A Computer-Assisted Lower Bound and an Explicit Scope-95 Witness for m(6,5)

An (n,k)-difference triangle set is a family of n rows of k+1 integers, each starting at 0, whose n*C(k+1,2) within-row differences are pairwise distinct; its scope is the largest difference, and m(n,k) is the minimum scope. The cell (6,5) is the first open case at k=5. This deposit contains the paper and its complete verification package, establishing 94 <= m(6,5) <= 95 with both ends resting on verifiable evidence. Lower bound: m(6,5) >= 94, by a computer-assisted exclusion of scope 93. A hand-checkable dual certificate settles m(6,5) >= 92 and provably exhausts the linear-programming relaxation; a finite evaluation of a Fourier obstruction of Ma and Yi gives >= 93; and a complete, independently audited exact-cover exhaustion over a finite universe of row-length sets, organised by an elementary partition bound, gives >= 94. Upper bound: m(6,5) <= 95, by an explicit witness — to the author's knowledge the first explicit scope-95 (6,5)-difference triangle set on record, which supplies a third-party-verifiable witness for the bound J. B. Shearer reported in 2008 without publishing one. The package (m65-dts-package.zip, 68 files) contains the nine-witness corpus with canonical hashes; the 114 hole-confinement certificates with an independent 50-digit verifier; the T=534, 535 and 536 elimination evidence with both solvers' sources; the 30-campaign global theorem artifact with its replay and an independent regeneration; the search and analysis programs; a documented literature check supporting the priority claim; and portable, fail-closed verification programs with positive and negative regression tests. Every verification command runs in a clean Python 3.8+ environment with nothing installed and no network access; a README maps each claim in the paper to the artifact and command that reproduce it. Integrity is checked with the included SHA256SUMS.txt. The paper also corrects and consolidates the k >= 5 lower-bound frontier, reports the verification methodology, and documents an explicit trust boundary: what is hand-checkable, what is audited computation, and what remains trusted.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-08-24
DOI
https://doi.org/10.5281/zenodo.22079671
Primary Topic
Cryptography and Residue Arithmetic
Type
preprint
Controls
|||
ALL TIME
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
preprint

A Computer-Assisted Lower Bound and an Explicit Scope-95 Witness for m(6,5)

Yuhao Liu
Zenodo (CERN European Organization for Nuclear Research)
Cryptography and Residue Arithmetic
preprint

A Computer-Assisted Lower Bound and an Explicit Scope-95 Witness for m(6,5)

Yuhao Liu
preprint en

Abstract

An (n,k)-difference triangle set is a family of n rows of k+1 integers, each starting at 0, whose n*C(k+1,2) within-row differences are pairwise distinct; its scope is the largest difference, and m(n,k) is the minimum scope. The cell (6,5) is the first open case at k=5. This deposit contains the paper and its complete verification package, establishing 94 <= m(6,5) <= 95 with both ends resting on verifiable evidence. Lower bound: m(6,5) >= 94, by a computer-assisted exclusion of scope 93. A hand-checkable dual certificate settles m(6,5) >= 92 and provably exhausts the linear-programming relaxation; a finite evaluation of a Fourier obstruction of Ma and Yi gives >= 93; and a complete, independently audited exact-cover exhaustion over a finite universe of row-length sets, organised by an elementary partition bound, gives >= 94. Upper bound: m(6,5) <= 95, by an explicit witness — to the author's knowledge the first explicit scope-95 (6,5)-difference triangle set on record, which supplies a third-party-verifiable witness for the bound J. B. Shearer reported in 2008 without publishing one. The package (m65-dts-package.zip, 68 files) contains the nine-witness corpus with canonical hashes; the 114 hole-confinement certificates with an independent 50-digit verifier; the T=534, 535 and 536 elimination evidence with both solvers' sources; the 30-campaign global theorem artifact with its replay and an independent regeneration; the search and analysis programs; a documented literature check supporting the priority claim; and portable, fail-closed verification programs with positive and negative regression tests. Every verification command runs in a clean Python 3.8+ environment with nothing installed and no network access; a README maps each claim in the paper to the artifact and command that reproduce it. Integrity is checked with the included SHA256SUMS.txt. The paper also corrects and consolidates the k >= 5 lower-bound frontier, reports the verification methodology, and documents an explicit trust boundary: what is hand-checkable, what is audited computation, and what remains trusted.

Zenodo (CERN European Organization for Nuclear Research)
Cryptography and Residue Arithmetic
AI Navigator

Ask Laika to Summarize, Analyze, and Connect papers live on the map.

Summarize Papers & Methodologies

Extract key findings, datasets, and comparative methods across publications.

Benchmark Rankings & Visual Analytics

Rank top research institutions, authors, funders, topics, and journals by Field-Weighted Citation Impact (FWCI) and paper volume with instant charts.

Connect Distant Disciplines

Bridge topological clusters on the map to find hidden collaborative intersections.

A Computer-Assisted Lower Bound and an Explicit Scope-95 Witness for m(6,5) — Yuhao Liu · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS