Role-Arity and the Structure of Emergence

Editor’s note: This essay predates Running Comes First and originally framed invariance as primitive. I now treat persistence, distinction, and selection as increasingly explicit resolutions of a more primitive self-sustaining process. The diagnostic claims of the essay remain intact; the ontological framing has been updated.

Some questions fail because they demand more functional structure than the target can supply, or because they demand less and force a false dichotomy.

The Claim

Structural capabilities become explicit at specific thresholds of role-arity — the minimum number of functionally distinct roles required for an operation to count as an instance of its operation class. Role-arity here is not argument-arity; it is the number of functionally distinct slots required for an operation to be well-typed.

Invariance is what self-sustaining process looks like at lowest resolution. Difference is what it looks like when distinction stabilizes. Selection is what it looks like when the process discriminates between positions under a criterion and produces a verdict. Role-arity tracks these thresholds of explicitness. These are not assembly stages. The arity levels are focal lengths, not construction steps. The numbering reflects descriptive resolution, not temporal or ontological priority. They are increasingly explicit articulations of one underlying activity.

A question’s role-arity is the minimum number of functional roles that must be instantiated to answer it. A target’s role-arity is the minimum roles it supplies without importing external structure. When a question’s arity exceeds its target’s, the surplus generates phantom structure — foils, criteria, modal spaces, alternative branches — that the target cannot ground. When a question’s arity falls short, it forces flattening: false dichotomies imposed on a subject with more internal structure than the question can represent.

Arity mismatch is a sufficient condition for specific failure modes: phantom depth from overspecification, false dichotomy from underspecification. It is not the whole of question well-formedness, but when the mismatch is present, the failure mode is predictable.

The Table

Notes on formal objects

Arity 1: Invariance is the core stable appearance at this level of description. The simplest formal expression is E(x) = x — a fixed point under a unary operator. But invariance is broader than literal fixed points. Limit cycles, attractors, conserved quantities, and symmetries are all ways self-sustaining process can appear as persistence at low resolution. What matters is not a particular formal shape, but that the substrate carries something through strongly enough for it to remain trackable as the same. The identity function of lambda calculus is a specific instance: I(I) = I, a fixed point of self-application in the untyped setting. But self-application is itself a binary operation (applying something to something), so it is an illustration of invariance, not its definition.

Arity 3: The claim is that explicit selection — the specific operation class that includes computation, classification, and self-reference — requires three functionally distinct roles: candidates for discrimination (input), a criterion that discriminates (process), and a determinate verdict (output). These are functional positions, not necessarily distinct entities. One object can occupy multiple positions — a Turing machine’s read head, rule table, and write head can be instantiated in the same physical substrate. The operation type still has three slots. What matters is that the slots are functionally irreducible: removing any one of the three causes the operation to degenerate into a lower arity. Without candidates, there is nothing to select from. Without a criterion, there is no selection, only a pair of things (arity 2). Without a verdict, selection produces no result and cannot feed back.

Arity 1: Invariance

E(x) = x. Something unchanged under an operation. A fixed point. More broadly: any structure that persists strongly enough across operation to count as the same. An attractor. A conserved quantity. A symmetry. At this resolution, self-sustaining process appears as persistence.

That is what invariance names here. Not a static ontological atom. Not a world made of unary objects. Invariance is the simplest stable appearance of successful carry-through. It is process holding form strongly enough to be tracked as itself.

A thing does not first exist as a finished self-identical unit and only later happen to persist. Persistence is already an achievement. To say that x remains x is already to say that something has happened and that the happening has not lost the thread. Invariance is therefore not prior to process. It is the lowest-resolution report of process succeeding.

At arity 1, only persistence is explicit. Difference is not yet explicit as partition. Selection is not yet explicit as directed verdict. What is available here is the minimal stable appearance of anything being carried through as itself at all.

What arity 1 cannot do

Produce explicit difference. For difference you need two roles: this and not-this. One role has no “not-this.” There is no outside. There is no boundary between inside and outside because boundary is a 2-arity concept. Arity 1 is persistence without explicit partition.

Arity 2: Difference

Two roles. What holds, and what does not hold against it. This and not-this. Inside and outside. 0 and 1. True and false as partition.

At this resolution, self-sustaining process appears as stable distinction. What was implicit in persistence becomes explicit as boundary — the visible face of where carry-through fails or diverges. Difference is not a second substance added onto invariance. It is the same underlying activity now exhibiting contrast: what persists here does not persist there. Arity 2 is the resolution at which non-holding becomes structurally visible against holding.

