E0018 proposition notation E0001

Affine reconstruction

A forward difference along a curved cube \(c\) E0016 decomposes as a sum over coverings of \([k]\), each term an affine difference along the Möbius coordinates \(c(A)\).

(1) Proposition. For \(k \geq 1\), \(f: X \to G\) and \(c \in CT_k(X; x)\),

\[ \Delta(f; x; c) = \sum_{\KC \in \Cov(k)} \Delta(f; x; (c(A))_{A \in \KC}). \]

Proof. Each vertex \(\zeta_x(c;T) = x + \sum_{\emptyset \neq R \subseteq T} c(R)\) is a vertex of the affine cube on the index set \(\KP_+(k)\) with legs \((c(R))_{R \in \KP_+(k)}\). By Taylor duality E0005, with the empty family \(\KH = \emptyset\) contributing the base value \(f(x)\),

\[ f(x + \sum_{\emptyset \neq R \subseteq T} c(R)) = \sum_{\KH \subseteq \KP_+(T)} \Delta(f; x; (c(R))_{R \in \KH}). \]

Insert into the alternating sum defining \(\Delta(f; x; c)\) and exchange sums: a family \(\KH \subseteq \KP_+(k)\) occurs in the term of \(T\) exactly when \(\bigcup \KH \subseteq T\), so its total coefficient is \(\sum_{\bigcup \KH \subseteq T \subseteq [k]} (-1)^{k - |T|} = [\bigcup \KH = [k]]\) by the Boolean sieve E0002. Since \(k \geq 1\), the empty family has \(\bigcup \emptyset = \emptyset \neq [k]\), so exactly the covers survive. (For \(k = 0\) the left side is \(f(x)\) while \(\Cov(0) = \emptyset\), so the hypothesis is necessary.)

(2) Validation (AI review, 2026-07-19, claude-fable-5, pass). Sieve exchange and Taylor step verified against E0002/E0005; hand-expanded \(k = 2\) with defect \(w\) (all five covers, coefficients match); probed \(k = 0, 1\) and affine \(c\). Fixed: added missing hypothesis \(k \geq 1\) (\(\Cov(0) = \emptyset\) but LHS \(= f(x)\)) and made the empty-family/base-value convention explicit in the proof.

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