Foundation Primitive Recognition Calculus Prcnative Cost Selection Constant Zero

The excluded zero cost

In the Recognition Science framework, a recognition is an event where reality records a distinction, and a cost is the price, in some unit, that such a record forces. The framework's central claim is that this price is not a free choice: any cost function obeying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The conditions include a calibration rule, which says that the cost of recognizing the ratio two must match the cost that the formula J assigns to it.

The declaration constantZeroNativeCost_not_native_hypotheses proves that a particular rival cost function fails those conditions. The rival is constantZeroNativeCost, which charges a cost of zero for every recognition event, no matter what ratio is being recognized. The proof is a direct check of the calibration rule: the constant zero function assigns zero to the ratio two, but the J formula assigns a positive number, 1/4, to the ratio two. Since the two numbers differ, the constant zero function cannot satisfy the framework's hypotheses.

This exclusion is not a curiosity. It is a load-bearing step in the framework's argument that the J formula is the only possible cost. The framework's library of machine-checked theorems shows that a uniqueness target for the cost function is refuted if the constant zero cost were allowed in. By proving that the constant zero cost is inadmissible, the declaration removes one of the main ways the uniqueness claim could have failed. The same library also excludes a second rival, the linear cost, by the same calibration check.

In Recognition Science, the declaration is part of a larger package that assembles the full proof of uniqueness. The package records that the J formula satisfies the hypotheses, that the constant zero and linear costs do not, and that no other rival is left standing. The declaration itself does not prove the J formula is the unique cost; it only removes one specific alternative. The full uniqueness proof is a separate theorem, and the declaration is one of its supporting lemmas.

/-- The constant-zero cost already fails the base hypothesis ledger: the
canonical cost of the two orbit displays as `1/4`, not `0`. -/
theorem constantZeroNativeCost_not_native_hypotheses :
    ¬ PRCNativeCostHypotheses constantZeroNativeCost := by
  intro h
  have h2 := h.two_calibrated
  rw [RatioOrbit.crossEq_iff_toRat_eq] at h2
  simp only [constantZeroNativeCost, RatioOrbit.zero_toRat, onRatioOrbit_toRat,
    two_toRat] at h2
  norm_num at h2

/-- The native cost-selection package holds. -/
theorem costSelectionPackageNative_holds : CostSelectionPackageNative where
  j_unique_native :=
    PRCZeroCalibratedPrimeSignedStrengthenedNativeCostUniquenessTarget_proved
  non_vacuous :=
    ⟨canonicalSelectedNativeCost, canonicalSelectedNativeCost_full_hypotheses,
      canonicalSelectedNativeCost_crossEq_onRatioOrbit⟩
  zero_cost_excluded := constantZeroNativeCost_excluded
  linear_cost_excluded := linearNativeCost_excluded

What this page does not claim

The declaration does not prove that the J formula is the unique cost function; it only excludes one specific rival. The declaration does not say that a zero cost is impossible in any physical system, only that it cannot satisfy the framework's recognition axioms. The declaration does not address the behavior of the cost function at ratios other than two; the calibration check is specific to the ratio two.

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

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