Encyclopedia Foundation Foundation Primitive Recognition Calculus Prcnative Cost Structural Ledger Canon

ARTICLE 4 claims 4 theorems

Foundation Primitive Recognition Calculus Prcnative Cost Structural Ledger Canon

A single rule about how cost changes when a ratio is flipped forces the entire cost function, and the proof is checked by machine.

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The sign-reversing cost

In mathematics, a cost function assigns a number to each possible state, and the sign-reversing rule considered here concerns ratios. The rule says that when you replace a ratio by its reciprocal, the cost changes sign in a precise way: the cost of the reciprocal is the negative of the original cost, minus 2. This is not a physical law but a definitional choice about what kind of cost function is being studied.

The Recognition Science framework models reality as maintaining a ledger, a discrete record of recognition events. The declaration in question, PRCSignReversingNativeCostUniquenessTarget_proved, is a proved theorem in the framework's machine-checked library of formal theorems. It states that any cost function satisfying the sign-reversing rule, along with a few other structural hypotheses, must be the specific function J(x) = (x + 1/x)/2 - 1. The proof is a chain of logical steps verified by a computer, with no gaps and no extra assumptions beyond the standard axioms of logic.

The theorem does not claim that the sign-reversing rule is the only way to arrive at J. The framework also proves a stronger result, PRCStructuralNativeCostUniquenessTarget_proved, which uses a different set of hypotheses and reaches the same conclusion. The sign-reversing version is one path among several. It also does not claim that the sign-reversing rule applies to all possible cost functions; it applies only to those that also satisfy the other structural conditions in the theorem's statement.

What the theorem establishes is a uniqueness result: within its stated assumptions, there is exactly one cost function, and it has the closed form J(x) = (x + 1/x)/2 - 1. This matters because it shows that a seemingly arbitrary rule about sign reversal pins down the entire cost structure. The framework's library also proves that without the sign-reversing condition, the uniqueness fails; there exist other cost functions, such as the cube-generated native cost, that satisfy the other hypotheses but not the sign-reversing rule. This contrast is what makes the sign-reversing assumption load-bearing.

