Resource-Bounded Distinction Systems

1. Definitions

1.1 Distinction Operator

Let $X$ be a set. A distinction operator on $X$ is a function $\Pi : X \to X$ such that:

  1. $\Pi(\Pi(x)) = \Pi(x)$ for all $x \in X$ (idempotence).
  2. $\Pi \neq \mathrm{id}_X$ (non-triviality).

The canonical partition is $X = \mathrm{Fix}(\Pi) \sqcup \mathrm{NonFix}(\Pi)$ where $\mathrm{Fix}(\Pi) = \{x \in X : \Pi(x) = x\} = \mathrm{Im}(\Pi)$ and $\mathrm{NonFix}(\Pi) = X \setminus \mathrm{Fix}(\Pi)$.

1.2 Bounded Distinction Operator

A bounded distinction operator is a function $\Pi_t : X \to X$ parameterized by $t \in \mathbb{R}_{\geq 0}$ such that:

  1. For all $0 \leq t < t' < B$, $\Pi_t(x) \neq \Pi_{t'}(x)$ (strict distinction before saturation).
  2. For all $t, t' \geq B$, $\Pi_t(x) = \Pi_{t'}(x)$ (saturation).
  3. $\lim_{t \uparrow B} \Pi_t(x) = \Pi_B(x)$ for all $x$ (right-continuity).

For discrete systems, $t \in \mathbb{N}$ and condition 1 specializes to "if $\Pi^{k-1}(x) \neq \Pi^k(x)$ then $\Pi^k(x) \neq \Pi^{k+1}(x)$ for all $k < B$." Every distinction operator is a bounded distinction operator with $B = 0$.

1.3 Operational Monoid

An operational monoid on $X$ is a monoid $(\mathcal{R}, \circ, e)$ of functions $X \to X$.

1.4 Resource Budget

A resource budget is a function $\mathcal{B} : \mathcal{R} \to \mathbb{R}_{\geq 0} \cup \{\infty\}$. A resource bound is a value $B \in \mathbb{R}_{\geq 0}$ such that only operations with $\mathcal{B}(r) \leq B$ are accessible.

1.5 Resource-Bounded Distinction System (RBDS)

An RBDS is a tuple $\mathcal{D} = (X, \Pi_t, \mathcal{R}, \mathcal{B}, B)$ where:

  • $X$ is a set (state space).
  • $\Pi_t : X \to X$ is a bounded distinction operator (distinction dynamics).
  • $\mathcal{R}$ is an operational monoid on $X$.
  • $\mathcal{B} : \mathcal{R} \to \mathbb{R}_{\geq 0}$ is a resource budget.
  • $B \in \mathbb{R}_{\geq 0}$ is a resource bound.

A distinction $\delta$ is realizable iff the cost of producing it via $\mathcal{R}$ does not exceed $B$.

1.6 Lax RBDS Morphism

A lax RBDS morphism $F : \mathcal{D}_1 \to \mathcal{D}_2$ is a function $f : X_1 \to X_2$ such that:

  1. $f \circ \Pi^{(1)}_B = \Pi^{(2)}_{B'} \circ f$ (distinction preservation).
  2. For every $r_1 \in \mathcal{R}_1$, there exists $r_2 \in \mathcal{R}_2$ with $f \circ r_1 = r_2 \circ f$ and $\mathcal{B}_2(r_2) \leq \mathcal{B}_1(r_1)$ (simulation with resource bound).

RBDSs and lax morphisms form a category $\mathbf{RBDS}$, provided the resource order is transitive and subadditive under composition, so that within-budget morphisms compose associatively. (Cf. the "good order structure" condition required for lax coalgebra morphisms to compose; Hughes-Jacobs 2003, Hasuo 2006.)

1.7 Indistinguishability: Tolerance vs Equivalence

The relation $j \sim_{\varepsilon, i} k \iff |d_{ij}(x) - d_{ik}(x)| \leq \varepsilon$ is reflexive and symmetric but not transitive. It is a tolerance relation, not an equivalence relation.

Fix (operational equivalence). Let $\mathcal{D}_{B,i}$ be the family of distinguishing tests available to observer $i$ at budget $B$. Then:

$$j \sim_{B, i} k \iff \forall D \in \mathcal{D}_{B, i} : D(j) = D(k)$$

This is an equivalence relation (intersection of equivalence relations). This convention is used throughout.

