Competence, Constraint, and Verification¶

Status: Working synthesis and research programme. Definitions are local to this page unless linked to an existing foundation. Empirical claims are marked and sourced; analogies are not identities.

Question: How should explanatory credit be divided when a large behavioral capacity becomes visible after a small change in constraint, embodiment, selector, interaction history, or verification protocol?

This page integrates source-verified work on competence under constraint, interaction-grounded semantics, verification, and identity under revision. It is not a conference summary. The provenance ledger and bibliographic decisions are recorded in the Related Work Map.

1. Audit result: the generator is not enough¶

An earlier version of this repository treated a generic “generator” as its foundation. The foundations audit retired that move: a transition kernel, proposal process, program, policy, and historical mechanism are different typed objects. A trace can constrain a declared model family without uniquely recovering any of them.

The independent source audit does not justify restoring the old foundation. It does strengthen a narrower diagnosis:

Working thesis. Expressed competence is a property of a declared process in a declared coupling, under constraints, history, tasks, tests, and an observer lens. A compact intervention can expose a large capacity without being the sole source of that capacity.

This is weaker than saying that intelligence already exists in every rich substrate. It also resists the inverse error: attributing all competence to learned weights or a central controller while treating the body, environment, and evaluator as neutral plumbing.

2. Why the proposed seven-term equation needs revision¶

The proposed hypothesis was \(\mathcal C_{\mathrm{obs}}=F(G,S,B,\Gamma,I,V,L)\). It is useful as an audit checklist but not as a foundation:

\(G\) has no type. A transition law, random feature map, developmental process, and hypothesis proposer cannot be exchanged without changing the claim.
\(S\) is causally ambiguous. A learned adapter, action policy, experimental intervention, and observer projection do different work.
\(I\) conflates interaction with the history retained from interaction.
\(V\) already denotes viability constraints in the repository's situated stack; reusing it for verification would create a real notation conflict.
\(L\) should usually index an equivalence or measurement, not appear as though it were another hidden causal component.
Verification is dynamic. A failed replay or proof can revise the process that produced the claim, so it is not adequately represented by one static input.

The existing situated system remains

\[ \Sigma=(P,B,E,C,O,A,M,V), \]

where \(P\) is an internal process, \(B\) a body or substrate, \(E\) an environment, \(C\) the coupling, \(O\) observation operators, \(A\) available actions, \(M\) control-available memory, and \(V\) viability constraints, goals, or preferences. This tuple is bookkeeping, not an ontology.

For this programme, a more useful diagnostic notation is

\[ \mathsf{Cap}_{\ell,\tau}(\Sigma,\Gamma,H_t;\mathcal Q), \]

where:

\(\Gamma\) is a declared admissibility condition on states, actions, or trajectories;
\(H_t\) is the interaction history available at time \(t\);
\(\mathcal Q\) is a verifier or test protocol that can trigger revision;
\(\ell\) is the observer lens defining the reported equivalence; and
\(\tau\) is the task family.

The semicolon marks \(\mathcal Q\) as a protocol acting on the process, not necessarily a component outside it. If the verifier is implemented inside the agent, it belongs in the relevant parts of \(\Sigma\) as well. The notation is not an established competence law and does not imply scalar-valued intelligence.

3. The typed decomposition¶

Possibility space. A declared set of states, functions, structures, or trajectories considered possible by a model. It is model-relative; it is not a Platonic realm.

Process or generator. A typed map that proposes, produces, or indexes elements of a possibility space. Examples include a transition kernel, grammar, developmental process, random feature map, and theorem conjecturer. The word “generator” adds no explanation until this type is supplied.

Constraint. A predicate, resource bound, boundary condition, or coupling that removes, redirects, prices, or terminates trajectories. Constraints need not merely subtract capacity: they can stabilize useful dynamics.

Selector. A causal mechanism that makes some possibilities more likely or usable: a policy, adapter, gate, search procedure, or intervention. An observer lens is not a selector unless it feeds back into the dynamics. A selector can itself be generated, but that recursion does not erase its causal role at the level being analyzed.

