Introduction
The ordinary chain rule is a composition law. For differentiable maps \(\phi:X\to Y\) and \(\psi:Y\to Z\) between affine Banach spaces, with \(y=\phi(x)\),
Higher derivatives mix orders under composition. The second derivative of a composite depends on both first and second derivatives of its factors. Faà di Bruno's formula is famously complicated in its explicit multi-index form. Constantine and Savits [CS1996] gave the general multivariate formula in this form, with all combinatorial coefficients.
The partition form, following Fraenkel [Fraenkel1978] and Lévy [Levy2006], is particularly suitable for applications: it works directly with symmetric multilinear derivatives, without choosing coordinates. Fix \(k\geq1\), and let \(\Part(k)\) denote the partitions of \([k]=\{1,\dots,k\}\). For composable \(C^k\) maps and \(v_1,\dots,v_k\in T_xX\), the formula reads
Each block \(A\) determines a derivative of \(\phi\); the resulting vectors are the arguments of a derivative of \(\psi\) of order \(|\pi|\).
In this article, we introduce the symmetric pushforward \(D_+(\phi;x)\) to assemble these derivatives into a single linear operator that composes. For a \(C^k\) map \(\phi\), it acts between the truncated symmetric tensor spaces
by \(D_+(\phi;x;1)=1\) and, for \(1\leq r\leq k\),
Each block contributes a derivative valued in \(T_yY\); the product of these vectors lies in the symmetric algebra. A partition with \(m\) blocks therefore contributes in symmetric degree \(m\). At order two,
The two terms occupy different symmetric degrees and are retained together.
For composable \(C^k\) maps, we prove the composition law
Taking the degree-one component of this identity gives the partition Faà di Bruno formula.
We also characterize \(D_+\) intrinsically: it is the unique counital coalgebra lift of the map collecting the positive-order derivatives of \(\phi\). Its adjoint \(D^+\) acts on symmetric forms by composition of truncated Taylor polynomials and hence pulls back Taylor jets. These descriptions give the higher differential both a covariant and a contravariant calculus. In finite dimensions, \(D_+\) is also realized as pushforward of point-supported distributions.
The discrete construction starts from a difficulty absent at first order: a nonlinear map generally sends an affine cube to a curved cube. Lists of edge directions are consequently insufficient for an exact calculus closed under composition. We use based cubes with coordinates \(c_A\) indexed by all nonempty subsets \(A\subseteq[k]\). These Möbius coordinates record edges and higher differences; together they determine every vertex.
Applying an arbitrary map \(\phi\) to the vertices and returning to Möbius coordinates defines the cubical pushforward
We obtain an exact composition law
without any regularity assumption. Its coordinate expansion is the covering Faà di Bruno formula: coverings of \([k]\) take the place of the partitions in the smooth formula. An adjoint pullback \(\Delta^+\) completes the discrete calculus.
We then construct the passage from \(\Delta_+\) to \(D_+\). Rescale the Möbius coordinates according to their order,
and conjugate the cubical pushforward:
For every \(t>0\), this family is exactly functorial for arbitrary maps. At \(t=1\) it is the cubical pushforward; on affine cubes its top coordinate is the usual normalized \(k\)-th forward difference. The family therefore places difference quotients inside a calculus that preserves composition at every positive scale.
Our main analytic theorem proves that, for a \(C^k\) map, this family extends continuously to \(t=0\) on the entire curved-cube space. Convergence is uniform on bounded sets of cubes at each base point, and the extension is jointly continuous in the base point, deformation parameter, and cube. The boundary has the explicit coordinates
We identify this operator with the fiber map of the \(k\)-fold iterated tangent functor,
Uniform convergence on bounded cubes allows exact functoriality to pass to the boundary.
The connection to the symmetric pushforward is furnished by the symbol
We prove the intertwining identity
The symbol sends an affine cube with legs \(v_1,\ldots,v_k\) to \(v_1\cdots v_k\); on curved cubes it retains contributions in every symmetric degree from one to \(k\). This identity explains how the symmetric higher differential arises from the cubical boundary. In the covering expansion, collapse suppresses the overlapping terms and selects the partitions.
We further determine the higher coefficients of this collapse. Explicit higher symbols \(\sigma_{k,m}\), beginning with \(\sigma_{k,0}=\sigma_k\), give
for \(f\in C^{k+M}\), uniformly on bounded sets of curved cubes. Thus the construction describes both the differential boundary and the successive finite-scale corrections to it.
Products and jets participate in the same deformation. We construct an associative product family whose coefficients weight each overlap by a power of \(t\). Its positive-scale fibers carry the discrete covering product; at zero, only disjoint decompositions remain. The symbol pullback relates this boundary product to multiplication of symmetric forms. Together with the adjoint operators, these products yield compatible naturality and Leibniz identities for discrete, quotient, zero, and smooth jets.
Two companion papers establish the coefficient identities used here: Taylor composition and the partition formula in [FDB], and the covering chain rule for finite differences in [DFDB]. The present paper introduces the symmetric pushforward as a higher differential, develops its exact cubical counterpart, and proves the operator identities, boundary extension, and symbol relations connecting them. Classical jets, point-supported distributions, and higher-order tangent bundles provide geometric realizations of this calculus. The final exhibits develop its use in inverse differentiation, finite stencils, the Laplace–Beltrami operator, polynomial and Newton expansions, and Richardson extrapolation.
Related work
Classical jet theory passes to quotients by powers of the maximal ideal [Ehresmann1951] [Cartan1971] [Saunders1989]. Their full duals are bundles of point-supported differential operators, whose counit kernels are the higher-order tangent bundles introduced by Pohl [Pohl1962] [KMS1993]. The cubical probe spaces used here admit no such quotient presentation (81) 🔗; the classical quotients and their duals are recovered at the boundary in The classical completion 🔗, Distributions 🔗, and Globalization 🔗.
Bertram, Glöckner, and Neeb define \(C^k\) maps over general topological fields by requiring the difference quotient \(f^{[1]}(x, v, t) = (f(x + tv) - f(x))/t\) to extend continuously through \(t = 0\), with higher orders by iterating the construction one direction at a time [BGN2004]; on Banach spaces the resulting class coincides with the Fréchet \(C^k\) class. This is the closest comparison for the extension problem studied here. Our family instead extends all Möbius coordinates of a \(k\)-cube simultaneously through one conjugated pushforward. The Newton expansion provides the scalar counterpart of this cubical organization, in the line of classical difference calculus with moving basepoints [Norlund1924] [Jordan1965] [RomanRota1978] [FL2007] [DT2008]. Bertram's later calculus develops the higher-order theory through simplicial divided differences and Weil functors [Bertram2013] and through cubic rings [Bertram2015b] [Bertram2017].
At order one, the coordinate change used here is the standard chart for deformation to the normal cone [Fulton1998]. This connects the family to Connes' tangent groupoid [Connes1994] and its higher-order variants [vEY19], although no groupoid structure is used in the proofs. The algebra fiber at \(t = 0\) is the Weil algebra of the iterated tangent functor [KMS1993] [BertramSouvay2012], compared with Bertram's cubic rings in (42) 🔗. Synthetic differential geometry takes nilpotent infinitesimals as primitive via the Kock--Lawvere axiom [Kock2006]; here the specific Weil algebra arises as the \(t = 0\) boundary of the deformation, with the Boolean vertex algebra at \(t = 1\).
The partition form of Faà di Bruno is classical [Fraenkel1978] [CS1996]; Johnson surveys the one-variable history [Johnson2002] [FaaDiBruno1855] [FaaDiBruno1857], and the Hopf-algebraic line treats the scalar formula [JR1979] [FGB2005] [FM2014]. The same partition lattice underlies the moment-cumulant transform [McCullagh1987] [LS1959] [Speed1983], the transform of Yates's algorithm is Möbius inversion on the Boolean cube [Yates1937], and the cubical alternating sums reappear as cross-effects in functor calculus [EML1954] [BJORT2018].
Point-supported distributions provide a second interpretation of the symmetric operators. We use the structure theorem for such distributions [Schwartz1966] [Hormander1983] and the pushforward viewpoint emphasized by Kontsevich [KDQ03]; the closing \(L_\infty\) exhibit compares the same coalgebra map with the conventions of deformation theory Structural dictionary: \(D_+\) as \(L_\infty\)-morphism 🔗. Murfet gives the partition formula for the cofree-coalgebra lift [Murfet2015], Thm. 2.22, and Clift and Murfet use the same construction in differential linear logic [CM2020]. Here the cogenerating map is the total Fréchet differential of a \(C^k\) map between affine Banach spaces. We identify \(D^+\) with Taylor pullback, realize \(D_+\) as distributional pushforward, and connect \(D_+\) to the cubical calculus through \(Q_k\) and the symbol map (124) 🔗.
Automatic differentiation gives the algorithmic counterpart. Forward mode transports truncated Taylor coefficients through composites and corresponds to the covariant direction; reverse mode transports covector arrays and corresponds to the contravariant direction [GriewankWalther2008] [Betancourt2018] [Sangha2025]. The order-condition exhibit treats finite-difference schemes through their coefficient probes Order conditions and Richardson extrapolation 🔗; Richardson's deferred limit [Richardson1911] [RichardsonGaunt1927] is expressed by cancellation of the leading error coefficient in the deformation parameter.
Setting
Throughout, \(X, Y, Z\) are affine spaces over real vector spaces \(E, F, G\); points are denoted \(x \in X\), \(y \in Y\), \(z \in Z\). Maps of affine spaces are \(\phi: X \to Y\) and \(\psi: Y \to Z\) with \(\phi(x) = y\), \(\psi(y) = z\); no regularity is assumed. Scalar functions are \(f, g: X \to \IR\). The exact theory is topology-free; norms and the Fréchet \(C^k\) convention enter only with the extension theory in Deformations and collapse 🔗.
(1) Remark (Notation). We prefer explicit arguments over subscript and bracket notation, writing \(\Delta(f; x; c)\) instead of \(\Delta_c f(x)\) and \(D(f; x; v_1, \dots, v_k)\) instead of \(D^k f(x)[v_1, \dots, v_k]\). Semicolons and commas are both argument separators; we use semicolons as a visual hint when arguments are of different kinds. Two currying conventions keep the formulas light. Tail currying: dropping trailing arguments denotes the resulting map, so \(\Delta_+(\phi; x)\) is the map \(c \mapsto \Delta_+(\phi; x; c)\); conversely, applying a curried operator appends to the argument list, \(\Delta_+(\phi; x)\, c = \Delta_+(\phi; x; c)\). Slot currying: a dash in an argument slot denotes the map in that slot, as in \(\Delta(-; x): f \mapsto \Delta(f; x)\).
One symbol may have several definitions, selected by the type of its argument. Where two readings share a type, the declared role of the argument selects between them. This distinction matters for the differential \(D\): its map reading and its observable reading agree in positive degrees and differ in degree zero, where a based target kills constants and an observable records the value \(f(x)\) (45) 🔗; proofs that switch between the readings say so.
The direction slot of \(D\) accepts a tuple or a symmetric tensor, so \(D(f; x; v_1, v_2)\) and \(D(f; x; v_1 v_2)\) denote the same value. The argument determines the order of differentiation, so no order superscripts appear.
Partiality. Maps need only be defined near the base point: for \(\phi: U \to Y\) on a subset \(U \subseteq X\), every operator built from \(\phi\) is defined on the locus where its finitely many required vertices lie in \(U\); for \(U = X\) every operator is total. We write \(\phi: X \to Y\) throughout; all statements localize by this convention.
For \(k \geq 1\) write \([k] = \set{1, \dots, k}\). For a finite set \(S\), \(\KP(S)\) is its power set and \(\KP_+(S) = \KP(S) \setminus \set{\emptyset}\) is the set of nonempty subsets. The set \(\Part(S)\) consists of the unordered partitions of \(S\) into nonempty blocks; for example, \(\set{\set{1,3}, \set{2}} \in \Part([3])\), and \(\Part(\emptyset) = \set{\emptyset}\), the empty partition. A cover of \([k]\) is a family \(H \subseteq \KP_+(k)\) with \(\bigcup H = [k]\); the set of covers is \(\Cov(k)\), and \(\wt(H) := \sum_{A \in H} |A|\) is the weight of a cover. Every partition of \([k]\) is a cover. We abbreviate \(\KP(k) := \KP([k])\), \(\KP_+(k) := \KP_+([k])\), and \(\Part(k) := \Part([k])\).
Discrete calculus
This chapter develops an exact operator calculus for arbitrary maps. It uses no topology; jets enter only after the covariant and contravariant operators have been constructed.
Cubes and forward differences
(2) Definition (Cubes). For \(k \geq 1\) define:
- \(CT_k(X, x) := \Map(\KP_+(k), E)\), the tangent cubes of order \(k\); its coordinates are indexed by the nonempty subsets of \([k]\).
- \(CT_*(X, x) := \bigsqcup_{k \geq 0} CT_k(X, x)\), where \(CT_0(X, x) := \set{0}\).
- \(\Cube_k(X, x) := \set{q: \KP(k) \to X,\; q_\emptyset = x}\) is the space of geometric cubes of order \(k\) based at \(x\).
- Coordinates are written as subscripts, \(c_A := c(A)\), with \(c_i := c_{\set{i}}\) for singletons: on cube letters, subscripts denote Möbius coordinates. The values \(c_i\) are the legs of \(c \in CT_k(X, x)\); the values \(c_A\) with \(|A| \geq 2\) are its Möbius defects.
- The affine cube \(c = \Aff(v_1, \dots, v_k)\) is defined by \(c_i = v_i\) and \(c_A = 0\) for \(|A| \geq 2\); a cube is curved if it is not affine.
- For \(T \subseteq [k]\), the restriction \(\del_T c := c|_{\KP_+(T)}\) is the face on \(T\), regarded as a cube of order \(|T|\) by increasing relabeling.
(3) Remark (Iterated tangent spaces). Tangent cubes are the fibers of iterated tangent spaces. If \(X\) is open in an affine Banach space with model space \(E\), then \(TX\isom X\times E\), and iteration gives \(T^kX\isom X\times\Map(\KP_+(k),E)\). Thus \(T(TX)\) has coordinates \((x,v_{\set{1}},v_{\set{2}},v_{\set{1,2}})\), and each further tangent step adds a direction for every existing coordinate. The original basepoint remains in \(X\), while all other coordinates range over \(E\); fixing that basepoint at \(x\) leaves precisely \(CT_k(X,x)\). We use the coordinate identification of Duarte and Torres [DT2008].
(4) Proposition (Boolean Möbius inversion). Let \(S\) be a finite set, \(G\) an abelian group, and \(a:\KP(S)\to G\). Define
Then \(\mu\circ\zeta=\zeta\circ\mu=\id\) on \(\Map(\KP(S),G)\).
Proof. The coefficient of \(a(A)\) in \((\mu\zeta a)(T)\) is \(\sum_{A\subseteq U\subseteq T}(-1)^{|T|-|U|} =(1-1)^{|T\setminus A|}=[A=T]\). The computation for \(\zeta\mu\) is the same.
(5) Corollary (Boolean sieve). For \(A\subseteq S\), \(\sum_{A\subseteq U\subseteq S}(-1)^{|S|-|U|}=[A=S]\).
(6) Definition (Zeta and Möbius transforms). The geometric realization and the tangent logarithm
are maps \(\zeta_x: CT_k(X, x) \to \Cube_k(X, x)\) and \(\mu_x: \Cube_k(X, x) \to CT_k(X, x)\).
