Cost Unit From Minimality Unit Is Selected By Minimality Over Powers

Why the unit is the cheapest power

The Recognition Science framework assigns a real number, called cost, to any positive base x. The cost measures how expensive it is to recognize that base as a distinct event in a discrete ledger, a record that keeps track of distinctions. The framework's central cost function is J(x) = (x + 1/x)/2 - 1, which is forced by five plain axioms. For any base x that is positive and not equal to 1, and for any integer n at least 2, the theorem unit_is_selected_by_minimality_over_powers states: J(x) < J(x^n). In plain words, raising the base to the second or higher power always increases the cost. The first power, x^1 = x, is cheaper than every higher power.

This is a strict inequality, not just a non-strict one. The proof splits into two cases. If x is greater than 1, then x^n is larger than x, and the cost function is strictly increasing on bases above 1, so the cost of x^n exceeds the cost of x. If x is between 0 and 1, the framework uses a symmetry: the cost of x equals the cost of its reciprocal 1/x, and the same argument applies to the reciprocal, which is above 1. The theorem therefore covers all positive bases except the neutral base 1, where the cost is zero and the comparison is trivial. The statement is machine-checked in the framework's library of formal theorems, with no unproved assumptions beyond the standard logical axioms.

The theorem has a direct consequence: the exponent 1 is the unique least-cost power. For any positive base not equal to 1, the cost of x^1 is strictly less than the cost of x^n for every n ≥ 2. The framework also proves a stronger statement for odd powers: for any k ≥ 1, J(x) < J(x^(2k+1)), so the first odd power is cheaper than all higher odd powers. These results do not say that the cost function is globally minimized at x = 1; in fact, J(1) = 0, but the theorem concerns powers of a fixed base, not the base itself. The theorem also does not claim that the cost function is bounded below by J(x) for all x; that would be false, since J(x) can be arbitrarily close to 0 as x approaches 1.

In the framework, this minimality result is what selects the unit as the canonical base for measuring recognition. The first power is the natural reference point because it is the cheapest way to represent a base as a power of itself. This is not a claim about physical units in the conventional sense; it is a structural fact about the cost function. The theorem does not say that the unit is physically preferred or that the framework derives the value of any physical constant. It only establishes an ordering among powers of a given base.

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

/-- Packaging: on the discrete gauge, least cost is a selection principle. Every gauge
member other than `J` is strictly more expensive at every base that is not the unit. -/
theorem unit_is_selected_by_minimality :
    ∀ x : ℝ, 0 < x → x ≠ 1 → ∀ k : ℕ, 1 ≤ k → Jcost x < Jcost (x ^ (2 * k + 1)) :=
  fun x hx hx1 k hk => jcost_lt_odd_power x hx hx1 k hk

/-- The same, for the natural-number powers the discrete gauge uses. -/
lemma jcost_pow_inv (x : ℝ) (hx : 0 < x) (n : ℕ) :
    Jcost (x ^ n) = Jcost ((x⁻¹) ^ n) := by
  have hpos : 0 < x ^ n := pow_pos hx n
  rw [inv_pow]
  exact Jcost_symm hpos

What this page does not claim

The theorem does not claim that the cost function is globally minimized at any particular base. It does not claim that the unit base is physically preferred or that any physical constant is derived from this minimality. It does not claim that the cost function is bounded below by J(x) for all x.

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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