slug: wu-lou-permutation-invariant-niven-digit-sum-bound
bibkey: wu2025pinn
doi: 10.3390/sym18010186
url: https://arxiv.org/abs/2508.01611
triage: theorem
motivation_gids:
  - D5/S1/Digit/Admissibility/WuLouPermutationInvariantNivenDigitSum

Wu-Lou permutation-invariant Niven digit-sum bound

Problem

Wu and Lou, Permutation-Invariant Niven Numbers, arXiv:2508.01611v3, section 8.4, state that the relevant nontrivial distinct-digit permutation-invariant Niven numbers have digit sums between 3 and 81 and conjecture that no exceptions exist beyond the cases excluded there.

Motivation

The formal target is the universal bound for the same decimal permutation-invariant object, with the hypotheses excluding the two elementary families named in the source sentence. No broader claim about the paper is made.

Gap

The source labels the no-exceptions assertion as a conjecture. The journal text was not read because the MDPI page returned HTTP 403; this is recorded as an ASSUMED-UNVERIFIED source boundary below.

Route

The Lean theorem result proves the bound for the source-faithful DecimalPINN structure whenever at least two nonzero digit occurrences and two distinct digit values are present. It proves 3 | digit_sum and 3 <= digit_sum <= 81. The divisibility-by-three assertion is literature-attested; the lower bound is its positive immediate consequence; the upper bound is the repository-derived settlement of the conjectural part.

Evidence

• Lean module: D5/S1/Digit/Admissibility/WuLouPermutationInvariantNivenDigitSum.lean.
• Public declarations: DecimalPINN and result.
• Positive control: the digit list [6, 2, 1] inhabits DecimalPINN; its six permutations have values 126, 162, 216, 261, 612, and 621, each divisible by 9.
• Exhaustive recheck: lengths 2 through 6 gave 14, 64, 289, 959, and 2999 admissible vectors respectively, 4325 total, with zero exceptions to the bound.

The 3 | digit_sum component is attributed to Wu and Lou’s Theorem 1 consequence in section 8.4. The digit_sum <= 81 component is derived in this repository from permutation invariance and the two-distinct-digit hypothesis.

Falsifier

A source-faithful DecimalPINN satisfying the two hypotheses with digit sum outside the proved interval, or not divisible by three, would falsify the result.

Triage

The delivery uses admission_basis: open-problem-resolution. The proof shape is bind-only, with no escape witness: the bound follows by instantiating the source permutation-divisibility field and normalizing the resulting arithmetic.

ASSUMED-UNVERIFIED

Crossref confirmed DOI 10.3390/sym18010186. The journal text itself was not read because the MDPI page returned HTTP 403; the arXiv version is the available source text for the quoted locator.