\section*{AI use, human understanding, and verification limits}
The research direction, scope, and revision requests were provided by Sida He.
OpenAI ChatGPT/Codex agents generated most of the mathematical derivations,
manuscript text, numerical implementation, and Lean formalization, and performed
automated literature searches and internal checks. This work is therefore
disclosed as primarily AI-generated text, rather than merely language editing.

At the preparation date of this version, the author reports that he does not
understand the mathematical proofs and has not independently verified them.
The intended Hexagon disclosure is \emph{Human understanding: no parts}, with
respect to the proofs. No independent human expert verification or external
peer review of the proofs is claimed. Internal reviews by additional AI agents
do not constitute independent human mathematical review.

The Lean checks described above cover selected algebraic identities and
finite-step arguments only. They do not formalize the full PDE convergence
theorems or establish their analytic hypotheses. Numerical experiments are
implementation checks and illustrations, not proofs. The original proof source,
dependency pins, numerical results, and recorded build logs are retained in the
companion reproducibility material. This preparation revision does not claim a
new Lean compilation or a new numerical experiment. The manuscript is offered
for critical scrutiny; its correctness and novelty are not certified by these
disclosures or by repository acceptance.

\paragraph{Relationship to companion manuscripts.} The other parts of this series are Part~II \citep{partii2026}, and Part~III \citep{partiii2026}. They are cited as unpublished companion manuscripts; no public identifier or acceptance is claimed. Each part has its own hypotheses and scope. The series is not asserted to be a chain of special-case theorems.

\paragraph{Availability of reproducibility material.} A separate versioned package has been prepared with the Lean source, dependency lockfiles, numerical code, and archived results. No public repository URL or archive identifier has been assigned to that package at the preparation date of this manuscript. The mathematical source submitted to Hexagon does not itself contain the Lean project.