Foundation Primitive Recognition Calculus Prcnative Cost Minimality Constant Zer

The zero-cost exclusion

A cost function assigns a price to each possible event. The simplest candidate is the constant zero function: every event costs nothing. In the Recognition Science framework, where reality keeps a ledger, a discrete record of recognition events, the question is whether this trivial pricing scheme can satisfy the framework's defining conditions. The declaration constantZeroNativeCost_slim_excluded settles it: the constant zero cost function is excluded. It cannot meet the framework's hypotheses, which require a cost that is reciprocal, calibrated, and composed under combination, not merely one that happens to be cheap.

The proof works by contradiction. The framework's hypotheses include a zero-calibration condition, which fixes the cost of a doubled trace at a specific value. The constant zero function assigns that doubled trace a cost of zero. The calibration condition demands a nonzero value, so the constant zero function fails. The theorem is not a standalone curiosity; it is one of several exclusions that together force the unique nonzero cost function the framework derives. The linear cost function, which prices events in direct proportion to their size, is likewise excluded, as is the zero-flat function that passes a weaker test but fails the calibrated one.

What the declaration does not claim is just as important. It does not say that a zero-cost event can never occur within the framework; it says the global pricing scheme that charges zero for everything is inconsistent with the framework's axioms. It does not claim that the constant zero function is the only excluded candidate, nor does it by itself establish what the correct cost function is. That uniqueness result, which selects the canonical cost function, is a separate theorem in the same machine-checked library of formal theorems. The exclusion is a negative result that clears the field, narrowing the search to functions that actually charge a price.

/-- Decoy exclusion 1 against the slim class (dies at the base
two-calibration). -/
theorem constantZeroNativeCost_slim_excluded :
    ¬ PRCZeroCalibratedSignedStrengthenedNativeCostHypotheses
        constantZeroNativeCost :=
  fun h =>
    constantZeroNativeCost_not_native_hypotheses
      h.signed_strengthened.strengthened.native

/-- Decoy exclusion 1 against the slim class (dies at the base
two-calibration). -/
theorem constantZeroNativeCost_slim_excluded :
    ¬ PRCZeroCalibratedSignedStrengthenedNativeCostHypotheses
        constantZeroNativeCost :=
  fun h =>
    constantZeroNativeCost_not_native_hypotheses
      h.signed_strengthened.strengthened.native

/-- Decoy exclusion 2 against the slim class (dies at the base
two-calibration). -/
theorem linearNativeCost_slim_excluded :
    ¬ PRCZeroCalibratedSignedStrengthenedNativeCostHypotheses linearNativeCost :=
  fun h =>
    linearNativeCost_not_native_hypotheses
      h.signed_strengthened.strengthened.native

What this page does not claim

The constant zero function is the only excluded cost function. A zero-cost event can never occur within the framework. This declaration by itself establishes what the correct cost function is.

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)

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