Foundation Primitive Recognition Calculus Prcnative Cost Minimality

Cost minimality

In Recognition Science, a ledger is a discrete record of recognition events, and the cost of each recognition is forced, not chosen. The framework's central theorem states that any cost function satisfying five plain conditions (reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity) must equal J(x) = (x + 1/x)/2 - 1. This is a proved theorem in a machine-checked library of formal theorems, with no hidden assumptions beyond the standard axioms of logic.

The module PRCNativeCostMinimality.lean tests a stronger version of this uniqueness claim. It asks: if we add a zero-calibration condition and a signed-strengthened ledger, does the same J still win? The answer is yes, but only after a key adjustment. The library first refutes a naive target: a signed-strengthened ledger without a zero field admits a countermodel called the zero-flat cost, which satisfies all the strengthened hypotheses yet assigns zero cost to every event.

The fix is to add a zero-calibration condition, which forces the cost of the zero orbit to respect the doubled trace. With that condition in place, the library proves the strengthened uniqueness target: any function meeting the zero-calibrated signed-strengthened hypotheses must equal J on every ratio orbit. This is the theorem PRCZeroCalibratedSignedStrengthenedNativeCostUniquenessTarget_proved.

The module also shows what the strengthened axioms exclude. The constant zero cost fails because it violates the native hypotheses. The linear cost fails for the same reason. The zero-flat cost, which passes the signed-strengthened hypotheses, is excluded specifically by the zero-calibration condition. The canonical selected cost, which is J itself, satisfies all the strengthened hypotheses, so the uniqueness theorem is not vacuous.

What this establishes in plain language is that the cost function J is not an accident of the weakest axioms. Even when the ledger is made stricter, with signed fields and zero calibration, the same cost function is the only one that survives. The framework's chain from J to the golden ratio, the eight-tick cycle, and three spatial dimensions therefore rests on a cost function that is stable under strengthening, not a fragile artifact of a single axiom set.

/-- **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)

/-- **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 round-1 non-vacuity witness inhabits the slim class. -/
theorem canonicalSelectedNativeCost_slim_hypotheses :
    PRCZeroCalibratedSignedStrengthenedNativeCostHypotheses
      canonicalSelectedNativeCost :=
  (slim_class_iff_full_class canonicalSelectedNativeCost).mpr
    canonicalSelectedNativeCost_full_hypotheses

What this page does not claim

This module does not derive the fine-structure constant or any specific particle mass. The zero-flat cost is excluded only under the zero-calibration condition, not under the weaker signed-strengthened hypotheses alone. The uniqueness theorem applies to ratio orbits, not to arbitrary functions on the real line.

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.

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