E0031 theorem notation E0020 → E0001

Collapse theorem

Under weighted rescaling \((\lambda_t c)(A) = t^{|A|} c(A)\) of a tangent cube E0016, the forward difference \(t^{-k} \Delta(\phi; x; \lambda_t c)\) converges as \(t \to 0\). The limit is the cubical differential, a partition sum that bridges discrete and smooth calculus.

(1) Theorem. Let \(\phi\) be \(C^k\) near \(x\) and \(c \in CT_k(X; x)\). The limit

\[ D^\square(\phi; x; c) := \lim_{t \to 0} t^{-k} \Delta(\phi; x; \lambda_t c) \]

exists uniformly on bounded subsets of \(CT_k(X; x)\), and

\[ D^\square(\phi; x; c) = \sum_{\pi \in \Part(k)} D(\phi; x; (c(A))_{A \in \pi}). \]

On affine cubes, \(D^\square(\phi; x; \Aff(v_\bullet)) = D(\phi; x; v_1, \dots, v_k)\).

Proof. By the coordinate formula E0016 and affine reconstruction E0018, \(t^{-k} \Delta(\phi; x; \lambda_t c) = t^{-k} \sum_{\KC \in \Cov(k)} \Delta(\phi; x; (t^{|A|} c(A))_{A \in \KC})\).

For a cover \(\KC\) with \(r = |\KC|\) blocks and \(r \leq k\), the iterated fundamental theorem gives \(\Delta(\phi; x; (t^{|A|} c(A))_{A \in \KC}) = t^{\mathrm{wt}(\KC)} (D(\phi; x; (c(A))_{A \in \KC}) + \eta_\KC(t))\) where \(\eta_\KC(t) \to 0\) uniformly on bounded \(c\), by continuity of \(D^r(\phi; \cdot)\) at \(x\). For \(r > k\), the term is bounded by \(C t^{k+1}\) (at least one block has \(|A| \geq 2\), producing excess weight).

After dividing by \(t^k\): covers with \(r > k\) are \(O(t)\); covers with \(\mathrm{wt}(\KC) > k\) carry \(t^{\mathrm{wt}-k}\) and vanish; by the weight bound E0008, \(\mathrm{wt}(\KC) = k\) iff \(\KC \in \Part(k)\). Only partitions survive with coefficient \(1\).

(2) Definition (Cubical differential pushforward). The map \(D^\square_+(\phi; x): CT_k(X; x) \to CT_k(Y; y)\) with \(D^\square_+(\phi; x; c)(T) := D^\square(\phi; x; \del_T c)\) is the cubical differential pushforward.

(3) Theorem (Differential functoriality). \(D^\square_+(\psi \circ \phi; x) = D^\square_+(\psi; y) \circ D^\square_+(\phi; x)\).

Proof. Set \(c_t(A) := t^{-|A|} \Delta_+(\phi; x; \lambda_t c)(A)\), so \(\Delta_+(\phi; x; \lambda_t c) = \lambda_t c_t\), and \(c_t \to D^\square_+(\phi; x) c\) by the collapse theorem on each face. By exact functoriality E0017, \(t^{-k} \Delta(\psi \circ \phi; x; \lambda_t c) = t^{-k} \Delta(\psi; y; \lambda_t c_t)\). By uniformity of the collapse for \(\psi\) at \(y\) on the bounded family \((c_t)\), the right side converges to \(D^\square(\psi; y; c')\) with \(c' = D^\square_+(\phi; x) c\).

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