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

Abstract: 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)=c\,d\log_2 n $. With the interface frozen, we develop an information-theoretic lower-bound toolkit: Budget counting, $ \Psi $-Fooling, and $ \Psi $-Fano, with worked examples $ L_k $ and $ L_k^{\mathrm{phase}} $. We prove an oracle-relative separation $ P^{\Psi} \neq NP^{\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.