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We prove, under a computational complexity hypothesis, that it is consistentwith the true universal theory of p-time algorithms that a specific p-timefunction extending $n$ bits to $m \geq n^2$ bits violates the dual weakpigeonhole principle: every string $y$ of length $m$ equals the value of thefunction for some $x$ of length $n$. The function is the truth-table functionassigning to a circuit the table of the function it computes and the hypothesisis that every language in P has circuits of a fixed polynomial size $n^d$.

Small circuits and dual weak PHP in the universal theory of p-time algorithms