Lines with ordered slopes have ordered breakpoints, allowing a monotone deque of candidates. A small computation will anchor the general statement before the abstraction takes over.
Set-up
Fast algebraic algorithms exploit structure in a transform, matrix, or convex objective. The Fourier transform evaluates \(A(x)\) at roots \(\omega_n^k\), while linear-programming duality supplies certificates.
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
The calculation
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
What survives abstraction
A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.
The boundary
An asymptotically fast method still needs algebraic preconditions: an NTT modulus needs suitable roots, and convex-hull optimization needs monotone slopes or queries for its simplest form.
A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.