Phase-Bypass Geometry: Resolved Zero Sectors, Persistent First-Exit Dynamics, and Register-Free Spectral Memory

We formulate a conditional geometric framework for a phase bypass of a zero sector. The calibrated order parameter is a smooth section of a Hermitian complex line bundle, not a complete normalized quantum state. At a transverse zero, its normalized section has no canonical continuous extension to a full neighbourhood in the original space, whereas its pullback extends uniquely and smoothly to the exceptional boundary of the real oriented blow-up. The boundary value is determined by the vertical derivative and records the incoming normal direction; it does not select an outgoing path. On a controlled scattering component, a Hamiltonian flow supplies the missing dynamical datum under explicit transversality, two-sided non-trapping, completeness, and uniqueness assumptions. The resulting first-exit map is a symplectomorphism and is C1 persistent under small C2 perturbations of shifted Hamiltonians with uniform C3 control and quantitative regularity, flight-time, boundary-separation, and intersection-isolation margins. A Koopman–cocycle operator transports an internal fibre over this map but remains a classical construction unless a separate quantum realization is supplied. For a symmetry-constrained cubic exceptional-point slice, analytic continuation gives exact C3 monodromy, and a companion generator admits a formal GKSL-compatible no-jump embedding after a uniform loss shift. An engineered retain/erase benchmark provides programmed positive control and independently calibrated erase and lifted-EP3 nulls. The register-free bridge forbids deliberate injection of the orientation label, defines a natural carrier as minimal process memory across a certified causal break, and separates Hilbert-space spectral subspaces from Liouville-space operator modes. Exact memory identification is stated as a reversible full-algebra sufficiency criterion. A separate approximate theorem converts finite recovery and predictive-factorization errors into quantitative information and output-distribution bounds. A unified analytic benchmark derives the section, first-exit dynamics, resolved-to-control map, spectral family, and Kato transport from one calibrated joint model. These results define a falsifiable certification programme; they do not claim a completed hardware realization or a universal law of phase bypass.