Finite-Window Shock Diagnostic Certificates: Jump-Corrected Euler Defects, Uniform Subsolution Gates, and Fixed-Scale Telemetry

Abstract

## Overview This record releases Version v4.2r3 of a three-paper research set on finite-window PDE diagnostics, conditional shock certificates, and uncertainty-audited finite-scale telemetry. The package separates three logically distinct layers: 1. a proved scalar comparison and normalization layer;2. unresolved PDE closure gates for Navier–Stokes and compressible Euler shock problems;3. a reproducible engineering protocol for CFD and Schlieren-assisted wedge/Mach-reflection studies. The common certificate mechanism is \[\text{uniform AC/integral subsolution gate}+\text{initial safety margin}+\text{gap-relative residual and curvature budgets}\Longrightarrow\text{finite-window sub-barrier control}.\] The package does not identify finite-scale engineering telemetry with an analytic zero-scale limit unless additional convergence and closure gates are verified. ## Included papers ### Paper I **Finite-Scale Gradient Diagnostics for Navier–Stokes Flows: Riccati Comparison, Uniform Residual Gates, and Classical-Criterion Compatibility** Version v4.1r1 This paper defines a mollified analytic gradient diagnostic and proves an audit-ready finite-window comparison theorem for absolutely continuous or integral subsolutions. It records compatibility with the Beale–Kato–Majda, Prodi–Serrin, and Caffarelli–Kohn–Nirenberg mechanisms once the analytic limsup is bounded. Uniform pressure, inversion, weak-solution, and small-scale residual closure remain explicit gates. ### Paper II **Conditional Finite-Window Certificates for Compressible Euler Shock Diagnostics: Jump-Corrected Defects, Trace Gates, and Curvature Budgets** Version v4.2r1 This paper separates the reference shock load, residual jump, jump-corrected regular perturbation, low-frequency amplitude channel, and finite-scale observed sensor. The residual jump is removed from the regular-gradient channel by an explicit shock-chart corrector and is measured separately through a trace channel. The proved common layer is scalar finite-window comparison algebra. Relative-entropy compatibility, trace and modulation construction, jump-corrector regularity, curved-front control, vanishing-viscosity transfer, and Mach-reflection implications remain declared analytic gates. ### Paper III **Finite-Scale Margin Telemetry for Wedge and Mach-Reflection Studies: A Reproducible CFD–Schlieren Protocol** Version v2.4r3 This paper specifies an executable reporting protocol based on the fixed-physical-scale sensor \[D_h^{\mathrm{obs}}(t)=\frac{h}{U_{\mathrm{ref}}}\left\|\nabla \bigl(u_h * \varphi_h\bigr)(\cdot,t)\right\|_{L^\infty}.\] The protocol separates prospective `CERT-AHEAD` decisions from retrospective `OBSERVED-PASS` decisions and reports nominal and confidence-adjusted versions. An explicitly incomplete uncertainty ledger preserves nominal reporting but returns `NOT_EVALUATED` for robust decisions. A ledger declared complete but internally inconsistent invalidates the packet. The executable release supports the reconstructible uncertainty methods `linear_bound` and `rss_with_coverage`. The window registry is authoritative for window endpoints and referenced files. Positive and adversarial mutation tests audit the validator semantics. ## Package contents The release includes: - the three main papers and a public overview;- LaTeX source files;- claim-status, revision, review-response, and final-check documents;- JSON schemas and CSV templates;- complete and explicitly incomplete synthetic sample packets;- a registry-authoritative reporting-packet validator;- positive and mutation-test logs;- deterministic synthetic reporting figures;- SHA-256 manifests and a complete reproducibility bundle. ## Validation and claim boundary The mathematical results rigorously establish the scalar finite-window comparison mechanism, exact step normalization of the convolution-before-differentiation sensor, fixed-filter continuity under \(L^1\) convergence, and conditional transfer algebra under explicitly stated gates. The following are not claimed: - a proof of global regularity for the three-dimensional Navier–Stokes equations;- unconditional multidimensional Euler shock stability;- unconditional vanishing-viscosity selection;- ULS-free stability, Type-II exclusion, or convex-integration exclusion;- a complete regular-reflection/Mach-reflection transition law;- physical validation of the CFD or Schlieren protocol. All engineering figures and sample packet values are synthetic templates. Validator success establishes package-internal consistency and reproducibility, not physical validation. ## Current research directions The nearest theorem-facing target is closure of the jump corrector, trace topology, and a uniform finite-scale subsolution estimate in a scalar convex conservation law or a one-dimensional viscous Lax shock. The nearest application-facing target is a held-out wedge/Mach-reflection pilot using fixed-physical-scale grid ladders, uncertainty-audited margins, and independent Mach-stem detection.