Global bounds, conditional subcritical Hopf bifurcation and numerically supported bistability in a regularized processing–memory feedback model

This preprint analyzes a dimensionless processing-memory feedback model with a continuous overload switch and a positive denominator floor. It presents written proofs of forward completeness, positivity, boundedness and the existence of exactly one positive equilibrium, in addition to a boundary equilibrium at the origin. For the smooth overloaded branch, the study derives strict admissibility conditions for a Hopf crossing and reduces the dynamics to a classical quadratic family. A positive focal coefficient makes the admissible crossing subcritical. A separate planar trapping argument establishes the existence of at least one periodic orbit under the stated source-equilibrium conditions. The analysis distinguishes the zero-floor branch limit, finite-floor junction, local stability crossing and unit raw-load boundary. For the specified reference parameters at lambda = 0.36, archived shooting calculations with DOP853 and Radau support coexistence of a stable equilibrium and an attracting outer cycle, together with a repelling inner cycle in the invariant plane. The reported outer and inner periods are approximately 7.1688707014 and 5.4202427352. Their planar multipliers are approximately 0.0669035997 and 1.0702835848. Exact algebraic checks cover seven polynomial identities; numerical results remain finite-precision evidence rather than interval-certified proofs. The record includes the article, supporting information, editable sources, and a focused data/code archive with verification scripts, stored JSON results, trajectory tables, reproduction instructions and checksums. All numerical data are generated from the model equations. No empirical or laboratory observations are analyzed. Complete cycle counting, global continuation, physical interpretation and priority of the full regularized model remain open. AI assistance is disclosed in the manuscript; no independent human peer review is asserted.