This is why information becomes explicit at arity 2. A difference that makes a difference requires at least two stable positions — one where something holds and one where it does not, or where it holds differently. At arity 1, something persists. At arity 2, the boundary between persisting and not-persisting becomes a structure in its own right.

A - A = 0 bridges the arities. The invariance operation (arity 1) expressed across a difference (arity 2) yields zero — the measurement that invariance holds. A - B ≠ 0 is the measurement that it doesn’t. Every nonzero value is a degree of divergence from carry-through. Mathematics lives in the gap between zero and not-zero.

Two states can now hold as a partition. But nothing at this level yet determines membership in that partition. Determination is arity 3.

Partition vs. classification

This distinction matters and protects the framework from an internal contradiction. Arity 2 supports explicit partitions — structural differences held strongly enough to count. True and false as two states. Inside and outside as two regions. Food and not-food as two categories. The partition is there. The difference is real.

What arity 2 does not support is determination — reading the partition, applying a criterion, and producing a verdict about which side a thing falls on. “Does A match B?” introduces an irreducible third functional position: a criterion or rule slot. R(A, B) → result. Difference is a 2-role relation. Classification is a 3-role schema because it requires a rule that operates on the two things and produces a result. This is not a claim that the world must contain a third object. It is a claim that the operation class adds a third functional slot.

A bit has two states but no rule that reads them. A membrane has an inside and an outside but no process that decides which molecules belong where — the differential behavior is a structural property, not an evaluation. The moment something reads the bit, classifies the molecule, or determines which side of a boundary a thing falls on, you have arity 3.

Paraconsistency as arity inflation

Priest’s dialetheia — the claim that some propositions are both true and false — is interesting precisely because it reveals an arity constraint.

Arity 2, in its simplest form, is an exclusive partition: two roles, disjoint. A thing occupies one role or the other. True or false. This or that. The structural content of arity 2 is that the roles don’t overlap.

The moment you allow overlap — a proposition that is both true and false — you have introduced a third functional position: overlap membership, or consistency status, or whatever you want to call the track that determines whether a given proposition sits in one role, the other, or both. That third position is not acknowledged in the paraconsistent framework, but it is present. It is doing work. It is what distinguishes a “merely true” proposition from a “true-and-also-false” one. Without it, you cannot tell dialetheia apart from ordinary truths, which means you cannot operate the system.

So paraconsistency does not break arity 2. It inflates beyond it. The phenomenon Priest is modeling genuinely requires more than two functional roles, and his system smuggles in the extra role without acknowledging it as a structural commitment. Even if the extra status is definable within the formalism, its functional role is irreducible: it is exactly what makes “both” behave differently from “true-only” and “false-only” under consequence. This is not a refutation of paraconsistent logic as a formal tool. It is the observation that the formal tool is operating at arity 3 while claiming to extend arity 2.

Arity 3: Selection

Three functionally distinct roles in directed relation: candidates, criterion, verdict. This is the minimum structure required for explicit selection.

At this resolution, self-sustaining process no longer appears merely as persistence or partition. It appears as directed discrimination: candidates are taken up, processed against a criterion, and a verdict is returned. This is the threshold at which distinction becomes operative rather than merely structural.

This is also the threshold at which self-reference becomes possible in the stronger sense. The verdict can feed back as a new candidate. The system can take up its own products as material for further selection. Computation, feedback, explanation, and reflexive structure all become possible here because the process is no longer merely holding a form or a boundary. It is now explicitly operating on what it carries through.

This is where meaningful complexity becomes operationally visible:

• A Turing machine: takes input, applies rules, produces output, and can continue by taking its own evolving state as further input.
• A compiler: takes source, checks it against a criterion, and returns an executable or rejection.
• The halting limit: arises when evaluation is forced back onto its own process in a way that generates an ungrounded request structure.
• Gödel’s incompleteness: arises when a formal system is made to evaluate a sentence whose provability dependencies route through the system’s own proving behavior.

(Both cases are revisited below as applications of the role-arity diagnostic. The stronger claim — that these results arise from non-executable input reaching an ungated engine — is developed in Passarelli 2026, The Executability Gate.)

None of this is explicit at arity 2. A partition can hold at arity 2, but nothing in arity 2 yet reads the partition, applies a criterion, and returns a verdict. Determination is not just difference. It is directed operation on difference. That is why selection requires arity 3. Classification is selection where the verdict is a membership determination with respect to a partition — it is the simplest arity-3 operation, the one that reads arity-2 structure.

