Cost Unit From Minimality Exponent Zero Undercuts Everything

The zero exponent

The Recognition Science framework models recognition as a discrete record of events, and assigns each event a cost, a nonnegative number measuring how much the event stands out from a baseline. The framework's central cost function, written J(x), is proved to be J(x) = (x + 1/x)/2 - 1. A natural question is whether this cost can be generated by a power of the base, say x^n, for some integer n. The declaration exponent_zero_undercuts_everything answers that question with a theorem: for any positive base x and any integer exponent n, the cost of x^0 is less than or equal to the cost of x^n.

The proof is immediate from two facts. First, x^0 equals 1 for any x, and the cost of 1 is zero by the framework's normalization condition. Second, the cost function is always nonnegative. Therefore the cost of x^0, being zero, is always the smallest possible cost. This is why the theorem is named "undercuts everything": the zero exponent would trivially minimize cost for every base, making it useless as a way to distinguish any base from any other. The framework's definition of a meaningful exponent therefore requires the exponent to be at least one, a restriction stated in the companion theorem isLeastPower_iff_canonical.

What the theorem does not claim is just as important. It does not say that the zero exponent is a valid choice in the framework's definition of cost. The framework's definition of a meaningful power starts at exponent one, so the zero exponent is excluded from consideration. The theorem is a boundary observation: it shows why that exclusion is necessary, not that the zero exponent is a competing candidate. It also does not say that the continuous analog, where the exponent is a real number, has a minimum; in fact, the framework proves the opposite, that the continuous gauge has no least member, with costs approaching zero as the exponent approaches zero.

The practical consequence is that the framework's unit of recognition is not a power of the base, but the base itself. The theorem unit_is_selected_by_minimality_over_powers proves that for any base x not equal to 1, the cost of x is strictly less than the cost of x^n for any n at least 2. This is the minimality property that selects the base as the unit, and the zero-exponent theorem is the reason the minimality property must be stated with the exponent at least one. The framework's unit is therefore not chosen by a power, but by the base itself, and the zero exponent is the boundary case that makes this choice meaningful.

theorem exponent_zero_undercuts_everything (n : ℕ) (x : ℝ) (hx : 0 < x) :
    Jcost (x ^ (0 : ℕ)) ≤ Jcost (x ^ n) := by
  rw [exponent_zero_charges_nothing]
  exact Jcost_nonneg (pow_pos hx n)

/-- Exponent zero is the degenerate member: it charges nothing at every ratio. It is least in
the enlarged family, and for a reason that has nothing to do with cost, which is exactly why
selection needs nondegeneracy rather than leastness alone. -/
theorem exponent_zero_charges_nothing (x : ℝ) : Jcost (x ^ (0 : ℕ)) = 0 := by
  rw [pow_zero]
  exact Jcost_unit0

/-- **The canonical cost is the unique least nondegenerate member, at every exponent and not
merely the odd ones.** As before, the condition never names `J`. -/
theorem isLeastPower_iff_canonical (n : ℕ) (hn : 1 ≤ n) :
    IsLeastPowerCost n ↔ n = 1 := by
  constructor
  · intro h
    by_contra hne
    have hn2 : 2 ≤ n := by omega
    have hle := h 1 (le_refl 1) 2 (by norm_num) (by norm_num)
    have hlt := jcost_lt_pow (2 : ℝ) (by norm_num) (by norm_num) n hn2
    rw [pow_one] at hle
    exact absurd hle (not_le.mpr hlt)
  · rintro rfl
    intro m hm x hx hx1
    rw [pow_one]
    rcases Nat.lt_or_ge m 2 with hm2 | hm2
    · have hm1 : m = 1 := by omega
      subst hm1
      simp
    · exact le_of_lt (jcost_lt_pow x hx hx1 m hm2)

/-- And the descent runs all the way to nothing: the gauge family's pointwise limit as
the scale vanishes is the zero cost. So the infimum is not merely unattained, it is the
degenerate cost that charges nothing for anything. -/
theorem gauge_tendsto_zero (x : ℝ) (hx : 0 < x) :
    Filter.Tendsto (fun l : ℝ => Jcost (x ^ l)) (nhds 0) (nhds 0) := by
  have hrw : (fun l : ℝ => Jcost (x ^ l))
      = fun l : ℝ => Real.cosh (Real.log x * l) - 1 := by
    funext l
    rw [Real.rpow_def_of_pos hx, Jcost_exp_cosh]
  rw [hrw]
  have hcont : Continuous (fun l : ℝ => Real.cosh (Real.log x * l) - 1) := by
    fun_prop
  have h0 : Real.cosh (Real.log x * (0 : ℝ)) - 1 = 0 := by simp
  simpa [h0] using hcont.tendsto (0 : ℝ)

theorem unit_is_selected_by_minimality_over_powers :
    ∀ x : ℝ, 0 < x → x ≠ 1 → ∀ n : ℕ, 2 ≤ n → Jcost x < Jcost (x ^ n) :=
  fun x hx hx1 n hn => jcost_lt_pow x hx hx1 n hn

What this page does not claim

The zero exponent is a valid choice in the framework's definition of cost. The continuous gauge has a minimum cost member. The theorem proves that the zero exponent is the only way to achieve zero 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/UnitFromMinimality.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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