(7) Lemma (Zeta-Möbius inversion). \(\zeta_x\) and \(\mu_x\) are inverse bijections.
Proof. Apply Boolean Möbius inversion (4) 🔗 to the function \(T \mapsto q_T - x\) on \(\KP(k)\), which vanishes at \(T = \emptyset\) for based cubes. Conversely, for \(c \in CT_k(X, x)\) extend \(c\) to \(\KP(k)\) by \(c_\emptyset := 0\) and apply (4) 🔗 again.
(8) Definition (Forward difference). For \(k \geq 1\) and an observable \(h: X \to F\) with values in a real vector space \(F\) define:
- \(\Delta(h; x; c) := \sum_{T \subseteq [k]} (-1)^{k - |T|}\, h(x+\sum_{\emptyset\neq A\subseteq T}c_A) \in F\) is the forward difference of \(h\) along \(c\in CT_k(X,x)\).
- \(\Delta(h;x;(v_i)_{i\in S}):= \sum_{T\subseteq S}(-1)^{|S|-|T|}h(x+\sum_{i\in T}v_i)\), where \((v_i)_{i\in S}\) is a finite family in \(E\).
- \(\Delta(h; x; v_1, \dots, v_k) := \Delta(h; x; \Aff(v_1, \dots, v_k))\) is the difference along the affine cube with legs \(v_1,\dots,v_k\).
- \(\Delta(h; x; 0) := h(x)\) on the unique order-zero cube \(0 \in CT_0(X, x)\).
- \(\Delta(\phi; x; c) \in F\) for \(\phi: X \to Y\) and \(k \geq 1\) is defined by the same alternating sum in the translation space of \(Y\).
(9) Remark. On affine cubes, \(\Delta(f; x; v_1, \dots, v_r) = (\Delta_{v_1} \circ \cdots \circ \Delta_{v_r} f)(x)\) is the classical iterated forward difference, where \(\Delta_v f := f(\cdot + v) - f(\cdot)\); the operators \(\Delta_v\) commute.
(10) Proposition (Discrete Taylor duality). For an observable \(h: X \to F\) and \(c \in CT_k(X, x)\), the functions \(T \mapsto \Delta(h; x; \del_T c)\) and \(T \mapsto h(\zeta_x(c)_T)\) on \(\KP(k)\) are Möbius and zeta transforms of each other:
For affine cubes this is Newton's formula \(h(x + \sum_{i \in T} v_i) = \sum_{A \subseteq T} \Delta(h; x; v_A)\), where \(v_A := (v_i)_{i \in A}\).
Proof. \(\Delta(h; x; \del_T c)\) is by definition the Möbius transform of the vertex function; invert with (4) 🔗.
The cubical pushforward
(11) Definition (Cubical pushforward, cf. [DT2008] Sec. 3). For a map \(\phi: X \to Y\) define:
- \((\phi_* q)_T := \phi(q_T)\) is the geometric pushforward \(\phi_*: \Cube_k(X, x) \to \Cube_k(Y, y)\).
- \(\Delta_+(\phi; x) := \mu_y \circ \phi_* \circ \zeta_x\) is the cubical pushforward \(\Delta_+(\phi; x): CT_k(X, x) \to CT_k(Y, y)\).
In grade zero, all cubical pushforwards are the unique maps between the singletons \(CT_0 = \set{0}\), and their pullbacks act identically on the coefficient space.
(12) Lemma (Coordinates of the pushforward). For \(\emptyset \neq T \subseteq [k]\),
and on affine cubes \(\Delta_+(\phi;\, x;\, \Aff(v_1, \dots, v_k))_T = \Delta(\phi;\, x;\, v_T)\), where \(v_T = (v_i)_{i \in T}\).
Proof. \(\mu_y(\phi_* \zeta_x c)_T = \sum_{A \subseteq T} (-1)^{|T| - |A|} (\phi(\zeta_x(c)_A) - y)\); the terms \(-y\) cancel since the coefficients sum to zero, and \(\zeta_x(c)_A\) for \(A \subseteq T\) depends only on \(\del_T c\).
(13) Theorem (Exact functoriality). For arbitrary maps \(\phi: X \to Y\), \(\psi: Y \to Z\) and every \(k \geq 1\),
and the pushforward is compatible with faces: for \(T \subseteq [k]\),
the right side taken in order \(|T|\).
Proof. \((\psi \circ \phi)_* = \psi_* \circ \phi_*\) holds vertexwise. Insert \(\zeta_y \circ \mu_y = \id\) (7) 🔗 between the two factors in \(\mu_z \circ \psi_* \circ \phi_* \circ \zeta_x\). Face compatibility: by (12) 🔗, both sides have coordinates \(\Delta(\phi;\, x;\, \del_A c)\) for \(\emptyset \neq A \subseteq T\), before the conventional increasing relabeling. Both sides are therefore the same restriction to \(\KP_+(T)\).
Adjunction and pullback
(14) Definition (Functions and measures). We use the following spaces, maps, and pairing.
- \(F(X) := \Map(X, \IR)\) is the space of functions on \(X\), with the topology of pointwise convergence; the bare letter \(F\) remains the model space of \(Y\), the argument distinguishing the two.
- \(F'(X) := \mathcal{L}(F(X), \IR) \isom \Map_{\mathrm{fin}}(X, \IR)\) is its continuous dual, the finitely supported signed measures \(\nu = \sum_i \lambda_i \delta(z_i)\).
- A map \(\phi: X \to Y\) acts by pullback \(\phi^* f := f \circ \phi\) on functions and by pushforward \(\langle \phi_* \nu, f \rangle := \langle \nu, \phi^* f \rangle\) on measures.
- \(\langle \nu, f \rangle := \sum_i \lambda_i f(z_i)\) is the duality pairing between \(F'(X)\) and \(F(X)\). For a real vector space \(F\), the same finite sum defines \(\langle \nu, f \rangle \in F\) for \(f: X \to F\).
The cube measure records the forward difference as a pairing in \(F'(X)\) (14) 🔗.
(15) Definition (Cube measure). The cube measure of \(c \in CT_k(X, x)\) is
The forward difference is the pairing of a cube measure against an observable:
(16) Definition (Co-cubes and graded observables). For \(k \geq 0\) define:
- \(CT^k(X, x) := \Map(CT_k(X, x), \IR)\) is the space of co-cubes of order \(k\); in order zero, \(CT^0(X, x) \isom \IR\).
- \(CT^*(X, x) := \Map(CT_*(X, x), \IR) = \prod_{k \geq 0} CT^k(X, x)\) is the space of graded observables.
- \(CT^*(X, x; F) := \prod_{k \geq 0} \Map(CT_k(X, x), F)\) is the space of \(F\)-valued graded observables, for a real vector space \(F\); \(CT^*(X, x) = CT^*(X, x; \IR)\). Every pullback below acts on \(F\)-valued observables by the same formula.
- The pointwise product on \(CT^k(X, x)\) and, gradewise, on \(CT^*(X, x)\) is \((\omega\eta)(c) := \omega(c)\, \eta(c)\), with unit \(\mathbf 1_{\mathrm{pt}} = (1, 1, 1, \dots)\). This differs from the unit \((1, 0, 0, \dots)\) of the covering products below.
- For \(\omega \in CT^*(X, x)\), the truncation is \(\omega^{\leq k} := (\omega_0, \dots, \omega_k)\).
- \(\langle \omega, c \rangle := \omega(c)\) is the evaluation pairing of \(CT^k(X, x)\) with \(CT_k(X, x)\), and of \(CT^*(X, x)\) with \(CT_*(X, x)\).
(17) Definition (Cubical pullback). For an arbitrary map \(\phi: X \to Y\), the cubical pullback is precomposition with the cubical pushforward,
By construction, pushforward and pullback are adjoint under the evaluation pairing,
(18) Theorem (Discrete adjunction). For arbitrary \(\phi: X \to Y\), \(f: Y \to F\), and \(c \in CT_k(X, x)\),
Proof. By (7) 🔗, \(\zeta_y(\Delta_+(\phi; x)c) = \phi_*(\zeta_x c)\); the first identity follows vertexwise, the second by pairing against \(f\).
(19) Theorem (Contravariant functoriality). For arbitrary maps \(\phi: X \to Y\) and \(\psi: Y \to Z\),
(20) Proposition (The pullback is multiplicative). \(\Delta^+(\phi;\, x)\) is a unital algebra morphism for the pointwise product of co-cubes.
Proof. Precomposition with any map is multiplicative for pointwise products: \((\omega \eta) \circ \Delta_+(\phi; x) = (\omega \circ \Delta_+(\phi; x))\,(\eta \circ \Delta_+(\phi; x))\); the unit \(\mathbf 1_{\mathrm{pt}}\), constant \(1\), pulls back to itself.
Affine reconstruction and Faà di Bruno
With respect to the difference pairing, a curved cube expands as a sum of affine cubes indexed by covers.
(21) Proposition (Affine reconstruction). For an observable \(h: X \to F\) and \(c \in CT_k(X, x)\),
Proof. Each vertex is a vertex of the affine cube on the index set \(\KP_+(k)\) with legs \((c_A)_{A \in \KP_+(k)}\): by Newton's formula (10) 🔗,
Insert this into the definition of \(\Delta(h; x; c)\) and exchange sums: a family \(H \subseteq \KP_+(k)\) occurs in the term of \(T\) exactly when \(\bigcup H \subseteq T\), so its total coefficient is \(\sum_{\bigcup H \subseteq T \subseteq [k]} (-1)^{k - |T|} = [\bigcup H = [k]]\) by the Boolean sieve (5) 🔗. Only covers remain.
(22) Theorem (Covering formula, [DFDB]). For arbitrary maps \(\phi: X \to Y\), observables \(f: Y \to F\), and \(v_1, \dots, v_k \in E\),
and for arbitrary cubes the one-term form
Proof. The one-term form is the discrete adjunction (18) 🔗. For the covering form let \(c = \Aff(v_1, \dots, v_k)\): by the coordinate formula (12) 🔗, \(\Delta_+(\phi;\, x)\, c\) has coordinates \(\Delta(\phi;\, x;\, v_A)\), and affine reconstruction (21) 🔗 expands the one-term form over covers.
(23) Remark. The coefficient systems and iterated covering formulas are developed in [DFDB]. The order-two formula is computed in the exhibits An order-two cube through the deformation 🔗.
Deformations and collapse
This chapter constructs the deformation on the cubical \(Q\) and \(\Delta\) rungs and extends it to the cubical zero fiber. The next chapter applies the symbol map and passes to symmetric tensors.
The Q-family
(24) Definition (Rescalings). For \(c \in CT_k(X, x)\) define:
- \((\lambda_t c)_A := t^{|A|}\, c_A\) is the (isotropic) weighted rescaling, where \(t \in \IR\);
- \((\lambda_{\mathbf t}\, c)_A := \bigl(\prod_{i \in A} t_i\bigr) c_A\) is the anisotropic rescaling (overloaded, selected by argument type), where \(\mathbf t = (t_1, \dots, t_k) \in (0,1]^k\).
(25) Remark. \(\lambda_t = \lambda_{(t, \dots, t)}\). Then \(\lambda_t \Aff(v_1, \dots, v_k) = \Aff(t v_1, \dots, t v_k)\), \(\del_T \lambda_t = \lambda_t \del_T\), and \(\lambda_t^{-1} = \lambda_{1/t}\) for \(t \neq 0\). The isotropic rescaling defines the deformation below. We make no multiscale claim for the anisotropic form; it reappears only in the jet ladder of Jet calculus 🔗.
(26) Definition (The Q-family). For \(t \in (0,1]\) and arbitrary \(\phi: X \to Y\), the deformed cubical pushforward is the conjugation
with coordinates, for \(\emptyset \neq T \subseteq [k]\), \(Q_k(\phi;\, x;\, t;\, c)_T = \frac{1}{t^{|T|}}\, \Delta(\phi;\, x;\, \lambda_t\, \del_T c)\). Across all grades we write \(Q_*(\phi;\, x;\, t): CT_*(X, x) \to CT_*(Y, y)\).
(27) Remark (Affine cubes). On affine cubes the conjugation produces the difference quotients: since \(\lambda_t \Aff(v_1, \dots, v_k) = \Aff(t v_1, \dots, t v_k)\) (24) 🔗, the top coordinate is
the grade-\(k\) difference quotient of \(\phi\) along the legs \(v_i\) at scale \(t\); the lower coordinates are the same expressions on the faces.
(28) Definition (Deformed pullback). For \(t \in (0,1]\), the deformed pullback is precomposition,
acting on \(F\)-valued observables by the same formula.
The superscript \(0\) marks the boundary value at \(t = 0\), where it exists: the zero fiber \(Q^0_*\) is defined in the Extension section below.
(29) Proposition (The punctured deformation). For arbitrary maps \(\phi: X \to Y\), \(\psi: Y \to Z\) and every \(t \in (0,1]\):
1) \(Q_k(\phi;\, x;\, 1) = \Delta_+(\phi;\, x)\).
2) \(Q_k(\psi \circ \phi;\, x;\, t) = Q_k(\psi;\, y;\, t) \circ Q_k(\phi;\, x;\, t)\) and \(Q_k(\id_X;\, x;\, t) = \id\).
3) The family is compatible with faces: for \(T \subseteq [k]\), \(\del_T\bigl(Q_k(\phi;\, x;\, t)\, c\bigr) = Q_{|T|}(\phi;\, x;\, t)\bigl(\del_T c\bigr)\).
4) \(\lambda_t\) trivializes the family: \(\lambda_t \circ Q_k(\phi;\, x;\, t) = \Delta_+(\phi;\, x) \circ \lambda_t\), so the family is isotrivial over \((0,1]\).
5) Contravariantly, \(Q^*(\psi \circ \phi;\, x;\, t) = Q^*(\phi;\, x;\, t) \circ Q^*(\psi;\, y;\, t)\), \(Q^*(\id_X;\, x;\, t) = \id\), and \(Q^*(\phi;\, x;\, 1) = \Delta^+(\phi;\, x)\).
Proof. For 1), use \(\lambda_1=\id\). For 2), conjugate exact functoriality (13) 🔗 by the invertible map \(\lambda_t\), whose inverse is \(\lambda_{1/t}\). The identity \(\del_T\lambda_t=\lambda_t\del_T\) (24) 🔗, together with the face clause of (13) 🔗, gives 3). Part 4) is the defining conjugation written without the inverse. Finally, 5) follows from 1) and 2) because precomposition reverses composition.
(30) Lemma (Successor identities). For an arbitrary map \(\phi: X \to Y\) and \(v_0, v_1, v_1', v_2, \dots, v_k \in E\),
Additivity defect and base movement in grade \(k\) are grade-\((k{+}1)\) differences. Normalized, for \(t \in (0,1]\),
Thus each passage to the successor grade contributes one factor of \(t\) in the normalized family.
Proof. The difference operators commute, and \(\Delta_{v_1 + v_1'} = \Delta_{v_1} + \Delta_{v_1'} + \Delta_{v_1} \Delta_{v_1'}\), since \(\phi(\cdot + v_1 + v_1') - \phi = [\phi(\cdot + v_1) - \phi] + [\phi(\cdot + v_1') - \phi] + [\phi(\cdot + v_1 + v_1') - \phi(\cdot + v_1) - \phi(\cdot + v_1') + \phi]\). Applied to \(\Delta_{v_2} \cdots \Delta_{v_k} \phi\) at \(x\), this is the first identity; the second is \(\Delta_{v_0}(\Delta_{v_1} \cdots \Delta_{v_k} \phi)(x)\) written out. The normalized forms divide by \(t^k\).
