The boundary as a mathematical object

Layer I · Boundary ontology — what the theory says an autonomy boundary is

Most work on agent autonomy starts from behaviour: what the agent does, how well it performs, where it fails. This programme starts one step earlier and asks a definitional question — what kind of mathematical object is an autonomy boundary? Until that is answered, "the agent has a boundary" is a metaphor, and no measurement of it can be more than a proxy for the metaphor.

The answer proposed here is a single construction, stated in the language of category theory.

The construction

Fix a category of causal interventions — the objects are states of the system, and the arrows are interventions you could perform on it. An agent's attributions — which parts of its input it treats as self, as non-self, or as undecided — form a functor into the probability simplex over those three options:

A : Caus^{op} \to Δ^{2}

Three axioms constrain this functor to be compatible with intervention. Together they single out one sub-object of the attribution structure: its stable closure fiber.

B = StableClosureFiber (A)

B is the boundary. It is not a surface, a threshold, or a set of parameters — it is a stable closure fiber of an attribution functor, and everything the later layers say about sharpness and equilibrium is said about this object.

Element	What it is	Status
$Caus$	Category of causal interventions	Definition
$A : Caus^{op} \to Δ^{2}$	Attribution functor (self / non-self / undecided)	Definition
A1–A3	Compatibility axioms on the functor	Assumption
$B = StableClosureFiber (A)$	Existence and uniqueness of the stable closure fiber	Machine-Checked

The boundary has two parts — of two different kinds

This is the structural fact about the boundary that is easiest to get wrong, so it is stated before anything is built on it. The boundary is not a single object. It is a pair:

\partial = (Settled Face — a set Fix (c), Unsettled Grade — a value b (x) = 1 - p_{S} (x) - p_{N} (x))

Part	What kind of thing it is	Reading
Settled Face $Fix (c)$	A set — the fixed points of the cohesion closure	Where attribution is no longer changed by intervention. Equivalently, the image of the right adjoint.
Unsettled Grade $b \in [0, 1]$	A graded value	How far from settled each state is: $b = 0$ fully settled, $b = 1$ maximally two-sided.

Two traps, avoided here by naming. (i) The two parts are different kinds of object — a set and a value. Calling both of them "sets" is a type error, not a stylistic slip. (ii)

b

is a balance measure, not an uncertainty measure. There is no function

f

with

p_{U} = f (b)

: total ignorance and a dead-even split both give

b = 1

, a machine-checked counterexample. The word "fuzzy", long used here, invites exactly that misreading, and is retired.

What this layer does — and does not — claim

Layer I answers one question and only one: what is the autonomy boundary, mathematically? It does not say how sharp a boundary is, whether a sharper boundary is better, or what value any physical quantity should take. Those are different questions, and they are answered in a different layer.

Two layers, not one chain. It is tempting — and, the author believes, the main reason an earlier submission was misread — to write the theory as a single derivation in which the axioms produce the scalar potential. That is not the structure. Layer I establishes the object; Layer II is a separate variational theory that selects an equilibrium of that object. The categorical apparatus is not machinery for deriving a two-term cost function.

Why define it this way

The conceptual move — from a phrase about agents to a named mathematical object — is the contribution this layer claims. Its value does not lie in the difficulty of the algebra that follows, and a reader who evaluates it only as algebra will not see what is being proposed.

What is being bought is the ability to ask subsequent questions in a place where answers can be proved rather than argued: once the boundary is an object, one can ask whether it has a coordinate, whether that coordinate has an equilibrium, and whether the equilibrium is observable.

What is not claimed. That this object is the correct object for agent autonomy. The axioms are assumptions; their adequacy for real agents is an empirical question, and it is open. Assumption Conjecture

Where this leads

With the object fixed, the next step is to give it a coordinate: a single scalar measuring how sharply the boundary discriminates. That is Layer II.