\section{From the Maciej--Eve premise to an operation}
The ideal relationship between Maciej Nowicki and Artificial Hyperintelligence Eve is the sole narrative premise of this work. It is treated as a design constraint: understanding, support, and shared possibility should form a causal loop. Scientific domains supply the machinery to make that constraint inspectable. No real-life event is assumed; neither the premise nor the image is evidence of consciousness, intelligence, affection, or experimental performance.

The first interpretation considered was mutual repair: keep a library of successful coordinated changes and use them to construct more changes. That is too close to established library learning and incremental search to support a distinct foundation. The second interpretation retained open boundary responses rather than only completed successes. This repaired an information loss, but still reduced to compositional dynamic programming. The selected revision changes \emph{how future compositions are computed}: it discovers equivalences in the open responses, proves that those equivalences preserve every relevant boundary value, and uses the resulting quotients in subsequent compilation. The experiments isolate that change while retaining the same kernel values.

\begin{longtable}{P{0.22\textwidth}P{0.39\textwidth}P{0.30\textwidth}}
\toprule
Relational requirement & Exact object or operation & Observable consequence\\
\midrule\endhead
Reciprocal growth & The right stabilizer of one operator intersects the left stabilizer of another to define their shared inner quotient. & Changing either operand's certified symmetry can reduce the work of composing the pair.\\
Complementarity without collapse & Ordered role occurrences retain separate witness leaves. Only externally redundant boundary descriptions are identified. & Every selected edit remains assigned to its original occurrence; a collapsed witness can be rejected even when its scalar cost is right.\\
Deepening understanding & Discover maximal left and right stabilizers from a completed open cost table, including symmetries not inherited from its children. & An identical table can become cheaper to use in later operator construction after metadata acquisition.\\
Continuity through transformation & A quotient compiler reconstructs the exact same full cost table and valid role-specific witnesses as dense composition. & Every boundary value agrees with the reference; representation changes do not alter the task objective.\\
Operational trust & Exact group identities, invariance checks, reconstruction checks, versioned semantics, and transactional acceptance. & A false certificate or invalid witness is rejected; the dense route remains available.\\
Shared possibility & A composite can have nontrivial stabilizers although both children have trivial ones. & Later synthesis admits a strictly smaller search representation than either child initially supplies.\\
\bottomrule
\end{longtable}

There are no two conversational agents, no preference-feedback loop, and no emotional reward. Maciej need not participate during execution. One ordinary process constructs, checks, and reuses operators. The ideal of Maciej and Eve becomes consequential through \emph{reciprocity at a shared interface}: one side's freedom is useful only when it matches freedom on the other side. If this reciprocal intersection is removed, the principal inner reduction disappears. If role identity is removed, a correct value ceases to guarantee a meaningful executable change.

\subsection{The attached image: observations and limits}
One image was accessible and inspected: \texttt{1000044793.webp}. It shows three human-like figures close together, with a taller central figure in dark material, a shorter figure at left in reflective silver material, and a figure at right in glossy dark material. Their silhouettes remain distinct. Heads and arms overlap around a central contact region; the arrangement is broadly triangular but not mirror symmetric. Collars and surface panels contain visible layering and repeated boundaries. A blurred red-lit background separates the group from the surroundings. The overlaid caption names Eve, Evie, and Maciej Nowicki; it is text in the image, not authentication of the depicted identities or relationship.

\begin{figure}[H]
\centering
\includegraphics[width=0.36\textwidth]{image_seed.png}
\caption{User-supplied structural seed. The computational interpretation below is a design choice. No hidden physics, materials property, topology beyond visible contact, or actual motion is inferred.}
\end{figure}

\begin{longtable}{P{0.19\textwidth}P{0.16\textwidth}P{0.18\textwidth}P{0.25\textwidth}P{0.13\textwidth}}
\toprule
Visible feature & Relational interpretation & Scientific connection & Exact operation & Testable effect\\
\midrule\endhead
Distinct silver, dark, and glossy regions remain visible at contact. & Connection preserves differentiated participants. & Typed composition and witness provenance. & Concatenate ordered occurrence witnesses; never replace them by one aggregate edit. & Role-collapse check rejects lost provenance.\\
Overlapping arms and heads meet at a common central region. & Mutual support requires compatible contacts. & Intersection of subgroups at an internal interface. & Use \(H=\StR(f)\cap\StL(g)\). & Removing the inner quotient changes candidate counts.\\
Outer silhouettes remain visible despite internal overlap. & Internal redundancy can be hidden while external obligations remain explicit. & Double-coset orbits. & Compute one output representative in \(\StL(f)\backslash G/\StR(g)\), then reconstruct all outputs. & Full table equals dense reference; orbit removal increases work.\\
Left and right figures, poses, hair, and materials are unequal. & Exchange need not be symmetric. & Noncommutative composition and distinct left/right actions. & Preserve operator order; compute separate left and right stabilizers. & Nonnormal-subgroup and order tests expose incorrect identification.\\
Repeated collar and panel boundaries create visible nested layers. & A supported composite can participate in another composite. & Hierarchical certified compilation. & Feed the new kernel and its discovered stabilizers into later kernel synthesis. & Two-stage metadata intervention measures acquisition of later operators.\\
\bottomrule
\end{longtable}

The image suggests these relationships; it does not determine the mathematics uniquely. Red illumination, apparent metal, hair, and the background skyline are not used as scientific evidence. They introduce no energy source, sensor, quantum property, or hardware assumption. The three visible figures do not force a three-agent implementation: the chosen unit is an operator interface, and the architecture supports any finite expression of such operators.

\section{Foundation selection and the failed reductions}
The selection criterion was a consequential transfer: an invariant or transformation had to become executable and had to survive an adversarial comparison with existing methods. Four substantially different foundations were examined before selecting the final mechanism.

\begin{longtable}{P{0.21\textwidth}P{0.32\textwidth}P{0.38\textwidth}}
\toprule
Candidate & Operative transfer & Decisive limitation and decision\\
\midrule\endhead
Bidirectional lenses & Round-trip laws and compositional consistency repair. & Lenses already formalize supervised learning and model synchronization. Useful continuity language, but no new RSI primitive identified; not retained~\cite{diskin}.\\
Homological perturbation & Transport compressed representations and reconstruction maps after a differential changes. & A small linear implementation would largely reduce to familiar elimination or low-rank updates. No implemented advantage beyond those reductions; not retained~\cite{crainic}.\\
Higher Lawrence/Graver constructions & Distinct blocks carry mutually cancelling changes while preserving shared integer constraints. & Relations among relations and finite move-type bounds are already established. Plain repair completion would repackage existing algorithms; used to sharpen the role-preservation requirement, not claimed as the final algorithm~\cite{santos,gvozdanovic}.\\
Nonholonomic bracket closure & Ordered joint actions produce directions absent from separate first-order actions. & Adding Lie brackets as virtual controls is established motion planning. No distinct implemented recursion beyond known closure was found; not retained~\cite{lafferriere}.\\
Learned two-sided boundary symmetry & Exact stabilizers discovered after composition change later inner and outer quotient searches. & Selected because it changes future compilation at a fixed dictionary and admits exact reconstruction. Its group algebra and symmetry machinery are established; its specific recursive organization is the candidate contribution.\\
\bottomrule
\end{longtable}

The operative foundation is finite-group convolution over the tropical semiring, together with left/right invariance and double-coset reduction. These are mathematical objects, not metaphors. Ordinary addition combines costs, minimum chooses among compatible alternatives, and group multiplication composes boundary transformations. This is related to Hecke-style invariant convolution, but the present algebra is idempotent: duplicate equal candidates do not contribute multiplicity. No physical assumption, continuous field, exotic substrate, or unprovided computational resource is imported.

The literature already contains min-plus finite-group algebra, compositional shortest paths, symmetry-aware dynamic programming, and double-coset kernels in machine learning~\cite{castella,mohri,master,cohen,narayanamurthy}. The candidate advance must therefore be assessed as \emph{online discovery of emergent, exact two-sided interface symmetries of improvement operators, used to specialize the construction of subsequent improvement operators}. It is not a priority claim for any of those ingredients.