Extension
Norms enter here and remain in force for the rest of the paper: \(E, F, G\) are now Banach spaces because the integrals below require complete targets. We say \(\phi\) is \(C^k\) near \(x\) if it is \(k\) times Fréchet differentiable with continuous derivatives on a neighborhood of \(x\) [LangRFA]. The derivative \(D^r(\phi; x; v_1, \dots, v_r)\) is evaluated on the directions \(v_i\) and is symmetric and continuous multilinear; when the arguments determine the order, we drop the superscript:
Cubes are measured coordinatewise, \(\|c\| := \max_{\emptyset \neq A \subseteq [k]} \|c_A\|\) on \(CT_k(X, x)\); faces of bounded sets are then bounded.
(31) Lemma (Iterated fundamental theorem). Let \(\phi\) be \(C^r\) on an open set \(U \subseteq X\), and let \(x \in U\) and \(v_1, \dots, v_r \in E\) be such that the parallelotope \(x + [0,1] v_1 + \dots + [0,1] v_r\) lies in \(U\). Then
Proof. Set \(g(\theta) := \phi(x + \sum_i \theta_i v_i)\) on \([0,1]^r\). Since the parametrization is affine, \(g\) is \(C^r\) with \(\del_{\theta_1} \cdots \del_{\theta_r} g(\theta) = D(\phi; x + \sum_i \theta_i v_i; v_1, \dots, v_r)\). For a continuous Banach-space-valued function, the fundamental theorem of calculus in one variable and Fubini give, one variable at a time, \(\int_{[0,1]^r} \del_1 \cdots \del_r g \, d\theta = \sum_{T \subseteq [r]} (-1)^{r - |T|} g(1_T)\), the alternating sum over the corners of the cube, which is \(\Delta(\phi; x; v_\bullet)\).
(32) Corollary (Difference asymptotics). Let \(\phi\) be \(C^k\) near \(x\) and \(R > 0\).
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For \(1 \leq r \leq k\), \(t_1, \dots, t_r \in \IR\), and \(\|v_j\| \leq R\),
\[ \Delta(\phi;\, x;\, t_1 v_1, \dots, t_r v_r) = t_1 \cdots t_r ( D(\phi; x; v_1, \dots, v_r) + \eta(t, v) ), \]where \(\sup_{\|v_j\| \leq R} \|\eta(t, v)\| \to 0\) as \(\max_j |t_j| \to 0\).
-
For \(r > k\) there are \(\rho > 0\) and \(K < \infty\) such that for all \(w_1, \dots, w_r \in E\) with \(\|w_j\| \leq \rho\) and every \(k\)-element subset \(M \subseteq [r]\),
\[ \|\Delta(\phi;\, x;\, w_1, \dots, w_r)\| \;\leq\; K \prod_{j \in M} \|w_j\|. \]
Proof. 1) By (31) 🔗 with directions \(t_j v_j\) and multilinearity of the \(r\)-th derivative, the difference equals \(t_1 \cdots t_r \int D(\phi; x + \sum_j \theta_j t_j v_j; v_\bullet)\, d\theta\), and \(\|\eta\| \leq R^r \sup_{\xi} \|D^r(\phi; \xi) - D^r(\phi; x)\|\), the supremum over \(\xi\) in a ball that shrinks to \(x\); conclude by continuity of \(D^r(\phi; \cdot)\) at \(x\).
2) Since \(\Delta\) is symmetric in its directions we may take \(M = [k]\). The commuting operator factorization \(\Delta_{w_1} \cdots \Delta_{w_r} = (\prod_{j > k} \Delta_{w_j}) (\prod_{j \leq k} \Delta_{w_j})\) expands into
By continuity of \(D^k(\phi;\, \cdot)\) at \(x\), choose \(\rho\) small enough that all basepoints \(x + w_T\) and their attached parallelotopes lie in a ball \(B\) around \(x\) on which \(\phi\) is \(C^k\) and \(\|D^k(\phi;\, \cdot)\| \leq \|D^k(\phi;\, x)\| + 1 =: M_k\); by (31) 🔗, each summand is bounded by \(M_k \prod_{j \leq k} \|w_j\|\), and there are \(2^{r - k}\) summands. Take \(K = 2^{r-k} M_k\).
(33) Lemma (Weight bound). For \(k\geq1\), every \(H \in \Cov(k)\) satisfies \(\wt(H) \geq k\), with equality if and only if \(H \in \Part(k)\).
Proof. \(\sum_{A \in H} |A| \geq |\bigcup H| = k\), with equality iff the blocks are pairwise disjoint.
(34) Definition (Zero fiber). The zero fiber of the family is the gradewise limit
defined wherever the limit exists. Where it exists for every \(c \in CT_k(X, x)\), this yields the operator \(Q^0_k(\phi; x): CT_k(X, x) \to CT_k(Y, y)\); where every grade exists, these assemble into \(Q^0_*(\phi; x)\). Observables pull back by precomposition, gradewise.
Statements under a \(C^k\) hypothesis are made at the top grade \(k\): a \(C^k\) map is also \(C^{k-1}, \dots, C^1\), so the lower grades are the same statement with \(k\) decreased, and we use them without comment.
(35) Theorem (Boundary extension). Let \(k \geq 1\) and let \(\phi\) be \(C^k\) near \(x\). The zero fiber \(Q^0_k(\phi; x)\) exists on all of \(CT_k(X, x)\), with coordinates the partition formula
The convergence \(Q_k(\phi;\, x;\, t;\, c) \to Q^0_k(\phi;\, x;\, c)\) as \(t \downarrow 0\) is uniform on bounded subsets of \(CT_k(X, x)\), and the right side is continuous in \(c\). On affine cubes the top coordinate is the derivative,
Moreover, the extended family is jointly continuous through the boundary: for \(x'\) near \(x\), the map \((x', t, c) \mapsto Q_k(\phi;\, x';\, t;\, c)\), extended by \(Q^0_k(\phi;\, x';\, c)\) at \(t = 0\), is continuous wherever defined.
Proof. The proof runs in three steps: reduction of all coordinates to the top-coordinate limit; the exact treatment of covers with at most \(k\) blocks; the bound for covers with more blocks. Collecting the terms proves the fixed-base statement; joint continuity is proved separately at the end.
Reduction. The coordinates of the family are \(Q_k(\phi;\, x;\, t;\, c)_T = \frac{1}{t^{|T|}}\, \Delta(\phi;\, x;\, \lambda_t\, \del_T c)\) (26) 🔗: the \(T\)-coordinate is the top coordinate of the grade-\(|T|\) family on the face \(\del_T c\), and faces of bounded sets are bounded. It therefore suffices to treat the top coordinate, at grade \(k\) and, by decreasing \(k\), in the lower grades: we show
uniformly for \(\|c\| \leq R\). For small \(t\) all vertices of \(\lambda_t c\) lie in the domain of \(\phi\). By affine reconstruction (21) 🔗,
Covers with at most \(k\) blocks. Consider a cover \(H\) with \(r := |H| \leq k\) blocks: part 1 of (32) 🔗, applied with scales \(t^{|A|}\) and directions \(c_A\) for \(A \in H\), gives
with \(\eta_H(t) \to 0\) uniformly for \(\|c\| \leq R\).
Covers with more than \(k\) blocks. If \(r > k\), then \(H\) contains a block of size \(\geq 2\): the blocks are distinct nonempty subsets of \([k]\) and there are only \(k < r\) singletons. Choose a \(k\)-element subset \(M \subseteq H\) containing such a block; then \(\sum_{A \in M} |A| \geq k + 1\), and part 2 of (32) 🔗 bounds the term by \(K R^k t^{k+1}\).
Collection. Divide by \(t^k\). Covers with \(r > k\) are \(O(t)\). Covers with \(r \leq k\) appear with the factor \(t^{\wt(H) - k}\), where \(\wt(H) \geq k\) (33) 🔗; those of weight \(> k\) vanish in the limit with bounded cofactor, and the partitions, exactly the covers of weight \(k\) (33) 🔗, converge to their leading term. All estimates are uniform for \(\|c\| \leq R\). Each partition summand is continuous multilinear in the coordinates \((c_A)_{A \in \pi}\) and hence polynomial in \(c\). This proves continuity in \(c\).
Joint continuity. Let \(x_n \to x_0\) near \(x\), \(t_n \downarrow 0\), and \(c^{(n)} \to c^{(0)}\). In the covering expansion at base \(x_n\), the error of each cover with \(r \leq k\) blocks is bounded, as in the proof of (32) 🔗, by \(R^r \sup_\xi \|D^r(\phi;\, \xi) - D^r(\phi;\, x_n)\|\) with \(\xi\) within distance \(O(t_n)\) of \(x_n\); since \(\xi\) and \(x_n\) both converge to \(x_0\), continuity of \(D^r(\phi;\, \cdot)\) at \(x_0\) makes it vanish, while the leading terms \(D(\phi;\, x_n;\, (c^{(n)}_A)_{A \in H})\) converge to \(D(\phi;\, x_0;\, (c^{(0)}_A)_{A \in H})\) by continuity of the derivatives in the base and multilinearity. Covers with \(r > k\) are bounded by \(K t_n\), with \(K\) uniform for base points near \(x_0\): by continuity of \(D^k(\phi;\, \cdot)\) at \(x_0\), one ball around \(x_0\) has the bound \(\|D^k(\phi;\, \cdot)\| \leq \|D^k(\phi;\, x_0)\| + 1\) and, for large \(n\), contains every attached parallelotope. Continuity at points with \(t_0 > 0\) is composition of continuous maps.
The zero fiber agrees with the iterated tangent functor. Recall from (3) 🔗 that for open \(X \subseteq E\) the tangent bundle is \(TX = X \times E\) and its iteration is \(T^k X = X \times \Map(\KP_+(k), E)\), with the tangent cubes \(CT_k(X, x)\) as the fiber over \(x\). On maps, the tangent functor is defined for \(\phi \in C^1\) by
and its iterate \(T^k\phi := T(T^{k-1}\phi)\) is defined iteratively for \(\phi \in C^k\), by considering \(T^{k-1}\phi\) as a differentiable map between affine spaces. Each tangent step takes the total derivative of the previous one, so \(T^k\phi\) is the \(k\)-fold iterated total derivative of \(\phi\). Under the identification \(T^k X \isom X \times \Map(\KP_+(k), E)\) above, this map corresponds to
as the induction in the proof below shows. Functoriality \(T^k(\psi \circ \phi) = T^k\psi \circ T^k\phi\) is the chain rule, applied \(k\) times.
(36) Proposition (Iterated tangent functor). For \(\phi \in C^k\) near \(x\),
the fiber map over \(x\) of the \(k\)-fold iterated tangent map: the zero fiber is the differential calculus of the iterated tangent bundle, written in cube coordinates.
Proof. By the partition formula (35) 🔗, \(Q^0_k(\phi;\, x;\, c)_T = \sum_{\pi \in \Part(T)} D(\phi;\, x;\, (c_A)_{A \in \pi})\); we show by induction on \(k\) that \(T^k\phi\) has the same coordinates. For \(k = 1\), \(T\phi(x;\, v) = (\phi(x);\, D(\phi;\, x;\, v))\). For the step, split the coordinates of an order-\(k\) cube by whether the index set contains \(k\): the base point \(x\) together with the \(c_A\), \(\emptyset \neq A \subseteq [k-1]\), is a point of \(T^{k-1}X\), and the coordinates containing \(k\), re-indexed as \((c_{A \cup \set{k}})_{A \subseteq [k-1]}\) with \(A\) possibly empty, are a tangent direction at it. In this splitting, \(T^k\phi = T(T^{k-1}\phi)\) is the tangent map of \(\Phi := T^{k-1}\phi\) at that point in that direction. Its base component \(\Phi(x;\, c)\) returns the coordinates indexed by \(T \subseteq [k-1]\) unchanged, and its derivative component differentiates each coordinate \(\Phi_T\), producing the coordinates containing \(k\). The base derivative in direction \(c_k\) appends the singleton block \(\set{k}\) to a partition, and the derivative in the \(c_A\)-slot, in direction \(c_{A \cup \set{k}}\), replaces the block \(A\) by \(A \cup \set{k}\). Together these produce every partition of \(T \cup \set{k}\) exactly once.
(37) Remark (Three levels of extension). Extension through \(t = 0\) comes in three strengths. For fixed \(x\) and a single grade-\(k\) cube \(c\), the limit \(Q^0_k(\phi;\, x;\, c)\) may exist: the pointwise extension. If it exists for every cube, \(Q^0_k(\phi; x)\) is an operator (34) 🔗, the operator extension, with no implied continuity in \((x,c)\). The strongest form is the continuous extension supplied by the boundary-extension theorem for \(C^k\) maps (35) 🔗: convergence uniform on bounded cube sets at each base point, and joint continuity of \((x', t, c) \mapsto Q_k(\phi;\, x';\, t;\, c)\) through \(t = 0\). Existence alone does not imply \(C^k\). It does not force additivity of the boundary values: for \(f=|\cdot|\) on \(\IR\) at \(0\), the grade-one fiber exists on every cube and has value \(|v|\). Nor does it force continuity in the base: \(f(x) = x^2 \sin(1/x)\), \(f(0) = 0\), is differentiable everywhere with discontinuous derivative, so the grade-one fiber exists at every base point and is linear there, yet \(f\) is not \(C^1\). The converse problem, which growth conditions on the family recover \(C^k\) regularity, is not treated in this paper.
Collapse
Exact functoriality holds for every \(t>0\). The continuous extension allows the same identity to pass to \(t=0\).
(38) Theorem (Zero-fiber functoriality). For \(\phi \in C^k\) near \(x\) and \(\psi \in C^k\) near \(y = \phi(x)\),
Proof. For \(t \in (0,1]\) the family is exactly functorial (29) 🔗,
with no regularity assumptions; it remains to pass to the boundary. Fix \(c \in CT_k(X, x)\). The inner factor converges, \(c^{(t)} := Q_k(\phi;\, x;\, t;\, c) \to c' := Q^0_k(\phi;\, x;\, c)\) (35) 🔗, and is in particular bounded. The outer convergence \(Q_k(\psi;\, y;\, t) \to Q^0_k(\psi;\, y)\) is uniform on bounded cubes with continuous limit (35) 🔗, so \(Q_k(\psi;\, y;\, t;\, c^{(t)}) \to Q^0_k(\psi;\, y;\, c')\). The left side converges to \(Q^0_k(\psi \circ \phi;\, x;\, c)\) by (35) 🔗 applied to the composite, which is \(C^k\) near \(x\).
The algebra of the deformation
The deformed pushforward conjugates \(\Delta_+\) by the rescaling \(\lambda_t\) (26) 🔗; the deformed product arises the same way, by conjugating the \(t = 1\) product with the grade rescaling \(\Lambda_t(\omega) := (t^k \omega_k)_{k \geq 0}\). Term by term, the covering pair \((I, J)\) with \(I \cup J = [k]\) acquires the factor \(t^{|I|}\, t^{|J|}\, t^{-k} = t^{|I \cap J|}\). Thus each overlap contributes one power of \(t\). The definition below extends this conjugate through \(t = 0\), where only the disjoint pairs remain. Its \(t = 1\) fiber is the product in the discrete Leibniz rule (93) 🔗.
