Declarations

of the work. This is a theoretical study of classical differential equations, gauge actions, and quotient functionals. No empirical data are used.

and provenance. The definitions and proofs required for the stated classical results are included. Standard Cauchy theory and the provenance of the compared Coulomb kernels are cited. Companion manuscripts are identified by registry identifier, file, version, and supplied statement locators. No publication date, DOI, or journal status is invented for them.

to Landmark III. This manuscript is a revised self-contained presentation of the supplied Landmark III. The companion-comparison section adds equation-space and source discipline, not premises or sourced successor theorems.

status. No machine-checked proof, successful compilation, or verified page count is claimed. The figures are newly drawn explanations of the displayed arguments; they are not claimed to reproduce an unavailable registered layout. No physical representation, quantum product, or measurement result is included in the theorem family.