bibkey: narayan2026deltaset
authors: Ashwin Narayan
year: 2026
title: The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence
doi: 10.48550/arXiv.2610.02495
url: https://arxiv.org/abs/2610.02495v1
claim: "Conjecture 18.3 asserts that the expanded generating tuple has delta set E(h−1,d)."
strata_touched:
  - D5/S3/Arith/Additive/NarayanExpandedGeneratorDeltaRefutation
license: citation-only
triage: anchor

Expanded generating tuples and delta sets

Verified locator

DOI: 10.48550/arXiv.2610.02495

Source: https://arxiv.org/abs/2610.02495v1

Source statements

Conjecture 18.3, p. 22, states: “For Sₕ =⟨a,ah,ah+d,ah+2d,…ah+kd ⟩, Δ(Sₕ) = E(e,d)”. Its source TeX is:

For \[S_h =\langle a,ah,ah+d,ah+2d,...ah+kd \rangle,\]
\[\Delta(S_h) = E(e,d) \]

Section 3, p. 2, assumes “gcd(a,d) =1, 1<k<a, h > 1” and sets “e =h−1.” It states: “Suppose the set of factorizations of n ∈ S is given by Z(n) = { (z₁, z₂, …, zₚ) ∈ ℕᵖ ∣ z₁ n₁ + z₂ n₂ + … + zₚ nₚ = n }”. On p. 3: “If z = (z₁, z₂, z₃, …, zₚ) is a factorization of n, then the length of z is given by |z| = z₁ + z₂ + z₃ + … + zₚ”. It continues: “The set of lengths of n is defined by L(n) = {|z| | z ∈ Z(n)} ⊂ ℕ”.

Definition 3.1, p. 3, states: “We define the delta set as Δ(Sₕ) = { l₂ − l₁ : l₂,l₁ ∈ L(s), L(s) ∩ (l₁,l₂) = ∅ }”. Definition 3.2, p. 3, states: “We define the subtractive Euclidean Algorithm in this way: Let x₀ and y₀ = (e,d)” and “Recursively define” the next pair as (xᵢ−yᵢ,yᵢ) if xᵢ>yᵢ, (xᵢ,yᵢ−xᵢ) if yᵢ>xᵢ, and “terminate” if xᵢ=yᵢ. The abstract, p. 1, states: “Writing e=h−1 and letting E(e,d) denote the set of values arising from the subtractive Euclidean algorithm applied to e and d”.

Mathematical scope

Factorizations retain all coordinates of the displayed tuple, including the redundant ah. Coordinates are nonnegative integers as in the source’s subsequent factorization inequalities. The delta set is the union, over represented natural numbers, of positive gaps between distinct successive lengths. This supplies the unbound s and the omitted endpoint ordering in Definition 3.1. The initialization is (x₀,y₀)=(e,d). Keeping the terminal diagonal pair stationary adds no new visited positive coordinate. These readings are fixed in issue #23185.

For (a,h,d,k)=(3,2,2,2), the expanded tuple is (3,6,8,10). The repository’s argument proves that for every positive a coprime to 2, all represented length sets for (a,2a,2a+2,…,4a−2) are integer intervals and its delta set is {1}. The relation 2a=a+a and the pair-replacement rules yield complete paths with unit length increments. Residues modulo a determine their common terminal endpoint. The initial value 2 belongs to E(1,2), contradicting Conjecture 18.3. The tuple omitting ah has different factorizations; the paper’s main theorem for it and catenary Conjectures 18.1–18.2 are outside this conclusion.

Experiment and literature boundary

The exact-arithmetic checker is in the-omega-institute/trureturing-experiments, commit 23c744d471f53c63a530ec05520a55c2902f2050, at docs/reports/numerical-monoid-delta-set/check.py, SHA-256 6e39c9873b672007c385e46c4af6973fd500dbe17f6b37dc47fe46a2da1ab89a. The command python3 docs/reports/numerical-monoid-delta-set/check.py is reported in https://github.com/the-omega-institute/trureturing/issues/23185 with exit 0 and final line ALL_CHECKS_OK. Its finite windows do not supply the unbounded argument.

The source assertion is literature-attested; the family-specific completeness argument is repo-derived, with novelty only suspected in the screened scope. Issue #23185 records the source, the earlier Zenodo preprints, nonminimal-generator and presentation literature, and the literature screen. Completeness of later-citation, web and MathDB coverage, and the unrecovered Banks et al. work remain ASSUMED-UNVERIFIED. No worldwide priority claim is made.