Cost Gauge Orbit From Real Character Gauge Orbit Is Sign Or Odd Power Family Ref

What the refutation means

The declaration GaugeOrbitIsSignOrOddPowerFamily_refuted is a machine-checked proof that a proposed classification of cost functions is false. In plain language, the claim it refutes said: every cost function satisfying the framework's structural axioms must be one of two simple types, either a sign gauge (a step function that returns 0, -1, or -2 depending on the sign of the input) or an odd power family (a function of the form x times an even power of the absolute value, such as x, x^3, or x^5). The refutation shows this is not true.

The proof works by exhibiting a concrete counterexample: the function signedPowerNativeCost 1, which maps x to x times the absolute value of x, that is x|x|. This function satisfies all the structural axioms, yet it is neither the sign gauge nor any odd power. The machine-checked proof verifies this in two steps. First, it shows the function is not the sign gauge by checking its value at the input 2: the sign gauge returns 0 there, while x|x| returns 4. Second, it shows the function is not any odd power by a separate argument, signedPowerNativeCost_one_not_oddPower, which rules out equality with every odd power family member.

This refutation is a theorem in the framework's machine-checked library of formal theorems. It is not a conjecture or a heuristic; it is a proved result with no unverified assumptions. The theorem's statement is simply the negation of the proposed classification: GaugeOrbitIsSignOrOddPowerFamily_refuted : ¬ GaugeOrbitIsSignOrOddPowerFamily. The proof uses the counterexample function and its verified properties, so the conclusion is airtight within the framework's logic.

What the declaration does not claim is just as important as what it proves. It does not claim that the framework's axioms are inconsistent or that the framework fails. The structural axioms themselves remain intact; only the proposed classification of their consequences is false. It also does not claim that no classification of cost functions exists. The refutation only rules out this particular two-family classification. A different, broader classification might still hold. Finally, it does not claim anything about the physical interpretation of the counterexample function. The function x|x| is a mathematical object that satisfies the axioms; the theorem says nothing about whether it represents a physically realizable cost.

/-- **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 cost generated by the sign-extended power character `χ(x) = sgn(x)·|x|^(m+1)`. -/
def signedPowerNativeCost (m : ℕ) (q : RatioOrbit) : RatioOrbit :=
  if q.toRat = 1 then RatioOrbit.zero
  else onRatioOrbit (ratioOrbitOfRat (signedPow m q.toRat))

/-- `signedPow m x = sgn(x)·|x|^(m+1)`, written as `x·|x|^m` so that multiplicativity is one
`abs_mul` away and needs no sign case analysis. -/
def signedPow (m : ℕ) (x : ℚ) : ℚ := x * |x| ^ m

/-- **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

theorem signedPowerNativeCost_one_not_signGauge :
    ¬ ∀ q : RatioOrbit,
      RatioOrbit.crossEq (signedPowerNativeCost 1 q) (signGaugeNativeCost q) := by
  intro h
  have htwo := crossDisp (h two)
  rw [signedPowerNativeCost_one_two_toRat, signGaugeNativeCost_toRat,
    two_toRat] at htwo
  norm_num [signGaugeCostDisplay, jq] at htwo

theorem signedPowerNativeCost_one_not_oddPower (k : ℕ) :
    ¬ ∀ q : RatioOrbit,
      RatioOrbit.crossEq (signedPowerNativeCost 1 q)
        (oddPowerGeneratedNativeCost k q) := by
  intro h
  have htwo := crossDisp (h two)
  rw [signedPowerNativeCost_one_two_toRat, oddPowerGeneratedNativeCost_toRat,
    two_toRat] at htwo
  have hone : (1 : ℚ) ≤ (2 : ℚ) ^ (2 * k + 1) := one_le_pow₀ (by norm_num)
  have heq : (4 : ℚ) = 2 ^ (2 * k + 1) :=
    jq_inj_ge_one (by norm_num) hone htwo
  rcases Nat.eq_zero_or_pos k with hk | hk
  · subst hk
    norm_num at heq
  · have h8 : (8 : ℚ) ≤ 2 ^ (2 * k + 1) := by
      calc (8 : ℚ) = 2 ^ 3 := by norm_num
        _ ≤ 2 ^ (2 * k + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
    rw [← heq] at h8
    norm_num at h8

/-- **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 framework's axioms are inconsistent or false. No classification of cost functions is possible at all. The counterexample function represents a physically realizable 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.

Derived articles

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