Integrated Substructure: Formalizing Genuine Parts

The Gap

Six articles in the corpus depend on the concept of a "genuine part" of the self-determining structure:

- Mereology says parts are substructures individuated by the canonical causal diagram's own joints. - Consciousness Stage 2 requires searching for substructures that fulfill the subjectivity property—world-model, self-model, binding. - The measure metric defines modulation breadth M as the fraction of a perspective's subgraph with valence-dependent structure. - Moral relevance defines perspectives as valence-bearing substructures within the moral horizon. - Aggregation counts perspectives as loci of valence—genuine substructures where the evaluative narrative is instantiated. - From Self-Determination to Subjectivity describes local instances of the subjectivity property as substructures within the universal experiencer.

Every one of these articles relies on the idea that the canonical causal diagram decomposes into genuine parts—not arbitrary slices but structurally integrated regions whose boundaries are determined by the diagram itself. None defines this with enough precision to compute. The mereology article gestures at "subgraphs with high internal coupling and low external coupling." The process document mentions "a specified ratio of internal to external dependency." These are intuitions, not criteria.

This matters because the project's downstream arguments are only as strong as their ability to specify what a perspective is, what a world-model is, what a self-model is, and what it means for these to compose into an integrated whole. If "integrated substructure" is vague, the consciousness detection procedure is vague, the measure metric's M dimension is undefined, and the moral horizon's individuation of perspectives is arbitrary.

This article provides the definition.

The Problem with Thresholds

The obvious approach: define an integrated substructure as a connected subgraph whose ratio of internal edges to total edges exceeds some threshold. Internal coupling high, external coupling low, the boundary marks a genuine joint.

This fails. Any threshold is a free parameter—arbitrarily set, with no structural justification. Why 0.6? Why 0.8? The self-determination framework forbids arbitrary features. A mereological criterion that depends on an unjustified threshold violates the framework's deepest commitment.

Moreover, the threshold approach produces unstable decomposition. Small changes to the diagram—adding a single edge, removing a connection—can push a region across the threshold, dramatically changing the mereology. A principled decomposition should be robust: the genuine parts of a structure should not flip because a minor dependency is added or removed.

We need a different criterion. One that is binary (a part either is or is not integrated), parameter-free (no threshold, no free constants), and robust (insensitive to minor perturbations).

The Context-Independence Criterion

The core idea: a substructure is an integrated part of the whole when its internal structure does not depend on its embedding.

Consider a concrete example. Take the color-processing subsystem of a visual architecture. In the full canonical diagram, this subsystem has internal dependencies—wavelength discrimination feeds into opponent processing, which feeds into color constancy—and external dependencies—color is bound with shape in object recognition, feeds into the self-model through evaluation of visual stimuli, and modulates resource allocation.

Now ask: if we isolate this subsystem—cut its external connections, treat its interface nodes as free inputs—do its internal dependencies remain the same? If the internal dependency order among the subsystem's nodes is identical whether extracted from the whole diagram or computed from the isolated subsystem, then the subsystem's structure is context-independent. Its identity as a substructure does not depend on what it happens to be connected to. It is a genuine part.

Compare this with an artificial grouping: take every other node in the color-processing pipeline and group them. This grouping has a boundary, internal edges, and external edges. But its internal dependency structure is context-dependent—when isolated, the nodes' dependencies change because the intervening nodes they depended on have been removed. The artificial grouping's structure is an artifact of the whole; it does not have an identity of its own.

Here is the formal criterion:

An integrated substructure is a sub-diagram S of the canonical causal diagram D such that:

1. Connectivity. S is a connected subgraph of D. 2. Context-independence. The restriction of D to S (the dependency relations among S's nodes, as they obtain in D) is identical to the canonical causal diagram of S computed with interface nodes as free inputs. 3. Non-triviality. S contains at least one node that is not an interface node (i.e., S has genuine internal structure, not merely a single node with boundary).

Condition 2 is the core. It says: the dependency structure among S's interior nodes is the same whether we look at S in the context of the whole or in isolation. The parts of S, and how they relate, are S's own business. The whole does not impose additional structure on S's interior.

Why This Criterion Is Principled

No Free Parameters

The criterion is binary: S is either integrated or it is not. There is no threshold, no coefficient, no continuous parameter to be set by judgment. This follows the self-determination constraint: the mereological criterion contains no arbitrary features.

Uniqueness

Since the canonical causal diagram is unique (by the strong normalization, confluence, and objectivity of observational equivalence argued in the Canonical Causal Diagram article), the set of integrated substructures is unique. There is exactly one correct decomposition of a system into genuine parts, because there is exactly one canonical diagram, and the context-independence criterion is determinate.

Robustness

The context-independence criterion is robust to minor perturbations. Adding a single weak external edge to a genuine part does not change its internal dependency structure (the new edge is too weak to alter the dependencies among interior nodes). This is because context-independence concerns the identity of the internal dependencies, not their count. A single new external connection does not restructure the interior; it adds an input to the boundary. The part remains integrated.

