Discrete and Differential Calculus

We develop a functorial calculus for higher finite differences and
derivatives on affine Banach spaces. Cubical probes in Möbius coordinates
carry exact covariant and contravariant operators $\Delta_+$ and $\Delta^+$
for arbitrary maps; symmetric tangent and cotangent probes carry their
smooth counterparts $D_+$ and $D^+$. Composition becomes functoriality:
the coordinate formulas for $\Delta_+$ and $D_+$ are the cover- and
partition-indexed Faà di Bruno formulas. The pullbacks $\Delta^+$ and
$D^+$ are algebra morphisms and are adjoint to their respective
pushforwards. In the smooth sector, $D_+$ is the corresponding coalgebra
morphism, uniquely determined by the total differential. These
algebra-coalgebra adjunctions encode the higher product rules.

The central construction is a weighted collapse of cubical probes. Under
this collapse, covers of excess weight vanish and partitions remain. The
resulting symbol map

$\sigma_k(c) = \sum_{\pi\in{Part}(k)} \prod_{A\in\pi}c(A)$

intertwines the collapsed cubical pushforward with $D_+$. In finite
dimensions, rescaled cube measures converge weakly to point-supported
distributions; contravariantly, the collapse extracts Taylor coefficients
from cubical jets. The smooth higher chain and product rules follow from
the corresponding exact discrete identities.

Applications include vertexwise inversion of cubical pushforward, a finite
Neumann formula for inverse differential pushforward and the derivatives of
local inverses, higher-symbol expansions for collapsing stencils, and the
transport of point differential operators through arbitrary coordinates.
The latter recovers the Laplace--Beltrami and polar biharmonic operators and
produces an exact polar form of the five-point stencil. We give
self-contained proofs of the four-functor calculus and symbol map.

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