(39) Definition (Deformed product). For \(t \in [0,1]\), the deformed product on \(CT^*(X, x)\) is
the sum over ordered pairs \((I, J)\) of subsets with \(I \cup J = [k]\), not necessarily disjoint, with \(t^0 := 1\); the unit is \(\mathbf 1 = (1, 0, 0, \dots)\). At the endpoints:
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\(\cstar := \star_1\) is the covering product: every covering pair enters with weight one,
\[ (\omega \cstar \eta)_k(c) = \sum_{I \cup J = [k]} \omega_{|I|}(\del_I c)\, \eta_{|J|}(\del_J c). \] -
\(\pstar := \star_0\) is the partition product: only disjoint pairs remain,
\[ (\omega \pstar \eta)_k(c) = \sum_{I \sqcup J = [k]} \omega_{|I|}(\del_I c)\, \eta_{|J|}(\del_J c), \]the sum over ordered disjoint decompositions, with possibly empty parts.
(40) Theorem (The algebra family). For every \(t \in [0,1]\), the product \(\star_t\) is commutative, associative, and unital. For \(t \in (0,1]\), the grade rescaling \(\Lambda_t(\omega) := (t^k \omega_k)_{k \geq 0}\) is a unital algebra isomorphism
the family is trivial over the punctured interval. At the boundary it degenerates: \((CT^*(X, x), \pstar)\) is a local algebra. Its non-units are exactly the elements with \(\omega_0 = 0\), which form an ideal. By contrast, \((CT^*(X, x), \cstar)\) is not local, so the zero fiber is isomorphic to no other fiber. At \(t = 0\) the overlapping pairs disappear and only disjoint decompositions remain: this overlap-to-disjoint degeneration is the two-factor instance of the cover-to-partition collapse of the extension theory above.
Proof. Swapping \(I\) and \(J\) proves commutativity. If one factor is \(\mathbf 1=(1,0,0,\dots)\), only the pair with its index set empty contributes, which proves the unit law. For associativity, expand the triple product over triples \(I \cup J \cup K = [k]\): the left bracketing has the factor \(t^{|I \cap J| + |(I \cup J) \cap K|}\), the right bracketing \(t^{|J \cap K| + |I \cap (J \cup K)|}\), and both exponents equal \(|I| + |J| + |K| - k\) by inclusion-exclusion. This expression is symmetric in the three factors.
Multiplicativity of \(\Lambda_t\): in grade \(k\), the pair \((I, J)\) has the factor \(t^k \cdot t^{|I \cap J|}\) on the left and \(t^{|I|}\, t^{|J|}\) on the right, and \(|I| + |J| = k + |I \cap J|\); \(\Lambda_t\) fixes the unit and is invertible for \(t > 0\). Degeneration: in every \(\star_t\), \((\omega \star_t \eta)_0 = \omega_0\, \eta_0\), so elements with \(\omega_0 = 0\) are non-units, and they form an ideal. In \(\pstar\) they are the only non-units: for \(\omega_0 \neq 0\), the grade-\(k\) equation of \(\omega \pstar \eta = \mathbf 1\) determines \(\eta_k\) triangularly with leading coefficient \(\omega_0\). Hence \(\pstar\) is local. For \(\cstar\), Taylor duality makes every vertex evaluation a character:
since the pairs \((I, J)\) with \(I \cup J \subseteq [k]\), grouped by \(T = I \cup J\), enumerate exactly the product of two vertex sums. Kernels of characters consist of non-units; taking grade-one components constant, \((1, -\tfrac12, 0, \dots)\) is killed by \(\chi_c\) at every cube of order two, and \((0, \tfrac12, 0, \dots)\) by the order-zero evaluation, yet their sum is the unit: \(\cstar\) is not local.
(41) Remark (Weight principle). In the covering formula (22) 🔗, weight each cover \(H\) by the deformation monomial of its excess. On the diagonal this is \(t^{\wt(H) - k}\); per leg it is \(\prod_i t_i^{m_i(H) - 1}\), where \(m_i(H)\) is the number of blocks containing \(i\). No such scalar-weighted covering formula is an identity of the difference calculus: the outer difference \(\Delta(\phi;\, x;\, \cdot)\) is not multilinear in its slots, and the weights cannot be moved out of the arguments. The weight principle is exact precisely in multilinear readings: in the algebra family (40) 🔗, whose structure constants \(t^{|I \cap J|}\) realize it for pairs; in the cube-ring realization (42) 🔗 below; and at the boundary \(t \downarrow 0\), where the excess weights select the partitions. In the nonlinear calculus it holds only asymptotically, as the collapse rates of the extension theory.
(42) Remark (Cube rings and Weil algebras). The family has a finite free ring model of rank \(2^k\) over \(\IR[t]\); every fiber is \(2^k\)-dimensional. Set
the monomials \(X_A := \prod_{i \in A} X_i\), \(A \subseteq [k]\), form a basis over \(\IR[t]\), with
the structure constants of the algebra family (40) 🔗. For \(a \neq 0\), the rescaling \(\lambda_a\) acts as the substitution \(X_i \mapsto a\, X_i\) and identifies the fiber at \(1\) with the fiber at \(a\): from \(X_i^2 = X_i\) one gets \((a X_i)^2 = a\, (a X_i)\). This is the ring-level form of the trivialization over the punctured interval.
At \(t = 1\), the coordinate algebra of the Boolean cube, \(\IR[X_\bullet]/(X_i^2 - X_i)\), is the algebra of functions on the vertex set \(\set{0,1}^k\), the monomial and vertex-indicator bases related by the zeta and Möbius transforms (4) 🔗. At \(t = 0\), the square-zero ring \(\IR[X_\bullet]/(X_i^2)\) is the \(k\)-fold dual-numbers algebra, the Weil algebra of the iterated tangent functor \(T^k\) [KMS1993], matching the zero fiber's identification with \(T^k\) (36) 🔗. The family \(C_k\) can be made explicit as a Rees algebra: writing \(B_k := \IR[e_1, \dots, e_k]/(e_i^2 - e_i)\) for the vertex-function algebra with its monomial-degree filtration, \(C_k\) is the Rees algebra of \(B_k\), with \(X_i\) corresponding to the Rees element \(t\, e_i\). This is the isotropic case of Bertram's cubic rings [Bertram2017]; the per-leg version is given by his scaleoid rings [Bertram2015b]. The companion paper develops the realization theory relating the cube calculus to these rings [DFDB]. The present paper uses \(CT_*\), \(CT^*\), \(ST_*\), \(ST^*\) alone.
The zero fiber is so far a cubical object; the next chapter builds the symmetric multilinear calculus and identifies the two under the symbol map.
Differential calculus
This chapter constructs the symmetric multilinear calculus and compares it with the cubical zero fiber. It does not depend on the later jet calculus; the required coalgebra facts are collected in the appendix.
Symmetric probes and pushforward
(43) Definition (Symmetric tangent space). For \(r, k \geq 0\) define:
- \(\SYM^r(E)\) is the \(r\)-th algebraic symmetric power of \(E\), with \(\SYM^0(E) := \IR \cdot 1\).
- \(ST_r(X, x) := \SYM^r(T_x X)\) is the space of probes of exact degree \(r\); in particular \(ST_1(X, x) = T_x X\).
- \(ST_{\leq k}(X, x) := \SYM_{\leq k}(T_x X) = \Vsum_{r=0}^{k} \SYM^r(T_x X)\) is the symmetric tangent space of order \(k\).
- \(ST_*(X, x) := \SYM(T_x X)\) is the symmetric tangent space of all orders.
Elements of \(ST_{\leq k}\) are called probes; monomials are written \(v_1 \cdots v_r\). On spaces, operators, and forms, sub- and superscripts denote exact degree, and truncation is always marked \(\leq k\); on cube letters, subscripts remain Möbius coordinates (2) 🔗.
Besides the truncated symmetric product \(\xi \cdot \eta := \pi_{\leq k}(\xi \eta)\), the symmetric tangent space has a coproduct.
(44) Definition (Coproduct). The coproduct on \(ST_{\leq k}(X, x)\) is the linear map \(\Delta^{\times}: ST_{\leq k}(X, x) \to ST_{\leq k}(X, x) \tensor ST_{\leq k}(X, x)\) with \(\Delta^{\times}(1) = 1 \tensor 1\) and
where \(v^I := \prod_{i \in I} v_i\) is the monomial (the subfamily is \(v_I\), with the index down); the right side is symmetric and multilinear, so \(\Delta^{\times}\) is well defined. The counit is the linear map \(\varepsilon: ST_{\leq k}(X, x) \to \IR\) with \(\varepsilon(1) := 1\) and \(\varepsilon := 0\) in positive degrees.
The truncated product and the coproduct are paired structures on \(ST_{\leq k}(X, x)\): truncation makes it an algebra quotient and the degree filtration makes it a subcoalgebra. We do not regard the truncated space as a bialgebra. The untruncated bialgebra and the coalgebra facts used below are collected in the appendix Appendix: The symmetric bialgebra 🔗.
(45) Definition (Differential). Let \(\phi: X \to Y\) be \(C^k\) near \(x\), with \(y = \phi(x)\).
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The differential of \(\phi\) at \(x\) is the linear map \(D(\phi; x): ST_{\leq k}(X, x) \to ST_1(Y, y)\) defined on monomials by iterating the directional derivative \(\del_v \phi(x) := \tfrac{d}{ds}\big|_{s=0}\, \phi(x + s v)\):
\[ D(\phi;\, x;\, v_1 \cdots v_r) = \del_{v_r} \cdots \del_{v_1} \phi(x); \]on the degree-zero probe, \(D(\phi; x; 1) := 0\): the target has no degree zero, and the base point is transported by \(\phi\) itself, \(\phi(x) = y\).
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For an observable \(f: X \to F\) with values in a Banach space \(F\) the same formula defines a differential \(D(f; x): ST_{\leq k}(X, x) \to F\), with the degree-zero convention \(D(f; x; 1) := f(x)\). The two readings differ only in degree zero: a based target kills constants, an observable records \(f(x)\). A map into the affine space \(F\) and an \(F\)-valued observable have the same type; the declared role of the first argument selects the reading.
(46) Definition (Symmetric pushforward). For \(\phi \in C^k\) near \(x\), the symmetric pushforward \(D_+(\phi; x): ST_{\leq k}(X, x) \to ST_{\leq k}(Y, y)\) is the linear map with \(D_+(\phi; x; 1) := 1\) and
The right side is symmetric and multilinear in \((v_1, \dots, v_r)\), so it defines a linear map on \(\SYM^r(E)\).
The preceding partition formula is determined by a coordinate-free property: \(D_+(\phi;x)\) is the unique coalgebra lift of the differential \(D(\phi;x)\). The next two statements establish this characterization.
(47) Proposition (Coalgebra property). For \(\phi \in C^k\) near \(x\), \(D_+(\phi; x)\) is a coalgebra morphism:
Proof. Evaluate on \(v_1\cdots v_r\) and set \(w_A:=D(\phi;x;v^A)\). Applying \(\Delta^{\times}\) to each partition term splits its block set:
The data \((\pi,\pi_1,\pi_2)\) correspond bijectively to \((I\sqcup J=[r],\pi_1\in\Part(I),\pi_2\in\Part(J))\), with \(I=\bigcup\pi_1\) and \(J=\bigcup\pi_2\). The resulting sum is \(\sum_{I\sqcup J}D_+(v^I)\tensor D_+(v^J) =(D_+\tensor D_+)\Delta^{\times}(v_1\cdots v_r)\). For the counit, \(D_+(\phi;\, x;\, 1) = 1\), and every positive-degree probe lands in positive degrees, each partition term being a product of at least one degree-one factor.
(48) Corollary (Coalgebra lift). \(D_+(\phi; x)\) is the unique coalgebra morphism \(ST_{\leq k}(X, x) \to ST_{\leq k}(Y, y)\) whose degree-one component is the differential \(D(\phi; x)\).
Proof. The coalgebra property is (47) 🔗. Uniqueness is the truncated coalgebra-lift corollary (124) 🔗 of the appendix, applied to \(D(\phi; x)\); its partition expansion is (46) 🔗.
The symbol map
(49) Definition (Symbol map). The symbol map is
where the products are taken in the symmetric algebra. On affine cubes, \(\sigma_k(\Aff(v_1,\dots,v_k))=v_1\cdots v_k\). In grade zero, \(\sigma_0(0) := 1\). Although indexed by the cube order \(k\), the symbol is not homogeneous in symmetric degree: a partition with \(r\) blocks contributes degree \(r\).
(50) Lemma (Spanning). For every \(1 \leq r \leq k\) and \(v_1, \dots, v_r \in T_x X\), the monomial \(v_1 \cdots v_r\) lies in the image of \(\sigma_k\). Consequently the image of \(\sigma_k\), together with \(1\), spans \(ST_{\leq k}(X, x)\).
Proof. Choose a partition \([k] = B_1 \sqcup \dots \sqcup B_r\) into nonempty blocks and define \(c \in CT_k(X, x)\) by \(c_{B_j} := v_j\) and \(c_A := 0\) for all other \(A\). A partition \(\pi \in \Part(k)\) contributes to \(\sigma_k(c)\) only if all its blocks lie in \(\set{B_1, \dots, B_r}\), which forces \(\pi = \set{B_1, \dots, B_r}\); hence \(\sigma_k(c) = v_1 \cdots v_r\).
(51) Proposition (Symbol factorization). Let \(k \geq 1\) and let \(\phi\) be \(C^k\) near \(x\). For \(c \in CT_k(X, x)\),
Proof. The top coordinate of the partition formula (35) 🔗 reads \(Q^0_k(\phi;\, x;\, c)_{[k]} = \sum_{\pi \in \Part(k)} D(\phi;\, x;\, (c_A)_{A \in \pi})\). Each summand is the differential (45) 🔗 evaluated on the monomial \(\prod_{A \in \pi} c_A\), so by linearity the sum is \(D(\phi;\, x;\, \cdot)\) applied to \(\sigma_k(c)\).
(52) Remark (The classical symbol). The Rees deformation of a filtered ring has the ring as generic fiber and the associated graded as special fiber, reached through the leading-form map; for differential operators with the order filtration, \(\mathrm{gr}(\mathcal D) = \mathrm{Sym}(T)\) and the leading-form map is the principal symbol.
The deformation here is Rees-like. The spaces \(CT_k\) are constant, \(\lambda_t\) is the dilation action, and every overlap has positive deformation weight in the algebra family (40) 🔗. At the boundary, only the disjoint terms remain. By (33) 🔗, the smallest weight in grade \(k\) is \(k\), attained exactly by partitions. Thus \(\sigma_k\) is the corresponding leading-form expression, and symbol factorization (51) 🔗 states that the zero fiber factors through it. Throughout, "leading" refers to deformation weight, not to symmetric degree: the symbol mixes all degrees \(1, \dots, k\).
(53) Remark (Order one: deformation to the normal cone). In order one, \((x,v,t) \mapsto (x,x+tv,t)\) gives the standard affine coordinates for deformation to the normal cone of the diagonal \(X\subset X\times X\) [Fulton1998]. The same coordinates occur in Connes' tangent groupoid [Connes1994]. Restriction to its special fiber \(TX\) gives the principal cosymbol, a translation-invariant operator on the tangent fibers; fiberwise Fourier transform gives the pseudodifferential principal symbol on \(T^*X\) [vEY19]. We use only the coordinate comparison, not the groupoid structure. None of the subsequent results depends on this remark.
