16.8 Ontology extension

A transformational episode may require a new latent variable or intervention surface not exposed by the original interface. Such proposals must distinguish extending a scientific theory from merely reverse-engineering simulator code.

Ontology extension is a joint construction. CLIC proposes why a latent object would resolve intervention failures. OPTIC supplies an estimator or probe that makes the object operational. RELIC tests whether conditioning a decision on it changes admitted behavior. The proposed object is admitted only if these roles agree on its scope and held-out consequences.

SGTE–4: constructing a mediator-shaped latent factor.

SGTE–4 tests this requirement in a controlled sketch-extension world. The base sketch has twelve intervention objects, ten outcome objects, and sixty measured direct edges sampled from the 120 possible intervention–outcome responses. It has no object through which unmeasured responses can factor. Thirty further edges are sealed for admission and thirty for confirmation. All measurements contain independent Gaussian noise with standard deviation \(0.15\).

The registered extension language adds one object \(Z\) and arrows

The vectors \(a\) and \(b\) are constructed from discovery edges by alternating least squares; no candidate factor is enumerated in advance. This is an algebraic factorization semantics: the experiment does not establish that \(Z\) is a causal mediator, nor that either displayed arrow denotes a causal mechanism. Three world families separate construction from habitual extension. Closed worlds contain no systematic residual and should retain the old sketch. Positive worlds have one withheld rank-one factorization. Unsupported worlds have two independent components and therefore require at least two factors, outside the registered language.

The registered nonrealizability gate is correspondingly narrow. On sealed admission edges, the old zero-residual sketch and a capacity-matched additive row-plus-column model must both fail by the preregistered margin before the one-factor extension is opened. This rules out the registered fixed-theory repairs; it does not prove that every possible model of the old ontology is inadequate.

Across 200 worlds from each family, the learner makes the correct structural decision in \(98.7\% \) of the 600 trials. It constructs \(Z\) in \(96.0\% \) of positive worlds, chooses no change in every closed world, and abstains in every rank-two world. The false-extension rate on closed worlds is zero. On sealed confirmation edges, the admitted factor improves prediction by a median \(78.1\% \) over the stronger of the zero-residual base sketch and a capacity-matched additive row-plus-column model. The recovered intervention and outcome factors have median absolute cosines \(0.9968\) and \(0.9974\) with the withheld factors, up to the unavoidable sign and scale gauge.

The certificates distinguish three claims. Syntactic conservativity is exact because the new response branch is added without changing any old object, arrow, equation, or base prediction; this is not a claim of causal conservativity. Irreducibility is relative to the registered language: deleting \(Z\) or replacing it by the capacity-matched additive model loses the sealed prediction gain, while a single \(Z\) is rejected when the world requires two independent factors. Predictive value is measured only on edges unavailable during both construction and admission.

Experiment: SGTE–4: latent-factor construction. Withheld structure: one object \(Z\) and a rank-one factorization \(I\to Z\to O\).
Data: 60 discovery, 30 admission, and 30 confirmation edges; 600 worlds.
Structural decisions: \(98.7\% \) overall accuracy; \(96.0\% \) construction; \(100\% \) closed specificity; \(100\% \) unsupported abstention.
Confirmation: \(78.1\% \) median gain over the stronger fixed baseline; factor cosines \(0.9968/0.9974\).
Certificates: exact conservativity, zero closed-world false extension, and rank-two rejection.
Verdict: admitted as a mediator-shaped factorization inside a supplied one-factor linear grammar, not as a discovered causal mediator or unrestricted ontology discovery.