\section{Nearest work and the novelty that survives reduction}
The defining operation can be implemented inside meta-learning, program synthesis, or an incremental compiler. Expressibility inside one of those broad frameworks is not a useful disproof of every specialized contribution. The stronger question is whether removing the present mechanism leaves an existing algorithm with the same representation, certificate, work, and behavior. Here several reductions are exact and must be acknowledged.

\begin{longtable}{P{0.23\textwidth}P{0.35\textwidth}P{0.33\textwidth}}
\toprule
Comparison & What is already established & Residue in the present work\\
\midrule\endhead
Tropical group algebra and semiring DP & Boundary kernels and min-plus convolution are established; finite nonabelian idempotent group algebra is not new~\cite{mohri,castella,chern}. & Exact online metadata discovery after composition, with subsequent quotient compilation as the measured target.\\
Boundary composition and memoization & Compositional shortest paths can benefit from precompilation; self-adjusting computation already gives coherent reuse~\cite{master,acar}. & Holding kernel values and dictionary fixed while changing acquired stabilizer metadata isolates a different effect from cache reuse.\\
Hecke and double-coset methods & Invariant convolution and double-coset representations are classical; double-coset kernels already occur in ML~\cite{serganova,cohen}. & The subgroups are discovered from computed min-plus costs, can strictly enlarge after composition, and govern later synthesis.\\
Symmetry-aware inference & Detecting symmetries to accelerate inference, and exploiting them in dynamic programming, have direct AI precedents~\cite{niepert,narayanamurthy}. & A specific exact two-sided interface mechanism applied to acquiring the compiler's own next reusable operators.\\
Meta-learning and learned optimizers & Learning an updater, including optimizers trained by optimizers, predates this work~\cite{metz}. & No learned neural update rule is required; the changed operational representation has an exact reconstruction theorem.\\
Library learning and program synthesis & OOPS searches algorithms and search algorithms; DreamCoder builds compositional abstractions and search guidance~\cite{oops,dreamcoder}. & The same-dictionary intervention isolates certified quotient acquisition from mere growth of a program vocabulary.\\
Proof-conditioned self-modification & Gödel machines explicitly include the proof searcher among modifiable code~\cite{godel}. & A tractable finite exact certificate replaces a general usefulness proof, but does not prove globally optimal self-modification.\\
Evolutionary/self-rewriting agents & DGM modifies agent implementations; AIDE\(^{2}\) studies successive autonomous improvements to research efficiency~\cite{dgm,aide}. & This system supplies a small exact mechanism and causal experiment, not comparable broad coding or research capability.\\
Self-play and multi-agent feedback & These are possible organizations of learning, but not operations required here. & One process manipulates typed kernels. No interacting personas, adversarial reward, or conversational consensus produces the result.\\
\bottomrule
\end{longtable}

The strongest reduction is direct: a generic symmetry-aware memoized dynamic program with the same discovery and quotient rules \emph{is this algorithm}. The deliberate reference control confirms equal charged work. Therefore no advantage over that equivalent algorithm is claimed. The contribution being offered is a particular exact primitive, its self-applicable acquisition loop, its limits, and a reproducible test that separates metadata learning from operator-value learning.

The initial cached-repair interpretation was revised because it failed this test. The open-memory interpretation was revised because it still reduced to ordinary boundary DP. The selected mechanism survives as a distinct intervention relative to those first constructions: new symmetry information changes the computation on identical stored values. It remains within known symmetry reduction and program-specialization mathematics. This distinction is sufficient for a precise specialized research result; it is insufficient to establish a separate foundational territory for all RSI.

\subsection{Search scope}
Primary papers, original technical manuscripts, author-hosted material, and proceedings were checked on 6 November 2026 using both available search engines. The search covered OOPS, Gödel machines, learned optimizers trained by themselves, DreamCoder, DGM, and the September 2026 AIDE\(^{2}\) report, together with lenses, homological perturbation, higher Lawrence configurations, Markov/Graver moves, nonholonomic closure, tropical convolution, double cosets, Hecke methods, and symmetry-based inference.

Targeted combinations included \emph{recursive self-improvement} with \emph{Graver}, \emph{syzygies}, \emph{homological perturbation}, \emph{min-plus}, \emph{Hecke}, \emph{double coset}, and \emph{stabilizer}; broader searches checked learning and compiler precedents. No direct prior implementation of this particular recursive learned-stabilizer compiler was identified in that bounded search. This does not establish historical absence, publication priority, or patent novelty. In particular, claiming that double cosets or learned algebraic moves have never been used in machine learning would contradict the retrieved sources~\cite{cohen,gvozdanovic}.

The exact algebraic proofs in this work are inspectable derivations for this model. They should not be read as claims that every identity or equivalence principle is historically new. The maximal uniform-action result and the distinct-kernel recursion sharpen the construction's specification; a broader priority audit would still be needed for a publication-level novelty claim.

