A yes/no certificate can be exactly right and still be worthless one step away. That is the subject of this paper. There is a standard sign-vector existence certificate — a test that decides whether a prescribed pattern of signs can be realized — and the natural question is not whether it says yes, but how much room the yes has. Does it survive a small perturbation of the data, or does it hold only at the point?
The finding is pointwise validity without structural robustness: the certificate can be exactly true at a point while having zero margin around it, so that an arbitrarily small change in the data destroys it. The paper computes the exact margin — the precise distance to the nearest data at which the certificate fails — and shows it can vanish exactly where intuition says it should be safely positive.
The witness is a specific, named prior example rather than a constructed one: the Deshpande–Müller Example 22, taken from the sign-vector framework of Müller and Regensburger, with the margin pinned exactly on it. Using a case the literature already trusts is the point — the fragility is in the certificate, not in a pathological input.
Why it matters beyond the geometry: this kind of existence certificate underwrites robustness claims in chemical reaction network theory, where a “robust” conclusion that is only pointwise-valid is one that can fail silently under measurement noise. The exact margin is what tells you when the certificate is load-bearing and when it is decoration.
It sits alongside the chirotope-transition work — same sign-vector world, a different question — and it is a preprint; the arXiv version of record (math.AG) follows once endorsed. The load-bearing results are proved analytically and interval-certified; the objects carry descriptive names, not mine; the reproduction packet is non-load-bearing and self-verifying against a SHA-256 manifest with sha256sum -c. You don’t trust the author; you trust the trail.
→ The paper + reproduction packet (Zenodo, CC BY) · the sign-pattern distance law next door
Method note: language-model tools assisted with prose and analysis-code generation under author-defined statements, checks, and reporting constraints; I reviewed and take responsibility for every analysis and claim.