15.13.11 Discovering a withheld composition law

15.13.11 Discovering a withheld composition law

DIAL-URL–3 synthesized expressions from supplied constructors and supplied composition laws. DIAL-URL–4 withholds one level more: the seed sketch still contains Delay, Time, and Mix, but treats their pairwise interaction as ordinary and order-insensitive. Some generated worlds instead contain an oriented comparison cell

The learner is not told which pair \((A,B)\) is exceptional or which orientation is supported. It receives only the generic admissible edit form Law(A>B), together with ordered compound probes. Thus the task is to detect that the present sketch is inadequate, localize the defect to two known constructors, instantiate a new comparison cell, and orient it.

The notation is deliberately presentation-level. In this experiment \(\chi _{A,B}\) is a typed generator whose semantics is the registered directional counter-witness oracle. The checker does not quantify over all objects and morphisms of a category, verify naturality squares, or establish the coherence axioms of a distributive law. Calling it a “law” is therefore shorthand for an admitted relation in this finite experimental sketch.

The preregistered experiment crossed ten worlds, three probe budgets, and 400 replications per cell. Four controls required no law; six targets contained a withheld oriented law, including two three-constructor composites. A theory was admitted only at posterior probability at least 0.95 and after its expression transport and comparison cell type-checked. Table 15.13.11 separates exact recovery, wrong admission, abstention, false-law admission, constructor-pair recovery, and orientation recovery across the registered probe budgets.

Table 15.9. DIAL-URL–4 macro recovery. Exact, wrong, and abstention rates average over all ten worlds; false-law rates average over the four no-law controls; law-pair and orientation rates average over the six law-bearing worlds.

Seven of eight preregistered endpoint families passed. At fifty probes, exact theory recovery exceeded 90% in every world, wrong admission stayed below 5% in every cell, false-law admission stayed below 5% in every control, and all certificates were valid. The single miss was deliberately strict: every law-bearing world had to recover the correct unordered constructor pair at least 95% of the time. The hardest target, Time(Mix(Delay(Delay(Base)))) with a Time>Mix law, reached 93.5% pair recovery and 93.25% exact recovery. It abstained in 5.75% of trials and made a wrong admission in 1%.

This boundary is informative. In two-constructor targets the missing pair and orientation were recovered between 99.5% and 100% of the time. With three constructors, however, six oriented comparison cells compete while the system must also recover the base expression. Active evidence selection remained conservative, converting uncertainty mostly into abstention rather than false law invention, but did not meet the uniform 95% pair-recovery requirement.

Experiment: DIAL-URL–4: Withheld Comparison-Law Discovery. Input: known coalgebraic constructors, a generic one-cell repair schema, and marginal plus ordered compound probes.
Construction: select the constructor pair and orientation of a new comparison cell.
Admission: posterior at least 0.95 plus typed expression-transport and comparison-cell certificates.
Outcome: 98.58% macro exact recovery at fifty probes, with 0.13% false-law admission on controls; seven of eight endpoint families passed.
Epistemic status: not admitted at the experiment level because one preregistered endpoint family failed. Individual trials passing their posterior and type gates construct a finite presentation generator with registered oracle semantics, not a verified natural transformation or distributive law.
Boundary: the repair schema is supplied, and one three-constructor world missed the preregistered 95% pair-recovery threshold.

DIAL-URL shifts the object of evaluation beyond comparing RL algorithms on a fixed benchmark. The reported ladder reaches registered type selection, grammar-term construction, and calibrated trial-level recovery of a finite comparison generator; URL–4 remains an experiment-level near miss under its frozen endpoint conjunction. Its intended stronger artifact is a conservative extension of the theory of sequential behavior: one that retains established models, introduces a justified coalgebraic constructor or composition law, verifies the required categorical laws, and makes additional classes of dynamical systems representable, experimentally distinguishable, and reusable for subsequent prediction or control. No reported rung yet establishes that full artifact.