\section{Autonomous architecture and continuity} The implementation uses one process. Its state is \[ \Theta_t=(G,\mathcal A,D_t,S_t,T_t,B_t), \] where \(G\) is the fixed interface group; \(\mathcal A\) contains immutable primitive option tables; \(D_t\) stores kernels and their witness DAGs; \(S_t\) stores left/right stabilizers and generators; \(T_t\) is a trie indexing known fragment words; and \(B_t\) is the remaining work grant. The trie is a lookup representation, not an additional learned neural model. \begin{longtable}{P{0.26\textwidth}P{0.65\textwidth}} \toprule Fixed in this implementation & Allowed to change autonomously\\ \midrule\endhead Group law, boundary meaning, primitive options, integer objective, verifier, convolution and subgroup algorithms, work-unit definition. & Newly acquired composite kernels; their witness DAGs; discovered stabilizer subgroups and generator lists; dictionary/trie entries; the decomposition selected for a later word; the orbit representatives used in later convolution.\\ \bottomrule \end{longtable} The parser's algorithm stays fixed. Its selected decomposition changes when the dictionary changes. Separately, symmetry metadata changes the actual convolution search even at the same dictionary and decomposition. This second effect is the defining mechanism; the first is ordinary reusable-library support. \subsection{Complete execution cycle} \begin{enumerate} \item \textbf{Receive a new operator-construction task.} An ordered word of primitive types specifies a fresh composition; a boundary query specifies the eventual task-level use. No human feedback is requested. \item \textbf{Choose a decomposition.} Dynamic programming over the trie chooses a segmentation into the fewest cached atoms. It does not minimize a perfectly known future wall-clock cost; that limitation is measured rather than hidden. \item \textbf{Construct each missing composite.} Compute the reciprocal intersection, enumerate inner cosets and output double-coset orbits using subgroup generators, and minimize over representatives. Reconstruct all boundary costs and child-witness references. \item \textbf{Verify the local construction.} Every finite output split must multiply to its claimed boundary and match its stored cost. The fixed quotient theorem supplies the optimality induction. Independent test oracles check the implementation during the experiment. \item \textbf{Deepen the representation.} Discover exact output stabilizers, starting from inherited ones and using rare-value filtering plus subgroup closure. This is the autonomous metadata update \(S_{t+1}\); it may discover no new symmetry. \item \textbf{Commit or reject.} Charge dictionary publication before making it visible. If construction exceeds its grant, the incomplete acquisition is not published. A consistency assertion rejects the construction and stops that run. Previously committed kernels remain the last valid state. The system does not silently accept an unverified approximation. \item \textbf{Use and retain.} Expand a selected witness at the requested boundary, check every primitive choice against the current role occurrence, and use the retained kernel plus metadata when acquiring later operators. \end{enumerate} The cycle stops on exhaustion of the task stream, a work grant, or a consistency failure. A fully acquired exact kernel is retained even if its discovery overhead later proves uneconomic. There is no implemented guarantee that every retained update improves elapsed time. This is why asymmetric and constant-input controls are essential. A performance-aware admission policy would be an additional mechanism, not something credited to the present code. \subsection{Memory, identity, and repair} Continuity has an explicit meaning. The primitive type, its option table, its role occurrence, and the group-product contract remain fixed. A composite remembers an ordered word and a DAG of child boundary choices. Reusing the same primitive type twice creates two occurrences; it does not make the two choices one event. A quotient may identify equal boundary \emph{costs} while retaining all occurrence-specific evidence needed to execute the result. The prototype uses immutable primitive tables within a run, so word identifiers are sufficient cache keys in that scope. If option costs, available actions, or boundary semantics change, the whole affected dependency subgraph must be invalidated or keyed by semantic version. That cross-version migration is not implemented. Reusing a cache across changed primitives is a concrete failure condition, not an accepted form of identity preservation. For Maciej and Eve's ideal relationship, preserving what matters through transformation becomes neither sentimental wording nor a claim of a persistent conscious self: it becomes exact equivalence of externally relevant values together with a reconstructible history of separate actions. Trust is justified by these checks. The verifier itself remains fixed; this work does not claim autonomous improvement of its verifier or detector-repair procedure. \subsection{What is generated, rejected, or reversed} Candidate improvements are finite composite kernels and their discovered subgroup metadata. They are generated by actual representative minimization and full-table equality checks, not by placeholders or an oracle returning a preferred answer. An impossible boundary retains infinity. A budget-limited transaction is discarded before publication. A false certificate is a correctness failure and must not be used; the diagnostic suite injects such a certificate to verify that independent invariance checking detects it. For a future deployment, rollback means returning to the last committed dictionary and recomputing without the rejected specialization. The prototype demonstrates transaction rejection and exposes the unspecialized exact route; it does not implement a general unattended fault-recovery service. This distinction matters: the experiment measures acquisition efficiency under a fixed trustworthy evaluator. It does not change the definition of success to excuse a faulty update, retrain an evaluator on its own answers, or select a model using final held-out scores. \section{Reference implementation and evaluation protocol} The full benchmark is \texttt{relational\_compiler.py}, approximately 510 lines of standard-library Python. It runs on a CPU and needs no model weights, network connection, GPU, or optional package. The independent minimal implementation in Appendix~\ref{app:core} implements the defining primitive and the successive subgroup ladder directly. The audit program does not import the benchmark and supplies a separate implementation of the algebraic calculation. The final benchmark source was frozen before held-out execution. Development used seeds 1--3. The locked test used seeds 101--120, \(D_{16}\) with 32 elements, and three conditions. In each seed and condition there were 12 training words and 24 distinct full held-out words, with six each of lengths 4, 8, 16, and 32. No held-out full word occurred in training. Repeated primitive pairs and reusable subfragments are intentionally part of the task distribution, not an undisclosed accidental overlap. The test demonstrates transfer to new compositions within this finite interface; it is not transfer to an unseen scientific domain or a newly invented interface. \begin{longtable}{P{0.22\textwidth}P{0.69\textwidth}} \toprule Condition & Generating rule and purpose\\ \midrule\endhead Structured, nonconstant & Primitive costs are automorphic variants of the dihedral construction: zero on more than half the rotations, a unique inexpensive reflection, and larger cost elsewhere. Paired primitives acquire a nontrivial rotation stabilizer. Fresh longer words test reuse on new compositions.\\ Asymmetric & Primitive group costs are sampled without replacement. This removes simple primitive translation symmetries and tests overhead where the selected explanation predicts little gain. Composites are still checked for any symmetry that actually emerges.\\ Constant/saturation & Primitive zero-cost supports contain more than half the group. Paired products reach every boundary at zero cost. This tests whether a simple constant-kernel formula beats the more general method.\\ \bottomrule \end{longtable} \subsection{Controls and rival explanations} \begin{longtable}{P{0.23\textwidth}P{0.68\textwidth}} \toprule Method & What it isolates\\ \midrule\endhead Recursive learned symmetry & Cached exact kernels, discovered stabilizers, quotient composition, and later reuse.\\ Dense, same dictionary & Same word acquisition and segmentation; ordinary exact finite-support pair enumeration. Isolates all symmetry specialization.\\ Inherited symmetry only & Uses the quotient routine but never discovers symmetries beyond inherited metadata. On the main primitive families, initial stabilizers are trivial. Isolates emergent discovery.\\ Fast simple compiler & Same dictionary and segmentation; smaller-support enumeration, certified lower-bound early stopping, and a constant-factor closed form. A stronger comparator than dense search.\\ Frozen compiler & Primitive dictionary only; no persistent acquired composites or discovered composite symmetries.\\ Flat acquisition & New full results can be retained, but learned atoms cannot assist later fresh acquisitions. Within one acquisition, exact composition still operates. Tests cross-task recursive reuse, not the whole symmetry mechanism.\\ Closed-only memory & Retains only the identity-boundary value of acquired composites. Tests the loss of future support information; it can return feasible but suboptimal answers.\\ Sequential target DP & Exact role-by-role dynamic programming; only the requested target is needed in its final layer. It receives and solves the same task stream.\\ Symmetry-aware reference & Deliberately the identical algorithm under a generic description. Equality is an explicit reduction control, not a claimed competitor defeated by renaming.\\ \bottomrule \end{longtable} A further simple best-first pair-enumeration baseline was added after the principal study. The coordinating author had access to the principal results when requesting it. The implementing agent was not supplied those numerical outcomes and did not inspect the principal result files; it developed and checked the comparator only on seeds 1--3, then froze its source before its own evaluation on seeds 101--120. This was a supplementary comparison, not part of the original locked comparator set. It orders candidate pairs by cost, so the first pair reaching an output proves that output's minimum. The locked main implementation was not altered or retuned. \subsection{Resources, evaluator, and acceptance} Each algorithm receives the same total algorithm-operation grant, including setup, all 12 training tasks, discovery, parsing, kernel construction, metadata, witness materialization, validation, and dictionary transactions. The two grants are 250,000 and 2,000,000. The latter completes the intended workload for the principal exact methods and is used to compare acquisition costs. The former tests throughput at a matched grant. A failed acquisition is not counted as success. The meter records named abstract operations, not processor instructions. It covers the major algorithmic work categories; Python allocation internals, hash implementation details, and instrumentation overhead are not individually modeled. Elapsed time covers the instrumented implementation and is reported descriptively. It is not an uninstrumented production benchmark. Retained memory is counted in scalar slots across the full retained DAG, including uncached ancestors, rather than bytes or peak temporary memory. An independent sequential DP computes exact target optima before method evaluation. Its work is recorded separately and an equal full share is also added to each method's total-with-oracle field. The algorithm grant excludes this common benchmark infrastructure; adding the same oracle allowance gives a matched total experimental grant. The verifier, objective, generated instances, and truth values do not change between methods. Failures of the closed-only ablation are reported as failures, not used to revise the oracle. Finally, the subgroup-ladder experiment holds all acquired costs fixed while erasing, inheriting, or rediscovering metadata. It evaluates construction of \(h_2,h_3,h_4\), whose actual values differ. This separates rate of \emph{future operator acquisition} from merely scoring better on a task already learned.