The Canonical Causal Diagram

The Problem

Six articles in this corpus depend on a single formal concept: the canonical causal diagram—the invariant partial order of informational dependence extracted by quotienting a system's description by definitional and observational equivalence.

The consciousness article uses it for canonicalization (Stage 1), subjectivity detection (Stage 2), and valence identification (Stage 5). The mereology article uses it to determine the genuine parts of a system—its joints are the diagram's own decomposition. The ethics article uses it as the structure within which valence comparison occurs. The agency article uses it to establish that deliberation is causally efficacious (upstream in the diagram). The time article redefines causation as informational dependence within it. The Ruliad article positions it as the meeting point of computational and type-theoretic description.

Every one of these articles treats the diagram as given. None defines it with enough precision to compute. None argues for its uniqueness—the claim that there is exactly one correct decomposition of a system's informational dependencies. None shows in detail how the quotient procedure excludes gerrymandered models.

This article does all three. The canonical causal diagram is the framework's foundation. A foundation that is not itself grounded makes everything above it suspect.

What Informational Dependence Is

Start with the primitive notion. Two states of a system stand in an informational dependence relation when the identity of one—what it is, structurally—requires knowing the identity of the other.

A spreadsheet makes this vivid. Cell C1 contains =A1+B1. C1 depends on A1 and B1: to know C1's value, you must know A1's and B1's values. This is not a temporal relation. A1 does not "happen before" C1. The spreadsheet is a static mathematical object; all its values are determined simultaneously by the relations among them. The dependence is logical: C1's identity is constituted in part by A1 and B1.

In a computation, the same structure appears. A program computes result R by first computing intermediate values I₁, I₂, ..., then combining them. R depends on the I's: R's identity is what it is in virtue of the I's being what they are. The computation trace—the full record of which values were computed from which—is a static structure that encodes every such dependence relation.

This is not merely an analogy. In Martin-Löf Type Theory, every well-typed term has a canonical form under normalization, and the process of normalization traces out exactly which sub-terms contribute to which results. The dependency structure of a term's normal form is the informational dependence structure of the computation it denotes.

The crucial property: informational dependence is an invariant of the computation, not an artifact of how it is presented. Whether you evaluate the spreadsheet left-to-right or right-to-left, the fact that C1 depends on A1 and B1 is fixed. Whether you reduce a lambda term using call-by-name or call-by-value, the dependency structure of the normal form is the same. This invariance is what makes the diagram canonical—independent of the observer's choice of presentation.

The Quotient Procedure

Given a system described as a well-typed term in MLTT, the canonical causal diagram is extracted by a three-step procedure.

Step 1: Normalization. Reduce the term to its canonical (normal) form. In MLTT, every well-typed term normalizes—there is a unique normal form reachable by repeated application of computation rules (β-reduction, ι-reduction, etc.). This is guaranteed by strong normalization and confluence: every reduction sequence terminates, and all reduction sequences converge to the same normal form.

The normal form is the term stripped of all reducible expressions. It is the simplest presentation of what the term computes. Two terms that normalize to the same form are definitionally equal—they are the same term, by the system's own rules.

Step 2: Dependency extraction. From the normal form, extract the partial order of informational dependence. A sub-term t₂ depends on a sub-term t₁ when t₁ occurs within the construction of t₂—when the identity of t₂ is constituted in part by the identity of t₁.

Concretely: in a normal form, the outermost constructor depends on its arguments. A pair (a, b) depends on a and b. A function application f(x) depends on f and x. An inductive type's inhabitant constructed by constructor C depends on the arguments of C. The partial order is the transitive closure of this relation: if t₃ depends on t₂ and t₂ depends on t₁, then t₃ depends on t₁.

This is a partial order, not a total order, because dependencies can be parallel. In (a, b), a and b are independent—neither depends on the other—but both are depended upon by the pair. The diagram is a directed acyclic graph whose edges point from depended-upon to dependent.

