Foundation Primitive Recognition Calculus Prcnative Cost Selection

Choosing the cost

The primitive recognition calculus is the framework's starting ledger: a discrete record of events, each event being a recognition of one ratio by another. The framework models the cost of such a recognition as a function that takes a ratio and returns another ratio, and it asks which costs are possible at all. The answer, proved in a machine-checked library of formal theorems, is that any cost satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This is not a numerical fit; it is a forced conclusion from the conditions themselves.

The five conditions are reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. The composition law, written in full, is F(xy) + F(x/y) = 2F(x)F(y) + 2F(x) + 2F(y). The theorem that this forces J is the core result of the module. The proof also shows that the conditions are not vacuous: a specific cost function, the canonical selected native cost, satisfies all five. This witness function sends the unit ratio to the literal zero representative and otherwise equals J, so the uniqueness claim has a real inhabitant.

The module then tests the boundary of the uniqueness claim. It shows that two plausible but wrong candidates fail the conditions. The constant zero cost, which assigns zero to every ratio, fails the calibration condition. The linear cost, which subtracts one from every ratio, also fails. Both are excluded by the same calibration requirement. These exclusions are not optional; they are proved as theorems, and they sharpen what the uniqueness result actually covers.

The module also records a structural wall. A weaker hypothesis class, without the full strengthening, does not force uniqueness: the target statement is refuted. The full strengthened class is needed. The module packages this as a continuum price residue wall, a named structure that marks where the naive uniqueness claim breaks. The wall is tagged as a classical extension, meaning it goes beyond the strongest proved result.

What a reader can now see is the exact shape of the forcing argument. The cost is not chosen; it is the only function that survives the five conditions. The module proves the uniqueness, exhibits the witness, and names the false candidates. The framework's later results, such as the golden ratio and the eight-tick cycle, rest on this selection.

/-- 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

/-- 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

/-- Decoy exclusion 1 against the full frozen class. -/
theorem constantZeroNativeCost_excluded :
    ¬ PRCZeroCalibratedPrimeSignedStrengthenedNativeCostHypotheses
        constantZeroNativeCost :=
  fun h =>
    constantZeroNativeCost_not_native_hypotheses
      h.prime_signed.signed_strengthened.strengthened.native

/-- Decoy exclusion 2 against the full frozen class. -/
theorem linearNativeCost_excluded :
    ¬ PRCZeroCalibratedPrimeSignedStrengthenedNativeCostHypotheses
        linearNativeCost :=
  fun h =>
    linearNativeCost_not_native_hypotheses
      h.prime_signed.signed_strengthened.strengthened.native

/-- **Zero-orbit calibration is irreducible.** Without it, the prime-signed
strengthened ledger admits the zero-flat countermodel: uniqueness fails at the
zero orbit, where the canonical cost displays `-1` and the countermodel
displays `0`. -/
theorem PRCPrimeSignedStrengthenedNativeCostUniquenessTarget_refuted :
    ¬ PRCPrimeSignedStrengthenedNativeCostUniquenessTarget := by
  intro h
  have hzero :=
    h zeroFlatNativeCost zeroFlatNativeCost_prime_signed_strengthened_hypotheses
      RatioOrbit.zero
  rw [zeroFlatNativeCost_zero, RatioOrbit.crossEq_iff_toRat_eq,
    RatioOrbit.zero_toRat, onRatioOrbit_toRat, RatioOrbit.zero_toRat] at hzero
  norm_num at hzero

What this page does not claim

The module does not prove that the five conditions are the only possible ones. The module does not show that the J cost is the only function satisfying any subset of the five conditions. The module does not derive the fine-structure constant or any other physical constant.

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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Derived articles

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