Comparator note: canonical divisors for FPRD Boolean future-update vectors

Scope. Comparator search for the live intrinsic rooted-atlas attack. The question is whether the vector of all legal future substrate/Look updates from a prefix admits a canonical epi-GCD whose residual vector supports a computable, right-invariant quotient while preserving causal reachability, head bonds, and the observable archive.

Bottom line

The closest primary source is Bianchini–Puppis (LICS 2026), and it supplies a nearly exact proof template, not a ready-made theorem for FPRD. Their abstract streaming-transducer minimization factors each Hankel row of future updates

$$U=(U[t])_{t\in\Sigma^*}=g;V$$

into an epi greatest common divisor $g$ and a residual vector $V$. Under constrained-domain closure and existence of GCDs, the residuals give the right-invariance needed for a minimal transducer. They prove EGCD uniqueness up to isomorphism, compositional behavior under successor sub-vectors, and that GCD existence is necessary for minimal models over their “standard” data structures. Effectivity still requires model-specific algorithms for finite GCDs, residuals, reachable images, equation entailment, and isomorphism.

This is exactly the right comparator for the proposed FPRD epi-GCD. But an FPRD proof must first define an update category in which:

1. composition models actual sequential substrate change;
2. epis make residuals unique;
3. legal head bonds and reachability survive the factorization;
4. equality includes every legally observable future Look row; and
5. the already-emitted archive is either part of the update object or explicitly quotiented only by the legal observation kernel.

The recovered prefix-parity cycle proves that (5) cannot be dropped: two prefixes can have the same normalized residual dynamics but different addressable output cocycles.

The most important warning is that treating update vectors merely as elements of the Boolean semiring does not solve the problem. In $(\{0,1\},\lor,\land)$, idempotence and non-cancellation make scalar factorizations too coarse and residual witnesses non-unique. For vectors under coordinatewise AND, divisibility degenerates to support inclusion. A canonical meet/join of supports may exist as a lattice operation, but it does not encode causal substrate factorization, legal geometry, or archive addressability.

Primary comparators and what transfers

1. Streaming and subsequential transducer minimization

• Christian Bianchini and Gabriele Puppis, “Minimization of Streaming Transducers,” LICS 2026 / arXiv:2605.11190v2. https://arxiv.org/pdf/2605.11190

  Standard machinery. Updates form a typed composition monoid/category. A common inner divisor satisfies $u=g;v_u$; an epi divisor gives a unique residual. An EGCD is greatest among epi common divisors and is unique up to isomorphism. Hankel rows are factored into divisor and residual vectors. Constrained domains plus GCDs yield algebraically minimal models; effective minimization needs additional computability hypotheses. The paper explicitly identifies longest common prefixes in classical sequential transducers as the string instance of this GCD pattern.

  FPRD instantiation still needed. Prove that restricted two-hinge/D30 substrate updates form an admissible data structure; identify its epis; establish constrained-domain closure; prove EGCD existence (or exhibit failure); and show that the residual equivalence respects legal head bonds, stationary transitions, and archive observations.

• Emmanuel Filiot and Pierre-Alain Reynier, “Transducers, Logic and Algebra for Functions of Finite Words,” SIGLOG News 2016. https://pageperso.lis-lab.fr/~pierre-alain.reynier/publis/siglog16.pdf

  Standard machinery. Choffrut’s syntactic congruence has finite index exactly for sequential functions; the congruence compares continuations only after normalizing common output delay. This supports the distinction between finite normalized control and accumulated output.

  FPRD boundary. A WFL late Look can inspect the delay/archive itself, so ordinary output-normalized syntactic congruence is unsound unless archive equality is weakened precisely by the legal observation kernel.

• Yahia Idriss Benalioua, Nathan Lhote, and Pierre-Alain Reynier, “Minimizing Streaming String Transducers: An Algebraic Approach,” 2026. https://arxiv.org/pdf/2604.11567

  Standard machinery. State and register minimization are different coordinates; bimachines and asynchronous bimachines expose that minimizing finite control does not automatically minimize persistent data resources.

  FPRD use. This is a close resource-accounting comparator for the residual/archive split, but it does not impose physical fibers, head bonds, or global Look.

2. Semirings, Boolean factorization, and residual failure modes

• Mehryar Mohri, “Weighted Automata Algorithms.” https://research.google.com/pubs/archive/35076.pdf

  Standard machinery. Weighted determinization/minimization depends on algebraic hypotheses such as weak divisibility and consistent residual choice. The Boolean semiring is idempotent, with $\lor$ as addition and $\land$ as multiplication.

  Consequence for the live attack. Boolean truth-vector factorization alone is not automatically canonical in the sense Bianchini–Puppis require. Non-cancellative residual choices must either be quotiented, rigidified by an epi notion, or shown irrelevant by a diagonal-fill property.

• Jason Eisner, “Simpler and More General Minimization for Weighted Finite-State Automata,” HLT-NAACL 2003. https://aclanthology.org/N03-1009.pdf

  Standard machinery. Weighted minimization over general semirings needs more than unweighted partition refinement; semiring properties determine whether normalization and pushing are available.

• Pauli Miettinen and Stefan Neumann, “Recent Developments in Boolean Matrix Factorization,” IJCAI 2020. https://www.ijcai.org/proceedings/2020/0685.pdf

  Standard machinery. Boolean products express a binary relation as an OR of rank-one AND rectangles. This is the natural comparator for factorizing future Look answer matrices and for counting fibers/terms.