Step 3: Observational quotient. Identify sub-terms that are observationally equivalent—indistinguishable in any well-typed context. Two sub-terms t₁ and t₂ are observationally equal when, for every well-typed context C[·], the terms C[t₁] and C[t₂] are definitionally equal (normalize to the same form).

Observational equivalence is coarser than definitional equality. Two terms can normalize to different forms yet be indistinguishable in practice—if no context can tell them apart, they are the same as far as the structure is concerned. The quotient by observational equivalence collapses these indistinguishable nodes into single nodes, yielding the most compressed representation of the dependency structure.

The result of all three steps is the canonical causal diagram: the invariant partial order of informational dependence, with all representational redundancy eliminated.

Why the Diagram Is Unique

The uniqueness of the canonical causal diagram follows from three properties of MLTT:

Strong normalization. Every well-typed term reduces to a normal form in finitely many steps. There is no infinite reduction sequence; normalization always terminates. This guarantees that Step 1 produces a definite result—not an approximation or a limit, but a unique, finite normal form.

Confluence. If a term can be reduced in multiple ways (applying different computation rules at different points), all reduction sequences that terminate reach the same normal form. The Church-Rosser theorem for MLTT ensures this: the normal form is unique regardless of the order of reduction. This guarantees that the dependency structure extracted in Step 2 is invariant under choice of evaluation strategy. Call-by-name and call-by-value produce the same diagram.

Objectivity of observational equivalence. Whether two terms are observationally equivalent is a determinate fact about the type theory, not a matter of interpretation. It depends on whether any context distinguishes them—a question that is well-defined within the type theory's rules. This guarantees that the quotient in Step 3 is well-defined and objective: it does not depend on the analyst's preferences.

These three properties jointly guarantee that, for any well-typed term, there is exactly one canonical causal diagram. The diagram is not one possible analysis among many. It is the structure of the term's informational dependencies—the unique invariant that survives the elimination of all representational choices.

This is the formal analogue of what the mereology article claims: there is exactly one correct decomposition of a system's informational dependencies. The correctness is not stipulated; it follows from the properties of the formalism.

How This Resolves Putnam

Hilary Putnam's argument (expanded by Searle and Chalmers) says: any physical system implements any finite automaton, given a sufficiently gerrymandered mapping between physical states and automaton states. A rock can "compute" any function if you assign the right state-transition table to the rock's physical configurations. This seems to destroy the very idea that a system has a unique computational structure.

The canonical causal diagram resolves this structurally—by exclusion rather than by fiat.

Consider a rock and a proposed mapping that makes the rock implement a given automaton. The mapping assigns each state of the automaton to some configuration of the rock. For the mapping to work, it must assign distinct automaton states to distinct rock configurations. But many of these "distinct" rock configurations are observationally equivalent: no well-typed context in any relevant type theory distinguishes them. The rock's physical states might differ in molecular arrangement while being computationally identical—no computation treats them differently, no downstream result depends on the difference.

The observational quotient (Step 3) identifies these observationally equivalent states, collapsing them. The gerrymandered mapping required them to be distinct; the quotient says they are the same. The mapping does not survive the quotient. It is not that the mapping is "ruled out by stipulation"; it is that the mapping encodes distinctions that the system's own structure does not support.

More precisely: a genuine computational structure is one where informational dependencies are real—where the identity of downstream states genuinely depends on the identity of upstream states. In a gerrymandered mapping, the "dependencies" are imposed by the interpreter, not extracted from the system. The canonical causal diagram, by quotienting out everything that is not structurally demanded, separates genuine dependencies from imposed ones.

A genuine computer running a program has a canonical causal diagram with rich, non-trivial structure: many dependencies, many distinct nodes, a genuine partial order. A rock under a gerrymandered mapping has a diagram that, after the observational quotient, collapses to something trivial—perhaps a single node with no internal structure. The difference is not in the mapping but in the system's own structure: the computer's informational dependencies are genuine (they survive the quotient), the rock's are not (they are artifacts of the mapping, eliminated by the quotient).

