Foundation Phi Closure Selection Closure Poly Strict Mono

The closure polynomial

The golden ratio, usually written φ, is the positive number that satisfies r² = r + 1, approximately 1.618. It appears throughout mathematics, from the regular pentagon to the Fibonacci sequence. One way to see it is through the polynomials xⁿ − x − 1, one for each whole number n starting at 2. For n = 2, the polynomial is x² − x − 1, whose positive root is exactly φ. For n = 3, the polynomial x³ − x − 1 has a positive root too, known as the plastic constant, about 1.325. Each polynomial has exactly one root above 1, and these roots form a ladder of candidate scaling ratios.

The theorem named closurePoly_strictMono establishes a simple monotonicity fact about this family: for any n at least 2, if x is at least 1 and x is less than y, then xⁿ − x − 1 is strictly less than yⁿ − y − 1. In other words, each polynomial is strictly increasing on the interval [1, ∞). This is a purely analytic statement, proved in the machine-checked library of formal theorems. It is a supporting lemma, not a headline result, but it does real work: it guarantees that each closure polynomial has at most one root above 1, so the ladder of roots is well defined and ordered.

In Recognition Science, the framework models a discrete record of events, called the ledger, in which scales are posted and combined. A ladder closes at level k when 1 + r = rᵏ. The monotonicity theorem helps show that among such ladders, the one at level 2, with r = φ, is special: it is the only one where every posted scale is earned by composing smaller ones. This selection is structural, not a matter of minimizing cost; in fact, the framework proves that cost-minimization over closure levels selects no finite level at all, because higher levels are always cheaper. The monotonicity lemma is a small but necessary piece of that argument.

The theorem does not claim that φ is the root of every such polynomial, nor that the golden ratio is the only possible scaling ratio. It says nothing about why the ladder form 1 + r = rᵏ should hold in the first place; that remains a premise. The lemma is also not a statement about cost: it concerns the polynomials xⁿ − x − 1 alone, not the cost function J. Its role is auxiliary, a clean analytic fact that the framework's larger structural results rely on.

/-- The closure polynomial `x^n - x - 1` is strictly increasing on `[1, ∞)`
for `n ≥ 2`. -/
theorem closurePoly_strictMono {x y : ℝ} (hx : 1 ≤ x) (hxy : x < y) {n : ℕ}
    (hn : 2 ≤ n) :
    x ^ n - x - 1 < y ^ n - y - 1 := by
  have h := geom_sum_factor_bound hx hxy hn
  linarith

/-- The closure polynomial `x^n - x - 1` is strictly increasing on `[1, ∞)`
for `n ≥ 2`. -/
theorem closurePoly_strictMono {x y : ℝ} (hx : 1 ≤ x) (hxy : x < y) {n : ℕ}
    (hn : 2 ≤ n) :
    x ^ n - x - 1 < y ^ n - y - 1 := by
  have h := geom_sum_factor_bound hx hxy hn
  linarith

What this page does not claim

The theorem does not state that φ is the root of every closure polynomial. The theorem does not prove that the ladder form 1 + r = rᵏ holds in any physical or mathematical setting. The theorem makes no claim about the cost function J or about cost-minimization selecting a closure level.

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

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