The Boundary Between Exact and Local Sufficiency

Conceptual map

I-03.01Global fiber conditions
I-03.02Local tangent conditions
I-03.03Neighborhood factorization
I-03.04Connected components and branches
I-03.05Value, sign, and ordering targets

1. Pointwise derivative factorization

A target derivative can factor through an observation derivative at one state while the target still varies within nearby fibers. Pointwise, neighborhood, and global claims use different quantifiers.

Let $\mathcal{M}$ and $\mathcal{N}$ be smooth finite-dimensional manifolds, with report $h:\mathcal{M}\to \mathcal{N}$ and target $\tau:\mathcal{M}\to \mathbb{R}^{r}$ . At $x_{0}$ , vertical directions form $V_{x_{0}}=\operatorname{ker}(Dh_{x_{0}})$ .

Definition 1 · First-order sufficiency

The report is first-order sufficient for the target at $x_{0}$ when $D\tau_{x_{0}}$ vanishes on $V_{x_{0}}$ . Equivalently, a linear map $L:\operatorname{range}(Dh_{x_{0}})\to \mathbb{R}^{r}$ satisfies $D\tau_{x_{0}}=L Dh_{x_{0}}$ .

The equivalence is Proposition I-02 applied to tangent spaces. It describes derivatives at one point and supplies no neighborhood decoder by itself.

2. A local decoder under a neighborhood condition

Proposition 1 · Constant-rank local factorization

Suppose $$h$$ and $\tau$ are continuously differentiable near $x_{0}$ , $$h$$ has constant rank $$k$$ there, and $D\tau_{x}$ vanishes on $\operatorname{ker}(Dh_{x})$ at every point in that neighborhood. After shrinking to a constant-rank coordinate neighborhood, a continuously differentiable decoder $\delta$ exists on the local attained report manifold with $\tau=\delta\circ h$ .

Proof. The constant-rank theorem supplies local coordinates ( $$u,v$$ ) in which $$h(u,v)=(u,0)$$ . Vertical directions are the $$v$$ directions. The assumed derivative condition gives $D_{v}\tau=0$ throughout a sufficiently small coordinate box. Integrating along line segments in each $$v$$ -slice shows that $\tau(u,v)$ is independent of $$v$$ . Define $\delta(u,0)=\tau(u,0)$ . ∎

3. A pointwise condition with no local decoder

On $\mathbb{R}^{2}$ , set $$h(x,z)=x$$ and $\tau(x,z)=x^{2}+xz$ . At (0,0), $\operatorname{ker}(Dh)$ is the $$z$$ -axis and $D\tau$ =(0,0), so first-order sufficiency holds. For every nonzero $$x$$ near zero, changing $$z$$ changes $\tau$ while leaving $$h$$ fixed. A local exact decoder therefore does not exist.

The omitted condition is visible in the vertical derivative $\partial \tau/\partial z=x$ : it vanishes at the reference point and fails throughout every open neighborhood.

4. Disconnected fibers defeat global factorization

Let $\mathcal{M}=(\mathbb{R}\setminus\{0\})\times\mathbb{R}$ , define $h(x,z)=x^{2}$ , and set $\tau(x,z)=\operatorname{sign}(x)$ . The report has rank one everywhere. Each positive report $$y$$ has two connected fiber components, the lines $x=\sqrt{y}$ and $x=-\sqrt{y}$ .

The target is smooth on $\mathcal{M}$ and has zero derivative everywhere. It is constant on each connected component and takes values +1 and $$-1$$ on the two components of the same fiber. Local decoders exist around every state; a global decoder from $$h$$ to $\tau$ does not.

Failure case · Local derivative evidence promoted to exact sufficiency

Vanishing vertical derivatives provide componentwise constancy when suitable paths connect points inside a fiber. Exact global sufficiency still requires equality across disconnected components and any other globally separated states sharing the report.

5. Implementation, exercises, and sources

Evaluate the rank of $$Dh$$ across a neighborhood, construct a basis for its kernel, and test $D\tau$ on that basis. Then trace or enumerate connected fiber components and compare component-level target constants. Numerical continuation should record branch labels.

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Exercises

Verify first-order sufficiency and failure of local factorization in Section 3.
Compute the fibers and derivatives in Section 4 and prove the absence of a global decoder.
Strengthen Proposition 1 to a global result under a stated connected-fiber condition.

Partial solutions

1. $$Dh$$ =[1,0], so the vertical space is span{(0,1)}. The vertical target derivative equals $$x$$ and vanishes only on the line $$x=0$$ . 2. Equal positive reports pair the two signs of $$x$$ . A decoder value at $$y$$ would have to equal both +1 and $$-1$$ .

John M. Lee, Introduction to Smooth Manifolds, sections on the rank theorem, submersions, and level sets.Local coordinates and tangent spaces of fibers.
Victor Guillemin and Alan Pollack, Differential Topology, Chapter 1.Constant-rank maps and regular level sets.
III-02 · State Spaces, Fibers, Tangent Directions, and Metrics.Frames and metric-aware tangent calculations.

6. Audit checkpoint

Label the claim as pointwise first-order, local exact, componentwise, or global exact. Report the rank neighborhood, vertical derivative, fiber components, decoder domain, and any branch labels.

7. Scope boundary

The chapter treats smooth finite-dimensional constant-rank maps. Singular strata, nonsmooth fibers, and probabilistic approximate decoders require additional analysis.