2. Instantiations

2.1 Turing Machines (DTIME)

Theorem 2.1. Let $$M$$ be a deterministic Turing machine. For any time bound $T : \mathbb{N} \to \mathbb{N}$ , there exists an RBDS $\mathcal{D}_M^T$ whose realizable distinctions are exactly the configurations reachable in $$T(n)$$ steps on input of length $$n$$ .

Construction.

$X_M = \Sigma^* \times \mathbb{Z} \times Q$ (configurations).
Totalize $\delta$ by setting $\delta(c) = c$ at halting configurations.
$\Pi_M^T(w, p, q) = \delta^{\min(T(|w|), \tau_{\text{halt}}(w, p, q))}(w, p, q)$ (halt-at-or-budget-whichever-comes-first).
$\mathcal{R}_M$ generated by single-step transition $\delta$ .
$\mathcal{B}_M(\delta^k) = k$ .
$$B = T(n)$$ .

Proof of correctness.

( $\Rightarrow$ ) If $$c$$ is reachable in $$T(n)$$ steps from $$c_0$$ , then $c = \Pi_M^T(c_0)$ by construction; cost $T(n) \leq B$ , so $$c$$ is realizable.

( $\Leftarrow$ ) Let $\delta^k(c_0) = c$ for some $k \leq T(n)$ . Two cases:

$\tau_{\text{halt}}(c_0) \leq k$ : machine halted before step $$k$$ , $\Pi_M^T(c_0)$ sits at the halting sink, and $$c$$ is also at the sink (sink is a fixed point). Realizable.
$\tau_{\text{halt}}(c_0) > T(n)$ : machine has not halted by step $$T(n)$$ , $\Pi_M^T(c_0) = \delta^{T(n)}(c_0)$ , and $c = \delta^k(c_0)$ is reachable in $k \leq T(n)$ steps. Realizable.

$\blacksquare$

Corollary 2.1.1. $\mathbf{DTIME}(T(n))$ is the class of languages recognized by RBDSs with single-step operational monoid and unit cost. The time hierarchy theorem becomes the statement that distinct resource bounds yield non-equivalent RBDS morphisms.

2.2 Quantum Measurement

Theorem 2.2. For a finite-dimensional quantum system with Hilbert space $\mathcal{H}$ and observable $A = \sum_a a P_a$ with non-degenerate spectrum, there exists an RITE whose realizable distinctions are the post-measurement eigenstates.

Construction.

$X_A = \mathbb{CP}(\mathcal{H})$ (pure-state projective space).
$\Pi_A : \rho \mapsto \sum_a P_a \rho P_a$ (Lüders projection; idempotent on pure states for non-degenerate $$A$$ ).
$\mathcal{R}_A$ generated by unitary evolution and measurement.
$\mathcal{B}_A(r) =$ minimum measurement operations or decoherence time $\tau_d$ .

Status note. For degenerate observables, the Lüders map sends pure states to mixed states, violating the RITE requirement that experiments land in a common outcome space. Full RBDS formulation of general quantum measurement requires a stochastic-RBDS extension (open work).

2.3 Persistent Homology (Vietoris–Rips)

Theorem 2.3. Let $X = \{x_1, \ldots, x_n\} \subset \mathbb{R}^d$ and $$W > 0$$ . There exists an RBDS whose realizable distinctions are the simplices of the Vietoris–Rips complex $$K_W(X)$$ .

Construction.

$X_X = \binom{[n]}{\leq n}$ (subsets).
$\Pi_W(\sigma) = \sigma$ if $\sigma \in K_W(X)$ , else marked unrealized.
$\mathcal{R}_X$ generated by "test simplex for membership."
$\mathcal{B}_X(\sigma) = \binom{n}{|\sigma|} \cdot c_{\mathrm{dist}}$ .

Proof. By definition, $\sigma \in K_W(X) \iff \|x_i - x_j\| \leq W$ for all $i, j \in \sigma$ . The test is realizable iff the budget suffices. $\blacksquare$

Corollary 2.3.1. Persistent homology is the study of realizable distinctions in a one-parameter family of RBDSs $\{K_W\}_{W > 0}$ with shared state space and monotonically increasing budget.

2.4 Renormalization Group

Theorem 2.4. For a QFT $\mathcal{L}_\Lambda$ at UV cutoff $\Lambda$ , there exists an RBDS whose realizable distinctions are the operators accessible below the cutoff. RG flow is the operational monoid.

Construction.

$X_\Lambda$ = space of couplings / correlation functions.
$\Pi_b$ = Wilsonian coarse-graining (integrate out modes with $b\Lambda > |k| > \Lambda$ , rescale).
$\mathcal{R}_\Lambda$ generated by RG flow steps at scale $$b > 1$$ .
$\mathcal{B}(b) = \log b$ .

