Cost Monotone Multiplicative Power Eq One Of Two Eq One

The degenerate branch

In mathematics, a function on the positive integers is called completely multiplicative if the value at a product equals the product of the values, and nondecreasing if larger inputs never give smaller outputs. A simple example is the identity function, which sends each number to itself. The theorem eq_one_of_two_eq_one concerns such functions that also send 1 to 1. It states: if such a function sends 2 to 1, then it sends every positive integer to 1.

The proof is a squeeze. For any positive integer n, the number n is strictly less than 2 raised to the power n. Since the function is nondecreasing, the value at n is at most the value at 2^n. But complete multiplicativity forces the value at 2^n to be the nth power of the value at 2, which is 1. So the value at n is at most 1. Since the function is nondecreasing and sends 1 to 1, the value at n is also at least 1. The only number both at most and at least 1 is 1 itself.

This is a degenerate branch of a larger classification. The companion result, exists_exponent, shows that any such function is a power: there is some nonnegative real exponent c such that the value at n is n raised to the power c. The theorem here covers the case where that exponent is 0. The constant function that sends every positive integer to 1 is a real example, so the hypothesis class is not empty.

In the Recognition Science framework, this classification describes the possible cost functions that a ledger, a discrete record of recognition events, can assign. The theorem establishes that a cost function which values the number 2 at 1 must value everything at 1, meaning no distinction is made between any two positive integers. This is a boundary case within the framework's study of cost functions, not a central result about the golden ratio or the forcing chain.

The theorem does not claim that such a function exists, only that if it does, it is constant. It also does not claim anything about the value at 0, which is left unconstrained. And it does not claim that the constant function is the only possibility when the value at 2 is greater than 1; that is the subject of the companion exponent theorem.

/-- The degenerate branch. If the value at two is one then every value is one, because
every integer is below a power of two and the values in between are squeezed. -/
theorem eq_one_of_two_eq_one (hf : MonotoneMultiplicative f) (h2 : f 2 = 1)
    {n : ℕ} (hn : 1 ≤ n) : f n = 1 := by
  have hlt : n < 2 ^ n := Nat.lt_two_pow_self
  have hle := hf.mono n (2 ^ n) hn hlt.le
  rw [pow_eq hf (by norm_num) n, h2, one_pow] at hle
  exact le_antisymm hle (one_le hf hn)

theorem monotoneMultiplicative_const_one : MonotoneMultiplicative (fun _ : ℕ => (1 : ℝ)) where
  unit := rfl
  mul := by intro m n _ _; norm_num
  mono := by intro m n _ _; exact le_rfl

What this page does not claim

The theorem does not claim that such a function exists, only that if it does, it is constant. The theorem does not claim anything about the value at 0, which is left unconstrained. The theorem does not claim that the constant function is the only possibility when the value at 2 is greater than 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/MonotoneMultiplicativePower.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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