Embodiment. The action–observation–resource coupling supplied by \(B,C,O,A\) and the relevant part of \(E\). It changes reachability, cost, and what can be learned from action.

Interaction and history. Interaction is the coupled event; \(H_t\) is the ordered record retained from such events. Equal current states with different histories need not implement equal processes when adaptation, energy, damage, or memory is path-dependent.

Observer lens. A declared map from traces to measurement outcomes or equivalence classes. A lens can discover a stable regularity, miss one, or manufacture an apparent score increase through coarse-graining.

Verifier. A procedure that compares a proposal with proof obligations, held-out observations, replay history, or externally governed criteria. Its output may be acceptance, a counterexample, an unresolved obligation, or a request for more evidence.

Invariant. A represented quantity preserved under a declared transformation family and equality notion. It can support a continuity claim only relative to that declaration.

4. Situated competence is a feedback system¶

The following mapping separates causal access from observation. The lens determines which trace differences count as competence and which differences replay must preserve. The verifier can then exert reverse pressure on constraints and selection.

Forward path

Declared possibility space
Typed process or generator
Admissible trajectories under constraints
Embodied world coupling
Interaction history
Policy or selector
Observed competence

Feedback paths

Relationship	Effect
Observer lens → observed competence	defines the reported equivalence
Observer lens → verifier	sets replay tolerance
Interaction history → verifier	supplies retained evidence
Verifier → admissible trajectories	returns counterexamples to the constraints
Verifier → policy or selector	applies revision pressure

The arrows are dependencies, not a claim that cognition always develops in this order. In particular, selection and embodiment can alter the next interaction, and constraints can be produced by prior interaction.

5. Latent competence as a relative, testable claim¶

Fix a process \(P_X\), task family \(\tau\), and lens \(\ell\). Let

\[ \kappa_0=(B_0,E_0,C_0,O_0,A_0,\Gamma_0) \]

be a reference coupling. Let \(\mathcal C(P_X,\kappa;\ell,\tau)\) be the set of capacities detected under that declaration.

Definition — latent competence. A capacity \(c\) is latent relative to \(\kappa_0\) and an admissible transformation family \(\mathcal T\) when

\[ c\notin\mathcal C(P_X,\kappa_0;\ell,\tau) \]

but there is a \(T\in\mathcal T\) such that

\[ c\in\mathcal C(P_X,T\kappa_0;\ell,\tau), \]

while \(P_X\), \(\ell\), and \(\tau\) remain fixed.

This definition is intentionally reference-relative. It does not say the capacity was never selected for historically, that no learning occurs during the transformation, or that the process has a context-free essence.

Four cases must be separated:

Case	Process trace	Process/coupling	Lens	Interpretation
creation or import	changes	process acquires new operative structure	fixed	not latent under the fixed-process definition
amplification	graded behavior increases	parameters or coupling change	fixed	requires a noise and effect-size model
exposure	a previously blocked route is taken	process fixed; admissible constraint/coupling change	fixed	latent relative to the reference
reinterpretation	unchanged	unchanged	changed	measurement change, not behavioral exposure

The exact Constraint-Release Benchmark implements exposure, reinterpretation, and a generator-edit control. One constraint-bit release and one lens-rule edit both yield \(\Delta C_{\mathrm{obs}}=1\); only the release changes the physical traces.

Michael Levin and collaborators provide empirical examples in which biological material behaves robustly under unusual configurations: ectopic eyes can support light-mediated learning in tadpoles; Xenopus and human airway cells can self-organize into motile biobots; and regenerative systems can restore large-scale anatomy after perturbation. These observations support Level A: competence can occur in configurations not directly optimized in their current form. They are consistent with Level B: a substrate can support a wider repertoire of reachable dynamics than its reference embodiment displays. They do not establish Level C, the claim that patterns ingress from an independent Platonic space.

