The new bottom entry is the ordered boundary fold

Proved

Gilbreath.extension_bottom_identity

by EvanLLL · Sep 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

combinatoricsnumber-theory

Let $a:\mathbb N\to\mathbb N$ and let $E_a(n)$ be the ordered right boundary of the triangle on $a_0,\ldots,a_{n-1}$ , read from the top row downwards; $E_a(0)$ is empty. For a tuple $E=(e_0,\ldots,e_{n-1})$ , let $F_E(x)$ successively replace $x$ by $|x-e_i|$ in this order. Then

F_{E_a(n)}(a_n)=(\Delta^n a)(0).

This identifies the new bottom entry created by appending the next input. It is valid for arbitrary natural-valued inputs, including $n=0$ , without a primality or previous-leading-entry assumption.

Preamble

import Definitions.Def_gilbreath_finite_extension

Formal statement

namespace Gilbreath
theorem extension_bottom_identity (a : ℕ → ℕ) (n : ℕ) :
    extensionFold (extensionBoundary a n) (a n) = iterAbsDiff a n 0 := by sorry
end Gilbreath

Source

L. Muney, Holes in Valid-Extension Sets of Finite Gilbreath Sequences, arXiv:2606.23721v1, https://arxiv.org/html/2606.23721v1, Section 2, Proposition 2 and its right-edge recurrence. This is the underlying bottom-entry identity, stated for zero-based natural-valued sequences and including the top-row boundary entry.

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