Small diameter does not control hitting-time robustness

Unrefereed research candidate. A random walk is robust if an adversary who may change every transition probability by a small relative amount cannot make it much slower to reach its targets. Zuckerman (Markov chain robustness, ITCS 2026, LIPIcs 362, 118) conjectured that the largest safe relative change is at least a constant divided by the graph's diameter (Conjecture 1.16).

This package shows the conjecture is false. A graph can have short routes that the walk almost never uses; a small fixed bias away from a target accumulates between rare visits to them and depletes the target's stationary mass, and Kac's formula turns this into a long hitting time. For explicit cubic graphs J_b on N vertices with diameter O(log N), every effective resistance is at most 684 (proof), and a fixed perturbation of size 1/b, with error times diameter tending to zero, multiplies the worst-case hitting time by a factor of order b; the robustness is O(N^(-1/3)). For explicit 3m-regular graphs of diameter exactly three, the mixing time is at most 50(m+1), so with Zuckerman's mixing-time theorem the robustness lies between constant multiples of N^(-1/2) and N^(-1/4) (proof; not matching). An exact computation shows a 5.80-fold slowdown on a 5,995-vertex member with error times diameter 0.74. A diameter-two example, a reversible-perturbation bound and a comparison with Kemeny's constant are included.

The archive contains the manuscript (PDF and LaTeX), seven verification scripts with a driver that reproduces the reference JSON results byte for byte under python -B -O, semantic negative controls, figures and SHA-256 sums. Assurance: written proofs; exact checks; producer-side replay including a clean replay in a separate copy; internal AI-assisted review, AI-assisted adversarial reviews (OpenAI gpt-5.6-sol; Anthropic Claude Opus 5.5; OpenAI gpt-6-astra) and a verification re-review. This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.