A δ-Functional Riccati–Barrier Interface for Finite-Horizon Regularity Certificates (v3.8)

# Overview This record releases **δ-functional v3.8**, a referee-facing note that isolates a reusable**Riccati–barrier proof interface** for finite-horizon regularity certificates in PDE problems. The note is intentionally modular: it reduces the analytic chain to a small set of named checkpoints(Definition → RNF/Riccati inequality → Residual gate), so that downstream implementations (e.g., shock programs)can cite a stable, auditable interface. # Core interface (TCB = 3) **TCB checkpoints (label-first; numbers are secondary):**1) **(D) Diagnostic definition:** a scale-covariant diagnostic \( \delta(t) \) and its mollified version \( \delta_\varepsilon(t) \).2) **(RNF) Riccati normal form:** an inequality of the schematic form \[ \delta_\varepsilon'(t) \le a\,\delta_\varepsilon(t)^2 + b\,\delta_\varepsilon(t) + c + r_\varepsilon(t), \] with explicitly named coefficients \(a,b,c\) and residual \(r_\varepsilon\).3) **(RES) Residual closure gate:** an explicit, window-uniform bound controlling \(r_\varepsilon\) (the only remaining hard input). **Certificate statement (finite horizon, conditional):**Assuming (RES), the sub-barrier condition \(\sup_{t\in[0,T]}\delta(t)