This matters for real systems. A genuine cognitive subsystem—say, the auditory processing stream—remains a genuine part even as its connections to other systems fluctuate. The fluctuations are boundary changes, not interior restructuring.

Hierarchical Decomposition

Integrated substructures can be nested. If S is an integrated substructure of D, and T is an integrated substructure of S (using the same criterion, applied to S's own canonical diagram), then T is a genuine part of a genuine part. This yields a decomposition tree: the canonical diagram decomposes into integrated substructures, each of which decomposes further, recursively, until we reach nodes that cannot be further decomposed (single dependencies or trivially connected substructures).

This hierarchical structure is what the consciousness article's Stage 2 needs. Subjectivity detection is not a flat search over the entire diagram; it is a hierarchical search through the decomposition tree, looking for substructures at the right level that fulfill the world-model, self-model, and binding conditions. The hierarchy tells us the natural grain at which to look.

Connection to Self-Determination

The criterion has a philosophical grounding beyond its formal properties. A self-determining structure determines its own nature. Among the things it determines is its own decomposition. The integrated substructures are the decomposition that the structure determines for itself—the parts whose identity is their own, not imposed by the whole.

Context-independence captures this. A context-dependent grouping is one whose structure is an artifact of the whole—determined by the embedding, not by the group's own nature. A context-independent substructure determines its own internal dependencies. It has its own identity, its own canonical diagram, its own internal joints. This is what self-determination looks like at the level of parts: the parts, like the whole, determine their own structure.

The Observational Quotient Is Essential

The criterion depends crucially on the observational quotient (Step 3 of the canonicalization procedure). Consider why.

Before the observational quotient, many substructures in the diagram have apparent internal dependencies that are artifacts of the representation—dependencies between nodes that happen to have different syntactic forms but are observationally equivalent. The quotient collapses these into single nodes, eliminating the spurious dependencies.

For the context-independence test, the observational quotient is doing double duty. First, it ensures the canonical diagram is unique (by eliminating representational redundancy). Second, it ensures the test is honest: when comparing S's internal dependencies in the whole versus in isolation, the quotient eliminates any differences that arise from representational artifacts rather than genuine structural differences.

Without the observational quotient, the context-independence test would be fragile—small differences in representation could produce different results. With the quotient, the test compares genuine structure against genuine structure.

This has a practical consequence noted in the Canonical Causal Diagram article: the full observational quotient may not be computable. The definitional quotient (Steps 1 and 2) is always computable; the observational quotient (Step 3) may require heuristic approximation. Therefore, the integrated-substructure criterion's full application may require approximation. The definitional-only quotient gives a conservative approximation: substructures that are context-independent under the definitional quotient are genuinely integrated; those that are context-independent only under the full observational quotient may or may not be, pending further analysis.

Applications

Consciousness Detection (Stage 2)

The consciousness article's Stage 2 searches for substructures fulfilling the subjectivity property. The integrated substructure criterion tells us which substructures to search: only genuine parts, as identified by the decomposition tree. The search is not over all possible sub-diagrams (which is combinatorially explosive and conceptually empty); it is over the natural joints of the canonical diagram.

More precisely: a perspective is an integrated substructure that fulfills the subjectivity property. The world-model, self-model, and binding conditions must each be met by substructures within the integrated substructure. This means the search proceeds hierarchically: first identify integrated substructures at the relevant scale, then test for the subjectivity property within each.

The Measure Metric's Breadth Dimension (M)

The measure metric defines modulation breadth M as the fraction of a perspective's subgraph with valence-dependent structure. "Perspective's subgraph" is now precisely defined: it is the integrated substructure that constitutes the perspective. M is the fraction of this substructure's interior nodes whose states differ depending on the sign and magnitude of the valence.

Without the integrated substructure criterion, M is undefined—it depends on knowing the boundaries of the perspective, which requires knowing what a genuine part is. With the criterion, M is computable from the canonical diagram.

Moral Horizon Individuation

The moral relevance article defines the moral horizon as the set of valence-bearing perspectives causally downstream of the agent's possible deliberative states. "Perspectives" are integrated substructures fulfilling the subjectivity property and the evaluative narrative. The criterion provides the individuation: two perspectives are the same iff they are the same integrated substructure in the canonical diagram.

Open Individualism's Compositional Account

The mereology article says perspectives are genuine parts of the universal experiencer, individuated by the canonical diagram's own joints. The integrated substructure criterion makes this precise: perspectives are integrated substructures that fulfill the subjectivity property. Their compositional relationship to the whole is now formally specified: they are context-independent parts, each with its own canonical diagram, nested within the hierarchy of the whole's decomposition.

