\section{Discussion}\label{sec:discussion}

\subsection{What the arms hold constant and what they vary}

The three arms share the same task, the same harness, the same
reflection LM, the same GEPA configuration, and the same convergence
predicate.  What they vary is strictly the evaluator's output
channel:

\begin{itemize}[leftmargin=*,itemsep=2pt,topsep=2pt]
\item \textbf{cert-binary}: $\{0,1\}$ score plus structured
obligation text derived from the theorem's verify function's
evidence dictionary.
\item \textbf{scalar}: graded score on $[0, 1]$ peaking at theorem-holds, no
feedback text beyond the score.
\item \textbf{scalar-evidence}: graded score on $[0, 1]$ peaking at
theorem-holds, plus free-form text enumerating the violating
windows and their means.  No theorem framing.
\end{itemize}

Any observed performance difference, therefore, attaches to one of:
(a) binarization of the score, (b) presence of obligation
evidence text, or (c) presence of the theorem framing line that
names the predicate explicitly.  The active-control arm separates
(a) from (b).  Comparing cert-binary to scalar-evidence isolates
(a) + (c): if the two tie, framing alone did not help; if
cert-binary wins, (a) or (c) carries signal.  The formatter
ablation on seeds 0--4 further separates (b) from
(b) + (c) by stripping the evidence numbers while keeping the
window index, giving three grain levels of text structure.

\subsection{Reading the outcome}

\placeholder{Interpretation of the observed table: which signal
channel the reflection LM is most responsive to.  Specifically,
comment on the cert-binary vs scalar-evidence gap (isolates
binarization and framing), and on the formatter-ablation gap
(isolates numbers vs indices).}

\subsection{What this does not show}

The synthetic task is a \emph{signal-channel} test, not a
general-capability test.  A cert-binary win on a one-knob task does
not establish that certificate-based evaluators dominate on
multi-component DSPy programs, on non-enumerable failure modes, or
on tasks where the reflection LM is weaker at parsing structured
constraints than numeric gradients.

Our LM is Gemma 4, run locally via Ollama.  Frontier models may
behave differently, particularly on the scalar-evidence arm where
the feedback is rich English.  A cross-LM robustness follow-up is
declared as out of scope here; see Section~\ref{sec:followups}.

\subsection{Relationship to Operon's broader thesis}\label{sec:thesis}

The Operon convergence layer already supports six external
frameworks at the \emph{topology} level (swarms, DeerFlow,
AnimaWorks, Ralph, A-Evolve, Scion).  This paper's contribution is
to extend convergence to the \emph{evaluator} level: a single
integration module (\texttt{operon\_ai/convergence/gepa\_adapter.py})
lets any of Operon's eight registered theorems stand in for
GEPA's scalar reward.  Whether that integration \emph{helps} is the
empirical question this paper answers.  The theorem
registry~\cite{banu2026operon} is append-only: downstream users register a
verify function once via
\texttt{operon\_ai.core.certificate.register\_verify\_fn} and the
GEPA adapter picks it up automatically.