The selection loop

The criterion is not alien to the process it operates on. It emerges within the same running as the things it selects. The compiler is itself a program. The criterion is itself a proposition. Self-reference is what you get when the output includes the cycle itself as a possible input. And ungated undecidability is what you get when that self-reference becomes unavoidable.

The Diagnostic: Question-Arity vs. Target-Arity

Definitions

A question’s role-arity is the minimum number of functional roles that must be instantiated to count as answering it.

A target’s role-arity is the minimum roles it supplies without importing structure from outside itself.

Overspecification: question arity > target arity. The surplus roles (foils, criteria, modal spaces, alternative branches) are generated by the question’s own structure, not found in the subject. The apparent depth is a projection of the question’s complexity, not the subject’s.

Underspecification: question arity < target arity. The question forces the subject into fewer roles than it actually occupies. False dichotomies. Forced flattening.

Arity mismatch is a sufficient condition for these specific failure modes. It is not the only way a question can be ill-posed, and matching arity does not guarantee a question is well-formed. Additional conditions matter: the foil must be a live alternative in the target’s own modal space, and the criterion slot must be grounded by the target’s internal structure rather than imported. The diagnostic catches a specific class of failures, not all failures.

Application: “Why is there something rather than nothing?”

This is a contrastive why-question by construction. “Rather than” explicitly introduces a foil. A contrastive why-question has the following role structure:

1. Explanandum — the thing to be explained (something exists)
2. Foil — the alternative (nothing)
3. Criterion — what selects between them (the demanded “why”)
4. Justificatory output — the answer
5. Modal space — the domain in which the foil is a live alternative

That is arity 5 at minimum. The target, at lowest resolution, is persistence: some structure carried through strongly enough to count as the same. Arity 1.

The question does not just exceed the target by one role. It exceeds it by four. The foil (“nothing”), the criterion (“why this rather than that”), the justificatory output (“because…”), and the modal space (“in a world where either could obtain”) are all generated by the question’s own contrastive structure. The target — persistence, invariance, something unchanged under operation — supplies exactly one role. It has no foil, no criterion, no modal alternatives. It does not have the structural content to ground any of those things.

The claim is not that “invariance is the subject.” The claim is that the only candidate for a non-derivative stopping point — the only thing that cannot be further reduced without the act of reduction presupposing it — is 1-role persistence. And a contrastive why-question demands more structure than any 1-role stopping point can supply. The foil (“nothing”) is not found at the target level. It is fabricated by the contrastive structure of the question itself — the question introduces an alternative that the target’s own domain does not contain and cannot adjudicate.

The question is not unanswerable. It is overspecified relative to its target. The depth people sense is the question’s own structural complexity, mistaken for depth in the subject.

This is distinct from the earlier “the question presupposes its own answer” argument. That argument is valid but reads as circular to a hostile reader. The arity-mismatch argument is structural: the question has more moving parts than the thing it’s asking about. The gap between them is the gap between the question’s architecture and the target’s, and no operation within the question’s architecture can reduce its own arity to match.

Application: A contrastive why that resolves (the “passes” case)

“Why does water expand when it freezes rather than contract?”

Contrastive. Has a foil (contraction). Demands a criterion (what selects expansion over contraction). So it deploys at minimum arity 4: explanandum, foil, criterion, output.

The target — molecular structure and thermodynamics — has rich internal structure. Hydrogen bonding geometry, crystalline lattice formation, energy states, phase transition dynamics. The target supplies distinct functional roles: molecular configuration (input), thermodynamic constraints (criterion), resulting volume change (output). The foil (contraction) is a genuine alternative within the same physical domain — most liquids do contract when they freeze. The modal space is licensed: the question asks about this substance relative to the class of substances where the foil is the norm.

The question’s arity does not exceed the target’s. The foil is grounded. The criterion is available. The answer resolves: ice forms a relatively open hydrogen-bonded lattice, lowering density compared to liquid water. The contrastive why is well-formed because the target has enough internal structure to host the comparison.

This is what a well-formed contrastive why looks like: the target supplies enough roles to ground the foil and criterion the question demands. The diagnostic does not dissolve this question. It predicts it should resolve. And it does.

Application: Underspecification — “Is the universe deterministic or random?”

This is an arity-2 question. Two roles, presented as exhaustive: deterministic or random. The binary conflates ontic rule type with epistemic computability.

The subject requires at least a 2×2 description — two independent axes, not one binary:

• Axis 1 (generative rule): deterministic vs. stochastic. Is the process rule-governed with unique successor states, or does it involve genuine probability?
• Axis 2 (epistemic access): decidable vs. undecidable. Can an embedded observer compute the system’s trajectory, or does self-reference create principled limits on prediction?

