U24 Programme | Bryan Daugherty, Gregory Ward, Shawn Ryan | March 2026
We falsify the claim R(8,8) > 293 by exhaustive max-clique verification, confirm R(8,8) > 281 via Paley construction, and verify the Zero-Core Theorem for R(5,5). The central methodological finding is that stochastic sampling becomes unreliable for Ramsey verification at k = 8, where random sampling covers only ~6.6 x 10^-9 of the search space.
| Result | Value | Status |
|---|---|---|
| R(8,8) > 293 claim | Falsified | Proved (exhaustive Bron-Kerbosch) |
| Red max-clique in K_293 | omega = 8 (monochromatic K_8) | Proved |
| Blue max-clique in K_293 | omega = 11 (monochromatic K_11) | Proved |
| Red K_8 witness | {3, 44, 87, 130, 165, 219, 234, 285} | Proved (28 edges verified) |
| Stochastic sampling coverage | 6.6 x 10^-9 | Computational |
| Detection probability P(miss|V=1) | ~1.0 | Computational |
| R(8,8) > 281 (Paley) | Confirmed | Proved (exhaustive K_8 enumeration) |
| Paley(293) violations | 2,310,012 monochromatic K_8 | Computational (31.1s) |
| Zero-Core Theorem (R(5,5)) | Essential core = empty set | Proved (2,480 DPLL proofs) |
| Confirmed bound | 282 <= R(8,8) <= 1,870 | Proved |
12/12 checks PASS
Run the verification suite:
# R(8,8) falsification + Paley confirmation (requires NumPy)
python scripts/verify_r88.py
# Zero-Core Theorem verification (~4 min)
python scripts/verify_zerocore.py
# Generate all figures (requires matplotlib)
python scripts/generate_figures.py- Python 3.10+
- NumPy
- matplotlib (for figures only)
| File | Description |
|---|---|
data/K293_best_20260315_101534.npy |
GPU-optimized K_293 coloring (42,778 spins) |
data/verification_certificate_r88.json |
Falsification certificate |
data/k8_landscape.json |
Paley(281) K_8-free proof |
data/paley_293_k8.json |
Paley(293) 2.31M violations |
data/r88_gpu_sparse_20260315_101534.json |
GPU campaign results |
data/r88_gpu_log.txt |
GPU campaign log |
data/zerocore_certificate.json |
Zero-Core verification certificate |
| Claim | Falsified if... |
|---|---|
| R(8,8) > 293 is false (for this coloring) | Independent verifier shows omega(red) <= 7 AND omega(blue) <= 7 in the coloring file |
| R(8,8) > 281 | A monochromatic K_8 is found in either color of Paley(281) |
| Zero-Core Theorem | A constraint index i is found where S \ {c_i} is feasible |
| Stochastic sampling unreliable | A poly-time sampler detects K_8 cliques with high probability at n=293 |
| Figure | Description |
|---|---|
fig1_scale_comparison |
Log-scale: search space vs sampling budget |
fig2_paley_landscape |
Paley K_8 violation count by prime |
fig3_search_space_scaling |
Coverage gap grows super-exponentially with k |
fig4_witness_spacing |
Red K_8 witness vertices on K_293 ring + spacing analysis |
fig5_detection_probability |
P(miss) as function of violation count |
fig6_ramsey_bounds |
Diagonal Ramsey number bounds overview |
fig7_zero_core |
Zero-Core Theorem schematic |
This paper extends the Ramsey theory campaign from Papers 02 and 03 (R(5,5) bounds) to k = 8, demonstrating the methodological boundary where stochastic optimization fails. The Zero-Core Theorem connects to the S_4 stagnation structure (Paper 17): the distributed constraint obstruction mirrors permutation-invariant energy barriers in |S_4| = 24-dimensional landscapes.
@article{DaughertyWardRyan2026r88,
title = {Falsification of {R}(8,8) > 293: Exhaustive Max-Clique
Verification and the Structural Limits of Stochastic Sampling},
author = {Daugherty, Bryan and Ward, Gregory and Ryan, Shawn},
year = {2026},
note = {U24 Programme Paper 14},
url = {https://github.com/OriginNeuralAI/Papers}
}