Signed and unsigned Gallai homothety numbers of three-point sets
Abstract
For a finite set S ⊂ ℤ and an integer r ≥ 1, the Gallai homothety number Gᵣ(S) is the least N such that every r-colouring of N consecutive integers contains a monochromatic homothet b+kS with k ≥ 1 lying inside them. Letting the scale k range over all nonzero integers instead defines a second threshold G^(±)ᵣ(S), which for a three-point set S=Sset(0,s,s+u) is the Rado number of ux+sz=(s+u)y in the convention that asks for solutions with distinct coordinates. We compute G^(±)₃(Sset(0,2,5))=48, G^(±)₃(Sset(0,3,7)) ≥ 70 and G^(±)₃(Sset(0,1,7)) ≥ 69; the last of these is one row past the end of the published table for this family of equations, and an exhaustive search puts its value at 69. We have found no value of the signed threshold at three colours published for any three-point set other than Sset(0,1,m) with m ≤ 6. We prove that the threshold G^(±)₃(Sset(0,1,6))=60 has at least two extremal colourings that no permutation of the colours identifies, and that the window reflection carries one to the other with no relabelling at all: a statement about the extremal objects, which a refutation certificate cannot supply. The exhaustive halves proved here are discharged in one of two ways: by a refutation certificate re-checked inside the proof, or by a complete backtracking search that is proved sound in the one direction a refutation needs and stores nothing. The search closes two published cells the certificate route had priced out, G^(±)₃(Sset(0,1,6))=60 and G₃(Sset(0,1,5))=70, whose refutation certificates are 137 and 298 megabytes. We prove G^(±)₃(Sset(0,2,5))<G₃(Sset(0,2,5)) as a comparison of the two thresholds rather than of two numerals, with a gap of at least 29. We give an elementary two-colouring proof of the uniform lower bound G^(±)₂(Sset(0,s,s+u)) ≥ 4(s+u)+1 for coprime s,u with 4 ∤ s and 4 ∤ u, and we record exactly where the two readings separate at two colours: by one on the family {s,u}={1,4m} and by two at the sporadic pair {s,u}={3,4}. Finally we correct the record. The two-colour signed values are a closed form published by Gupta, Thulasi Rangan and Tripathi in 2015; the recent literature that tabulates four of them, and that observes the drop at {1,4} empirically, does not cite that theorem. Every theorem and proposition below is machine-checked in Lean 4; the numbers reported as measurements are labelled where they occur.
Open review
This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.
Archived files
- Version 2 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
10df319c2a7054739ccebae8316b19e6f1f55f74ae6503ef2c9adccb244f3961
Claim ledger
Stated results
G1routine2026-08-22
The definition, the threshold packaging, and the fidelity theorem that the base may be read in N rather than Z
G2known2026-08-22
The whole (2, r) row with no search: Gᵣ(0,1) = r + 1 for every r
G3known2026-08-22
Validation: the three van der Waerden cells, reproduced by the same pipeline before anything else is believed
G4known2026-08-22
The four values arXiv:2607.02226 calls new: G₃(0,1,3) = 42, G₃(0,1,4) = 57, G₂(0,2,3,5) = 67, G₂(0,1,5,6) = 80
G5known2026-08-22
The (3,2) closed form in full: thirteen primitive three-point sets of diameter at most 9, both branches
G6known data2026-08-22
All eighteen exact two-colour Gallai numbers of four-point sets from arXiv:2606.22155v2 Table 4
G7known data2026-08-22
The dispatching source's open cell: its own certificate verified, giving G₃(0,2,5) >= 77; the exact value 77 measured but not stored
G8routine2026-08-22
Negative controls: the scale bound, the normalization, the non-vacuity of the encoding, and the too-large/too-small pair
G9routine2026-08-22
The symmetry break is licensed, not assumed: gallaiCnfZ has its own fidelity theorem and its own bridge
S1routine2026-08-28
The signed (two-shape) Gallai number G^±ᵣ(S): the ThresholdProblem packaging, the Rado-reading fidelity theorem, G^± ≤ G, reflection invariance, and the CNF bridge
S2known2026-08-28
Validation: symmetric patterns transfer, giving G^±ᵣ(0,1) = r+1 for every r and the three van der Waerden anchors with no new search
S3known2026-08-28
The two-colour signed table: thirty-two primitive three-point cells, reproducing Gupta-Thulasi Rangan-Tripathi's 2015 closed form; 139 coprime pairs with D <= 30 checked against it
S4known2026-08-28
Where the signed and unsigned readings separate, proved rather than compared: by one at 0,1,5, by two at the sporadic 0,3,7, not at all at 0,2,5
S5known data2026-08-28
arXiv:2606.22155v2's Table 5 beyond two colours: R₃ = 29, 54, 55 proved with certificates, R₄ >= 59 and G₄(0,1,3) >= 94 from the source's own printed colourings, R₃(k=5) >= 60 with the value 60 measured
S6known2026-08-28
A uniform lower bound with no search: 4 ∤ s and 4 ∤ u imply G^±₂(0,s,s+u) >= 4(s+u)+1, for infinitely many coprime pairs, kernel-clean
S7routine2026-08-28
Negative controls for the signed reading, including the source's own claim that its four-colour Gallai certificate is not solution-free
S8candidate2026-08-28
Three-colour signed values off the published line: G^±₃(0,2,5) = 48 proved, G^±₃(0,3,7) >= 70 with the value 70 measured
S9routine2026-08-28
A compiled complete search replaces the certificate: the max-element decomposition, the forward-check prune proved sound, and the first-occurrence canonical form at three colours
S10known data2026-08-28
G^±₃(0,1,6) = 60, the cell parked over the certificate cap, proved at zero stored bytes
S11known2026-08-28
The search validated against the certificate route: both van der Waerden anchors from scratch, and the two largest stored certificates made redundant
S12routine2026-08-28
Negative controls for the search route
S13candidate2026-08-28
At least two extremal colourings at G^±₃(0,1,6) = 60, exchanged by the window reflection with no relabelling
S14known data2026-08-28
G₃(0,1,5) = 70, the Table 3 row this family never had, at zero stored bytes
S15candidate2026-08-28
G^±₃(0,1,7) = 69, a new value one row past the end of arXiv:2606.22155v2's Table 5; the lower half >= 69 proved
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- For a finite set S ⊂ ℤ, a homothet of S is b + k·S = b + k·s: s ∈ S with scale k ≥ 1. The Gallai homothety number Gᵣ(S) is the least N such that every r-colouring of N consecutive integers contains a monochromatic homothet of S lying inside them. It is finite for every S and r — that is the one-dimensional case of Gallai's theorem (Gallai; first published by Rado, independently by Witt).
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7