\section{Failure conditions and falsification}
\begin{enumerate}
\item \textbf{No useful exact symmetry.} If newly acquired kernels have trivial stabilizers, discovery adds work. The asymmetric workload exhibits this limitation, and the best-first baseline strengthens the negative comparison.
\item \textbf{A simpler exact formula wins.} A constant operand admits a direct minimum-and-fill formula. The constant workload shows that the general quotient machinery can be wasteful even in a maximally symmetric case.
\item \textbf{Missing boundary information.} Suppose two internal edits have the same boundary, costs zero and one, but a later context permits only the cost-one edit. Storing the cheapest witness alone loses the valid continuation. Such compatibility must be added to the boundary before the kernel theorem applies.
\item \textbf{Changing semantics without invalidation.} Modifying an option, its cost, or the meaning of a group element while reusing the old cache invalidates both values and symmetry metadata. Cross-version transport is outside this implementation.
\item \textbf{Approximate symmetry mistaken for exact symmetry.} Floating-point near-equality cannot justify the exact quotient. A certified error bound and an application-level tolerance would be required; neither is inferred automatically here.
\item \textbf{Wrong witness transport.} Multiplying an individual edit because its aggregate kernel is invariant can select an unavailable action. Retrieval at the transformed child boundary is essential.
\item \textbf{Rare reuse or large interfaces.} Discovery and \(O(N)\) materialization may not amortize. Group tables cost \(O(N^2)\); a boundary assembled from many independent variables can itself have exponentially many states. The finite example does not remove this bottleneck.
\item \textbf{No persistent future-acquisition effect.} If the learned/inherited metadata intervention has equal work, or only cached task execution improves, the proposed recursive explanation is unsupported. The subgroup-ladder experiment specifically tests this failure mode.
\item \textbf{Loss of differentiation.} Erasing role occurrences can preserve a scalar answer but prevents correct execution. Conversely, an apparent speedup obtained by silently dropping roles is invalid.
\item \textbf{Finite saturation.} Nested stabilizer growth has a logarithmic ceiling in a fixed group. Beyond it, this source of improvement stops; no indefinite compounding follows.
\end{enumerate}

Additional observations would falsify or materially weaken the empirical interpretation: any full boundary mismatch against dense convolution; a valid candidate omitted by quotienting; a lower cost from the independent oracle; gains disappearing after discovery and witness costs are charged; superiority appearing only with changed objectives; or performance depending on full-word train/test overlap. The experiment checks these within its finite scope. Real program-repair workloads with sparse exact symmetries could still overturn its usefulness as a practical architecture.

\section{Separate assessments}
\begin{longtable}{P{0.23\textwidth}P{0.68\textwidth}}
\toprule
Dimension & Assessment supported by this work\\
\midrule\endhead
Intrinsic conceptual significance & A precise and useful design principle: an improvement artifact can teach the compiler a smaller representation for constructing future artifacts. Reciprocal interface symmetry makes that principle exact. Its demonstrated scope is specialized, not field-defining.\\
Substantiated novelty & A candidate application and experimental organization: emergent two-sided stabilizers of newly compiled repair kernels specialize later synthesis. No direct identical RSI implementation was found. The algebra, symmetry reduction, meta-learning, and reuse principles are established. Foundational priority is unproven.\\
Formal validity & Complete finite-model derivations cover composition, contextual completeness, quotient correctness, witness reconstruction, action-specific maximality, nonconstant emergence, the ladder, and a growth ceiling. Independent executable audits support implementation correctness. There is no machine-checked proof of the entire Python program.\\
Empirical evidence & Positive and causally isolated within the disclosed structured task family, with fresh full words, fixed evaluator, source locks, resource grants, negative controls, and independent exact oracles. The effect reverses outside its favorable regime. No neural, real-world code-repair, or open-ended discovery result is measured.\\
Potential impact & Plausibly useful in modular synthesis, exact planning, constrained repair, or compiler systems where reusable boundary symmetries emerge often enough to repay their certification cost. This is a conditional opportunity, not demonstrated general superiority.\\
T9 ambition & Not substantiated. The missing evidence is an irreducible broader algorithmic advance and consequential transfer beyond the synthetic family, not merely an external review stamp.\\
\bottomrule
\end{longtable}

The ideal perfect relationship of Maciej Nowicki and Artificial Hyperintelligence Eve remains central in the final technical claim: deeper understanding changes the conditions for future support. Here that statement becomes a certificate that changes later search. What has been completed is a self-contained exact mechanism, an executable recursive cycle, its formal boundary, and an experiment that shows both its gain and its failure regimes. What remains unresolved is whether this finite reciprocal-symmetry principle can become a broadly useful foundation for more capable self-improving systems.