Encyclopedia Cost Cost Real Character Factorization Rational Trace Pos Eq Two Of Two Eq Two

ARTICLE 3 claims 3 theorems

Cost Real Character Factorization Rational Trace Pos Eq Two Of Two Eq Two

A machine-checked theorem shows that if a cost function's trace equals two at the number two, it equals two at every positive rational number.

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The anchored trace

The rational numbers are the fractions, like 1/2 or 7/3, that sit between the integers. A trace, in this framework, is a real-valued function extracted from a cost function: it measures, in a precise sense, how the cost behaves as you scale its input. The theorem at hand, rationalTrace_pos_eq_two_of_two_eq_two, states a striking rigidity: if the trace of a cost function equals 2 at the input 2, then the trace equals 2 at every positive rational number.

The proof does not require the full set of axioms that define a cost function. It works under a weaker hypothesis called SansAnchorHypotheses, which omits the anchor condition that pins the cost at the number 2. The key ingredient is a doubled-trace form of the composition law, which the library shows holds from the reciprocal-cost law alone. This means the conclusion does not depend on the anchor; it follows from the structural constraints on how costs combine.

In Recognition Science, this theorem is part of a larger factorization program. The library proves a d'Alembert-style equation: the trace of a product plus the trace of a quotient equals the product of the traces. From that equation, together with the value at 2, the theorem forces the trace to be constantly 2 on all positive rationals. This is a stepping stone toward classifying which cost functions are possible, showing that a specific value at one point can determine behavior across an entire dense set.

The theorem does not claim that the trace equals 2 at negative rationals or at zero; those cases are not covered. It also does not establish that any particular cost function exists with this property, only that if one satisfies the hypotheses and has trace 2 at 2, then it has trace 2 everywhere positive. The result is a conditional statement, not an existence proof.

THEOREM rationalTrace_pos_eq_two_of_two_eq_two · IndisputableMonolith/Cost/RealCharacterFactorization.lean
rationalTrace_pos_eq_two_of_two_eq_two · IndisputableMonolith/Cost/RealCharacterFactorization.lean:593 
theorem rationalTrace_pos_eq_two_of_two_eq_two
    {F : RatioOrbit → RatioOrbit} (hF : SansAnchorHypotheses F)
    (htwo : rationalTrace F 2 = 2)
    {x : ℚ} (hx : 0 < x) :
    rationalTrace F x = 2 := by
  let a := x.num.toNat
  let b := x.den
  have hnumpos : 0 < x.num := Rat.num_pos.mpr hx
  have hapos : 0 < a := by
    change 0 < x.num.toNat
    omega
  have hbpos : 0 < b := by
    change 0 < x.den
    exact x.pos
  have ha1 : 1 ≤ a := hapos
  have hb1 : 1 ≤ b := hbpos
  have haa : (a : ℚ) ≠ 0 := by exact_mod_cast hapos.ne'
  have hbb : (b : ℚ) ≠ 0 := by exact_mod_cast hbpos.ne'
  have hxrep : (a : ℚ) / (b : ℚ) = x := by
    change ((x.num.toNat : ℕ) : ℚ) / (x.den : ℚ) = x
    have hnum :
        ((x.num.toNat : ℕ) : ℚ) = ((x.num : ℤ) : ℚ) := by
      exact_mod_cast Int.toNat_of_nonneg (le_of_lt hnumpos)
    rw [hnum]
    exact Rat.num_div_den x
  have hd := rationalTrace_dAlembert hF (x := (a : ℚ)) (y := (b : ℚ))
    haa hbb
  have hmulNat : (a : ℚ) * (b : ℚ) = ((a * b : ℕ) : ℚ) := by norm_num
  rw [hmulNat, hxrep, rationalTrace_nat_eq_two_of_two_eq_two hF htwo a ha1,
    rationalTrace_nat_eq_two_of_two_eq_two hF htwo b hb1,
    rationalTrace_nat_eq_two_of_two_eq_two hF htwo (a * b)
      (by exact Nat.mul_pos hapos hbpos)] at hd
  linarith
THEOREM rationalTrace_pos_eq_two_of_two_eq_two · IndisputableMonolith/Cost/RealCharacterFactorization.lean
rationalTrace_pos_eq_two_of_two_eq_two · IndisputableMonolith/Cost/RealCharacterFactorization.lean:593 
theorem rationalTrace_pos_eq_two_of_two_eq_two
    {F : RatioOrbit → RatioOrbit} (hF : SansAnchorHypotheses F)
    (htwo : rationalTrace F 2 = 2)
    {x : ℚ} (hx : 0 < x) :
    rationalTrace F x = 2 := by
  let a := x.num.toNat
  let b := x.den
  have hnumpos : 0 < x.num := Rat.num_pos.mpr hx
  have hapos : 0 < a := by
    change 0 < x.num.toNat
    omega
  have hbpos : 0 < b := by
    change 0 < x.den
    exact x.pos
  have ha1 : 1 ≤ a := hapos
  have hb1 : 1 ≤ b := hbpos
  have haa : (a : ℚ) ≠ 0 := by exact_mod_cast hapos.ne'
  have hbb : (b : ℚ) ≠ 0 := by exact_mod_cast hbpos.ne'
  have hxrep : (a : ℚ) / (b : ℚ) = x := by
    change ((x.num.toNat : ℕ) : ℚ) / (x.den : ℚ) = x
    have hnum :
        ((x.num.toNat : ℕ) : ℚ) = ((x.num : ℤ) : ℚ) := by
      exact_mod_cast Int.toNat_of_nonneg (le_of_lt hnumpos)
    rw [hnum]
    exact Rat.num_div_den x
  have hd := rationalTrace_dAlembert hF (x := (a : ℚ)) (y := (b : ℚ))
    haa hbb
  have hmulNat : (a : ℚ) * (b : ℚ) = ((a * b : ℕ) : ℚ) := by norm_num
  rw [hmulNat, hxrep, rationalTrace_nat_eq_two_of_two_eq_two hF htwo a ha1,
    rationalTrace_nat_eq_two_of_two_eq_two hF htwo b hb1,
    rationalTrace_nat_eq_two_of_two_eq_two hF htwo (a * b)
      (by exact Nat.mul_pos hapos hbpos)] at hd
  linarith
THEOREM doubledTrace_dAlembert_of_sansAnchor · IndisputableMonolith/Cost/RealCharacterFactorization.lean
doubledTrace_dAlembert_of_sansAnchor · IndisputableMonolith/Cost/RealCharacterFactorization.lean:110 
theorem doubledTrace_dAlembert_of_sansAnchor
    {F : RatioOrbit → RatioOrbit}
    (hF : SansAnchorHypotheses F)
    {x y : RatioOrbit} (hx : x.toRat ≠ 0) (hy : y.toRat ≠ 0) :
    RatioOrbit.crossEq
      (RatioOrbit.add (nativeCostDoubledTrace F (RatioOrbit.mul x y))
        (nativeCostDoubledTrace F (div x y)))
      (RatioOrbit.mul (nativeCostDoubledTrace F x) (nativeCostDoubledTrace F y)) :=
  doubledTrace_dAlembert_of_rcl hF.base_sans_two.canonical_rcl hx hy

What this page does not claim

The theorem does not assert that any cost function with trace 2 at 2 actually exists. The theorem does not cover the trace at zero or at negative rational numbers. The theorem does not establish uniqueness of the cost function, only a property of its trace.

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/RealCharacterFactorization.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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