A certified reproduction of the multi-population Wardrop equilibria in Bakaryan, Aoun, de Lima Ribeiro, Hovakimyan & Gomes (AIMS Mathematics 11(5), 2026, doi:10.3934/math.2026623). The paper validates each equilibrium by watching a globally convergent flow settle; this reproduction adds the complementary thing a stopping rule cannot give — a certificate that the answer is there, alone, and positive — and where the instance does not support one, it REFUSES instead of picking a solution. The verifier is embedded in the published page; it was extracted and re-run during this build.
These four rows come out of a single published paper, and they are deliberately not all the same verdict. A reproduction that returned "confirmed" four times would be hiding the interesting row — the one where the object being reproduced is not unique, and the honest answer is to refuse.
A traffic equilibrium is the flow pattern in which no driver gains by switching routes. The paper computes each one with a globally convergent Hessian–Riemannian flow and validates it numerically — existence by algorithm plus a benchmark match. A stopping rule tells you WHERE the equilibrium is; it cannot tell you that what it converged to is the only solution nearby, or that the residual being small means the answer is right. The certificate is the complementary object: where the costs are affine, an EXACT rational equilibrium with the Wardrop gap identically zero; where they are smooth and nonlinear, a Krawczyk box holding exactly one solution; where the instance is degenerate, an honest refusal that exhibits the degeneracy instead of hiding it.
S2 (cars + trucks, affine costs): the equilibrium is solved over exact rationals — 38 unknowns, support slacks identically zero, so the Wardrop gap is EXACTLY 0 and the paper's Figure 2 reproduces with no arithmetic left over. S3 (emission cost, smooth nonlinear): an interval Krawczyk contraction proves a unique KKT zero in an explicit box of radius 7.319e-13 — the paper's convergent iterate is now the provably unique equilibrium of its neighbourhood. S1 (degenerate linear cost): the cost is monotone but NOT strictly, the paper's uniqueness theorem does not apply, and the reproduction proves why no split certificate can exist — the split-conservation operator has an exact null space of dimension six. The totals are certified; the split is refused; refusal is the correct verdict, not a failure to reach one.
Binding the certified S1 equilibrium to the paper's Table 1: twelve of fifteen printed totals match exactly after integer rounding; two differ by one unit of rounding; and edge (4,7) differs by two — the exact total is 1940/37 = 52.432…, which rounds to 52 where the table prints 54. The value is certified four independent ways in the reproduction, and it is reported as what it is: a rounding typo in a table, an erratum for the authors' benefit — not a defect in their method, whose scenarios all reproduce. This is the audit lane's contract: corrections carry certificates, and the correction and the confirmation come from the same arithmetic.
Reproducing the paper's Figure 4 forced this reproduction to find a bug in its OWN cost code — a dropped 10⁻³ emission factor and a congestion-flow slip. That is the strongest argument for certificates in reproduction work: they catch the implementer's mistakes, not only the author's. Under the corrected cost the certified equilibrium is invariant to the edge-length reading, so nothing hinges on measuring a figure. (This episode sits in the benign corner of the failure taxonomy this site maintains — caught before publication, by exactly the machinery built to catch it.)
The published page (byte-preserved in the repository, exactly as sent) embeds its complete verifier — plain Python, standard library only, MIT. This build extracted those bytes (sha256 5d5fca032ff5dbb7…), ran them, and required VERIFIED plus every scenario verdict and the erratum value above; any drift refuses the page. Do the same: download the extracted verifier and run python3 verify_wardrop.py — under a second, exact Fractions end to end.