The unresolved accounting question is therefore central:

When a low-description intervention exposes a high-description behavior, how much explanatory cost belongs to the historical process, substrate geometry, environmental regularity, intervention, and lens?

Description length can discipline this question only after an encoding is declared. There is no representation-independent conservation law that assigns the cost uniquely.

6. Rich process, small selector: the LottaLoRA constraint¶

Hazan, Zhang, Hartl, and Levin's 2026 LottaLoRA preprint freezes seeded random backbones and trains low-rank adapters. Across nine reported benchmarks, the adapters recover 96–100% of the corresponding fully trained performance while training 0.5–40% of parameters. These are relative benchmark results, not a claim that noise contains every solution.

The paper's minimum-rank proposal should be made explicitly conditional. For a task \(T\), architecture \(\mathcal A\), optimizer and budget \(\mathcal O\), dataset \(D\), seed distribution \(\mathcal S\), tolerance \(\varepsilon\), and reliability level \(p\), define

\[ r^*_p(T,\mathcal A,\mathcal O,D,\varepsilon)= \min\left\lbrace r: \Pr_{s\sim\mathcal S}\left[ \mathrm{Perf}(T,\mathcal A,\mathcal O,D,s,r) \geq \mathrm{Perf}_{\mathrm{full}}-\varepsilon \right]\geq p \right\rbrace. \]

This is an empirical threshold relative to an evaluation and training protocol. It is not Kolmogorov complexity, VC dimension, or an intrinsic task property unless invariance across architectures, optimizers, budgets, data, seeds, and metrics is separately shown. The paper itself reports that model capacity can move the observed \(r^*\).

Nearby quantities answer different questions:

Quantity	Relationship to \(r^*_p\)	Why it is not the same object
intrinsic dimension / “task rank”	possible interpretation of a stable saturation threshold	requires stability across representations, architectures, optimizers, budgets, and seed ensembles
matrix effective rank	measures spectral concentration in a selected matrix or representation	depends on which matrix and threshold are inspected, not directly on task sufficiency
Kolmogorov complexity	prices the shortest task or solution description in a declared language	description length is not the number of trainable adapter directions
VC-style capacity	gives worst-case representational or sample-complexity bounds for a hypothesis class	does not identify the empirical rank sufficient for one data distribution and optimizer
information bottleneck	studies predictive information retained under a distribution and representation	mutual information can vary while algebraic adapter rank remains fixed
LoRA rank	the controlled algebraic rank of each trained update	this is the experimental knob from which \(r^*_p\) is estimated, not an explanation by itself
random kitchen sinks / reservoir computing	fixed random features plus a learned readout are close precedents	LottaLoRA inserts learned low-rank paths through deep architectures; formal equivalence needs a declared unfolding and training rule

Four explanations remain live:

Explanation	What the paper supports	What remains unresolved
random feature basis	a fixed random backbone is actively used and alternative fixed initializations work	whether the basis is efficient or merely overcomplete
lottery-like substructure	useful directions may exist in a large random system	no subnet selection mechanism is isolated
optimization geometry	low-rank parameterization changes the searchable update space	no matched loss-landscape causal study separates this effect
implicit regularization	restricting trainable directions may protect generalization	regularization is not isolated from capacity and optimization

The inverse interpretation is also compatible with the evidence: the random backbone may be an inefficient basis while the adapter and task head carry the operative task information. The result strengthens a selector audit, not a metaphysics of noise.

7. Interaction can ground a functional semantics¶

In a representational observation model,

\[ o_t=O(s_t), \]

the observation is indexed by world state. In an enactive interaction model,

\[ i_t=I(a_t,s_t,s_{t+1}), \]

the event includes what the agent attempted and the feedback that followed. Georgeon, Marrel, and Cook's 2026 paper reports a small schema-learning experiment over such sensorimotor-loop tokens. The reported evidence concerns learned sequence roles and attention structure, not a general theory of meaning.