Proof. An operator of mass dimension $\Delta$ is accessible at scale $\mu$ iff its RG-flow cost from $\Lambda$ to $\mu$ does not exceed the budget. The Wilsonian relevant/marginal/irrelevant classification is exactly $\mathrm{Fix}(\Pi_b)$ vs $\mathrm{NonFix}(\Pi_b)$ . $\blacksquare$

Corollary 2.4.1. Triviality of $\phi^4$ in the continuum limit is the statement that $\mathrm{NonFix}(\Pi_b) = \emptyset$ as $\Lambda \to \infty$ .

2.5 Constructive Type Theory (Martin–Löf)

Theorem 2.5. For a Martin–Löf type theory $\mathcal{T}$ with normalization, the type inhabitants are the realizable distinctions of an RBDS $\mathcal{D}_\mathcal{T}$ .

Construction.

$X_\mathcal{T}$ = well-formed terms (modulo $\alpha$ -equivalence).
$\Pi_N(t) =$ normal form after $$N$$ reduction steps.
$\mathcal{R}_\mathcal{T}$ generated by reduction rules.
$\mathcal{B}(\text{reductions}) = N$ .

Proof. A term $$t : A$$ is a type inhabitant iff its normal form is the canonical introduction form of $$A$$ . The reduction sequence realizes the distinction. Strong normalization ensures $\mathcal{B}$ is bounded; classical (non-constructive) theories admit unbounded budgets. $\blacksquare$

2.6 σ-Algebras and Measure Theory

Theorem 2.6. For a measure space $(\Omega, \mathcal{F}, \mu)$ and sub-σ-algebra $\mathcal{G} \subseteq \mathcal{F}$ that is countably generated, there exists an RBDS whose realizable distinctions are the $\mathcal{G}$ -measurable events.

Construction.

$X_\mathcal{F} = \mathcal{F}$ .
$\Pi_\mathcal{G}(A) = A^*$ (measurable selection onto $\mathcal{G}$ , exists for countably generated $\mathcal{G}$ ).
$\mathcal{R}_\mathcal{G}$ generated by set operations.
$\mathcal{B}(r) =$ descriptive complexity.

Status note. Conditional expectation $\mathbb{E}[\cdot \mid \mathcal{G}]$ operates on $$L^1$$ / $$L^2$$ functions, not directly on σ-algebras. The construction requires either measurable selection (countably generated $\mathcal{G}$ ) or a measure-algebraic framework. The general arbitrary- $\mathcal{G}$ case is open.

2.7 Non-reciprocal Active Matter

Theorem 2.7. Let $$N$$ particles in $\mathbb{R}^d$ evolve under non-reciprocal forces $F_i(x) = G \sum_j w_{ij}(x)(x_j - x_i)$ with state-dependent weights $w_{ij}$ . The system is an RITE.

Construction.

$X = (\mathbb{R}^d)^N$ (configuration manifold).
$\Pi_B$ parameterized by softmax temperature $\varepsilon$ .
$\mathcal{R}$ generated by ODE flow.
$\mathcal{B}(r) =$ operation cost.
$B = \varepsilon$ .

The total COM force is

\dot{\mathbf{P}}_{\text{tot}} = 2G \sum_{i<j} \Omega_{ij}(x)(x_j - x_i) - \gamma \sum_i \dot{x}_i

where $\Omega_{ij}(x) = \frac{1}{2}(w_{ij}(x) - w_{ji}(x))$ is the antisymmetric attention tensor and $\mathbf{J}_{ij} := x_j - x_i$ is the spatial interaction. Component-wise Frobenius notation:

\dot{P}_{\text{tot}}^\alpha = 2G \langle \Omega, J^\alpha \rangle_{\mathrm{Frob}} - \gamma \sum_i \dot{x}_i^\alpha

where $J^\alpha_{ij} := (x_j - x_i)^\alpha$ .

Proof. $\sum_i F_i = G \sum_{ij} w_{ij}(x)(x_j - x_i) = G \sum_{i<j} (w_{ij} - w_{ji})(x_j - x_i) = 2G \sum_{i<j} \Omega_{ij}(x_j - x_i)$ . The symmetric part cancels because $\sum_{ij} S_{ij}(x_j - x_i) = \sum_{i<j}(S_{ij} - S_{ji})(x_j - x_i) = 0$ for $S_{ij} = S_{ji}$ . Adding the damping term gives the result. $\blacksquare$

Remark (What is distinctive about the witness-graph model). The active-matter literature provides the downstream physics; the witness- graph framework supplies an observer-selection mechanism generating the non-reciprocity.

Remark 4 (Perception-induced non-reciprocity: established mechanism, new formal placement). A state-dependent observation or ranking rule can turn reciprocal distances or pair forces into a directed interaction network. Vision-cone models, topological-neighbor models, adaptive networks, and perception-mediated active matter already establish this mechanism. The witness-graph model is therefore not presented as the first discovery that perception can induce non-reciprocity.

The precise specialization used here remains useful:

The base pair force $\mathbf{f}_{ij} = G(x_j - x_i)$ is reciprocal.
A ranking rule, for example $q_i(x) = \arg\max_j \|x_i-x_j\|$ , selects a state-dependent directed witness graph.
The effective attention matrix has an antisymmetric sector $\Omega(x) = (A(x)-A(x)^\top)/2$ , which is the source of aggregate momentum drift in Theorem 7.1.