1.8 Limit Procedures and Filippov Regularization

When $B$ falls below a critical threshold $\varepsilon_c$, the witness graph $A(x)$ becomes discontinuous in $x$, producing a hybrid system with discontinuous right-hand side.

The Filippov set-valued map is

$$F_{\mathrm{Fil}}(x) = \overline{\mathrm{co}}\left\{ \lim_{x_k \to x} F(x_k, A(x_k)) \right\}$$

where the convex hull is taken over limit points with $A$ fixed on a definite side of each switching surface. Filippov solutions $\dot{x} \in F_{\mathrm{Fil}}(x)$ admit existence and uniqueness under mild conditions away from switching surfaces.

The softmax weight $w_{ij}(x; \varepsilon)$ provides a one-parameter family of $\varepsilon$-regularizations, all Utkin-equivalent in the $\varepsilon \to 0$ limit.

1.9 Resource-Indexed Theory of Experiments (RITE)

RBDS is a special case. The general formulation is:

Let $\mathbf{C}$ be a fixed ambient category ($\mathbf{Set}$, $\mathbf{Meas}$, $\mathbf{Stoch}$, $\mathbf{Hilb}$, or category of $C^*$-algebras). A Resource-Indexed Theory of Experiments is

$$\mathfrak{D} = \bigl(X, (\mathcal{B}, \preceq), \{\mathcal{E}_B\}_{B \in \mathcal{B}}\bigr)$$

where:

  • $X$ is an object of $\mathbf{C}$ (state space).
  • $(\mathcal{B}, \preceq)$ is a partially ordered set (resource poset).
  • $\mathcal{E}_B \subseteq \mathrm{Mor}(\mathbf{C})$ is a set of admissible experiment morphisms at resource $B$, each landing in a common outcome space $(\mathcal{O}, d_{\mathcal{O}})$.

Degradation functor. For each pair $B_1 \preceq B_2$, specify $\mathcal{D}_{B_1 B_2} : \mathcal{E}_{B_2} \to \mathcal{E}_{B_1}$ such that:

  • $\mathcal{D}_{BB} = \mathrm{id}_{\mathcal{E}_B}$.
  • $\mathcal{D}_{B_1 B_3} = \mathcal{D}_{B_1 B_2} \circ \mathcal{D}_{B_2 B_3}$ when $B_1 \preceq B_2 \preceq B_3$.

This makes $\{(\mathcal{E}_B, \mathcal{D}_{B_1 B_2})\}$ a functor from the resource poset to $\mathbf{Set}$.

Distinguishability pseudometric.

$$d_B(x, y) = \sup_{E \in \mathcal{E}_B} d_{\mathcal{O}}(E(x), E(y))$$

For this to be a pseudometric:

  1. The supremum is finite (either $\mathcal{E}_B$ uniformly bounded, or $\mathcal{O}$ compact and image bounded).
  2. $d_{B_1}(x, y) \geq d_{B_2}(x, y)$ for $B_1 \preceq B_2$ (monotonicity).
  3. Triangle inequality inherited from $d_{\mathcal{O}}$.

Approximate indistinguishability. $x \sim_B^\varepsilon y \iff d_B(x, y) \leq \varepsilon$ is a tolerance relation. The exact indistinguishability $\sim_B := \sim_B^0$ is an equivalence relation.

Operational quotient. $Q_B = X / {\sim_B}$.

Distinction complexity.

$$C_D(x, y) = \inf\bigl\{ B \in \mathcal{B} : d_B(x, y) > \varepsilon_B \bigr\}$$

Proposition 1.1 (RBDS Recovery). RBDS is recovered from RITE when (i) the resource poset is $(\mathbb{R}_{\geq 0}, \leq)$, (ii) each $\mathcal{E}_B = \{\Pi_B\}$ contains one idempotent experiment, and (iii) the degradation functor acts as identity on $\Pi_B$.