  FPRD boundary. Boolean rank/factorization is extensional and static. It does not enforce a common rooted generator, sequential accessibility, head motion, nonduplication, or geometric realizability. It can support lower bounds or candidate cores, not by itself the atlas quotient.

3. Coalgebraic behavioral equivalence

• Jan J. M. M. Rutten, “Universal Coalgebra: A Theory of Systems,” TCS 249 (2000), 3–80. https://www.cs.cornell.edu/courses/cs6861/2024sp/Handouts/Rutten.pdf

  Standard machinery. Behavioral equivalence is equality under the unique map into a final coalgebra (when available); bisimulation/coinduction provide proof principles for equality of all future observations.

• Filippo Bonchi et al., “Algebra-Coalgebra Duality in Brzozowski’s Minimization Algorithm,” LMCS 2014. https://www.cs.mcgill.ca/~prakash/Pubs/BrzMin.pdf

  Standard machinery. Minimization separates reachability from observability and combines initial/forward and final/backward structure.

  FPRD instantiation. “Equality of all legal future fiber-activation and Look rows” is coalgebraic behavioral equivalence only after the coalgebra’s observation functor includes the observable cocycle/archive and its transition functor restricts successors to legal causal/head-bond moves. Coalgebra supplies the quotient principle; it does not prove finite index, computability, or physical realizability.

4. Canonical Boolean representations and prime implicants

• Randal E. Bryant, “Graph-Based Algorithms for Boolean Function Manipulation,” IEEE Transactions on Computers 35(8), 1986. https://www.cs.cmu.edu/~bryant/pubdir/ieeetc86.pdf

  Standard machinery. Reduced ordered BDDs are canonical for a fixed variable order and support equivalence and restriction algorithms.

  FPRD use and limitation. A fixed legal future-event order could give a canonical finite representation of each extensional Look row and efficient equality checking. BDD canonicity is relative to the imposed order and says nothing about whether the represented updates arise from one reachable stationary atlas. BDD blow-up also remains possible.

• Prime-implicant / Quine–McCluskey machinery is the standard exact route from a Boolean function to maximal implicant cubes and a covering problem. For the one-hinge image, implicants correspond to candidate depth-$\le2$ fibers. This can normalize a fixed terminal Look function, but minimal DNF covers need not be unique, so it is not a canonical divisor theory for causal updates.

5. Finite obstructions and dual certificates

• Farkas’ lemma gives an exact finite alternative for rational linear inequality systems: either the primal atlas/geometry inequalities are feasible or a nonnegative dual combination certifies infeasibility. A concise computational account is E. D. Andersen, “How to use Farkas’ lemma to say something important about infeasible linear problems” (2013): https://docs.mosek.com/whitepapers/infeas.pdf

  Standard machinery. Rational linear infeasibility has checkable rational certificates; extracting an irreducible infeasible subsystem gives a finite unsatisfiable core (minimality requires additional deletion checks).

  FPRD instantiation. Once a bounded atlas candidate is encoded as:

  1. finite Boolean/SAT constraints for transition, fiber incidence, and head bonds, plus (ii) rational linear constraints for hinge coordinates/order, a dual proof packet should be hybrid: a SAT/DRAT-style certificate for the discrete core and a Farkas vector for the linear geometric core. Farkas alone cannot certify failures caused purely by Boolean branching, stationarity, or archive mismatch.

Recommended theorem test for the restricted subclass

Define the future-update row $H_u[t]$ for each reachable prefix $u$ and legal continuation $t$ as a typed physical update, not merely its Boolean Look bit. Include the accumulated archive modulo the kernel of all legal late observations. Then test:

1. Factorization: enumerate epi common divisors $g_u$ of the defined entries of $H_u$, with unique residual vector $V_u=g_u/H_u$.
2. Successor law: verify that $V_{ua}$ is isomorphic to the residual of the $a$-subvector of $V_u$, with the actual hold/advance head bond recorded.
3. Finite quotient: merge prefixes only when divisor type, residual vector, reachable causal type, legal bonds, and archive-observation class agree.
4. Effectivity: use exact BDDs/truth tables for Boolean rows and exact rational geometry; do not infer computability merely from abstract GCD existence.
5. Failure certificate: if two incomparable maximal epi common divisors occur, save both as the smallest non-GCD witness. If no candidate survives reachability/geometry, minimize the SAT core and attach the Farkas vector for its linear part.

Status judgment

• Published standard machinery: output-normalized syntactic congruences; Hankel future rows; divisor/residual minimization; EGCD uniqueness up to isomorphism under suitable hypotheses; coalgebraic behavioral equivalence; canonical fixed-order BDDs; Boolean rectangle factorization; Farkas infeasibility certificates.
• Already established FPRD boundary: state-only/output-normalized congruence can erase legally addressable archive history.
• Still unproved: that the restricted two-hinge/D30 physical update structure has EGCDs; that such EGCDs are computable; that their residuals have finite index; or that the quotient is complete for rooted stationary atlas realizability.
• No novelty claim: the search identified close machinery but did not find a published theorem combining physical fiber updates, causal head bonds, and a legally queryable emitted archive.