The candidate contribution is the quotient-compatible resource placement of this mechanism (§1.10): attention is required to factor through $q_B : X \to Q_B$ , and the same column-imbalance statistic controls both mechanical COM forcing and aggregate representation drift. That is a formal synthesis and a testable correspondence; it is not a claim that generic attention asymmetry or perception-driven active matter is new. The distinction between a state becoming distinguishable and that distinction changing attention is measured by $$C_D$$ versus $$C_A$$ .

Remark 5 (Column-Imbalance Drift Principle — "Attention Is All You Need" as mechanics). The witness-graph ODE and Theorem 7.1 are the mechanical face of a crisp algebraic statement that also governs transformer representations. Fix a row-stochastic attention matrix $A \in \mathbb{R}^{N \times N}$ with $A\mathbf{1} = \mathbf{1}$ (the softmax weights of §2.7 are row-stochastic by construction). Define the column-mass vector $c := A^\top \mathbf{1}$ and the column imbalance $\delta := c - \mathbf{1} = A^\top \mathbf{1} - \mathbf{1}$ . Then $\delta = 0$ iff $$A$$ is doubly stochastic.

The same linear functional $\delta^\top(\,\cdot\,)$ couples $\delta$ to two systems:

Attention-Laplacian mechanics. For $\ddot{\mathbf{x}}_i = \kappa \sum_j A_{ij}(\mathbf{x}_j - \mathbf{x}_i)$ , $\sum_i \ddot{\mathbf{x}}_i = \kappa \sum_j (c_j - 1)\mathbf{x}_j = \kappa\,\delta^\top X, \qquad \ddot{\mathbf{x}}_{\mathrm{COM}} = \tfrac{\kappa}{N}\,\delta^\top X \ (\text{equal masses}).$
Transformer representation. For a self-attention head $$Y = AV$$ ( $$V$$ the value matrix, $A\mathbf{1} = \mathbf{1}$ ), $\mathbf{1}^\top Y = c^\top V, \qquad \Delta\mathbf{y}_\Sigma = \mathbf{1}^\top Y - \mathbf{1}^\top V = \delta^\top V.$

Rosetta-stone equation. Column imbalance is the common algebraic source of aggregate drift: $\delta = A^\top \mathbf{1} - \mathbf{1}$ acts as $\delta^\top X$ (COM forcing) in mechanics and as $\delta^\top V$ (representation drift) in the transformer — the same statistic, two carrier spaces.

Column imbalance is the divergence of the antisymmetric sector. Writing $A = S + \Omega$ with $\Omega = (A - A^\top)/2$ (Theorem 7.1's decomposition), row stochasticity gives $\delta = A^\top\mathbf{1} - A\mathbf{1} = (A^\top - A)\mathbf{1} = -2\,\Omega\mathbf{1}.$ So $\Omega$ is the edge-level circulation / non-reciprocity, and $\delta$ is its node-level net imbalance — the divergence of the antisymmetric attention flow. Edge-level antisymmetry is necessary but not sufficient for aggregate drift: a non-symmetric doubly-stochastic matrix has $\Omega \neq 0$ yet $\Omega\mathbf{1} = 0$ (zero column imbalance), so it reconfigures local states with no net translation — the linear-algebraic analogue of Corollary 7.1.1 (mutual pair cancellation).

Balanced-attention theorem. If $$A$$ is doubly stochastic ( $A\mathbf{1} = A^\top\mathbf{1} = \mathbf{1}$ ), then simultaneously there is no attention-induced COM drift ( $\ddot{\mathbf{x}}_{\mathrm{COM}} = 0$ ) and no attention-induced change in the aggregate representation ( $\Delta\mathbf{y}_\Sigma = 0$ ): redistribution without net translation. Conversely, $A^\top\mathbf{1} \neq \mathbf{1}$ endows attention with a global sink/source structure — some tokens become representation sinks, some bodies become interaction sinks, measured by the same functional $\delta$ .

This sharpens "Attention Is All You Need" (Vaswani et al. 2017) into a mechanical statement without claiming tokens are particles: the correspondence is an identity of the same matrix statistic ( $\delta$ ) acting on two different carrier spaces. The transformer-as-interacting- particle-system program is established (Geshkovski-Letrouit-Polyanskiy- Rigollet, BAMS 62, 427-479, 2025, arXiv:2312.10794; Castin-Ablin- Carrillo-Peyré, arXiv:2501.18322; Duerinckx-Geshkovski-Rossi, arXiv:2605.09213), but the column-imbalance drift principle — aggregate drift is sourced by the divergence of the antisymmetric attention sector — is the specific correspondence drawn here.

Remark. $\mathrm{NonFix}(\Pi)$ (subset of $$X$$ ) and $\Omega$ (matrix-valued function on $$X$$ ) are distinct mathematical objects, related by a construction but not literally equal. The claim " $\mathrm{NonFix}(\Pi)$ is the antisymmetric rank of $$A$$ " is incorrect; the correct statement is Theorem 7.1 below.