separately proves the uniqueness of the cost under the zero-calibrated signed-strengthened hypotheses, which add a zero-calibration condition. The zero-flat counterexample fails that extra condition, so it is excluded from the successful uniqueness proof. The refuted declaration is a negative result about a specific proof strategy, not about the framework's central result.

What the declaration does not claim is just as important. It does not claim that the cost function is not unique, nor that the framework's main theorem is wrong. It claims only that this particular factorization approach fails. The library records this as a settled question: the named target is refuted by inclusion, meaning the counterexample that defeats the stronger claim also defeats this weaker one. The path is closed, but the destination stands.

/-- Corollary: the signed-strengthened ledger cannot factor every inhabitant
through a signed-admissible character (the zero-flat cost cannot factor, since
character-generated costs are canonical at the zero orbit). -/
theorem PRCSignedStrengthenedNativeCostSignedAdmissibleCharacterFactorizationTarget_refuted :
    ¬ PRCSignedStrengthenedNativeCostSignedAdmissibleCharacterFactorizationTarget :=
  fun h =>
    PRCSignedStrengthenedNativeCostUniquenessTarget_refuted
      (PRCSignedStrengthenedNativeCostUniquenessTarget_of_signed_admissible_factorization
        h)

/-- **The launch prompt's named target is refuted.** The signed-strengthened
ledger (base + pairs + signed unit, no zero field) admits the zero-flat
countermodel: every one of its fields lives on nonzero orbits. -/
theorem PRCSignedStrengthenedNativeCostUniquenessTarget_refuted :
    ¬ PRCSignedStrengthenedNativeCostUniquenessTarget := by
  intro h
  have hzero :=
    h zeroFlatNativeCost
      zeroFlatNativeCost_prime_signed_strengthened_hypotheses.signed_strengthened
      RatioOrbit.zero
  rw [zeroFlatNativeCost_zero, RatioOrbit.crossEq_iff_toRat_eq,
    RatioOrbit.zero_toRat, onRatioOrbit_toRat, RatioOrbit.zero_toRat] at hzero
  norm_num at hzero

/-- **The slim ledger already forces J.** Factorization needs only base + zero;
the pair and sign fields transfer to the factor character; the new transport
lemma recovers per-prime calibration from the pair field and the base
two-calibration; the existing signed-admissible rigidity closes. -/
theorem PRCZeroCalibratedSignedStrengthenedNativeCostUniquenessTarget_proved :
    PRCZeroCalibratedSignedStrengthenedNativeCostUniquenessTarget := by
  intro F hF q
  rcases PRCZeroCalibratedNativeCostCharacterFactorizationTarget_proved F
      hF.signed_strengthened.strengthened.native hF.zero_calibrated with
    ⟨χ, hχ, hFχ⟩
  have hpair : PRCCharacterPrimePairProductCostConsistent χ := by
    intro p hp r hr
    exact RatioOrbit.crossEq_trans
      (RatioOrbit.crossEq_symm
        (hFχ (RatioOrbit.mul (primeDirection p hp) (primeDirection r hr))))
      (hF.signed_strengthened.strengthened.prime_pair_product_cost p hp r hr)
  have htwoCal :
      RatioOrbit.crossEq (costFromCharacter χ two) (onRatioOrbit two) :=
    RatioOrbit.crossEq_trans (RatioOrbit.crossEq_symm (hFχ two))
      hF.signed_strengthened.strengthened.native.two_calibrated
  have hprime : PRCCharacterPrimeDirectionCalibrated χ :=
    character_pair_two_calibration_forces_prime_calibration hχ hpair htwoCal
  have hsignCost :
      RatioOrbit.crossEq (costFromCharacter χ negativeOneRatio)
        (onRatioOrbit negativeOneRatio) :=
    RatioOrbit.crossEq_trans (RatioOrbit.crossEq_symm (hFχ negativeOneRatio))
      hF.signed_strengthened.signed_unit
  have hsign : PRCCharacterSignedUnitCalibrated χ :=
    costFromCharacter_negativeOne_forces_signed_unit hχ hsignCost
  exact RatioOrbit.crossEq_trans (hFχ q)
    (PRCNativeCostSignedAdmissibleCharacterRigidityTarget_proved χ
      ⟨⟨hχ, hprime, hpair⟩, hsign⟩ q)

What this page does not claim

The cost function is not unique. The framework's main theorem is wrong. The zero-flat cost satisfies the zero-calibrated hypotheses. The factorization approach is the only possible route to the uniqueness theorem.

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/PRCNativeCostMinimality.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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