(54) Theorem (Symbol intertwining). Let \(\phi\) be \(C^k\) near \(x\). Then
Proof. In grade zero both sides send the point cube to \(1\). For \(k \geq 1\), both sides are sums indexed by the same two-level partition data: a partition \(\rho\) of \([k]\) together with a partition \(\kappa\) of the block set of \(\rho\). For \(k=2\), these data give the three terms \(D(\phi;x;c_{12})\), \(D(\phi;x;c_1,c_2)\), and \(D(\phi;x;c_1)D(\phi;x;c_2)\). In the last two terms, \(\kappa\) either groups the singleton blocks of \(\rho\) together or keeps them separate. We now write both expansions in the same indexing for arbitrary \(k\). For \(c \in CT_k(X, x)\), expand the left side by the partition formula (35) 🔗:
Expanding the product, the index data is a pair \((\pi, (\rho_A)_{A \in \pi})\) with \(\rho_A \in \Part(A)\). Such pairs correspond bijectively to pairs \((\rho, \kappa)\) of a partition \(\rho \in \Part(k)\) and a partition \(\kappa\) of the block set of \(\rho\): put \(\rho := \bigsqcup_{A \in \pi} \rho_A\) and \(\kappa := \set{\rho_A : A \in \pi}\); conversely \(\pi = \set{\bigcup C : C \in \kappa}\). Under this bijection the term of \((\pi, (\rho_A))\) equals \(\prod_{C \in \kappa} D(\phi;\, x;\, (c_B)_{B \in C})\). On the other side, the symmetric pushforward (46) 🔗, applied to the monomial of the \(|\rho|\) vectors \((c_B)_{B \in \rho}\), gives
The two sums agree term by term.
Higher symbols and collapse expansion
The symbol gives the leading coefficient of the collapse function \(t \mapsto \Delta(f;\, x;\, \lambda_t\, c)\) of an observable: by boundary extension and symbol factorization, the function has leading asymptotic \(t^k\, D(f;\, x;\, \sigma_k(c))\). The same cover calculus computes every subsequent coefficient.
(55) Definition (Higher symbols). For \(H \in \Cov(k)\) and multiplicities \(\nu: H \to \IN_{>0}\) set \(\wt(\nu) := \sum_{A \in H} |A|\, \nu_A\) and \(\nu! := \prod_{A \in H} \nu_A!\). The higher symbols are
the products taken in the symmetric algebra.
Minimal weight forces \(\sigma_{k,0} = \sigma_k\): \(\wt(\nu) \geq \wt(H) \geq k\) (33) 🔗, with equality only for partitions with all multiplicities equal to one. On affine cubes only the singleton cover contributes, the multiplicities become multi-indices, and, writing \(\nu! := \prod_i \nu_i!\), \(v^\nu := v_1^{\nu_1} \cdots v_k^{\nu_k}\), and \(\wt(\nu) := \nu_1 + \dots + \nu_k\) for \(\nu \in \IN_0^k\),
homogeneous of symmetric degree \(k + m\).
(56) Proposition (Affine collapse). Let \(f: X \to F\) be \(C^n\) near \(x\), and let \(k \geq 1\) and \(R > 0\). As \(t \to 0\),
uniformly over \(\|v_1\|, \dots, \|v_k\| \leq R\). For \(n \geq k\), divided by \(t^k\) and grouped by weight, the normalized difference expands with the affine higher symbols as coefficients,
Proof. Writing \(n = k + m\), the claim \((\ast_{m,k})\) states that for every observable \(h: X \to F\) that is \(C^{k+m}\) near \(x\), the first display holds uniformly over \(\|v_1\|, \dots, \|v_k\| \leq R\); for \(k = 0\) read \((\ast_{m,0})\) as the exact identity \(\Delta(h;\, x) = h(x)\). We argue by lexicographic induction on \((m,k)\).
Base \(m = 0\). The normalized difference \(t^{-k}\, \Delta(h;\, x;\, t v_\bullet)\) is the top coordinate of the family on the affine cube (26) 🔗, and the cubes with \(\|v_i\| \leq R\) form a bounded family; by boundary extension (35) 🔗 it converges to the derivative \(D(h;\, x;\, v_1 \cdots v_k)\), uniformly over the family. This is \((\ast_{0,k})\): the only multi-index of weight \(k\) is \(\nu = (1, \dots, 1)\), and \(\nu! = 1\).
Step. Let \(m, k \geq 1\) and let \(h\) be \(C^{k+m}\) near \(x\). Write \(G := D(h;\, -): X \to \mathcal L(E, F)\) for the derivative map, an observable that is \(C^{k+m-1}\) near \(x\) with \(D(G;\, x;\, u_1, \dots, u_s)(u) = D(h;\, x;\, u_1, \dots, u_s, u)\). Read from right to left, the successor identity (30) 🔗 rewrites the last leg as a base displacement:
the map \(y \mapsto \Delta(h;\, y;\, t v')\) is a finite alternating sum of translates of \(h\), with derivative \(\Delta(G;\, y;\, t v')(u)\) in direction \(u\), so the fundamental theorem (31) 🔗 along the segment to \(x + t v_k\), followed by the same successor identity applied to \(G\) at the moving base \(x + \theta t v_k\), gives
all base points and segments lying in the neighborhood once \(t\) is small. Both differences on the right sit at the fixed base \(x\): the first has grade \(k - 1\) and excess \(m\), the second grade \(k\) and excess \(m - 1\). The induction hypothesis expands both, uniformly over legs in the \(R\)-ball; in the integral the legs \((v_1, \dots, v_{k-1}, \theta v_k)\) stay in that ball for all \(\theta \in [0, 1]\), so the remainder is uniform in \(\theta\). Evaluation at \(v_k\) increases the bound by at most a factor \(R\). The first term contributes, for each \(\mu \in \IN_{>0}^{k-1}\) with \(\wt(\mu) \leq k - 1 + m\),
exactly the multi-indices with \(\nu_k = 1\). In the integral the leg \(\theta v_k\) enters each summand homogeneously, through \(\theta^{\nu_k}\), and \(\int_0^1 \theta^{\nu_k}\, d\theta = \tfrac{1}{\nu_k + 1}\) raises the last exponent:
since \(\nu!\, (\nu_k + 1) = \tilde\nu!\): exactly the multi-indices with \(\tilde\nu_k \geq 2\). Together the two terms produce every \(\nu \in \IN_{>0}^k\) of weight at most \(k + m\) exactly once, with total remainder \(t\, o(t^{k+m-1}) = o(t^{k+m})\). The factorials are obtained from the factors \(1/(\nu_k+1)\) contributed by the \(\theta\)-integrals at each induction step.
If \(n < k\) the sum in the first display is empty and the claim reads \(\Delta(f;\, x;\, t v_\bullet) = o(t^n)\): reading the last \(k - n\) legs as an iterated difference of the map \(y \mapsto \Delta(f;\, y;\, t v_1, \dots, t v_n)\) expands
and by joint continuity through the boundary (35) 🔗 each summand is \(t^n\, (D(f;\, x;\, v_1 \cdots v_n) + o(1))\); the alternating signs sum to zero and cancel the common limit.
For the second display, divide by \(t^k\) and group by \(\wt(\nu) = k + m\); the inner sums are the affine higher symbols.
(57) Theorem (Curved collapse). Let \(M \geq 0\) and let \(f: X \to F\) be \(C^{k+M}\) near \(x\), with \(k \geq 1\). As \(t \downarrow 0\),
uniformly for \(c\) in bounded subsets of \(CT_k(X, x)\).
Proof. Affine reconstruction (21) 🔗 applied to \(\lambda_t\, c\) writes the collapse function as the finite sum
Fix \(R\) with \(\|c_A\| \leq R\) for all \(A\), and fix a cover \(H\) with \(r\) blocks. Each leg factors as \(t^{|A|} c_A = t \cdot t^{|A|-1} c_A\) with \(\|t^{|A|-1} c_A\| \leq R\) for \(t \leq 1\), so affine collapse (56) 🔗 applies at scale \(t\) to the bounded, \(t\)-dependent leg vectors \((t^{|A|-1} c_A)_{A \in H}\), uniformly. If \(r > k + M\) the expansion at order \(k + M\) is empty and the difference is \(o(t^{k+M})\). If \(r \leq k + M\) it reads, after collecting the powers of \(t\) leg by leg,
since the term of \(\nu\) has the factor \(t^{\sum_A \nu_A}\) from the scale and \(t^{\sum_A (|A|-1)\, \nu_A}\) from the legs, together \(t^{\wt(\nu)}\). The finitely many terms with \(\sum_A \nu_A \leq k + M\) but \(\wt(\nu) > k + M\) are \(O(t^{k+M+1})\), uniformly on the bounded set, and join the remainder. Summing over the covers and grouping the surviving terms by \(\wt(\nu) = k + m\) gives exactly the higher symbols (55) 🔗, with \(\wt(\nu) \geq k\) throughout (33) 🔗.
The term \(m=0\) recovers symbol factorization (51) 🔗. Under the stronger \(C^{k+M}\) hypothesis, the boundary extension has a Taylor expansion in the deformation parameter; its \(t^{k+m}\)-coefficient is \(D(f;\, x;\, \sigma_{k,m}(c))\).
Functoriality and Faà di Bruno
(58) Corollary (Smooth functoriality). For \(\phi \in C^k\) near \(x\) and \(\psi \in C^k\) near \(y = \phi(x)\),
Proof. For the identity, \(D^r(\id_X; x) = 0\) for \(r \geq 2\), so only the partition into singletons contributes in (46) 🔗. For the composite, both sides are linear and fix \(1\); by the spanning lemma (50) 🔗 it suffices to compare them on \(\sigma_k(c)\) for \(c \in CT_k(X, x)\), each occurrence of \(\sigma_k\) taken in the space indicated by its argument. The composite is \(C^k\) near \(x\), and combining the intertwining (54) 🔗 with zero-fiber functoriality (38) 🔗:
(59) Theorem (Smooth adjunction). Let \(\phi\) be \(C^k\) near \(x\) and let \(f: Y \to F\) be an observable that is \(C^k\) near \(y\). For \(\xi \in ST_{\leq k}(X, x)\),
Proof. Both sides are linear in \(\xi\) and agree at \(\xi = 1\): the observable reading (45) 🔗 gives \((f \circ \phi)(x) = f(y)\) on the left and, since \(D_+(\phi;\, x;\, 1) = 1\), the value \(f(y)\) on the right; for \(k = 0\) this is already the whole assertion. For \(\xi = \sigma_k(c)\), read \(f\) as a map into the affine space \(F\). The two readings agree in positive degrees, and \(\sigma_k(c)\) has no constant component. We may therefore pass through the zero fiber. By symbol factorization (51) 🔗 applied to \(f \circ \phi\) and to \(f\), zero-fiber functoriality (38) 🔗, and the intertwining (54) 🔗,
Conclude by the spanning lemma (50) 🔗.
(60) Corollary (Partition Faà di Bruno, [Fraenkel1978] [CS1996]). For \(\phi\), \(f\) as above and \(v_1, \dots, v_k \in T_x X\),
Proof. Apply smooth functoriality (58) 🔗 to \(\psi = f\), a \(C^k\) map into the Banach space \(F\), evaluate at \(v_1 \cdots v_k\), and compare degree-one components. By (46) 🔗, the degree-one component of \(D_+(f;\, y;\, \eta)\) is \(D(f;\, y;\, \eta)\) for \(\eta\) of pure positive degree: the left side contributes \(D(f \circ \phi;\, x;\, v_1 \cdots v_k)\), the right side the partition sum.
The cotangent space and pullback
(61) Definition (Symmetric cotangent space). For \(r, k \geq 0\) define:
- \(\mathcal L_s^r(E;\IR)\) is the Banach space of continuous symmetric \(r\)-linear forms \(E^r\to\IR\). Equivalently, it is the continuous dual of \(\SYM^r(E)\) equipped with the symmetric projective tensor norm.
- \(ST^{\leq k}(X, x) := \Vsum_{r=0}^k \mathcal L_s^r(T_xX;\IR)\) is the symmetric cotangent space of order \(k\). Symmetrized tensor product, truncated above degree \(k\), makes it an algebra.
- \(ST^*(X, x) := \prod_{r \geq 0} \mathcal L_s^r(T_xX;\IR)\) is its degree completion. Its product is defined degreewise, so every coefficient is a finite sum.
- \(ST^*(X, x; F) := \prod_{r \geq 0} \mathcal L_s^r(T_xX; F)\) is the space of \(F\)-valued forms, for a Banach space \(F\), with order-\(k\) truncation \(ST^{\leq k}(X, x; F) := \Vsum_{r=0}^k \mathcal L_s^r(T_xX; F)\); \(ST^*(X, x) = ST^*(X, x; \IR)\). The permanent pairing and the pullback below act on \(F\)-valued forms by the same formulas.
-
The symmetric product is given on homogeneous forms by
\[ (a \cdot b)(v_1, \dots, v_{r+s}) := \frac{1}{(r+s)!} \sum_{\tau \in S_{r+s}} a(v_{\tau(1)}, \dots, v_{\tau(r)})\; b(v_{\tau(r+1)}, \dots, v_{\tau(r+s)}). \] -
For \(a\in\mathcal L_s^r(E;\IR)\) and \(\xi\in\SYM^r(E)\), set \(\langle a,\xi\rangle:=r!\,a(\xi)\), and pair distinct degrees by zero. This is the permanent pairing between \(ST^{\leq k}\) and \(ST_{\leq k}\), and between \(ST^*\) and \(ST_*\). On decomposables,
\[ \langle \ell_1 \cdots \ell_r,\; v_1 \cdots v_r \rangle = \sum_{\tau \in S_r} \prod_{i} \ell_i(v_{\tau(i)}); \]in degree two, \(\langle \ell_1 \ell_2,\, v_1 v_2 \rangle = \ell_1(v_1)\, \ell_2(v_2) + \ell_1(v_2)\, \ell_2(v_1)\). We write \(\langle -, - \rangle_E\) when needed.
(62) Lemma (Separation). The pairing separates forms: if \(\alpha\in ST^*(X,x)\) pairs to zero with every \(\xi\in ST_*(X,x)\), then \(\alpha=0\).
Proof. Degreewise, \(\langle \alpha_r,\, v^r \rangle = r!\, \alpha_r(v, \dots, v)\) for \(v \in E\), and a symmetric multilinear form vanishing on the diagonal vanishes, by polarization.
(63) Lemma (Product and coproduct are adjoint). For \(\alpha,\beta\in ST^*(X,x)\) and \(\xi\in ST_*(X,x)\),
the tensor pairing taken factorwise.
Proof. Bilinearity reduces to \(\alpha \in \mathcal L_s^r(E;\IR)\), \(\beta \in \mathcal L_s^s(E;\IR)\), \(\xi = v_1 \cdots v_n\) with \(n = r + s\). In the defining sum of the symmetric product, there are \(r!\,s!\) permutations \(\tau\) for which \(\{\tau(1),\dots,\tau(r)\}\) is a fixed \(r\)-set \(I\). Thus, with \(J := [n]\setminus I\), \(\langle \alpha \cdot \beta, v_1 \cdots v_n \rangle = n! \, (\alpha \cdot \beta)(v_\bullet) = r!\, s! \sum_{|I| = r} \alpha(v_I)\, \beta(v_J) = \sum_{I \sqcup J = [n]} \langle \alpha, v^I \rangle \langle \beta, v^J \rangle\), which is the right side by (44) 🔗.