A system can be deterministic and undecidable (a Turing machine running a non-halting program — fully rule-governed, trajectory unpredictable from within). A system can be stochastic and decidable (a fair coin — genuinely random, but the distribution is fully characterizable). The two axes are independent.

The binary question “deterministic or random?” collapses both axes into one. It forces a subject that varies along two independent dimensions into a single 2-role slot. This is underspecification. The answer cannot be represented in the question’s format because the question does not have enough functional positions to capture the subject’s structure.

The result is a false dichotomy. People argue endlessly about whether the universe is “really deterministic” or “really random” because the question forces a choice between two options that do not exhaust the space. The answer may be “deterministic on one axis and undecidable on the other” — a description that requires more roles than the question provides.

Application: The Halting Limit as Overspecification

“Can we build a computable function that determines, for any arbitrary program, whether it halts?”

The target is a running program. What it supplies is one functional role: running or cessation at the level of the process itself. At lowest resolution, that is arity-1 persistence.

The question demands at minimum five functional roles:

1. The program — the thing being evaluated. The target supplies this.
2. An external evaluator — something distinct from the program, outside it, examining it from a position the target does not supply. Imported by the question.
3. A binary criterion — “halts” versus “loops forever.” Halting is grounded: it is a transition from running to not running. “Loops forever” is not grounded: no physical or mental process instantiates infinite non-termination. One leg of the criterion is real. The other is a non-executable verdict category (Passarelli 2026, Why Is There Something Rather Than Nothing).
4. A finiteness constraint — the evaluator must return its verdict in finite time. But finiteness is a property of completed processes, not of procedures considered in advance. The constraint is circular.
5. Universal scope — the evaluator must handle all programs, including adversarial self-referential constructions engineered to exploit the evaluator’s own process. The target is one program running. Universality is imposed by the question.

The target supplies one role. The question demands five. Four are fabricated by the question’s own structure. The perceived depth of the “halting problem” is the question’s structural complexity, mistaken for depth in the subject.

The well-formed version of the question is: “Is this program currently running?” Arity 2 at most. The target supplies both roles. It resolves trivially. Or: run the program and observe. F(x) = x. The program is its own halting oracle. The overspecification is the entire source of the “limit.”

The formal validator that catches the specific non-executable input Turing’s proof constructs — D(D), the adversarial program fed to itself — is specified in Passarelli (2026), The Executability Gate. The role-arity diagnostic identifies why the specification fails: it demands more roles than the target supplies. The executability gate identifies where the specification fails: the evaluative request ⟨D(D), halting, F⟩ generates a dependency cycle with no grounded exit.

Application: Gödelian Incompleteness as Overspecification

“Is there a consistent formal system, sufficiently powerful to express arithmetic, that can prove all true statements about arithmetic?”

The target is a formal system. What it supplies: axioms (input), inference rules (evaluator), theorems (output). Arity 3.

The question demands that the system evaluate its own completeness from within itself. This requires the system to simultaneously occupy two roles: the object being assessed and the assessor. Operator and operand collapse into one entity — the same structural move as Turing’s D(D). The completeness criterion further demands that the system contain a verdict on every sentence in its domain, including sentences constructed to reference the system’s own proving behavior.

The Gödel sentence G — “this sentence is not provable in S” — is the non-executable expression these demands generate. It is constructed such that its provability verdict routes through itself: proving G requires determining whether a sentence that denies its own provability is provable. The provability of G is defined as a function of the content of G, and the content of G is defined as a function of G’s provability. The specification cycles without ground.

The “incompleteness” is not discovered in the formal system. It is manufactured by the question’s demand that the system evaluate its own completeness — a demand that forces self-referential constructions the system cannot host without generating non-executable input.

The formal analysis — showing that the evaluative request ⟨G, provability, S⟩ fails V₂ by generating a dependency cycle via definitional expansion, negation, and the provability predicate — is specified in Passarelli (2026), The Executability Gate. The role-arity diagnostic identifies the structural reason: arity collapse between operator and operand generates specifications that cycle without ground.

The Unification: Non-Executable Input as a Single Pattern

Three canonical results across three domains:

These are not three separate results. They are three instances of one pattern: a formally correct derivation from non-executable input, producing a result that does not refer to anything, interpreted as a fundamental limit on the domain in which it was derived.

