The dual objective gives an upper bound on every primal feasible solution, and equality certifies optimality. A small computation will anchor the general statement before the abstraction takes over.
Statement
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.
The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.
Worked algebra
This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.
Conceptual compression
The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.
Caveat
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.
The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.