Categorical Entropy Obstruction Theory III Diagram-Collapse Obstructions for Finite Marked Sampleable Stochastic Presentations

This paper develops the finite presentation layer of Categorical Entropy Obstruction Theory.
The basic objects are joint-source finite sampleable DAG presentations equipped with an observed
marking. For such a marked presentation, observation is coordinate collapse: one retains selected
vertex coordinates and forgets the complementary coordinates. The induced joint law defines a
diagram-collapse posterior reconstruction profile
Postdiag(G, Vobs) = (POG , {PUG|OG=o}o∈supp(OG)
),
whose scalar entropy shadow is the diagram-collapse obstruction
Obdiag(G, Vobs) = H(UG | OG) = H(XV \Vobs | XVobs ).
Here “hidden” means unobserved relative to the chosen marking, not ontologically inaccessible
and not a latent-variable model to be statistically identified from samples. We prove zero-
obstruction, support-fiber, targeted reconstruction, monotonicity, computability, and minimal-
marking criteria in the finite setting. We also separate retained root-target composites from
genuine sink endpoint composites, distinguish strong from support-relative composite equality,
and distinguish coordinate-preserving profile equivalence from abstract observed-hidden law
equivalence. The categorical content is finite and groupoid-level: the obstruction factors through
the observed-hidden collapse-profile isomorphism groupoid and is invariant under bijective
law-preserving relabeling via the functorial factorization
JSampDAGmark
fin −→ CollProf ∼=
fin −→ Rdisc
≥0 .
Finally, we prove a quantitative non-completeness theorem: a fixed root-endpoint composite kernel
can support a continuum of hidden finite presentations whose posterior profiles vary and whose
hidden obstruction values form a nondegenerate interval. This gives the finite reconstruction
counterpart of the CEOT I bridge obstruction and provides a finite presentation-level obstruction
interface on which CEOT II-type scaling regimes can be formulated.