Small Circuits and Dual Weak PHP in the Universal Theory of p-time Algorithms

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