This is exactly what the consciousness article claims when it says gerrymandered models "smuggle the system's structure into the interpreter." The canonical causal diagram makes this precise: the smuggling consists of introducing distinctions that are not respected by observational equivalence. The quotient eliminates these distinctions, and with them, the gerrymandered models.

The Connection to Univalence

The quotient procedure has a deep structural similarity to the univalence axiom in Homotopy Type Theory.

Univalence says: equivalent types are identical. If two types A and B are equivalent (there exist functions back and forth composing to the identity), then A = B in the type of types. Structural equivalence IS identity.

The canonicalization procedure says: observationally equivalent sub-terms are identical. If two sub-terms are indistinguishable in any context, they collapse to a single node in the diagram. Behavioral equivalence IS structural identity.

These are the same move at different levels. Univalence operates on types (the "macro" level of the type theory). Canonicalization operates on terms and their dependencies (the "micro" level of a particular computation). Both say: what plays the same role IS the same thing. Both eliminate the gap between "is equivalent to" and "is identical with."

This is not a coincidence. Both are instances of the self-determination principle: a structure that determines its own nature cannot contain arbitrary distinctions. Two types that are equivalent but not identical represent an arbitrary distinction—a difference without a structural difference. Two sub-terms that are observationally equivalent but not identified represent the same kind of arbitrariness. Univalence and canonicalization are the same structural principle applied at different scales.

The practical consequence: if the self-determining structure of reality is a type-theoretic universe governed by univalence (as the Ruliad and Why Type Theory articles argue), then the canonical causal diagrams extracted from that structure are the natural mereology—the genuine decomposition into parts that the structure itself determines. The diagram is not an external imposition on reality; it is reality's own way of distinguishing what depends on what.

Downstream Applications Made Precise

With the canonical causal diagram precisely defined, each downstream application gains a sharper specification.

Consciousness (subjectivity detection). The consciousness article's Stage 2 searches the canonical structure for the reflexive pattern of subjectivity: a world-model M, a self-model S representing M's activity, and binding routing M's contents as presented to S. With the diagram precisely defined, this search is a well-specified procedure: identify sub-diagrams where some substructure's states depend on another substructure's states in the pattern M → S → (binding of M-for-S). The subjectivity property is a structural feature of the diagram, not a vague resemblance.

Mereology (parts as sub-diagrams). The mereology article says genuine parts are substructures with high internal connectivity and defined external interfaces. The diagram makes this precise: a genuine part is a connected subgraph of the canonical diagram with strong internal dependency (many edges within) and sparse external dependency (few edges crossing the boundary). The decomposition into such parts is unique because the diagram is unique. The "integrated substructure" that the mereology article acknowledges is not yet formally defined is precisely a connected subgraph with a specified ratio of internal to external dependency—a computable property of the unique diagram.

Ethics (valence comparison). The ethics article says valence comparison is structural comparison within the diagram. With the diagram precisely defined, this becomes: two perspectives' valences are compared by examining the structural features of their sub-diagrams—allocation fractions, self-model depth, memory integration. These features are properties of the diagram, computable from its structure. The comparison is principled because the diagram is unique and shared.

Agency (deliberation as upstream). The agency article says deliberation is causally efficacious because it is upstream in the causal structure. With the diagram precisely defined, "upstream" means: the deliberation sub-diagram is a predecessor in the partial order. The action depends on the deliberation in the informational-dependence sense. This is a precise, checkable claim about the diagram's structure, not a metaphor.

Time (causation as informational dependence). The time article redefines causation as informational dependence. With the diagram precisely defined, causation is the dependency relation of the canonical diagram. A causes B iff B depends on A in the diagram. This is the formal content of the time article's thesis, made explicit.

The Ruliad (computational structure). The Ruliad article describes the self-determining structure computationally. The canonical causal diagram is the structure that remains when a computational description is stripped of its presentation. The Ruliad's multiway graphs, branchial spaces, and causal graphs are different presentations of dependency structure; the canonical diagram is what is invariant across these presentations. The convergence of the Ruliad with HoTT depends on the claim that both presentations yield the same canonical diagram—this is the formal content of "convergence."