THEOREM PRCSignReversingNativeCostUniquenessTarget_proved · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/PRCNativeCostStructuralLedger.lean
PRCSignReversingNativeCostUniquenessTarget_proved · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/PRCNativeCostStructuralLedger.lean:302 
/-- Round 4 terminal: the sign-reversing ledger forces the canonical cost. -/
theorem PRCSignReversingNativeCostUniquenessTarget_proved :
    PRCSignReversingNativeCostUniquenessTarget := by
  intro F hF q
  refine PRCZeroCalibratedSignedStrengthenedNativeCostUniquenessTarget_proved F ?_ q
  exact
    { signed_strengthened :=
        { strengthened := hF.strengthened
          signed_unit :=
            signReversing_forces_signed_unit
              hF.strengthened.native.unit_zero hF.sign_reversing }
      zero_calibrated := hF.zero_calibrated }
THEOREM PRCSignReversingNativeCostUniquenessTarget_proved · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/PRCNativeCostStructuralLedger.lean
PRCSignReversingNativeCostUniquenessTarget_proved · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/PRCNativeCostStructuralLedger.lean:302 
/-- Round 4 terminal: the sign-reversing ledger forces the canonical cost. -/
theorem PRCSignReversingNativeCostUniquenessTarget_proved :
    PRCSignReversingNativeCostUniquenessTarget := by
  intro F hF q
  refine PRCZeroCalibratedSignedStrengthenedNativeCostUniquenessTarget_proved F ?_ q
  exact
    { signed_strengthened :=
        { strengthened := hF.strengthened
          signed_unit :=
            signReversing_forces_signed_unit
              hF.strengthened.native.unit_zero hF.sign_reversing }
      zero_calibrated := hF.zero_calibrated }
THEOREM PRCStructuralNativeCostUniquenessTarget_proved · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/PRCNativeCostStructuralLedger.lean
PRCStructuralNativeCostUniquenessTarget_proved · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/PRCNativeCostStructuralLedger.lean:687 
/-- **Round 5 terminal.** The structural ledger forces the canonical cost. -/
theorem PRCStructuralNativeCostUniquenessTarget_proved :
    PRCStructuralNativeCostUniquenessTarget := by
  intro F hF q
  obtain ⟨χ, hχ, hFχ⟩ :=
    PRCZeroCalibratedNativeCostCharacterFactorizationTarget_proved F hF.native
      hF.zero_calibrated
  have hcal := structural_character_calibrated_on_positive_integers hF hχ hFχ
  have hresp : PRCCharacterRespectsCrossEq χ :=
    PRCCharacterRespectsCrossEq_of_normalizeRatio_canonical hχ
      PRCNormalizeRatioCanonicalTarget_proved
  have hchi : ∀ a b : RatioOrbit, a.toRat = b.toRat →
      (χ a).toRat = (χ b).toRat := by
    intro a b hab
    exact crossDisp
      (hresp a b (dispCross hab))
  -- every positive integer orbit is calibrated
  have hposcal : ∀ (a : RatioOrbit), IsPosIntOrbit a →
      RatioOrbit.crossEq (costFromCharacter χ a) (onRatioOrbit a) := by
    rintro a ⟨n, hn, han⟩
    have hdisp : a.toRat = (natOrbit n).toRat := by rw [natOrbit_toRat, han]
    rw [RatioOrbit.crossEq_iff_toRat_eq, costFromCharacter_jq, jq_onRatioOrbit,
      han]
    have hn' := hcal n hn
    rw [← hchi a (natOrbit n) hdisp] at hn'
    exact hn'
  have hprime : PRCCharacterPrimeDirectionCalibrated χ := fun p hp =>
    hposcal _ (primeDirection_isPosInt hp)
  have hpair : PRCCharacterPrimePairProductCostConsistent χ := by
    intro p hp r hr
    refine hposcal _ ⟨p.toNat * r.toNat, ?_, ?_⟩
    · obtain ⟨a, ha, _⟩ := primeDirection_isPosInt hp
      obtain ⟨b, hb, _⟩ := primeDirection_isPosInt hr
      have hpn : 1 ≤ p.toNat := by
        have h := primeDirection_toRat_ne_zero p hp
        rw [primeDirection_toRat] at h
        exact Nat.one_le_iff_ne_zero.mpr (by exact_mod_cast h)
      have hrn : 1 ≤ r.toNat := by
        have h := primeDirection_toRat_ne_zero r hr
        rw [primeDirection_toRat] at h
        exact Nat.one_le_iff_ne_zero.mpr (by exact_mod_cast h)
      exact Nat.one_le_iff_ne_zero.mpr (Nat.mul_ne_zero (by omega) (by omega))
    · rw [RatioOrbit.mul_toRat, primeDirection_toRat, primeDirection_toRat]
      push_cast
      ring
  have hsignCost :
      RatioOrbit.crossEq (costFromCharacter χ negativeOneRatio)
        (onRatioOrbit negativeOneRatio) :=
    RatioOrbit.crossEq_trans (RatioOrbit.crossEq_symm (hFχ negativeOneRatio))
      (signReversing_forces_signed_unit hF.native.unit_zero hF.sign_reversing)
  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)
THEOREM PRCStructuralSansAnchorUniquenessTarget_refuted · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/PRCNativeCostStructuralLedger.lean
PRCStructuralSansAnchorUniquenessTarget_refuted · IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/PRCNativeCostStructuralLedger.lean:1080 
/-- **The anchor is a genuine unit gauge.** The anchor-free structural ledger
does NOT force the canonical cost. Everything else in the ledger is structure;
the last field is a choice of unit. -/
theorem PRCStructuralSansAnchorUniquenessTarget_refuted :
    ¬ PRCStructuralSansAnchorUniquenessTarget := by
  intro huniq
  exact cubeGeneratedNativeCost_two_not_canonical
    (huniq cubeGeneratedNativeCost cubeGeneratedNativeCost_sansAnchor two)

What this page does not claim

The sign-reversing rule is the only set of hypotheses that forces the cost function J. The theorem applies to all possible cost functions without additional structural assumptions. The sign-reversing rule is a physical law rather than a definitional choice within the framework.

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/PRCNativeCostStructuralLedger.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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