Psi-Turing Machines: Bounded Introspection for Complexity Barriers and Oracle Separations

We introduce Psi-Turing Machines (Psi-TM): classical Turing machines equipped with a constant-depth introspection interface $\iota$ and an explicit per-step information budget $B(d,n)=cd{\log }_{2}n$. With the interface frozen, we develop an information-theoretic lower-bound toolkit: Budget counting, $\mathrm{\Psi }$-Fooling, and $\mathrm{\Psi }$-Fano, with worked examples $L_k$ and $L_k^{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}$. We prove an oracle-relative separation $P^{\mathrm{\Psi }}\mathrm{\ne }N{P}^{\mathrm{\Psi }}$ and a strict depth hierarchy, reinforced by an Anti-Simulation Hook that rules out polynomial emulation of ${\iota }_k$ using many calls to ${\iota }_{k-1}$ under the budget regime. We also present two independent platforms (Psi-decision trees and interface-constrained circuits IC-AC${}^0$/IC-NC${}^1$) and bridges that transfer bounds among machine, tree, and circuit with explicit poly/log losses. The model preserves classical computational power outside $\iota$ yet enables precise oracle-aware statements about barriers (relativization; partial/conditional progress on natural proofs and proof complexity). The aim is a standardized minimal introspection interface with clearly accounted information budgets.