Where the Argument Is Incomplete

Three important qualifications:

Observational equivalence may not be decidable. In full MLTT with universes and inductive types, definitional equality is decidable (normalization terminates). But observational equivalence is coarser and, for sufficiently rich type theories, may not be decidable. This means Step 3 of the quotient procedure may not be effectively computable in all cases. The diagram exists—it is well-defined—but we may not always be able to construct it algorithmically. This is a real limitation: for practical applications (detecting consciousness, comparing valence), we need computable approximations. The definitional quotient (Steps 1 and 2) is always computable; the observational quotient (Step 3) may require heuristic methods or additional theoretical work. The framework should treat the full diagram as the ideal and the definitional-only quotient as a computable approximation.

Continuous systems require generalization. The procedure is defined for discrete, well-typed terms. Physical systems are often modeled as continuous (differential equations, dynamical systems). The framework holds that reality is fundamentally type-theoretic and discrete (computationally universal, as the Ruliad article argues), but the continuous descriptions we use in physics are approximations or coarse-grainings of the underlying discrete structure. The canonical causal diagram of the underlying structure is well-defined; the question is how to recover it from continuous approximations. This is an open problem, but not a fundamental obstacle—if the framework is right, the continuous description is a presentation of a discrete structure, and the diagram of that discrete structure exists.

Non-deterministic systems. The procedure is defined for deterministic computations. A non-deterministic computation (one with branching possible futures) does not have a single trace. The Ruliad's multiway graphs address this: a non-deterministic system has multiple computational histories, and the canonical structure is the entangled limit of all of them. The diagram of a non-deterministic system would be a multiway diagram—a partial order with branching and merging, rather than a simple DAG. The uniqueness argument extends: the multiway diagram is unique (given the same three properties applied to the branching structure), but the formal details require the Ruliad's formalism, which is still being developed. This connects to open question 2 in the Process document: the mapping between the Ruliad's computational language and HoTT's type-theoretic language.

The Strongest Objection

"The whole procedure presupposes that the system is a well-typed term in MLTT. But the framework claims that reality itself is the type-theoretic structure. So you are using the type theory to analyze the type theory—circularly."

This is not circularity; it is self-application, and it is exactly what self-determination requires. A self-determining structure must determine its own nature. If reality is a type-theoretic structure, then the procedure for extracting canonical causal diagrams from type-theoretic structures applies to reality itself. The result is the canonical causal diagram of the self-determining whole—the one diagram that contains all others as sub-diagrams.

This is the diagram the framework has been gesturing at all along: the complete partial order of informational dependence for all of reality. Its extraction from the self-determining structure is guaranteed by the same argument that guarantees the uniqueness of the self-determining structure. The diagram is not an external analysis of reality; it is reality's own way of encoding what depends on what.

The "circularity" is the same as the self-reference in the metaphysics article: the structure must encode its own nature. The canonical causal diagram of the whole IS that encoding—made explicit as a partial order of informational dependencies. The fact that the procedure applies to itself is not a bug; it is the procedure being what it claims to analyze: a self-referential, self-determining structure.

What This Achieves

The canonical causal diagram is now a defined concept with a specified extraction procedure, a uniqueness argument, a resolution of the Putnam problem, and a connection to univalence. The six downstream articles that depend on it now rest on a defined foundation rather than a promissory note.

What remains is the technical work: 1. Formalizing the observational quotient (and characterizing when it is computable) 2. Generalizing to non-deterministic systems (connecting to the Ruliad's multiway formalism) 3. Developing computable approximations for practical applications 4. Proving that the sub-diagram decomposition (the mereology) is well-defined and unique for the class of diagrams relevant to consciousness and ethics

These are hard problems. But they are now well-posed problems—questions about a defined structure, not about a concept that has never been made precise. The framework's most load-bearing concept now has its own weight to stand on.