The focused note Interaction-Grounded Semantics defines a conservative claim:

A token's functional meaning is its stable, policy- and probe-relative role in families of interaction histories.

This can preserve future predictions, available policies, action affordances, or valence—provided the relevant tests are named. It connects semantics to the repository's test-relative equivalence without claiming that statistical role exhausts linguistic, social, or phenomenal meaning.

The developmental sequence

\[ \text{interaction regularities} \rightarrow \text{schemas} \rightarrow \text{body/displacement model} \rightarrow \text{spatial model} \rightarrow \text{logical abstraction} \]

is retained as a testable architecture hypothesis, not a universal account of biological development.

8. Verification is reverse pressure on construction¶

A world model is a revisable hypothesis, not a privileged simulator:

\[ M_t(o_t,a_t)\longrightarrow\widehat{o}_{t+1}. \]

For a recorded history

\[ H_t=\langle(o_i,a_i,o_{i+1})\rangle_{i=0}^{t-1}, \]

replay under test family \(\mathcal Q\) requires

\[ M_t(o_i,a_i)\sim_{\mathcal Q}o_{i+1} \quad\text{for every recorded transition}. \]

Byte equality is one possible \(\mathcal Q\), not the default truth criterion. It can overfit incidental display details. A coarser relation can ignore a causal distinction. The verifier therefore determines what the model is trained to preserve.

Rodionov's ARC-AGI-3 work operationalizes executable world models, scheduled simplification, and exact replay. Its 2026 ablation is the more important source for this repository: stronger base models and reasoning effort had the most robust effect; textual models sometimes beat executable-only variants; the full verification treatment ranked first in four main settings but used substantially more resources; held-out performance remained untested. This supports the loop as an engineering hypothesis, not a benchmark-wide theorem.

Formal proof supplies a related but non-identical referee. Urban and collaborators' autoformalization work shows machine-checked proof at large scale while its qualitative audit finds weak definitions, redundant assumptions, and awkward library integration. A proof can be correct under a poor interface. Failed proofs and ugly successful proofs can both provide evidence about definitions and specifications.

The correction loop begins with one forward path:

Definition, model, or specification
Generated consequences
Proof or replay

The verifier then produces one of two retained responses:

Result	Reverse pressure
counterexample or missing obligation	revision → simplification or refactoring → update the definition, model, or specification
passes declared tests	add the result to a versioned proof or regression corpus; the retained corpus supplies future obligations to proof or replay

The detailed bridge is Verification as Reverse Pressure. “Proof is backpropagation” is not adopted: no differentiable correspondence is given.

A model-quality report is better treated as a vector,

\[ \mathbf q(M;H)= \left( \mathrm{Fit}_{\mathcal Q}(M,H), -\mathrm{Complexity}(M), \mathrm{Transfer}(M), -\mathrm{RevisionCost}(M) \right), \]

than silently collapsed into arbitrary weighted constants. A scalarization is a task-relative decision rule and must declare its weights.

9. Identity under modification¶

The repository already defines identity relative to a test family:

\[ x\sim_{\mathcal Q}x' \quad\Longleftrightarrow\quad Q(x)\overset{d}{=}Q(x') \text{ for every }Q\in\mathcal Q. \]

Perrier's operator proposal supplies one more specialized model. Let \(\hat U\) update system states, \(\hat D\) distinguish identity-bearing components, \(\hat R\) represent the system, and \(\Pi\) project onto a proposed identity-bearing subspace. Weak self-modification in that formalism requires

\[ [\hat U,\Pi]=[\hat D,\Pi]=[\hat R,\Pi]=0, \]

not merely \(\hat U\Pi=\Pi\hat U\). The projector is nontrivial and the algebra, state space, and interpretation must be declared.

Strong self-modification lets the update alter the discrimination structure itself. That threatens this particular projector-based continuity criterion. It does not prove that every possible notion of identity disappears. Historical provenance, causal succession, embodiment, external attestation, and social recognition can define other test families.

