Inspect the verification and its domain

Executable checks accompany the proofs. Their exact, numerical and negative-control roles remain separate, as do the local reconstruction theorem and questions about global geometry or physical selection.

20. Verification, limitations and reproducibility

The public package accompanies the mathematical proofs with executable symbolic and numerical checks. All six input archives were extracted and their recorded payload manifests checked; each nested predecessor archive matched the corresponding uploaded archive byte for byte. Their scientific checkers were then executed unchanged against the consolidated environment. Each complete result object reproduced its retained baseline object. The checks use SymPy [7]; the numerical integral diagnostics use mpmath.

The separately written final audit checks the consolidated section conventions, reverse and variation maps, action sign, finite boundary response, opposite exchange, original density normalization, electromagnetic energy, optical endpoint relation, mass join, turning factorization and approximation counterexamples. It is an additional implementation by the same AI-assisted workflow, not independent external review. Its forty groups overlap mathematically with earlier claims and are not forty new theorems. A negative control passes when an explicitly invalid shortcut is detected, not when an erroneous equation is accepted.

Eight additional high-precision diagnostics use $65$ decimal digits and strict inequalities comparing regularized light or orbit integrals with their leading values and the analytic bounds. They are separate from the $570$ exact groups. They are not interval-arithmetic certificates or empirical tests; the remainder proofs are in the article. The recorded precision and parameter values, rather than an unstated equality tolerance, specify these inequality diagnostics.

The supplied environment was Python 3.13.5, SymPy 1.14.0 and mpmath 1.3.0. Reproduction instructions, source hashes, pinned mathematical dependencies, complete results, logs, theorem and assumption registers, source provenance and a clean-extraction receipt are included. Another Python patch version may change environment metadata without changing scientific checks; the runner reports both full-object and scientific-content comparisons rather than concealing that difference.

The audit preserves substantive development corrections. An early fluid control incorrectly treated an off-normalization decomposition as failing, although only its projection interpretation fails. Another chosen frame/potential control accidentally gave zero bare antisymmetric derivative and was replaced by one that exposes the missing comparison term. A symbolic integration timeout in an earlier finite-flux computation was resolved by an explicit primitive and endpoint proof. These are recorded as development events, not relabelled as passing historical executions. The current final suites and final-publication audit all complete without failed groups.

The public package does not contain private research protocols, their prompts, inaccessible conversation references, or third-party page scans. The essential proofs and definitions are supplied here; the original scans remain cited sources, not redistributed supplements. Checksums establish byte integrity, not theorem validity. No result is described as proof-assistant certified or independently peer reviewed.

Limits of the reconstruction

The principal limits are mathematical and physical, not merely computational. Native admission does not select dimension, signature or action. All-state coverage is local or atlas-based on smooth nondegenerate fields; it is not a global coframe, horizon-extension or topology theorem. Matter solution correspondence is not general well-posedness for arbitrary equations of state. Electromagnetic stress does not determine its primitive field, and Maxwell’s equations alone do not supply a charge-motion law. The chart-tagged gravitational current is not a coordinate tensor or an observer-independent positive density. Bulk variational agreement does not identify arbitrary boundary ensembles, symplectic structures or charges.

The weak-field result is a controlled finite-order calculation with the stated derivative norms, not a general uniform-in-time convergence theorem. The exact radial completion is static and spherical. Its optical and orbital error bounds apply to that family and to the declared variables, not to unspecified stellar structure or fitted astronomical elements. Historical numerical claims are not new observations. None of the reconstructed nonlinear terms has been eliminated; no quantum compatibility or ultraviolet improvement is established or required for this paper.