ifc-0214

15.13.10 From named families to functor expressions

DIAL-URL–3 removes the four model-class names. The learner instead receives the recursive grammar

\[ E ::= \mathsf{Base} \mid \mathsf{Delay}(E) \mid \mathsf{Time}(E) \mid \mathsf{Mix}(E). \]

The delay constructor may recur to depth three; time and mixture may occur at most once. Registered distributive laws normalize equivalent construction orders to

\[ \mathsf{Time}^{?} \bigl(\mathsf{Mix}^{?} (\mathsf{Delay}^{d}(\mathsf{Base}))\bigr), \qquad 0\leq d\leq 3. \]

Best-first grammar expansion produces sixteen canonical expressions. Their prior mass decreases exponentially with constructor count, so unsupported structure is penalized rather than rewarded. Active lag, holding-time, and mixture probes update a posterior over expressions. Admission requires posterior probability at least 0.95 and a checked transport chain from the seed expression. Thus the output is not the label “partially observable semi-Markov model,” but an explicit construction such as

\begin{equation} \begin{aligned} \mathsf{Base} & \longrightarrow \mathsf{Delay}(\mathsf{Base}) \longrightarrow \mathsf{Delay}^{2}(\mathsf{Base})\\ & \longrightarrow \mathsf{Mix}(\mathsf{Delay}^{2}(\mathsf{Base})) \longrightarrow \mathsf{Time}(\mathsf{Mix}(\mathsf{Delay}^{2}(\mathsf{Base}))). \end{aligned} \end{equation}

15.1

Eleven evaluator worlds test direct and redundant base behavior, delay depths one through three, the two single constructors, three two-constructor composites, and the expression in Equation (15.1). The last expression is never supplied as a named alternative; it exists only because the grammar composes its constructors. Each world has 500 independent replications.

World	Synthesized target	Exact	Wrong	Abstain
Direct base	Base	98.6%	0.4%	1.0%
Redundant hidden alias	Base	99.0%	0.0%	1.0%
Delay depth 1	Delay(Base)	94.0%	0.0%	6.0%
Delay depth 2	Delay(Delay(Base))	93.0%	0.4%	6.6%
Delay depth 3	Delay(Delay(Delay(Base)))	97.8%	0.2%	2.0%
Time	Time(Base)	99.6%	0.2%	0.2%
Mixture	Mix(Base)	98.6%	0.2%	1.2%
Delay + time	Time(Delay(Base))	95.0%	0.0%	5.0%
Delay + mixture	Mix(Delay(Delay(Base)))	96.2%	0.6%	3.2%
Time + mixture	Time(Mix(Base))	98.8%	0.4%	0.8%
Unnamed triple	Time(Mix(Delay(Delay(Base))))	96.8%	0.2%	3.0%

Table 15.8. DIAL-URL–3 canonical expression recovery after thirty probes. Every row contains 500 replications. The target column displays generated grammar terms, not model-class labels.

All preregistered endpoints passed. Macro exact synthesis rose from 8.84% at ten probes to 79.38% at twenty and 97.04% at thirty; corresponding wrong admission was 0.29%, 0.71%, and 0.24%. Every world exceeded 90% exact recovery at the largest budget. The unnamed triple reached 96.8%, was never collapsed to Base, and every admitted expression carried a valid finite transport certificate. The redundant hidden-state world retained Base: simulator multiplicity alone did not induce theory extension.

This result is stronger than choosing a family name but remains bounded. The grammar, probe likelihoods, complexity prior, and distributive normalization laws are supplied. The certificate checks a typed expression chain, not naturality in an arbitrary category. Genuine transformational construction begins when a persistent obstruction cannot be expressed by this grammar and forces a new constructor or composition law.

Experiment: DIAL-URL–3: Behavior-Functor Grammar Synthesis. Input: a recursive typed grammar, probe semantics, and a seed behavior expression.
Construction: expand and normalize grammar terms; actively test the remaining structural distinctions.
Admission: posterior at least 0.95 plus a checked transport chain from Base.
Outcome: 97.04% macro exact synthesis at thirty probes; 96.8% on an unnamed three-constructor composite.
Epistemic status: all preregistered endpoints passed; admitted grammar-term construction inside a supplied language, not language extension.
Boundary: constructors and their composition laws are supplied; no new primitive or distributive law is invented.