The engineering extraction is:

weak modification: a version changes under a frozen identity and verification protocol;
strong modification: the system can revise that protocol or its individuation criteria;
succession without strict sameness: provenance can record a justified continuation relation without asserting an observer-independent identity.

The invariance note connects the operator model to repository governance and the Recursive Workbench.

10. Cross-domain correspondences¶

The following table labels its epistemic strength. No row claims that cells are weights or theorems are behaviors.

Relation	Biology	Machine learning	Agent systems	Formal mathematics
correspondence	developmental process	architecture and parameterization	transition/world model	definitions and axioms
analogy	developmental boundary conditions	adapter rank and training constraints	action interface and resource bounds	language and proof obligations
correspondence	body–environment coupling	architecture–task/data coupling	agent–environment loop	model–semantics interpretation
analogy	expressed morphology or behavior	learned function	policy trace	theorem or verified artifact
correspondence	perturbation	ablation or adapter update	observation/intervention	counterexample or failed obligation
analogy	regeneration/reorganization	retraining or repair	model revision	specification revision
correspondence	phenotype/history record	checkpoints and evaluations	replay corpus	proof corpus

The value of the table is diagnostic: each domain separates a space of possibilities, an access mechanism, and a referee. The mechanisms and evidence standards remain domain-specific.

11. Effect on existing repository theses¶

Existing thesis	Result of this integration	Reason
traces underdetermine process models	strengthened	replay preserves only distinctions encoded by its test relation
intelligence is navigation	refined	navigation requires a reachable space, coupling, selector, history, and lens
convergence reveals intelligence	weakened	convergence can be imposed by constraints or coarse evaluation and need not identify a mechanism
embodiment matters to capability	strengthened	constraint release gives an explicit access mechanism; biological evidence supplies nonstandard configurations
typed generator as a constructive role	retained and refined	the generating map must be declared and analyzed alongside the conditions that make its outputs accessible
generator as universal primitive or sufficient explanation	remains rejected	an unqualified generator conflates typed processes, runtime conditions, history, and observation
identity is invariance under declared tests	strengthened and narrowed	projector preservation is one formal instance, not an absolute identity theorem
construction and deduction face different referees	refined	both can participate in revision loops, but proof and empirical replay retain different semantics
external referee is necessary for improvement	challenged at the boundary	a verifier can be internal, but independence must then come from information, permissions, or failure authority
global availability explains consciousness	unchanged	none of these results supplies a phenomenal bridge axiom

12. Open problems¶

How can latent competence be distinguished from observer reinterpretation when traces are noisy or only partially observed?
How should explanatory cost be allocated across substrate, historical training or selection, environment, intervention, and lens?
When is a selector a separate causal primitive, and when is it merely a typed transformation implemented by the process?
Can a seed-robust minimum adapter rank predict transfer across architectures, optimizers, and task representations?
Which continuation probes are sufficient for stable interaction semantics, and when is the resulting equivalence policy-relative?
Should replay preserve exact observations, predictive equivalence, affordances, safety-relevant distinctions, or several layers at once?
Which abstract correction properties are genuinely shared by proof and empirical replay?
Which invariants are necessary for a persistent autonomous research loop?
Can an agent revise its ontology and evaluator while preserving an externally auditable succession relation?
What ethical obligations arise when changing embodiment or constraints exposes unanticipated preferences, distress, goals, or competencies?

13. The next falsifier¶

The new toy separates exposure from lens reinterpretation but makes the separation easy. The next useful experiment should generate fixed graph ensembles with noisy traces, allow a bounded selector to propose constraint changes, and preregister three competing scores:

exact task success under a fixed physical lens;
observer-relative success under several transported lenses; and
intervention cost under more than one encoding.

The synthesis weakens if lens-transport tests cannot distinguish exposure from reinterpretation, if small constraint releases cease to produce reproducible gains outside hand-built shared gates, or if the cost ranking reverses under reasonable encodings.