The observability caveat

The central claim contains a meta-level assumption that should be made explicit: a failure of compositionality must be observable through some available diagnostic. It is not enough for a declared diagram to fail in the mathematical model. The learning system, its designer, or an accompanying audit process must be able to obtain a witness that distinguishes the failure from ordinary variation.

Let \(O(D)\) denote the typed obstruction associated with a realized diagram \(D\). In practice the learner does not inspect \(O(D)\) directly. It sees an observation profile

where the maps \(\omega _\lambda \) may be norms, tests, probes, comparison maps, counterfactual outcomes, human judgments, or instrument readings. After quotienting decision-null variation, the observer family should separate the obstruction classes on which the system is authorized to act:

This is an observability condition, not a theorem supplied by LINCS. If every registered observer vanishes on a genuine actionable obstruction, the system has a structural blind spot. No repair rule driven only by those observations can be expected to correct it.

¹1. Observability is always relative to a declaration, a probe family, and a quotient. A diagnostic may detect that something failed while remaining unable to identify which typed obligation failed or what caused it. ↩

Reinforcement learning makes the distinction concrete. An agent playing backgammon may know the legal moves and eventually observe whether it won or lost. That terminal signal does not by itself reveal the compositional structure of a Bellman equation, much less localize a failed Bellman factorization. Bellman observability requires additional representational commitments: states or predictive histories, rewards, transition structure, and comparisons between one-step and continued value. The game supplies an outcome; the learning architecture supplies the structural observer.

Natural cognition raises the deeper question of where such observers and declarations originate. The debate surrounding universal grammar asks how much language-specific structure is available prior to, or constrains, language learning [ Chomsky , 1965 ] . This should not be identified uncritically with semantic compositionality, but it illustrates the general problem: a learner cannot diagnose every structural regularity unless some representational resources make that regularity expressible.

Developmental studies offer a related example. Spelke’s core knowledge program argues that human cognition is founded in part on early-developing, domain-specific representational systems [ Spelke , 2000 ] . A later synthesis identifies systems for objects, actions, number, and space, with a possible further system for social partners [ Spelke and Kinzler , 2007 ] . On this account, infants do not begin with an undifferentiated stream and learn every object and relation from scratch. Early structural resources constrain what can be tracked, compared, and combined. Whether a particular resource is innate, learned, culturally scaffolded, or engineered is an empirical question; LINCS requires only that the operative structural promise and its observers be declared.

There are consequently four possible sources of observability:

1. the environment supplies an explicit witness, such as a violated rule or failed test;

2. a system designer registers structural probes and audit maps;

3. a learner acquires observers from repeated intervention and repair;

4. an organism or architecture begins with structural priors that make certain failures salient.

The third possibility opens an important outer loop. A LINCS system can learn not only how to repair under a fixed observer family, but also where its current observers fail to predict admission outcomes. It may then propose a new probe, refine its cover, or enlarge its sketch. The new observer must itself be validated on held-out failures; otherwise the system merely changes what it is able to notice in order to certify its preferred repair.

The dual issue is controllability. Even if the probes reveal and localize an obstruction, the registered repair language may contain no admissible path from the current realization to one satisfying the declaration. A system can therefore be observable but uncontrollable: it can diagnose a failure yet must abstain from repairing it. Conversely, a system with many available interventions but no observer capable of discriminating their relevant effects is controllable only in a dangerous, unlicensed sense. Viewed as a control system, LINCS makes both assumptions relative and typed: the probe family determines which obstruction classes are observable, while the repair language determines which admissible classes are reachable. Admission then asks whether the reached realization is supported, stable, and semantically licensed.