1.10 Resource-Indexed Attention Kernel (RIAK)

Let $V = \{1,\ldots,N\}$ be entities, tokens, agents, or bodies, and let $X$ be a global state space. Let $(\mathcal{B},\preceq)$ be a resource poset. The resource need not be computational: it may encode measurement precision, visual field, sensing time, memory, information capacity, or another operational limit. RITE supplies, at each $B \in \mathcal{B}$, an operational equivalence $x \sim_B y$ and quotient $q_B : X \to Q_B := X/{\sim_B}$.

Define the row-stochastic kernel space

$$ \operatorname{Stoch}_N := \left\{A \in \mathbb{R}_{\geq 0}^{N\times N} : A\mathbf{1} = \mathbf{1}\right\}. $$

Definition (Resource-Indexed Attention Kernel). A RIAK is a family $A_B : X \to \operatorname{Stoch}_N$ satisfying the quotient-compatibility axiom

$$ x \sim_B y \quad\Longrightarrow\quad A_B(x) = A_B(y). \tag{RIAK-1} $$

Equivalently, attention factors through the distinctions available at the resource level:

$$ X \xrightarrow{q_B} Q_B \xrightarrow{\bar A_B} \operatorname{Stoch}_N, \qquad A_B = \bar A_B \circ q_B. \tag{1.10} $$

Thus a system cannot attend differently to states it cannot operationally distinguish. This is stronger than merely sparsifying or penalizing attention after a compute budget has been chosen.

For a stochastic observation model, agent $i$ has a resource-limited channel $O_{i,B}: X \rightsquigarrow Y_{i,B}$ and an allocation policy $\alpha_{i,B}: Y_{i,B} \rightsquigarrow \Delta(V)$. The composite $X \xrightarrow{O_{i,B}} Y_{i,B} \xrightarrow{\alpha_{i,B}} \Delta(V)$ generates row $i$ of $A_B$. This separates the question of what an agent can perceive from what it chooses to attend to.

Blackwell coherence. If $B_1 \preceq B_2$ and the poorer observation is a garbling of the richer one,

$$ O_{B_1} = K_{12} \circ O_{B_2}, $$

then a Blackwell-coherent RIAK requires its observation and allocation maps to respect these garblings. The family is consequently ordered by information quality, not only by a scalar attention budget.

For noisy or approximate quotients, replace (RIAK-1) by a Lipschitz condition such as

$$ \|A_B(x)-A_B(y)\| \leq L_B d_B(x,y), $$

where $d_B$ is the RITE distinguishability pseudometric.

Aggregate drift identity. Define the resource-indexed column imbalance

$$ \delta_B(x) := A_B(x)^\top\mathbf{1} - \mathbf{1}. \tag{1.11} $$

For every compatible feature or state matrix $Z$,

$$ \mathbf{1}^\top(A_B-I)Z = \delta_B^\top Z. \tag{1.12} $$

Hence $\delta_B=0$ exactly when attention preserves every linear aggregate. Since $A_B$ is row-stochastic, this is equivalent to $A_B$ being doubly stochastic. In mechanics,

$$ Nm\ddot{x}_{\mathrm{COM}} + N\gamma\dot{x}_{\mathrm{COM}} = G\,\delta_B(X)^\top X, $$

while for linear attention $Y=A_B(H)V$,

$$ \mathbf{1}^\top(Y-V) = \delta_B(H)^\top V. $$

The two equations have different carriers but the same imbalance operator. Writing $\Omega_B=(A_B-A_B^\top)/2$ gives

$$ \delta_B = -2\Omega_B\mathbf{1}. $$

The hierarchy is therefore edge-level antisymmetry $\Omega_B$, node-level imbalance $\delta_B$, and realized aggregate drift $\delta_B^\top Z$. Antisymmetry can exist without aggregate drift when $A_B$ is nonsymmetric but doubly stochastic.

Attention distinction complexity. In addition to the distinction complexity $C_D$ of RITE, define

$$ C_A(x,y) := \inf\left\{B \in \mathcal{B}: A_B(x) \neq A_B(y)\right\}, $$

and, for an edge or allocation entry,

$$ C_A^{ij}(x,y) := \inf\left\{B \in \mathcal{B}: [A_B(x)]_{ij} \neq [A_B(y)]_{ij}\right\}. $$

$C_D$ records when a distinction becomes operationally available; $C_A$ records when that distinction becomes behaviorally consequential through attention. Distinguishability therefore need not imply dynamic selection.