The pullback of forms is the transpose of the symmetric pushforward under the permanent pairing, as the adjunction below records; the factorial in the defining formula is the pairing normalization.
(64) Definition (Pullback). Let \(\phi:X\to Y\) be smooth near \(x\), with \(\phi(x)=y\), and let \(\alpha=(\alpha_r)_{r\geq0}\in ST^*(Y,y)\). Define \(D^+(\phi;x)\alpha\in ST^*(X,x)\) degreewise by
where the degree-zero monomial is \(1\). The right side is a continuous symmetric \(r\)-linear form. Moreover, it only uses \(\alpha_0,\dots,\alpha_r\), because \(D_+\) does not increase degree. Thus the formula defines a map on the degree completion and commutes with every truncation. Hence \(D^+(\phi;x): ST^*(Y,y)\to ST^*(X,x)\) is well defined. For \(\phi\) only \(C^k\) near \(x\), the same formula defines \(D^+(\phi;x): ST^{\leq k}(Y,y)\to ST^{\leq k}(X,x)\) on the truncations.
(65) Proposition (Pushforward-pullback adjunction). Let \(\phi: X \to Y\) be smooth near \(x\). For \(\alpha\in ST^*(Y,y)\) and \(\xi\in ST_*(X,x)\),
For \(\phi \in C^k\) near \(x\), the same identity holds with \(\alpha \in ST^{\leq k}(Y, y)\) and \(\xi \in ST_{\leq k}(X, x)\).
Proof. For \(\xi=v_1\cdots v_r\), the permanent pairing and (64) 🔗 give
the factor \(r!\) cancels the \(1/r!\) in the definition, leaving \(\langle\alpha,D_+(\phi;x;v_1\cdots v_r)\rangle_F\). The result follows for every \(\xi\in ST_*(X,x)\) by linearity.
(66) Proposition (The pullback is an algebra morphism). For \(\phi\) smooth near \(x\), \(D^+(\phi;x)\) is a unital algebra morphism on the completed symmetric cotangent algebra. For \(\phi \in C^k\) near \(x\), \(D^+(\phi;x): ST^{\leq k}(Y,y) \to ST^{\leq k}(X,x)\) is a unital algebra morphism on the truncations.
Proof. Pair with \(\xi\in ST_*(X,x)\). By the product-coproduct adjunction (63) 🔗, the defining adjunction (65) 🔗, and the coalgebra property of \(D_+\) (47) 🔗,
Separation (62) 🔗 gives multiplicativity. Unitality: \(D_+(\phi;\, x;\, 1) = 1\) and \(D_+\) sends every positive-degree probe to positive degrees, so \((D^+\mathbf 1)_0 = 1\) and \((D^+\mathbf 1)_r = 0\) for \(r \geq 1\): \(D^+\mathbf 1 = \mathbf 1\). The truncated clause is the same computation, which only uses degrees \(\leq k\).
(67) Corollary (Contravariant functoriality of the pullback). For \(\phi\) smooth near \(x\) and \(\psi\) smooth near \(y = \phi(x)\), on the completed algebras,
For \(\phi, \psi \in C^k\), the same identities hold on the truncations \(ST^{\leq k}\).
Proof. Pair with \(\xi \in ST_*(X, x)\): by the adjunction (65) 🔗 and smooth functoriality (58) 🔗, with base points suppressed,
and separation (62) 🔗 concludes. The identity clause follows the same way from \(D_+(\id_X;\, x) = \id\), and the truncated clause is the same computation in degrees \(\leq k\).
(68) Definition (Symbol pullback). The symbol pullback \(\sigma^*: ST^*(X, x) \to CT^*(X, x)\) is given gradewise by the permanent pairing against the symbol,
Since \(\sigma_k\) takes values in degrees \(1, \dots, k\) and \(\sigma_0 = 1\) (49) 🔗, the grade-\(k\) component uses only \(\alpha_1, \dots, \alpha_k\) for \(k \geq 1\), and \((\sigma^*\alpha)_0 = \alpha_0\): the symbol pullback is grade-triangular and commutes with every truncation. The same formula defines the \(F\)-valued symbol pullback \(\sigma^*: ST^*(X, x; F) \to CT^*(X, x; F)\), the pairing now taking values in \(F\).
(69) Proposition (Symbol pullback is an algebra morphism). \(\sigma^*: (ST^*(X, x),\, \cdot\,) \to (CT^*(X, x),\, \pstar)\) is a unital algebra morphism.
Proof. Unitality: the unit of \(ST^*\) is the constant form \(1\) in degree zero, and \(\sigma^* 1 = \mathbf 1\), since \(\sigma_k(c)\) has no constant component for \(k \geq 1\) and \(\sigma_0(0) = 1\). Multiplicativity rests on the splitting of the symbol under the coproduct: by the bijection of the coalgebra proof (47) 🔗, a partition of \([k]\) with a splitting of its block set is a disjoint decomposition \(I \sqcup J = [k]\) with partitions of the parts. Hence
so, by the product-coproduct adjunction (63) 🔗,
Distributions
Discretely, probes are the finitely supported measures \(\delta(x;\, c)\) and \(\Delta_+\) acts as their pushforward (18) 🔗. Under rescaling, these cube measures converge to the point distributions \(\delta(x;\xi)\) determined by their symbols (77) 🔗. Their pushforward is represented by \(D_+\) (75) 🔗. This section gives that distributional interpretation of the covariant calculus; the jet calculus below does not depend on it.
(70) Definition (Smooth functions and distributions). Let \(X\) be finite-dimensional. Define:
- \(\mathcal{E}(X) := C^\infty(X, \IR)\) with its standard Fréchet topology.
- \(\mathcal{E}'(X) := \mathcal{E}(X)'\) is the space of compactly supported distributions.
- \(\mathcal{E}'(X, x)\) is the space of distributions supported at \(x\).
- \(\mathcal{E}'(X, x)_{\leq k}\) is its subspace of distributions of order at most \(k\).
- \(\phi^* f := f \circ \phi\) is the pullback of smooth functions.
- \(\langle \phi_* u, f \rangle := \langle u, \phi^* f \rangle\) is the pushforward of distributions.
(71) Definition (Point distributions). For \(\xi \in ST_{\leq k}(X, x)\), the point distribution \(\delta(x;\, \xi)\) is the linear functional
It is supported at \(x\) and of order \(\leq k\), so the same formula is meaningful for any \(f\) that is \(C^k\) near \(x\), scalar- or Banach-valued, the pairing then taking values in the target. We keep the same notation in this case. In particular, \(\delta(x;\, 1) = \delta(x)\) is evaluation at \(x\).
(72) Lemma (Finitely supported measures are distributions). Restriction along \(\mathcal{E}(X) \subseteq F(X)\) embeds the finitely supported measures into the distributions, \(F'(X) \mono \mathcal{E}'(X)\), as the distributions of order \(0\) with finite support. In particular every cube measure \(\delta(x;\, c)\) of (15) 🔗 is a distribution.
Proof. Evaluation at a point is continuous on \(\mathcal{E}(X)\) and extends to \(C^0(X, \IR)\), so \(\nu = \sum_i \lambda_i\, \delta(z_i)\) defines a distribution of order \(0\) supported in the finite set \(\set{z_i}\); the map is injective since the \(\delta(z_i)\) are linearly independent as functionals on \(\mathcal{E}(X)\).
(73) Theorem (Structure of point-supported distributions, [Hormander1983], Theorem 2.3.4). On smooth scalar test functions, \(\xi \mapsto \delta(x;\, \xi)\) (71) 🔗 is a linear isomorphism
between the symmetric tangent space and the distributions of order \(\leq k\) supported in \(\set{x}\).
(74) Remark (Sign-free basis). The basis \(\delta(x;\, \xi)\) is sign-free (73) 🔗: it represents the operator datum \(\xi\) directly rather than through \(\del^\alpha \delta(x) = (-1)^{|\alpha|}\, \delta(x;\, e^\alpha)\)-type conventions; in one direction, \(\del_v\, \delta(x) = -\delta(x;\, v)\).
For smooth \(\phi\), the pushforward \(\langle \phi_* u,\, f \rangle := \langle u,\, \phi^* f \rangle\) (70) 🔗 is the usual operation on \(\mathcal{E}'(X)\). For \(\phi\) only \(C^k\) near \(x\), the composite \(f \circ \phi\) is only \(C^k\) and the general operation is unavailable; on point distributions of order \(\leq k\), the finite-order pairing above still applies, and the same formula, \(\langle \phi_*\, \delta(x;\, \xi),\, f \rangle := \langle \delta(x;\, \xi),\, f \circ \phi \rangle\), defines the pushforward there.
(75) Theorem (Distribution pushforward). For \(\phi \in C^k\) near \(x\) and \(\xi \in ST_{\leq k}(X, x)\),
Proof. \(\langle \phi_*\, \delta(x;\, \xi),\, f \rangle = D(f \circ \phi;\, x;\, \xi) = D(f;\, y;\, D_+(\phi;\, x)\, \xi) = \langle \delta(y;\, D_+(\phi;\, x)\, \xi),\, f \rangle\) by the smooth adjunction (59) 🔗.
(76) Proposition (Filtration and principal symbol). \(D_+(\phi; x)\) preserves the degree filtration \(ST_{\leq r}(X, x) \subseteq ST_{\leq k}(X, x)\), and on the associated graded it is the multiplicative extension of the first derivative:
Proof. A partition \(\pi \in \Part(r)\) contributes in degree \(|\pi| \leq r\), with equality only for the partition into singletons, whose term is \(\prod_i D(\phi;\, x;\, v_i)\).
(77) Corollary (Distributional collapse). For \(c \in CT_k(X, x)\), the rescaled cube measures converge weakly in \(\mathcal{E}'(X)\), that is, pointwise on \(\mathcal{E}(X)\):
The left side is a finitely supported measure; the right side is a point distribution in \(\mathcal{E}'(X, x)\).
Proof. For every \(f \in \mathcal{E}(X)\),
the limit of the top coordinate by boundary extension (35) 🔗, applied to \(f\) read as a map into \(\IR\), identified through symbol factorization (51) 🔗.
(78) Remark. Pairing with observables gives the dual statement: Newton's exact finite expansion converges degreewise to the corresponding Taylor jet. Under higher regularity, Higher symbols and collapse expansion 🔗 expands \(\Delta(f;\,x;\,\lambda_t c)\) to every finite order. Its \(t^{k+m}\)-coefficient pairs the derivatives of \(f\) with the cover-indexed higher symbol \(\sigma_{k,m}(c)\); on affine cubes these are the repeated-direction Taylor monomials.
Jet calculus
This chapter derives the discrete, quotient, zero, and smooth jets from the pullbacks \(\Delta^+\), \(Q^*\), and \(D^+\). Their naturality and product laws follow from the corresponding operator identities.
Jets and naturality
(79) Definition (Jets). For an observable \(f: X \to F\) with values in a Banach space \(F\) define:
-
the discrete jet \(j^\Delta(f; x) \in CT^*(X, x; F)\), the forward differences,
\[ j^\Delta_k(f; x)(c) := \Delta(f;\, x;\, c), \qquad k \geq 0; \] -
the quotient jet \(qj(f; x;\, t) \in CT^*(X, x; F)\) for \(t \in (0,1]\), the normalized differences at scale \(t\),
\[ qj_k(f; x;\, t;\, c) := \frac{1}{t^k}\, \Delta(f;\, x;\, \lambda_t\, c), \qquad k \geq 0; \] -
the smooth jet \(j(f; x) \in ST^{\leq k}(X, x; F)\) of an \(f\) that is \(C^k\) near \(x\), the Taylor expansion of \(f\) at \(x\),
\[ j(f; x) := \sum_{r = 0}^{k} \frac{1}{r!}\, D^r(f; x), \qquad D^0(f; x) := f(x). \]
Scalar jets are the case \(F = \IR\); the coefficient is suppressed there. The order of the smooth jet is set by the regularity of \(f\); for smooth \(f\) the jets of all orders are compatible and assemble into \(j(f; x) \in ST^*(X, x; F)\).
(80) Remark (Polynomial evaluation). The factorial normalization makes the permanent pairing agree with polynomial evaluation. If \(\alpha = \sum_{r=0}^{k} \alpha_r \in ST^{\leq k}(X, x)\) represents the polynomial
then, with \(\exp(v) := \sum_{r \geq 0} v^r / r!\), \(p_\alpha(x + v) = \langle \alpha,\, \exp(v) \rangle\): evaluation at \(x + v\) is represented on the symmetric tangent side by the exponential probe.
The discrete jet is not obtained from a quotient by powers of the point-vanishing ideal. The next proposition shows why that construction has no nontrivial discrete counterpart.
(81) Proposition (Prolongation, not quotient). Let \(I_x = \set{f \in F(X) : f(x) = 0}\) be the vanishing ideal of \(x\) (14) 🔗.
1) \(I_x\) is idempotent, \(I_x^2 = I_x\): the filtration by powers of \(I_x\) is constant, and \(F(X)/I_x^m = \IR\) for every \(m \geq 1\).
2) The discrete jet is injective: the values of \(j^\Delta(f;\, x)\) on cubes of order \(\leq 1\) already determine \(f\), via \(f(x + v) = f(x) + \Delta(f;\, x;\, v)\).
Proof. 1) For \(f \in I_x\) let \(g \in I_x\) take the value \(1\) away from \(x\); then \(f = fg \in I_x^2\). 2) The order-zero value is \(f(x)\), and the order-one values are the increments \(f(x + v) - f(x)\).
(82) Lemma (Jet pairing). For \(f \in C^k\) near \(x\) and \(\xi \in ST_{\leq k}(X, x)\), in the observable reading,
Proof. Degreewise: the permanent pairing (61) 🔗 contributes \(r!\) against the coefficient \(\frac{1}{r!}\) of \(D^r(f; x)\); in degree zero both sides record \(f(x)\).
(83) Theorem (Jet naturality). For arbitrary maps \(\phi: X \to Y\), observables \(f: Y \to F\), and \(t \in (0,1]\),
If \(\phi\) is \(C^k\) near \(x\), then for every \(f\) that is \(C^k\) near \(y = \phi(x)\) the differential jets satisfy
Proof. For the discrete clause, evaluate on \(c\). By the discrete adjunction (18) 🔗, \(\langle \Delta^+(\phi;\, x)\, j^\Delta(f;\, y),\; c \rangle = \Delta(f;\, y;\, \Delta_+(\phi;\, x)\, c) = \Delta(f \circ \phi;\, x;\, c)\). Quotient clause, in grade \(k\): \(\lambda_t\, Q_k(\phi;\, x;\, t) = \Delta_+(\phi;\, x)\, \lambda_t\) (29) 🔗, so
again by the discrete adjunction. In grade zero both sides record \(f(y)\). For the differential clause, pair with \(\xi \in ST_{\leq k}(X, x)\). The jet pairing (82) 🔗, smooth adjunction (59) 🔗, and the pushforward-pullback adjunction (65) 🔗 give
The permanent pairing separates \(F\)-valued forms degreewise, so the differential jets agree.
