A δ-Functional Riccati–Barrier Interface for Finite-Horizon Regularity Certificates (v3.8)
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2026
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| author | Lee, Byoungwoo |
| author_facet | Lee, Byoungwoo |
| contents | <p># Overview</p> <p>This record releases **δ-functional v3.8**, a referee-facing note that isolates a reusable<br>**Riccati–barrier proof interface** for finite-horizon regularity certificates in PDE problems.</p> <p>The note is intentionally modular: it reduces the analytic chain to a small set of named checkpoints<br>(Definition → RNF/Riccati inequality → Residual gate), so that downstream implementations (e.g., shock programs)<br>can cite a stable, auditable interface.</p> <p># Core interface (TCB = 3)</p> <p>**TCB checkpoints (label-first; numbers are secondary):**<br>1) **(D) Diagnostic definition:** a scale-covariant diagnostic \( \delta(t) \) and its mollified version \( \delta_\varepsilon(t) \).<br>2) **(RNF) Riccati normal form:** an inequality of the schematic form<br> \[<br> \delta_\varepsilon'(t) \le a\,\delta_\varepsilon(t)^2 + b\,\delta_\varepsilon(t) + c + r_\varepsilon(t),<br> \]<br> with explicitly named coefficients \(a,b,c\) and residual \(r_\varepsilon\).<br>3) **(RES) Residual closure gate:** an explicit, window-uniform bound controlling \(r_\varepsilon\)<br> (the only remaining hard input).</p> <p>**Certificate statement (finite horizon, conditional):**<br>Assuming (RES), the sub-barrier condition \(\sup_{t\in[0,T]}\delta(t)<\phi_+\) yields smoothness on \([0,T]\),<br>where \(\phi_+\) is the positive equilibrium barrier level determined by \(a,b,c\).</p> <p># What is new in v3.8</p> <p>- Added a **Skeptic’s Map / Audit Checklist** (label-first, TCB = 3) and a **Constant Dashboard**<br> (no unnamed constants may enter the barrier level).<br>- Labeled the barrier equilibria \(\phi_\pm\) explicitly (equilibrium formula fixed as a citable equation).<br>- Quarantined numerics/diagnostics as **optional** and **not used** in the proof interface.<br>- Disabled unused cross-document reference machinery to prevent “moving target” confusion.</p> <p># Scope</p> <p>- This note does **not** claim global regularity.<br>- The main remaining analytic difficulty is the unconditional verification of the residual closure gate (RES)<br> at the target solution class.<br>- Any numerical illustration is a **diagnostic only** and is not a proof input.</p> <p>#Implemented for compressible Euler shocks in: 10.5281/zenodo.18850356</p> <p># Author</p> <p>Lee Byoungwoo leeclinic@protonmail.com<br>```</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18850142 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | A δ-Functional Riccati–Barrier Interface for Finite-Horizon Regularity Certificates (v3.8) Lee, Byoungwoo Navier–Stokes equations, 3D incompressible flow, regularity criteria, blow-up diagnostics, δ-functional, Riccati inequality, barrier method, Beale–Kato–Majda, Prodi–Serrin, Caffarelli–Kohn–Nirenberg, mollification, commutator estimates, vorticity, energy methods, partial regularity, Leray–Hopf solutions, computational diagnostics, turbulence monitoring Navier–Stokes; regularity criterion; Riccati inequality; barrier method; diagnostic functional; mollification; residual closure; proof interface; verification; PDE blow-up <p># Overview</p> <p>This record releases **δ-functional v3.8**, a referee-facing note that isolates a reusable<br>**Riccati–barrier proof interface** for finite-horizon regularity certificates in PDE problems.</p> <p>The note is intentionally modular: it reduces the analytic chain to a small set of named checkpoints<br>(Definition → RNF/Riccati inequality → Residual gate), so that downstream implementations (e.g., shock programs)<br>can cite a stable, auditable interface.</p> <p># Core interface (TCB = 3)</p> <p>**TCB checkpoints (label-first; numbers are secondary):**<br>1) **(D) Diagnostic definition:** a scale-covariant diagnostic \( \delta(t) \) and its mollified version \( \delta_\varepsilon(t) \).<br>2) **(RNF) Riccati normal form:** an inequality of the schematic form<br> \[<br> \delta_\varepsilon'(t) \le a\,\delta_\varepsilon(t)^2 + b\,\delta_\varepsilon(t) + c + r_\varepsilon(t),<br> \]<br> with explicitly named coefficients \(a,b,c\) and residual \(r_\varepsilon\).<br>3) **(RES) Residual closure gate:** an explicit, window-uniform bound controlling \(r_\varepsilon\)<br> (the only remaining hard input).</p> <p>**Certificate statement (finite horizon, conditional):**<br>Assuming (RES), the sub-barrier condition \(\sup_{t\in[0,T]}\delta(t)<\phi_+\) yields smoothness on \([0,T]\),<br>where \(\phi_+\) is the positive equilibrium barrier level determined by \(a,b,c\).</p> <p># What is new in v3.8</p> <p>- Added a **Skeptic’s Map / Audit Checklist** (label-first, TCB = 3) and a **Constant Dashboard**<br> (no unnamed constants may enter the barrier level).<br>- Labeled the barrier equilibria \(\phi_\pm\) explicitly (equilibrium formula fixed as a citable equation).<br>- Quarantined numerics/diagnostics as **optional** and **not used** in the proof interface.<br>- Disabled unused cross-document reference machinery to prevent “moving target” confusion.</p> <p># Scope</p> <p>- This note does **not** claim global regularity.<br>- The main remaining analytic difficulty is the unconditional verification of the residual closure gate (RES)<br> at the target solution class.<br>- Any numerical illustration is a **diagnostic only** and is not a proof input.</p> <p>#Implemented for compressible Euler shocks in: 10.5281/zenodo.18850356</p> <p># Author</p> <p>Lee Byoungwoo leeclinic@protonmail.com<br>```</p> |
| title | A δ-Functional Riccati–Barrier Interface for Finite-Horizon Regularity Certificates (v3.8) |
| topic | Navier–Stokes equations, 3D incompressible flow, regularity criteria, blow-up diagnostics, δ-functional, Riccati inequality, barrier method, Beale–Kato–Majda, Prodi–Serrin, Caffarelli–Kohn–Nirenberg, mollification, commutator estimates, vorticity, energy methods, partial regularity, Leray–Hopf solutions, computational diagnostics, turbulence monitoring Navier–Stokes; regularity criterion; Riccati inequality; barrier method; diagnostic functional; mollification; residual closure; proof interface; verification; PDE blow-up |
| url | https://doi.org/10.5281/zenodo.18850142 |