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Theorem, exact count, and finite counterexample

FPRD-T81

A five-site obstruction to greatest causal factors

Exact statement

Complete future-row equality in the cyclic Boolean two-hinge carrier is a computable right congruence with 1+∏j(2∣Fj∣−1)1+\prod_j(2^{|F_j|}-1) classes for orbit fibres (F_j). Raw updates and fixed-frame row classes have meet GCDs, but on five sites the targets (C_1=\{0,1,2\}) and (C_2=\{0,1,3\}) have no greatest common causal semantic factor: the edge ({0,1}) and diagonal ({0,2}) are nonassociated maximal common factors.

StatusProved and internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Independent residual frames let each target align a proposed factor differently. On a pentagon this exposes two incomparable maximal shapes.

Hypotheses and scope

  • The carrier is the pinned cyclic two-reflection family on five sites.
  • Each target may use its own continuation frame; factors are compared up to temporal units.

Proof or evidence

A finite criterion identifies greatest factors with a single orbit of inclusion-maximal movable subpatterns. The two targets admit both an edge and a diagonal, admit no three-site common factor, and the two maximal factors lie in different dihedral orbits.

Five-site obstruction →

Verification notes

The complete quotient, congruence law, exact class count, factor preorder, residuals, orbit separation, and carrier census through five sites were independently rederived and checked exhaustively.

Limitations

  • Five-site minimality is proved only within the pinned cyclic two-reflection family.
  • The result is a semantic right-action obstruction, not a counterexample to typed GCD or epi-GCD theorems for streaming transducers.
  • Pinned alignment, or one frame shared by all targets, restores an intersection GCD.

Open work

Compile the certificate into the smallest typed irreversible incidence category and test whether legal composition forces a common alignment.

Notes

Direct quotient enumeration finds no failure through n=4 and the stated failure at n=5, proving computational minimality only within the pinned cyclic two-reflection family. Current-Look equality already fails to be a right congruence at n=3. Independent residual reframing is load-bearing; pinned alignment or one frame shared by the whole target vector restores an aligned intersection GCD. This is a semantic-GCD obstruction, not a Bianchini-Puppis EGCD counterexample, a physical atlas obstruction, or a novelty claim.