Cost Gauge Orbit From Real Character Structural Sans Anchor Real Character Facto

The structural bridge

A cost function, in the Recognition Science framework, is a rule that assigns a number to each possible ratio between two quantities, measuring the price of recognizing one as the other. The framework's central question is which cost functions are forced by basic structural principles. This declaration, structural_sansAnchor_realCharacterFactorization, is a bridge: it proves that any cost function satisfying a certain set of structural hypotheses also satisfies the hypotheses of a separate, more general factorization theorem.

The structural hypotheses, called anchor-free because they do not rely on a fixed reference point, include conditions like sign reversal and monotonicity. The factorization theorem, real-character factorization, expresses any such cost as a product of a sign and a power. The declaration shows that the structural conditions are strong enough to guarantee this factorization exists. It does not, however, say which sign and power, or even that the factorization is unique.

In Recognition Science, this is a step toward classifying all possible cost functions. The framework's library, a machine-checked collection of formal theorems, uses this bridge to study specific examples. For instance, it proves that a cost function based only on the sign of the ratio satisfies the structural hypotheses, and that its factorization has exponent zero. It also proves that a family of cost functions based on signed powers, where the cost is x times the absolute value of x raised to a power, also satisfies the hypotheses.

The declaration does not claim that these are the only possible cost functions, nor does it claim that the factorization is always the signed power family. In fact, the library contains a refutation of a statement that would say every structural cost is either the sign gauge or an odd power. The bridge is a necessary condition, not a complete classification. It tells you that a real-character factorization exists, but leaves open the task of finding it for each specific cost function.

/-- The structural ledger fields imply the light hypothesis pack used by the
real-character factorization theorem. -/
theorem realCharacterFactorizationHypotheses_of_structural
    {F : RatioOrbit → RatioOrbit}
    (hF : PRCStructuralNativeCostHypothesesSansAnchor F) :
    IndisputableMonolith.Cost.RealCharacterFactorization.SansAnchorHypotheses F where
  base_sans_two :=
    { reciprocal := hF.base_sans_two.reciprocal
      normalized_invariant := hF.base_sans_two.normalized_invariant
      canonical_rcl := hF.base_sans_two.canonical_rcl
      unit_zero := hF.base_sans_two.unit_zero }
  sign_reversing := fun q r hr => hF.sign_reversing q r hr
  monotone := by
    intro a b ha hb hab
    exact hF.monotone a b ha hb hab
  zero_calibrated := hF.zero_calibrated

/-- The zero-exponent sign member satisfies every field of the anchor-free
structural ledger. -/
theorem signGaugeNativeCost_sansAnchor :
    PRCStructuralNativeCostHypothesesSansAnchor signGaugeNativeCost where
  base_sans_two := signGaugeNativeCost_base_sans_two
  sign_reversing := signGaugeNativeCost_signReversing
  monotone := signGaugeNativeCost_monotone
  zero_calibrated := signGaugeNativeCost_zero_calibrated

/-- **The corrected classification is false too.** The exponent-two sign-extended member is
neither the sign cost nor any odd power. -/
theorem GaugeOrbitIsSignOrOddPowerFamily_refuted :
    ¬ GaugeOrbitIsSignOrOddPowerFamily := by
  intro h
  rcases h (signedPowerNativeCost 1) (signedPowerNativeCost_sansAnchor 1) with
    hsign | ⟨k, hk⟩
  · exact signedPowerNativeCost_one_not_signGauge hsign
  · exact signedPowerNativeCost_one_not_oddPower k hk

What this page does not claim

The declaration does not claim to classify all possible cost functions. The declaration does not claim that the real-character factorization is unique. The declaration does not claim that every structural cost function is either the sign gauge or an odd power.

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/Cost/GaugeOrbitFromRealCharacter.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.

Derived articles

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