Finite Categorical Entropy Obstruction Calculus I: Stochastic Factorizations and Bridge Reconstruction

We develop a finite state-relative categorical entropy reconstruction-obstruction calculus. The
ambient setting is the nonempty finite stochastic category FinStoch̸ =∅, whose objects are finite
nonempty sets and whose morphisms are finite Markov kernels. Composition is convolution. The
purpose is not to reaxiomatize probability theory, to introduce a new stochastic category, or
to claim new entropy identities. The contribution is more precise: we organize standard finite
conditional-entropy identities as exact numerical reconstruction certificates for explicitly specified
hidden-variable reconstruction problems attached to pointed stochastic morphisms, factorizations,
trajectories, and bridges. In this first finite paper, obstruction means a support-relative numerical
zero-set criterion for exact reconstruction, not a cohomological, derived, or metric obstruction
theory.
For a stochastic morphism K : X ⇝ Y and an input distribution μ, we define the kernel
entropy
Eμ(K) = H(Y | X).
We prove that Eμ(K) = 0 if and only if K admits a visible deterministic realization on the
support of μ, and we obtain genuine global deterministic realization in FinSet̸ =∅ when μ has
full support. For composable stochastic morphisms
X K
⇝ Y L
⇝ Z,
we define the entropy defect
∆μ(K, L) = Eμ(K) + EμK (L) − Eμ(L ⋆ K),
and prove
∆μ(K, L) = H(Y | X, Z).
Thus the defect is precisely the conditional entropy of the hidden intermediate state. In particular,
endpoint-reconstructible stochastic factorization need not be deterministic or invertible.
For finite stochastic categorical trajectories
X0
K0
⇝ X1
K1
⇝ · · · Kn−1
⇝ Xn,
we define accumulated local conditional uncertainty S(n)
μ0 , endpoint composite entropy C(n)
μ0 , and
hidden path defect D(n)
μ0 . The main result is the path entropy balance law
C(n)
μ0 + D(n)
μ0 = S(n)
μ0 .
We also prove the backward chain decomposition
D(n)
μ0 =
n−1∑
k=1
∆comp
k .

Here ∆comp
k is the two-step defect of the compressed pointed factorization
(X0, μ0) K0:k
⇝ (Xk, μk) Kk
⇝ Xk+1, μk = μ0K0:k.
Finally, we introduce categorical bridge entropy
Φ(n)
t,μ0 = H(Xt | X0, Xn),
a state-relative invariant of the marked two-block factorization detecting exact failure of
intermediate-state reconstruction and quantifying the residual bridge uncertainty. We show
by example that bridge entropy is not determined by the bare endpoint composite. For the
binary symmetric channel, we compute this invariant explicitly and prove a fixed-parameter
long-trajectory reconstruction discontinuity:
Φ∞(θ) =



0, θ = 0,
log 2, 0 < θ < 1,
0, θ = 1.
This is a fixed-parameter long-trajectory reconstruction discontinuity, not a finite-length singu-
larity. It is invisible to the endpoint-only normalized diagnostic n−1H(Xn | X0) on fixed finite
state spaces; this is not a criticism of the classical path entropy rate n−1H(X1, . . . , Xn | X0).