An independence of the MIN principle from the PHP principle

We extend the typical forcing of M. M\"uller and derive conditions on the forcing frame for which generic expansions preserve injective/bijective pigeonhole principle for polynomial-time computable graphs of functions. Applying this machinery, we show that the bounded arithmetic theory $\forall \textsf{T}^1_2(\textsf{PV}(\alpha))$ augmented by the polynomial-time injective pigeonhole principle does not prove the linear ordering, tournament, and dual weak pigeonhole principles.