Cost Gauge Orbit From Real Character Sign Gauge Native Cost Character Exponent Z

The sign gauge cost

In the Recognition Science framework, a recognition cost is a rule that assigns a price to each possible ratio between two quantities, with the price forced by structural conditions rather than chosen freely. The sign gauge cost is the simplest possible such rule: it looks only at the sign of the ratio and charges 0 for a positive ratio, -1 for a ratio of exactly zero, and -2 for a negative ratio. This is a definitional choice, a model of what a minimal cost could look like.

The machine-checked library of formal theorems proves that this sign gauge cost satisfies all the structural hypotheses that a recognition cost must satisfy, except for one calibration condition that is deliberately omitted. It is sign-reversing, meaning that flipping the sign of the ratio flips the sign of the cost; it is monotone, so that a larger ratio never costs more than a smaller one; and it is zero-calibrated in a doubled-trace sense. The library also proves that the sign gauge cost is not equal to any member of the odd-power family of costs, and it is not equal to the signed-power family either. These are theorems, not conjectures.

The declaration signGaugeNativeCost_characterExponent_zero establishes a specific fact about the sign gauge cost: its real character exponent is zero. In plain language, the real character factorization theorem associates to each structural cost a real-valued character, and the exponent of that character for the sign gauge cost is 0. This means that, in the factorization, the sign gauge cost behaves like a character with exponent zero, which is the simplest possible case. The theorem is proved in the machine-checked library, so it is a theorem, not a hypothesis.

What this declaration does not claim is equally important. It does not claim that the sign gauge cost is the unique cost with exponent zero, nor does it claim that the sign gauge cost is a member of the odd-power or signed-power families. The library explicitly refutes both of those claims. It also does not claim that the sign gauge cost is the cost function J(x) = (x + 1/x)/2 - 1 that the framework's central theorem forces; the sign gauge cost is a different, simpler object that satisfies only the structural hypotheses, not the full set that forces J. The declaration is a narrow, precise result about one cost function, not a sweeping statement about all costs.

The significance of this result is that it shows the structural hypotheses alone do not pin down a unique cost. The sign gauge cost is a genuine structural cost, but it is not the forced cost J. This means the full forcing chain, which derives J and then the golden ratio and three dimensions, requires the additional calibration condition that the sign gauge cost omits. The sign gauge cost is a counterexample that sharpens the boundary of what the structural hypotheses can and cannot force.

/-- 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

theorem signGaugeNativeCost_signReversing :
    PRCNativeCostSignReversing signGaugeNativeCost := by
  intro q r hr
  rw [signGaugeNativeCost_toRat, signGaugeNativeCost_toRat, hr]
  rcases lt_trichotomy q.toRat 0 with hn | hz | hp
  · have hnegpos : 0 < -q.toRat := neg_pos.mpr hn
    norm_num [signGaugeCostDisplay, hn.not_gt, ne_of_lt hn, hnegpos]
  · rw [hz]
    norm_num [signGaugeCostDisplay]
  · have hnegneg : -q.toRat < 0 := neg_neg_of_pos hp
    norm_num [signGaugeCostDisplay, hp, hnegneg.not_gt, ne_of_lt hnegneg]

theorem signGaugeNativeCost_monotone :
    PRCNativeCostMonotone signGaugeNativeCost := by
  rintro a b ⟨m, hm, ha⟩ ⟨n, hn, hb⟩ _
  rw [signGaugeNativeCost_toRat, signGaugeNativeCost_toRat]
  have hapos : 0 < a.toRat := by
    rw [ha]
    exact_mod_cast (show 0 < m by omega)
  have hbpos : 0 < b.toRat := by
    rw [hb]
    exact_mod_cast (show 0 < n by omega)
  simp [signGaugeCostDisplay, hapos, hbpos]

theorem signGaugeNativeCost_zero_calibrated :
    PRCDoubledTraceZeroCalibrated
      (nativeCostDoubledTrace signGaugeNativeCost) := by
  rw [PRCDoubledTraceZeroCalibrated, RatioOrbit.crossEq_iff_toRat_eq]
  simp only [nativeCostDoubledTrace, doubledTraceValue, RatioOrbit.mul_toRat,
    RatioOrbit.add_toRat, two_toRat, RatioOrbit.one_toRat,
    signGaugeNativeCost_toRat, RatioOrbit.zero_toRat]
  norm_num [signGaugeCostDisplay]

/-- 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 extracted character has exponent zero. -/
theorem signGaugeNativeCost_characterExponent_zero :
    IndisputableMonolith.Cost.RealCharacterFactorization.exponentOfCharacter
      (IndisputableMonolith.Cost.RealCharacterFactorization.realCharacterCandidate
        signGaugeNativeCost) = 0 := by
  rw [IndisputableMonolith.Cost.RealCharacterFactorization.exponentOfCharacter,
    signGaugeNativeCost_realCharacterCandidate,
    IndisputableMonolith.Cost.RealCharacterFactorization.natOrbit_toRat]
  have hsign :
      IndisputableMonolith.Cost.RealCharacterFactorization.rationalSignCharacter
        (((2 : ℕ) : ℚ)) = 1 := by
    norm_num
      [IndisputableMonolith.Cost.RealCharacterFactorization.rationalSignCharacter]
  rw [hsign]
  norm_num

What this page does not claim

The sign gauge cost is not the unique cost with character exponent zero. The sign gauge cost is not a member of the odd-power or signed-power families. The sign gauge cost is not the forced cost J(x) = (x + 1/x)/2 - 1.

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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