The naturality property characterizes the differential pullback. Since the jet map is surjective, the naturality square determines \(D^+(\phi;\,x)\) on every form.
(84) Lemma (Jet realization). For \(\alpha \in ST^{\leq k}(X, x)\), the polynomial \(p_\alpha\) (80) 🔗 satisfies \(j(p_\alpha;\, x) = \alpha\). In particular the jet map \(j(-;\, x)\) is surjective onto \(ST^{\leq k}(X, x)\).
Proof. The \(r\)-th derivative at \(v = 0\) of the \(s\)-homogeneous term vanishes for \(r \neq s\) and equals \(r!\, \alpha_r\) for \(r = s\), by polarization.
(85) Corollary (Characterization of jet pullback). For \(\phi \in C^k\) near \(x\), the pullback \(D^+(\phi; x)\) is the unique linear map \(ST^{\leq k}(Y, y) \to ST^{\leq k}(X, x)\) such that
for every scalar \(f\) that is \(C^k\) near \(y = \phi(x)\).
Proof. Existence is the differential clause of jet naturality (83) 🔗. For uniqueness, let \(L\) be another such map and let \(\alpha \in ST^{\leq k}(Y, y)\). By jet realization (84) 🔗, \(\alpha = j(p_\alpha;\, y)\), whence
(86) Corollary (Taylor pullback on generators). For \(\ell \in F^*\), let \(\ell_y(z) := \ell(z - y)\). For \(\phi \in C^k\) near \(x\),
the Taylor expansion at \(x\) of the centered observable \(\ell_y \circ \phi\), and, for \(1 \leq q \leq k\), with the product taken in the truncated algebra \(ST^{\leq k}(X, x)\),
on generators, the pullback is composition of Taylor expansions.
Proof. In degree \(r\), pairing with \(\ell\) selects the degree-one part of \(D_+(\phi;\, x;\, v_1 \cdots v_r)\), the term of the one-block partition, giving the Taylor coefficient \(\frac{1}{r!}\, \ell \circ D^r(\phi;\, x)\); the constant term \(\ell_y(\phi(x)) = 0\) vanishes. The product formula is multiplicativity of the pullback (66) 🔗.
(87) Remark. Expanding the composition of Taylor expansions into partition data recovers the coefficient calculus of Faà di Bruno; the combinatorial development is the companion's [FDB].
(88) Corollary (Jets as pullbacks of the identity). On the target space \(F\), write \(u_k(c) := c_{[k]}\) for the top-coordinate observable on \(CT_k(F, y)\). The cube fibers \(CT_k(F, y) = \Map(\KP_+(k), F)\) do not depend on the base point. Write \(\iota := \id_F \in \mathcal L_s^1(F; F)\) for the identity form. The jets of the identity observable \(\id_F\) at \(y\) are the scale-invariant universal elements
and every jet is a pullback of the identity's jet: for an observable \(f: X \to F\) with \(y = f(x)\),
the cubical clauses for arbitrary \(f\) in all grades, the smooth clause in degrees \(\leq k\) for \(C^k\) \(f\).
Proof. Jets of the identity: for \(k \geq 1\), the alternating vertex sum of \(\id_F\) retains only the top Möbius coordinate, \(\Delta(\id_F;\, y;\, c) = \sum_{T \subseteq [k]} (-1)^{k - |T|}\, \zeta_y(c)_T = c_{[k]}\), by the Boolean sieve (5) 🔗: the base point cancels, and among the coordinates only the one indexed by \(A=[k]\) contributes. Hence also \(qj_k(\id_F;\, y;\, t;\, c) = t^{-k} (\lambda_t\, c)_{[k]} = c_{[k]}\) for every \(t>0\), while grade zero records \(\id_F(y) = y\). Thus the quotient jets of the identity do not depend on \(t\). For the smooth jet, \(D^0(\id_F;\, y) = y\), \(D^1(\id_F;\, y) = \iota\), and the higher derivatives vanish. The three pullback clauses follow from jet naturality (83) 🔗, applied to the map \(f\) and the identity observable \(\id_F\).
The jet ladder
The jets of (79) 🔗 assemble into the ladder
whose rungs are defined and compared below.
(89) Definition (Zero jet). The zero jet is the gradewise limit
defined wherever the limit exists.
(90) Theorem (Zero jet on the \(C^k\) locus). Let \(f: X \to F\) be \(C^k\) near \(x\). The grade-\(k\) zero jet exists on all cubes and, reading \(f\) as a map into the affine space \(F\),
Proof. By (88) 🔗, \(qj_k(f;\, x;\, t) = u_k \circ Q_k(f;\, x;\, t)\) for \(t \in (0,1]\); boundary extension (35) 🔗 applies to the map \(f\) and gives, in the limit, existence on all cubes and the first identification. For the second, the jet pairing (82) 🔗 and symbol factorization (51) 🔗 apply because the two readings of \(D\) agree on \(\sigma_k(c)\), which has no constant component. Thus, for \(k \geq 1\),
in grade zero both sides record \(f(x)\) (68) 🔗.
(91) Corollary (Zero-fiber jet naturality). For \(\phi \in C^k\) near \(x\) and an observable \(f: Y \to F\) that is \(C^k\) near \(y = \phi(x)\),
Proof. By (90) 🔗, \(qj^0_k(f \circ \phi;\, x) = u_k \circ Q^0_k(f \circ \phi;\, x)\), the composite being \(C^k\) near \(x\); zero-fiber functoriality (38) 🔗 splits \(Q^0_k(f \circ \phi;\, x) = Q^0_k(f;\, y) \circ Q^0_k(\phi;\, x)\), and \(u_k \circ Q^0_k(f;\, y) = qj^0_k(f;\, y)\), again by (90) 🔗.
(92) Proposition (Symbol pullback intertwining). Let \(\phi\) be \(C^k\) near \(x\). For \(\alpha \in ST^{\leq k}(Y, y)\),
the symbol pullback intertwines the smooth and zero-fiber pullbacks.
Proof. Evaluate grade \(k\) on \(c \in CT_k(X, x)\): by the pushforward-pullback adjunction (65) 🔗 and the symbol intertwining (54) 🔗,
which is \((\sigma^*\, \alpha)_k(Q^0_k(\phi;\, x;\, c))\).
The scopes of the rungs differ. The left half of the ladder is exact and total: \(j^\Delta\) and \(qj(\cdot;\, t)\) are defined for every observable, with \(qj(f;\, x;\, 1) = j^\Delta(f;\, x)\). The zero jet is partial: it exists wherever the limit does, and its values need not be linear in the cube. For \(f = |\cdot|\) at \(0\), the grade-one zero jet is \(|v|\) (37) 🔗. The right rung requires regularity: for \(C^k\) observables the zero jet is total in grades \(\leq k\) and equals \(\sigma^*\, j(f;\, x)\) (90) 🔗.
Leibniz rules
(93) Theorem (Leibniz rules). For scalar functions \(f, g: X \to \IR\) and \(t \in (0,1]\),
Moreover, let \(c \in CT_k(X, x)\). If the zero jets of \(f\) and \(g\) exist on every face of \(c\), then the zero jet of \(fg\) exists at \(c\) and
the right side evaluating \(qj^0(f;\, x)\) and \(qj^0(g;\, x)\) only on faces of \(c\).
If \(f\) and \(g\) are \(C^k\) near \(x\), then in the truncated symmetric cotangent algebra,
Proof. The \(\cstar\) clause in grade \(k\) is the discrete product rule
by Taylor duality (10) 🔗, \(f(\zeta_x(c)_T) = \sum_{I \subseteq T} \Delta(f; x; \del_I c)\) and similarly for \(g\); multiply, insert into the alternating sum defining \(\Delta(fg; x; c)\), and exchange sums: the pair \((I, J)\) occurs for the vertices \(T \supseteq I \cup J\), with total coefficient \(\sum_{I \cup J \subseteq T \subseteq [k]} (-1)^{k - |T|} = [I \cup J = [k]]\) by the Boolean sieve (5) 🔗. The \(\star_t\) clause follows by transport: \(qj_k(fg;\, x;\, t;\, c) = \frac{1}{t^k}\, \Delta(fg;\, x;\, \lambda_t\, c)\); apply the \(\cstar\) clause at \(\lambda_t c\), use \(\del_I \lambda_t c = \lambda_t \del_I c\) (24) 🔗, and collect the powers of \(t\): the pair \((I, J)\) contributes \(t^{|I|}\, t^{|J|}\, t^{-k} = t^{|I \cap J|}\). For the zero-jet clause, let \(t \downarrow 0\) in the \(\star_t\) clause at \(c\): each factor converges by hypothesis, and \(t^{|I \cap J|} \to [I \cap J = \emptyset]\), so only the disjoint pairs remain. Their sum is the grade-\(k\) \(\pstar\) component (39) 🔗. Now suppose that \(f\) and \(g\) are \(C^k\) near \(x\). Their product is \(C^k\) near \(x\). In degree zero the claim is \((fg)(x) = f(x)g(x)\). For \(1 \leq n \leq k\), evaluate the zero-jet clause at the affine cube \(\Aff(v_1, \dots, v_n)\): faces of affine cubes are affine, \(\del_I \Aff(v_1, \dots, v_n) = \Aff(v_I)\), and for a \(C^k\) function the zero jet exists there with value the derivative, \(qj^0_{|I|}(f;\, x;\, \Aff(v_I)) = D(f;\, x;\, v^I)\) by part 1 of (32) 🔗, the grade-zero jet recording \(f(x)\). The clause becomes
with the ordered disjoint decompositions matching, term by term, the normalized symmetric product: its degree-\(n\) component satisfies
This is the degree-\(n\) component of \(j(fg;\, x)\).
(94) Remark (Algebra-valued coefficients). For observables \(f, g: X \to A\) with values in a Banach algebra, the pointwise product \(fg\) obeys the same clauses with the same proofs, including the differential-jet clause when \(f\) and \(g\) are \(C^k\). The order of the factors is kept as written. When \(A\) is noncommutative, the products \(\star_t\) on \(A\)-valued cochains are no longer commutative, and \(f g \neq g f\) propagates through every rung.
(95) Corollary (Smooth product rule). For scalar functions \(f, g: X \to \IR\) that are \(C^k\) near \(x\) and \(v_1, \dots, v_k \in T_x X\),
the sum over ordered disjoint decompositions with possibly empty parts.
The classical completion
Classically, the \(k\)-jet of a smooth function is its class modulo the \((k{+}1)\)-st power of the maximal ideal, and jet pullback is ideal-theoretic. This section realizes the quotients on symmetric cotangent spaces and identifies the formal pullback with \(D^+\). It is the contravariant counterpart of Distributions 🔗: there, probes were realized as point-supported distributions; here, function classes are realized as symmetric forms. Throughout, \(X\) is finite-dimensional and functions are smooth (70) 🔗; apart from the globalization Globalization 🔗, nothing below depends on this section.
(96) Definition (Maximal ideal, classical jets, completion). Define:
- \(\mathfrak m_x := \set{f \in \mathcal E(X) : f(x) = 0}\), the maximal ideal of \(x\);
- \(\mathcal E_k(X, x) := \mathcal E(X) / \mathfrak m_x^{k+1}\), the classical \(k\)-jet space at \(x\); the class \([f]_k\) of \(f\) is its classical \(k\)-jet, traditionally written \(j^k_x f\);
- \(\hat{\mathcal E}(X, x) := \varprojlim_k\, \mathcal E_k(X, x)\), the completion of the functions at \(x\), with the hat map \(f \mapsto \hat f := ([f]_k)_{k \geq 0}\) and the projective limit topology.
(97) Remark. Over arbitrary functions the vanishing ideal is idempotent and the quotient construction collapses (81) 🔗. Over smooth functions the powers \(\mathfrak m_x^{k+1}\) filter strictly, and the classical construction proceeds: the discrete jet is a prolongation, the classical jet a quotient, and the realization theorem below is the bridge on the smooth column.
(98) Proposition (Formal functoriality). For smooth \(\phi: X \to Y\) with \(\phi(x) = y\), the pullback \(\phi^* f = f \circ \phi\) satisfies \(\phi^*\, \mathfrak m_y \subseteq \mathfrak m_x\) and \(\phi^*\, \mathfrak m_y^{k+1} \subseteq \mathfrak m_x^{k+1}\): it descends to algebra morphisms \(\mathcal E_k(Y, y) \to \mathcal E_k(X, x)\) and \(\hat\phi^*: \hat{\mathcal E}(Y, y) \to \hat{\mathcal E}(X, x)\), with, for smooth \(\psi: Y \to Z\),
Proof. \(f(y) = 0\) implies \(f(\phi(x)) = 0\), and \(\phi^*\) is multiplicative, so the ideal containments hold and the quotient maps exist; functoriality is that of \(\phi^*\).
(99) Theorem (Taylor realization). For each \(k\), the jet map \(j(-;\, x): \mathcal E(X) \to ST^{\leq k}(X, x)\) (79) 🔗 is an algebra morphism onto \(ST^{\leq k}(X, x)\) with kernel \(\mathfrak m_x^{k+1}\); it descends to an isomorphism of algebras
Proof. Multiplicativity is the smooth clause of the Leibniz theorem (93) 🔗, and surjectivity is the jet realization (84) 🔗 on the smooth polynomials \(p_\alpha\). The kernel consists of the functions flat to order \(k\) at \(x\): all derivatives of order \(\leq k\) vanish. Flatness is a differential condition, membership in \(\mathfrak m_x^{k+1}\) an algebraic one; the two inclusions need separate arguments. \(\mathfrak m_x^{k+1} \subseteq \ker j(-;\, x)\): a derivative of order \(\leq k\) of a product of \(k+1\) functions vanishing at \(x\) leaves, in each Leibniz term, at least one factor undifferentiated. \(\ker j(-;\, x) \subseteq \mathfrak m_x^{k+1}\) is Hadamard's lemma iterated: by Taylor's formula with integral remainder, a smooth function flat to order \(k\) at \(x\) is a sum of products of \(k+1\) coordinate functions centered at \(x\) with smooth coefficients, hence lies in \(\mathfrak m_x^{k+1}\) [Nestruev2020]. This step uses smoothness and the finite dimension of \(X\).
(100) Corollary (Realization of the completion). The stagewise isomorphisms assemble to an isomorphism of algebras
where \(j_k\) is the stage-\(k\) Taylor realization (99) 🔗. Under this isomorphism the formal pullback becomes the operator: for smooth \(\phi\),
Proof. The isomorphisms (99) 🔗 commute with the quotient maps \(\mathcal E_{k+1}(X, x) \to \mathcal E_k(X, x)\) and the degree truncations, so they induce an isomorphism of the limits (61) 🔗. The pullback identity is the differential clause of jet naturality at every finite stage: if \(a_k=[f]_k\), then \(j_k(\phi_k^*a_k)=j(f\circ\phi;\,x) =D^+(\phi;\,x)j(f;\,y)\), truncated in degree \(k\) (83) 🔗. These identities are compatible with truncation, so they pass to the inverse limit. No single smooth lift of an arbitrary element \(a\in\hat{\mathcal E}(Y,y)\) is used.
