Cost Gauge Orbit From Real Character Sign Gauge Native Cost Real Character Candi

The sign gauge cost

In the Recognition Science framework, a ledger is a discrete record of recognition events, and a cost is a function that assigns a number to each ratio in that record. The declaration signGaugeNativeCost_realCharacterCandidate concerns one particular cost function, called the sign gauge cost. This function is defined by a plain rule: if the input ratio is positive, the cost is 0; if the input is zero, the cost is -1; if the input is negative, the cost is -2. It is a simple step function that only looks at the sign of the ratio, not its magnitude.

The declaration establishes a precise structural fact. For every ratio, the real character candidate of the sign gauge cost equals the rational sign character of that ratio. In plain language, the character, which is a derived quantity that summarizes how the cost behaves under the framework's operations, is exactly the sign function itself. This is a theorem in the framework's machine-checked library of formal theorems, meaning it has been verified by a computer proof checker. The same library also proves that the sign gauge cost satisfies all the structural hypotheses required of a native cost, including sign reversal, monotonicity, and zero calibration.

What this declaration does not claim is broader. It does not claim that the sign gauge cost is the only cost satisfying those structural hypotheses. In fact, the library contains a separate theorem showing that the sign gauge cost is not a member of the odd power family, and another showing that the family of structural costs is not exhausted by the sign gauge cost together with the odd power family. The signed power cost with exponent 1 provides a concrete counterexample: it also satisfies the structural hypotheses, but it is neither the sign gauge cost nor an odd power cost. So the declaration pins down one specific cost's character, not a classification of all possible costs.

The significance of this result is that it gives a clean example of how the framework's character machinery works on a simple, discontinuous cost. It shows that the character of a cost can be computed and identified explicitly, even when the cost itself is not a smooth power function. This is a useful calibration point for the framework's theory of real character factorization, and it demonstrates that the structural hypotheses do not force every cost to be a power function.

Within the framework, this result is a building block. It is not a physical prediction and it does not involve any empirical constants. It is a purely mathematical statement about the structure of cost functions. The framework's larger claims, such as the derivation of the golden ratio or the number of spatial dimensions, rest on other theorems in the library, not on this particular declaration.

/-- The factorization theorem extracts the sign character on this member. -/
theorem signGaugeNativeCost_realCharacterCandidate (q : RatioOrbit) :
    IndisputableMonolith.Cost.RealCharacterFactorization.realCharacterCandidate
        signGaugeNativeCost q =
      IndisputableMonolith.Cost.RealCharacterFactorization.rationalSignCharacter
        q.toRat := by
  rw [IndisputableMonolith.Cost.RealCharacterFactorization.realCharacterCandidate,
    if_pos signGaugeNativeCost_rationalTrace_two]

/-- 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 sign member cannot equal any positive odd-power cost. -/
theorem signGaugeNativeCost_not_oddPowerGeneratedNativeCost (k : ℕ) :
    ¬ ∀ q : RatioOrbit,
      RatioOrbit.crossEq (signGaugeNativeCost q)
        (oddPowerGeneratedNativeCost k q) := by
  intro h
  have htwo := crossDisp (h two)
  rw [signGaugeNativeCost_toRat, oddPowerGeneratedNativeCost_toRat,
    two_toRat] at htwo
  have hexp : 2 * k + 1 ≠ 0 := by omega
  have hp : (1 : ℚ) < 2 ^ (2 * k + 1) :=
    one_lt_pow₀ (by norm_num) hexp
  have hzero : jq ((2 : ℚ) ^ (2 * k + 1)) = 0 := by
    norm_num [signGaugeCostDisplay] at htwo
    exact htwo.symm
  have hone := jq_eq_zero (ne_of_gt (lt_trans zero_lt_one hp)) hzero
  exact (ne_of_gt hp) hone

/-- **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 sign gauge cost is not the only cost satisfying the structural hypotheses. The declaration does not classify all possible costs or prove a uniqueness theorem. The result does not involve any physical prediction or empirical constants.

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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