i \setminus \Phi][\Phi]$ is well-defined. By definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], $[\Psi \setminus \Phi][\Phi] = (\Psi \setminus \Phi) \cup \Phi = \Psi$.

The case $\Phi = \varnothing$ does not occur naturally in systems theory because no proper system is unresponsive to all possible stimuli. When $\Psi = \Phi$, the basis has no event space through which to receive transient external stimuli.

Uncoverable processes {#S:UNCOVERABLE_PROCESS_APPENDIX}
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Although any procedure does cover some process, some processes have no covering procedure. This disparity arises naturally through limiting the quantity of distinct functionalities participating in a procedure. Here the constraining mechanism is the catalog of functionality, whose membership must be finite. This stricture’s rationale is to emulate software, which is presumed to possess finite functionality.

Uncoverability of a process entails more than failure of definition \[D:COVERING\_PROCEDURE\] in the case of a particular procedure; uncoverability implies failure for *any* procedure constructed from a given catalog of functionality. With process $\lbrace {\mathbf{f}}_n \rbrace \colon {\mathbb{N}}\to {\prod{\Psi}} \times {\prod{\Phi}}$ and catalog of functionality ${\mathscr{F}}$, uncoverability requires an $i \in {\mathbb{N}}$ and term ${\mathbf{f}}_i = {\mathbf{f}}$ such that ${\mathbf{f}} \notin {\mathit{f}}$ for each ${\mathit{f}} \in {\mathscr{F}}$.

Pigeonhole principle
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The pigeonhole principle can verify uncoverability, but not coverability. Suppose two frames have the same abscissa but different ordinates. No single functionality can cover both frames, since functionalities are mappings. In more general analogy, let distinct equi-abscissa frames be pigeons, while functionalities be pigeonholes. If more than $N$ pigeons occupy $N$ pigeonholes, then some pigeonhole contains more than one pigeon, which is not allowed.

Underpigeonholing
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The *frame set* derived from the initial segment of