# Graffiti 284 is false: the Hoffman–Singleton graph

**Status:** independent exact verification of a refutation first found by the "Capy Build" agent run of 2026-07-22 (public transcript, tokiwa.space "CE10"; also posted on X by @justinsunyt, 2026-07-23). Priority for the counterexample belongs to that run. This note contributes the fully exact, float-free verification chain and the equality-case context.
**Date:** 23 July 2026.

## The statement

Verbatim from *Written on the Wall*, page 79 (vision transcription of the scan, artifact `wow-p79-bot.png`):

> **284.** If girth is ≥ 5 then the minimum dual degree ≤ − the smallest eigenvalue of distance matrix.

Dual degree of \(v\) = mean degree of the neighbors of \(v\). Aouchiche–Hansen (LAA 458 (2014), distance-spectra survey) list this conjecture, state that *"as far as we know, the above conjecture remains open,"* and observe that the **Petersen graph attains equality**: \(\delta^* = -\partial_n = 3\). Roucairol–Cazenave (arXiv:2409.18626, Table 1) also list it open; their searches to \(n = 50\) found nothing — unsurprisingly, as the unique counterexample in that range appears to be a single strongly regular graph their edge-flip searches would essentially never hit.

## The counterexample, verified exactly

Let \(G\) be the Hoffman–Singleton graph: \(n = 50\), \(m = 175\), 7-regular, girth 5, diameter 2 — the third Moore graph of girth five, after \(C_5\) and Petersen.

Every step below is integer arithmetic (script `verify_284_hoffman_singleton_exact.py`; independent construction in `independent_check_284_290.py`):

1. **SRG identity.** \(A^2 + A - 6I = J\) holds exactly. Hence on \(\mathbf{1}^\perp\) every adjacency eigenvalue satisfies \(\mu^2 + \mu - 6 = 0\), so \(\mu \in \{2, -3\}\); regularity gives eigenvalue 7 on \(\mathbf 1\). Trace conditions \(7 + 2a - 3b = 0\), \(a + b = 49\), \(49 + 4a + 9b = 350\) force multiplicities \(7^1\, 2^{28}\, (-3)^{21}\).
2. **Distance matrix.** BFS gives every off-diagonal distance in \(\{1,2\}\), and \(D = 2(J - I) - A\) exactly.
3. **Distance spectrum.** Therefore \(D\) has eigenvalues \(2(n-1) - 7 = 91\) on \(\mathbf 1\), and \(-2 - \mu\) on \(\mathbf 1^\perp\): \(-4\) (multiplicity 28) and \(1\) (multiplicity 21). So \(\partial_n = -4\) **exactly**.
4. **The conjecture fails.** Minimum dual degree \(= 7\) (7-regular). The conjecture asserts \(7 \le 4\), which is false. Margin: exactly \(3\).

## Remarks

- The conjecture is **tight on Petersen** (3 = 3, noted by Aouchiche–Hansen) and fails on the next Moore graph. For Petersen, \(D = 2(J-I) - A\) with \(\mu_{\min} = -2\) gives \(\partial_n = -3\) against dual degree 3. For a hypothetical 57-regular Moore graph (\(n = 3250\)), \(\mu_{\min} = -8\) would give \(57 \le 7\): false by margin 50. So among girth-5 Moore graphs the conjecture holds only for \(C_5\) and Petersen — the pattern "true on small Moore graphs, false from HS onward" is exact.
- General diameter-2 girth-5 graphs are exactly the Moore graphs (degree–diameter), so the counterexample family is as small as it could be; all other girth-5 graphs have diameter \(\ge 3\) where the distance spectrum spreads and the searches of Roucairol–Cazenave found nothing — consistent with HS being an isolated violation below \(n = 3250\).

## Provenance and honesty

- First finder: Capy Build agent (grok-4-5), 2026-07-22 18:14–18:22 UTC, transcript indexed at tokiwa.space ("CE10 · Refuting Graffiti 284"); floating-point + analytic verification therein.
- Public announcement: X posts of 2026-07-23 (@justinsunyt; commentary @RadishHarmers).
- This package: statement fidelity re-established from the page-79 scan; the refutation re-verified with a fully exact integer chain and an independent second construction. No priority claim is made for the counterexample itself.