The role-arity framework provides the diagnostic: when a question’s arity exceeds its target’s, or when operator and operand are collapsed into one entity, the surplus or collapsed roles generate specifications that cycle without ground. The executability gate (Passarelli 2026) provides the formal mechanism that catches these specifications before the engine processes them.

A note on the existing essay’s treatment of these results: the arity-3 section above describes the halting limit and Gödel’s incompleteness as cases where self-reference within the system generates ungrounded evaluative structures. That description is correct within an ungated framework — it is what happens when non-executable input reaches an engine with no executability validator. The stronger claim, developed in The Executability Gate, is that the ungated framework is architecturally wrong to admit those specifications at all. Undecidability is the symptom. Non-executable input reaching the engine is the cause.

The Arity Progression Applied to Artificial Life

The cell-to-dwarf design problem maps onto arity thresholds:

The design question — “how far from authored behavior can you get while keeping emergence legible?” — becomes: at what arity threshold does the system start producing arity-3 agents from arity-1 and arity-2 rules? That is the phase transition. That is where something happens that looks like a decision being made by an entity you did not design.

Compact Summary

Role-arity is the minimum number of functionally distinct roles required for an operation. At lowest resolution, self-sustaining process appears as persistence. At higher resolution, it appears as distinction. At higher resolution still, it appears as selection.

Contrastive why-questions have high arity (explanandum, foil, criterion, output, modal space). When aimed at a target whose arity is lower, the surplus roles are generated by the question, not the subject. The perceived depth is in the question, not the answer.

“Why something rather than nothing” is overspecified. The target, at lowest resolution, is persistence: some structure carried through strongly enough to count as the same. The question has arity ≥ 4. The foil and criterion cannot be grounded at the target level. The question dissolves not because it is too hard but because it does not fit.

“Is the universe deterministic or random” is underspecified. The subject varies along at least two independent axes (generative rule and epistemic access). The question provides one binary. The answer cannot be expressed in the question’s format.

Well-formed contrastive questions have targets with enough internal structure to ground the foil and criterion. “Why does water expand when it freezes?” resolves because the target’s molecular structure hosts the comparison.

Arity mismatch is a sufficient condition for these failure modes, not a universal test for question well-formedness.

One role: persistence is explicit. Two roles: distinction is explicit. Three roles: selection becomes explicit and can turn back on itself.

Formal Adjacencies

What’s Novel

The specific move is treating the mismatch between a question’s demanded structure and its target’s available structure as a diagnostic for predictable failure. Overspecification produces phantom depth: surplus roles generated by the question rather than grounded in the target. Underspecification produces false dichotomy: a subject with richer internal structure forced into an impoverished frame.

The role-arity framework names three thresholds at which one underlying activity becomes increasingly explicit: persistence at lowest resolution, distinction at higher resolution, selection at higher resolution still. This turns arity from a count of abstract positions into a structural diagnostic over what a target can actually sustain.

Applied to metaphysics, this dissolves “why is there something rather than nothing” as a contrastive why-question whose demanded structure exceeds that of its target. Applied to broader philosophical questioning, it predicts which contrastive questions resolve and which merely project their own architecture onto the subject.

Applied to logic and computation, the framework yields a boundary architecture rather than a post hoc repair strategy. Contradiction is treated not as a curiosity to be tolerated by weakening inference, but as substrate-cancellation that should be blocked before discrimination begins. Cyclic anchor-failure is treated not as a profound limit on thought, computation, or mathematics, but as non-executable evaluative structure that should never have reached the engine.

The unification claim is that logical explosion, the halting limit, and Gödelian incompleteness are not best understood as three separate monuments in three separate fields. They are three instances of the same general pattern: formally correct derivations from input that cannot be successfully carried through as evaluation.

The paraconsistency observation: that allowing overlap between truth values inflates beyond arity 2 by introducing an unacknowledged third functional role (overlap membership / consistency status), making paraconsistent logic evidence for the arity framework rather than evidence against it.

The arity thresholds as a structure for emergence in artificial life systems — each threshold marks a qualitative phase transition in what kind of agent behavior becomes structurally possible.

A note on precedent: Peirce’s phenomenological categories (Firstness, Secondness, Thirdness) independently identify the same irreducibility structure. Peirce proved formally that triadic relations cannot be decomposed into compositions of dyadic ones. What this project adds is the process-first grounding (the categories as resolutions of one self-sustaining running, not as irreducible phenomenological facts), the validator architecture (V₁ and V₂ as boundary machinery for non-executable input), and the diagnostic applications (overspecification, underspecification, and the dissolution of canonical formal results). The convergence from independent starting points is evidence that the three-role constraint is structural, not arbitrary.