3.1 Shared Skeleton
Each instantiation has the same abstract structure: state space, distinction operator, operational monoid, resource budget, bound. The table is:
| Component | TM | QM | TDA | RG | Type Theory | Measure | Active Matter |
|---|---|---|---|---|---|---|---|
| State space | Configurations | Pure states | Simplex candidates | Couplings | Terms | Events | Configuration manifold |
| Distinction op. | $T(n)$-step | Lüders | Simplex membership | Coarse-graining | Normalization | Conditional exp. | Witness selector |
| Operations | Step transition | Unitary + meas. | Distance test | RG flow | Reduction | Set ops | ODE flow |
| Budget | Time steps | Decoherence | Filtration param | Log scale | Reduction steps | Complexity | Time / $\varepsilon$ |
| Bound | $T(n)$ | $\tau_d$ | $W$ | $\log(\Lambda_{UV}/\Lambda_{IR})$ | Normalization | Observer | $\varepsilon$ |
3.2 Categorical Structure
$\mathbf{RBDS}$ supports three natural constructions:
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Coarsening. Define $\mathcal{D}'$ on $X/\!\sim$ where $\sim$ identifies configurations indistinguishable under $\Pi_B$. The morphism $q : X \to X/\!\sim$ is an RBDS morphism if $\mathcal{B}$ respects the quotient. Coarsening loses realizable distinctions.
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Refinement. Define $\mathcal{D}'$ lifting $\Pi_B, \mathcal{R}, \mathcal{B}$ via a refinement $X' \to X$. The morphism is resource-non-increasing only if the refinement is free (introduces no new distinctions).
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Monoidal product. $\mathcal{D}_1 \times \mathcal{D}_2$ has $X_1 \times X_2$, $\Pi^{(1)} \times \Pi^{(2)}$, and $\mathcal{B}(r_1, r_2) = \mathcal{B}_1(r_1) + \mathcal{B}_2(r_2)$.
3.3 Resource Hierarchies
For budgets $B_1 < B_2$, there exist distinctions realizable at $B_2$ but not at $B_1$. This generalizes the time hierarchy (TMs), filtration complexity (TDA), and scale dependence of relevance (RG).