Cost Gauge Orbit From Real Character Real Character Factorization Hypotheses Of

The structural bridge

A ledger, a discrete record of recognition events, can assign a cost to every ratio of two quantities. The Recognition Science library studies which cost functions are possible. One family of costs, the structural costs, obeys a set of plain conditions: the cost of a ratio and its reciprocal are related in a fixed way, the cost of one is zero, the cost grows as the ratio moves away from one, and a doubled-trace calibration holds. The theorem realCharacterFactorizationHypotheses_of_structural proves that any cost function meeting these structural conditions also satisfies the hypotheses of the real-character factorization theorem. That is the bridge: the structural conditions are enough to enter the factorization framework.

The real-character factorization theorem, in turn, provides a way to write a cost function as a product of a real character and a power. The declaration does not say which specific cost function emerges, only that the factorization applies. The library demonstrates this with two example families. The sign-gauge cost sends positive ratios to zero, zero to minus one, and negative ratios to minus two. The signed-power costs raise the ratio to a power while preserving its sign. Both satisfy the structural conditions, and the theorem guarantees both fit the factorization mold. The library also proves that the gauge orbit, the set of costs reachable from the structural conditions, is not exhausted by either the odd-power family or the sign-or-odd-power family. A third family, the signed-power costs, exists outside those two.

What the declaration does not claim is just as important. It does not identify the unique cost function. It does not say that the structural conditions force a particular member of the factorization family. It does not claim that the gauge orbit is a simple union of the sign-gauge cost and the odd-power costs; the refutation of that hypothesis shows the orbit is larger. The theorem is a bridge, not a destination. It opens the door to the factorization framework and leaves the exploration of the gauge orbit to further theorems.

/-- 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 proposed classification is false: it omits the zero-exponent sign
member already allowed by the structural ledger. -/
theorem GaugeOrbitIsOddPowerFamily_refuted :
    ¬ GaugeOrbitIsOddPowerFamily := by
  intro h
  obtain ⟨k, hk⟩ := h signGaugeNativeCost signGaugeNativeCost_sansAnchor
  exact signGaugeNativeCost_not_oddPowerGeneratedNativeCost k hk

/-- **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

/-- 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

/-- **Every nonnegative integer exponent inhabits the anchor-free ledger.** The
sign-extended power character satisfies every structural field, including orientation
reversal, at every index and so at both parities of exponent. -/
theorem signedPowerNativeCost_sansAnchor (m : ℕ) :
    PRCStructuralNativeCostHypothesesSansAnchor (signedPowerNativeCost m) where
  base_sans_two := signedPowerNativeCost_base m
  sign_reversing := signedPowerNativeCost_signReversing m
  monotone := signedPowerNativeCost_monotone m
  zero_calibrated := signedPowerNativeCost_zero_calibrated m

What this page does not claim

The structural conditions do not identify a unique cost function. The gauge orbit is not a simple union of the sign-gauge cost and the odd-power costs. The theorem does not claim that the real-character factorization is unique for each cost.

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.

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