Resource-Bounded Distinction Systems

4. Contributions

4.1 Candidate Structural Contributions

After the literature review, the honest boundary is:

  • Resource-aware attention, state-dependent neighbor selection, and perception-induced non-reciprocity are established adjacent literatures. They are included as foundations and mechanisms, not claimed as standalone inventions.
  • The proposed structural contribution is quotient-compatible RIAK (§1.10). The attention family is indexed by a resource poset and is required to factor through the corresponding operational quotient, $A_B = \bar A_B \circ q_B$. This adds a behavioral threshold $C_A$ to the observational threshold $C_D$.
  • The cross-domain column-imbalance calculus is the second proposed contribution. The identity $\mathbf{1}^\top(A_B-I)Z = \delta_B^\top Z$ gives the same aggregate-drift functional in attention mechanics and linear representation updates. It is a correspondence to test, not a claim that doubly-stochastic attention or antisymmetric interaction theory is new.
  • Resource-bounded / graded Karoubi envelope (§1.5–1.6). Idempotent splitting under a cost budget with resource-non-increasing morphisms has no direct prior art found in the research review. The closest programs (resource theories of Coecke-Fritz-Spekkens 2016, OPTs of Chiribella et al. 2010, graded monads of Fujii-Katsumata-Melliès 2016, Lawvere cost enrichment 1973) do not produce the same structure.

4.2 Honest Assessment of §7 Theorems

None of the theorems in §7 are mathematically new. They are clean derivations or unifications of established results.

  • Theorem 7.1 (Antisymmetric Momentum Principle). This is a one-line Newton's-third-law cancellation: pair $(i,j)$ contributes $G w_{ij}(x_j - x_i)$ to particle $i$ and $-G w_{ji}(x_j - x_i)$ to particle $j$; the symmetric part cancels exactly. The same result appears as $M\ddot{R} = 2F_n(r)$ in Ivlev et al. (PRX 5, 011035, 2015; arXiv:1403.2417) and explicitly for state-dependent nonreciprocal pair forces in Alston, Cocconi, Bertrand (PRR 5, 043032, 2023; arXiv:2304.07738). The theorem is recovered here as a special case of the framework's non-reciprocal active matter instantiation (§2.7).

  • Theorem 7.3 (Complete Energy Ledger). The three-channel decomposition unifies established components:

    • Nonreciprocal work corresponds to odd elasticity (Scheibner et al., Nat. Phys. 16, 475, 2020; arXiv:1902.07760).
    • Switching work corresponds to ratchet / information-engine physics and the coupling term in the Dechant–Sasa–Ito geometric decomposition of entropy production (PRE 106, 024125, 2022; arXiv:2202.04331).
    • Work-generating cycles in non-reciprocal living solids have been demonstrated experimentally by Tan et al. (Nature 607, 287, 2022) and Chao et al. (Nature Physics, 2026; arXiv:2410.18017).

    The decomposition is a useful unified treatment in the state-dependent witness-graph model, not a new physical result.

4.3 Synthesized Content (Not New)

Properly attributed:

  • Idempotent operators, canonical partitions — Karoubi envelope (1968/1970; textbook 1978).
  • Resource-bounded indistinguishability — Blackwell 1951/1953; Abramsky-Dawar-Wang pebbling comonad 2017; cryptographic indistinguishability.
  • Operational quotients — gauge theory, σ-algebras, conditional expectation.
  • Nonreciprocal active matter mechanics — Ivlev et al. PRX 2015; Fruchart et al. Nature 2021; Alston et al. PRR 2023; Bowick et al. PRX 2022; Scheibner et al. Nat. Phys. 2020.
  • Filippov regularization — Filippov 1988; Utkin 1992.
  • Vietoris-Rips persistent homology — Edelsbrunner-Harer 2010; Carlsson 2009.
  • RG coarse-graining — Wilson 1975; Mori-Zwanzig; Raju-Machta-Sethna.
  • State-test duality / formal contexts — FCA; Chu spaces (Barr LNM 752, 1979; Pratt naming); Abramsky's "Big Toy Models" Synthese 2012 and "Coalgebras, Chu Spaces" JPL 2013.
  • Relation-first foundations — Carboni-Walters 1987; Freyd-Scedrov allegories 1990; Shulman SEAR 2008. (Lawvere 1964 is function-first, not relation-first.)
  • Behavioural equivalence / coalgebra — Rutten; Baldan-Bonchi-Kerstan- König LMCS 2018 (quantitative coalgebraic metrics).
  • Information Bottleneck — Tishby et al. 2000.
  • Metrological resolution — BIPM / VIM (resolution 4.14 and discrimination threshold 4.16 are dual definitions, not identical).
  • Ashby's Law of Requisite Variety — Ashby 1956.
  • Operational theories / reconstruction — Chiribella et al. 2010/2011; Hardy 2011/2013.
  • Quantum Darwinism — Zurek 2009 (Nature Physics 5, 181), 2022. Note: trace distance satisfies $d_F \le d_{F \cup G}$ (more environment increases discrimination); QD-specific content is mutual-information saturation, not trace-distance dynamics.
  • Algorithmic distinguishability — Bennett-Gács-Li-Vitányi-Zurek 1998 (exact only up to O(log) corrections).
  • EF games / FMT — Ehrenfeucht 1961; Immerman 1999; Abramsky-Shah CSL 2018 (separate EF comonad; not to be confused with the pebbling comonad, whose resource is pebble number k, not quantifier rank).
  • Wasserstein / optimal transport — Villani 2009; Santambrogio 2015.
  • Rényi divergences — van Erven-Harremoës 2014 (with 2024 errata).
  • Self-adjointness / open quantum systems — Reed-Simon; Lindblad 1976; Breuer-Petruccione 2002.

4.4 Position

Blackwell experiments, metrological resolution, computational indistinguishability, behavioural equivalence, observability, coarse-graining, and discrimination resource theories all formalize different aspects of a common state-test-resource architecture.

The contribution is the explicit articulation of this common skeleton across 25 traditions (Appendix C), together with the proposed quotient-compatible RIAK, its column-imbalance drift calculus, and the resource-bounded / graded Karoubi envelope (§1.5–1.6). The mechanical theorems in §7 are useful derivations and unifications, not new physics.