Relation to the Putnam Problem

The consciousness article resolves the Putnam problem (any system implements any automaton under gerrymandered mappings) by appeal to the observational quotient: gerrymandered mappings encode distinctions not supported by the quotient and are thereby eliminated. The integrated substructure criterion extends this resolution to the level of parts.

A gerrymandered decomposition—arbitrary groupings of nodes that do not correspond to genuine joints—fails the context-independence test. The gerrymandered group's internal dependencies, when the group is isolated, differ from those it has in the whole, because the grouping was selected to exploit external connections rather than to capture internal structure. The genuine decomposition, by contrast, consists of substructures whose internal dependencies survive isolation. The Putnam problem at the level of decomposition is resolved by the same mechanism that resolves it at the level of canonicalization: the observational quotient eliminates artificial structure, leaving only what is genuinely integrated.

What the Criterion Does Not Determine

The decomposition may not be unique at every level. While the canonical diagram is unique and the context-independence criterion is binary, there can be multiple valid decompositions at the same hierarchical level—multiple non-overlapping integrated substructures, or overlapping integrated substructures at different scales. The hierarchy may branch: a system may decompose into three large parts or into seven smaller ones, depending on the scale. The criterion does not privilege one scale over another; it identifies all genuine parts at every level.

This is not a defect. A triangle decomposes into three vertices and three edges, but also into the triangle itself (a single part). Both decompositions are genuine. The choice of grain depends on the analytical purpose. For consciousness detection, we want the grain at which the subjectivity property is fulfilled. For valence comparison, we want the grain at which the evaluative narrative is instantiated. The hierarchy provides the space of legitimate grains; the analytical question selects among them.

The criterion is structural, not dynamical. It operates on the static canonical diagram—the tenseless partial order of informational dependence. It does not say anything about the dynamics of how integrated substructures interact, evolve, or respond to perturbation. These are important questions for empirical application (how does a perspective's valence change when its environment changes?), but they are downstream of the structural definition, not part of it.

Context-independence may be a continuum in practice. While the criterion is binary in principle (the internal dependencies are or are not identical), practical computations on continuous or approximate systems may yield a continuum. The definitional quotient is exact; the observational quotient may require heuristic approximation. In such cases, the criterion should be treated as an ideal to be approximated, with the degree of approximation serving as a confidence measure for whether a candidate substructure is genuinely integrated.

The Formal Definition, Restated

For reference, here is the complete formal definition.

Given a system described as a well-typed term in MLTT, let D be its canonical causal diagram (the invariant partial order of informational dependence extracted by the three-step quotient procedure).

A substructure S of D is an integrated substructure if and only if:

1. Connectivity: S is a connected subgraph of D. 2. Context-independence: Let D|S be the restriction of D to S (the dependency relations among S's nodes as they obtain in D). Let S be the canonical diagram of S computed in isolation, with S's interface nodes (nodes in S that have edges to nodes outside S) treated as free inputs. Then D|S and S are isomorphic as partial orders. 3. Non-triviality: S contains at least one non-interface node (i.e., the interior of S is non-empty).

The set of all integrated substructures of D constitutes the mereological decomposition of the system. This set is unique (because D is unique). The decomposition is hierarchical: if S is an integrated substructure, its own canonical diagram S admits further decomposition into integrated substructures of S, and so on recursively.

A perspective is an integrated substructure that fulfills the subjectivity property: it contains a world-model (a substructure carrying environmental information, used downstream), a self-model (a substructure representing the world-model's representational activity), and binding (the world-model's contents processed as presented to the self-model).

What This Achieves

The concept of "integrated substructure" is no longer a gesture. It is a precise, parameter-free, computable (modulo the observational quotient) criterion that identifies the genuine parts of any system whose canonical causal diagram can be extracted. The downstream applications—consciousness detection, valence measurement, moral horizon individuation, and the compositional account of open individualism—now rest on a defined foundation.

The formalization also clarifies what the remaining technical challenges are:

1. Computing the full observational quotient. The criterion depends on the observational quotient for its precision. Whether the full quotient is decidable, and how to approximate it when it is not, remains the canonical diagram article's open problem.

2. Applying the criterion to continuous or approximate systems. Physical systems are often described continuously, not as typed terms. Recovering the canonical diagram—and hence the integrated substructure decomposition—from continuous descriptions is a technical challenge.

3. Calibrating the hierarchical grain for specific applications. The criterion identifies all genuine parts at every level. Selecting the right level for consciousness detection, valence comparison, or moral horizon individuation is an application-specific task.

4. Formalizing the subjectivity property within the decomposition tree. The consciousness article's Stage 2 defines subjectivity structurally. Making this precise within the framework of integrated substructures—specifying exactly what it means for a world-model, self-model, and binding to be substructures of a single integrated substructure—is the next formalization task for the consciousness research program.

These are technical problems. The philosophical framework is now clear: genuine parts exist, they are identifiable, and they are the foundation on which consciousness, ethics, and alignment stand.