(101) Proposition (Duality with point distributions). Let \(a=(a_k)_{k\geq0}\in\hat{\mathcal E}(X,x)\). If \(u\in\mathcal E'(X,x)\) has order at most \(k\), choose any representative \(f_k\) of \(a_k\) and set \(\langle a,u\rangle:=u(f_k)\). This is independent of \(k\), of the representative, and of any choice of a global smooth lift, and it induces an isomorphism \(\mathcal E'(X, x) \xrightarrow{\sim} \hat{\mathcal E}(X, x)'\). Under the realizations (100) 🔗, (73) 🔗, it is the permanent pairing:
In particular, if \(a=\hat f\) for a smooth function \(f\), this value is \(D(f;\,x;\,\xi)=\langle j(f;\,x),\xi\rangle\).
Proof. Well-definedness: a point-supported distribution has finite order, and if \(u\) has order \(\leq k\) it vanishes on \(\mathfrak m_x^{k+1}\): such functions are flat to order \(k\) at \(x\) (99) 🔗, and flat functions are \(C^k\)-approximable by functions vanishing near \(x\) [Hormander1983], Theorem 2.3.3; so \(u\) factors through \(\mathcal E_k(X, x)\). Compatibility of the stages makes the value independent of every larger \(k\). Injectivity: if \(u \neq 0\) then \(u(f) \neq 0\) for some \(f\). Surjectivity: a continuous functional on the projective limit factors through a finite stage; composing with the quotient map gives a distribution that vanishes on \(\mathfrak m_x^{k+1}\), hence is supported in \(\set{x}\), since a function vanishing near \(x\) lies in every power of \(\mathfrak m_x\), and has order \(\leq k\). The pairing identity is the definition of the point distribution (71) 🔗 together with the jet pairing (82) 🔗.
(102) Remark (The two interpretations).
the covariant calculus is realized on point-supported distributions, with \(D_+\) their pushforward (75) 🔗; the contravariant calculus is realized on classes of smooth functions, with \(D^+\) the formal pullback (100) 🔗. The permanent pairing between \(ST_*\) and \(ST^*\) realizes the evaluation \(u(f)\) on both sides (101) 🔗.
Globalization
The classical spaces globalize. On a smooth finite-dimensional manifold both fibers have chart-free definitions, so the bundles can be named before any gluing: the function classes form the jet bundle, the point distributions its dual, the operators are pullback of functions and pushforward of distributions, and the cocycle condition for the chart trivializations is smooth functoriality itself (58) 🔗. The one point requiring care is what the bundles are not: chartwise every fiber is \(\SYM_{\leq k}\) of a tangent space, but the gluing is the full pushforward of the transitions, not the symmetric power of their differentials, and the glued bundle is not the symmetric algebra of the tangent bundle (106) 🔗. Throughout this section, \(M\), \(N\) are smooth finite-dimensional manifolds.
(103) Definition (Manifold jets and point probes). For \(p \in M\) set
with \(\mathfrak m_p\) and \(\mathcal E_k(M, p)\) as in (96) 🔗 and \(\mathcal E'(M, p)_{\leq k}\) the continuous functionals on \(\mathcal E(M)\) supported in \(\set{p}\) of order \(\leq k\). Write \(ST^{\leq k} M\) and \(ST_{\leq k} M\) for the disjoint unions fibered over \(M\), and \(j(f;\, p) := [f]_k\) for the jet of an observable.
(104) Proposition (Globalization).
-
Every chart \(\varphi: U \to \IR^n\) at \(p\) identifies the fibers with the affine models,
\[ ST^{\leq k}(M, p) \isom ST^{\leq k}(\IR^n,\, \varphi(p)), \qquad ST_{\leq k}(M, p) \isom ST_{\leq k}(\IR^n,\, \varphi(p)), \]by transfer of germs along \(\varphi\) followed by the realizations (99) 🔗, (73) 🔗. Two charts differ by \(D^+(\tau;\, x)\) and \(D_+(\tau;\, x)\) of the transition \(\tau\), whose components are the derivatives of \(\tau\); the identifications therefore make \(ST^{\leq k} M\) and \(ST_{\leq k} M\) smooth vector bundles. The multiplication of classes, the coproduct, and the pairing \(\langle [f]_k,\, u \rangle = u(f)\) are chart-free, and the identifications carry them to the model structures.
-
For smooth \(\Phi: M \to N\) and \(q := \Phi(p)\), pullback of functions and pushforward of distributions,
\[ D^+(\Phi;\, p)\, [f]_k := [f \circ \Phi]_k, \qquad D_+(\Phi;\, p)\, u := \Phi_* u = u(\,\cdot \circ \Phi), \]are bundle morphisms, mutually adjoint under the pairing, and functorial by composition: \(D_+(\Psi \circ \Phi) = D_+(\Psi) \circ D_+(\Phi)\), \(D_+(\id_M) = \id\). In charts they are \(D^+\) and \(D_+\) of the chart representative of \(\Phi\) (100) 🔗, (75) 🔗.
-
For smooth \(f: M \to \IR\), the jet \(p \mapsto j(f;\, p)\) is a smooth section of \(ST^{\leq k} M\), and naturality is tautological: \(j(f \circ \Phi;\, p) = D^+(\Phi;\, p)\, j(f;\, q)\).
Proof. 1) A function vanishing near the base point lies in every power of the maximal ideal, so both quotients depend only on germs; with a cutoff, every germ transfers along \(\varphi\), and composition with \(\varphi\) descends to an isomorphism of the quotients. Point-supported functionals transfer by \(\varphi_*\) for the same reason. On an overlap the composite identification is precomposition with the transition \(\tau\), respectively \(\tau_*\), which the realizations convert to \(D^+(\tau;\, x)\) (100) 🔗 and \(D_+(\tau;\, x)\) (75) 🔗; the components are the derivatives \(D(\tau;\, x;\, v^A)\), \(|A| \leq k\) (46) 🔗, smooth in \(x\), and the cocycle condition is smooth functoriality (58) 🔗. 2) Well-definedness: \(\mathfrak m_q \circ \Phi \subseteq \mathfrak m_p\) gives \(f \circ \Phi \in \mathfrak m_p^{k+1}\) for \(f \in \mathfrak m_q^{k+1}\), so the pullback descends; dually, \(\Phi_* u\) is supported in \(\set{q}\) and kills \(\mathfrak m_q^{k+1}\), hence has order \(\leq k\) (101) 🔗. Adjointness is \(\langle [f]_k,\, \Phi_* u \rangle = u(f \circ \Phi) = \langle [f \circ \Phi]_k,\, u \rangle\), and functoriality is associativity of composition. In charts, the pullback is the formal pullback of the representative and the pushforward its adjoint, which are \(D^+\) and \(D_+\) by the citations. 3) In a chart the section has components the derivatives of \(f \circ \varphi^{-1}\), smooth in the base point; naturality is the definition of the pullback in clause 2.
(105) Remark (The classical bundles). \(ST^{\leq k} M\) is the classical bundle of \(k\)-jets of functions [Ehresmann1951] [Saunders1989], with \(D^+(\Phi)\) its jet pullback. Its full (augmented) dual is \(ST_{\leq k}M\): a smooth field \(p\mapsto u_p\) of point distributions is a linear differential operator of order \(\leq k\) on \(M\), \((Lf)(p)=\langle j(f;\,p),u_p\rangle\). Evaluation on constants defines the counit \(\epsilon(u_p):=u_p(1)\). The higher-order tangent bundle of Pohl [Pohl1962] [KMS1993] is the subbundle
whose sections are precisely the differential operators of order \(\leq k\) that annihilate constants. At \(k=2\) this is the second-order tangent bundle of stochastic analysis, and \(D_+(\Phi)\) preserves the counit and restricts to the Schwartz morphism transporting second-order tangent vectors [Emery1989].
(106) Remark (Not the symmetric algebra of the tangent bundle). The symmetric algebra bundle \(\SYM_{\leq k}(TM)\), glued by the symmetric powers \(\SYM(D(\tau;\, x))\) of the transition differentials, is a different vector bundle. The transition of \(ST_{\leq k} M\) is the full pushforward \(D_+(\tau;\, x)\): its degree-preserving component is \(\SYM(D(\tau;\, x))\), the all-singletons term of the partition sum, while its degree-lowering components carry the higher derivatives of \(\tau\) (46) 🔗. Since the partition sum only lowers degree, \(D_+(\tau)\) preserves the degree filtration but not the grading: \(ST_{\leq k} M\) is a filtered bundle with associated graded \(\SYM_{\leq k}(TM)\), and dually the jet bundle is filtered by order, with graded pieces \(\mathfrak m_p^r/\mathfrak m_p^{r+1} \isom\) the symmetric powers of the cotangent space. The grading does not glue; choosing a connection splits the filtration, which is the mechanism by which covariant derivatives assemble higher jets. The top filtration quotient is natural, and for a field of point distributions it is the principal symbol of the corresponding differential operator: tensoriality of the classical symbol is naturality of the top graded piece.
(107) Remark (The Banach case and the affine layer). On general Banach manifolds the quotient-by-powers and point-distribution realizations above are not available without additional hypotheses. The symmetric spaces nevertheless glue directly by their transition operators: chart pairs \((\varphi, \xi)\) modulo \((\varphi,\, \xi) \sim (\psi,\, D_+(\tau;\, \varphi(p))\, \xi)\), and dually for forms, define the bundles directly, with smooth functoriality (58) 🔗 as the cocycle condition and \(D_+\) natural by the same conjugation. The exact discrete layer does not globalize even there: cubes take their legs in the model space and \(\lambda_t\) uses its linear structure, so \(\Delta_+\) and the family \(Q_*(t)\) are chart objects. Their boundary is intrinsic: tangent cubes glue along \(Q^0_k(\tau)\) of the transitions, by zero-fiber functoriality (38) 🔗, into the fibers of the iterated tangent bundle \(T^k M\) (36) 🔗, and the symbol intertwining transports this gluing to the symmetric layer (54) 🔗. The deformation is a chart construction whose \(t = 0\) fiber is intrinsic, the situation familiar from deformation to the normal cone (53) 🔗.
The ladder, assembled
The discrete, quotient, zero, and smooth jets each have a pullback and a product. Three comparison maps relate adjacent rungs. The following panorama collects these identities; every statement has been proved above.
- Evaluations. At \(t = 1\) the quotient rung is the discrete rung, \(qj(f;\, x;\, 1) = j^\Delta(f;\, x)\) and \(Q_*(\phi;\, x;\, 1) = \Delta_+(\phi;\, x)\) (29) 🔗. As \(t \downarrow 0\) it converges gradewise to the zero fiber, \(qj_k \to (\sigma^*\, j(f;\, x))_k\) and \(Q_k \to Q^0_k(\phi;\, x) = T^k\phi\,|_x\) (35) 🔗, (90) 🔗, (36) 🔗; for \(C^{k+M}\) observables the approach is a Taylor expansion in \(t\), its coefficients the higher-symbol pairings (57) 🔗.
- Naturality. Each rung is natural for its own pullback: \(\Delta^+\), \(Q^*(\cdot;\, t)\), precomposition with \(Q^0_*\), and \(D^+\) (83) 🔗, (91) 🔗.
- Multiplicativity. Each rung is multiplicative for its product: \(\cstar\), \(\star_t\), \(\pstar\), and the truncated symmetric product (93) 🔗; the pointwise product on \(CT^*\) (16) 🔗 remains distinct from these convolution products.
- Comparisons. The symbol pullback is a unital algebra morphism (69) 🔗 and intertwines the smooth and zero-fiber pullbacks (92) 🔗. The evaluations restrict the quotient squares and product rules to the discrete ones at \(t = 1\) and, by the uniform convergence of (35) 🔗, give the zero-fiber identities as \(t \downarrow 0\). The \(\sigma^*\) identities identify the smooth pullback and product with their zero-fiber counterparts.
(108) Remark (Per-leg jet). The ladder has an anisotropic variant. For \(\mathbf t = (t_1, \dots, t_k) \in (0,1]^k\), the per-leg jet (overloaded, selected by argument type) is
It refines the family off the diagonal. As with the anisotropic rescaling (24) 🔗, this is recorded as an export for the quantitative companion and receives no theory in this paper.
Exhibits
Each exhibit follows a concrete object through the deformation, from the exact finite fiber through the family to the zero fiber and its symbol; we call such a path an orbit. The propositions proved along the way are chapter-local tools; the body does not depend on them.
An order-two cube through the deformation
Fix an arbitrary map \(\phi: X \to Y\) with \(y = \phi(x)\) and directions \(v_1, v_2 \in E\). At order two, the deformation, its collapse, and the symbol can all be written out on the single orbit
with coordinates (26) 🔗
The exact fiber. At \(t = 1\) the orbit is the cubical pushforward, \(c(1) = \Delta_+(\phi;\, x)\, \Aff(v_1, v_2)\): the vertexwise image of the parallelogram with vertices \(x\), \(x + v_1\), \(x + v_2\), \(x + v_1 + v_2\), re-read in Möbius coordinates (12) 🔗. The legs are the image increments, and the defect \(c_{12}(1) = \phi(x + v_1 + v_2) - \phi(x + v_1) - \phi(x + v_2) + \phi(x)\) measures the failure of the four image points to close a parallelogram: an arbitrary map does not preserve affinity, and the pushforward records this failure without any regularity assumption. For every \(t>0\), conjugation by \(\lambda_t\) identifies this fiber with the fiber at \(t=1\) (29) 🔗. The new fiber is therefore the boundary value at \(t=0\).
The five covers. Pair an observable \(f: Y \to \IR\) against the orbit. Jet naturality (83) 🔗 and affine reconstruction (21) 🔗 expand the composite quotient over the five covers of \([2]\):
five cover terms at finite scale, against two partition terms in the smooth formula. For \(\phi\) that is \(C^2\) near \(x\), the orbit \(c(t)\) is bounded as \(t \downarrow 0\) (35) 🔗. If \(f\) is \(C^3\) near \(y\), the iterated fundamental theorem (31) 🔗 bounds each cover term by \(O(t^{\wt(H)})\):
| cover | \(\wt(H)\) | bound | role |
|---|---|---|---|
| \(\set{\set{1}, \set{2}}\) | 2 | \(O(t^2)\) | principal partition term |
| \(\set{\set{1,2}}\) | 2 | \(O(t^2)\) | principal partition term |
| \(\set{\set{1}, \set{1,2}}\) | 3 | \(O(t^3)\) | first correction |
| \(\set{\set{2}, \set{1,2}}\) | 3 | \(O(t^3)\) | first correction |
| \(\set{\set{1}, \set{2}, \set{1,2}}\) | 4 | \(O(t^4)\) | higher error |
After division by \(t^2\), the two partitions are exactly the covers of weight two (33) 🔗, and every other cover has positive excess weight.
The boundary. Now let \(\phi\) be \(C^2\) near \(x\). Boundary extension (35) 🔗 closes the orbit at \(t = 0\):
so \(c(t) \to c^0 := Q^0_2(\phi;\, x;\, \Aff(v_1, v_2))\), the cube with the first derivatives as legs and the second derivative as defect. For \(f\) that is \(C^2\) near \(y\), the three excess covers are \(O(t^3)\): part 1 of (32) 🔗 gives this bound for the two-block covers; for the three-block cover, part 2 gives it by choosing the leg \(t^2 c_{12}(t)\) and one of the legs \(t c_i(t)\). Thus all three terms vanish after division by \(t^2\), and the expansion collapses onto the partitions. This boundary conclusion requires only \(C^2\) regularity of \(f\), although the table's \(O(t^4)\) bound requires \(C^3\):
The symbol. The two partition terms assemble in the symmetric algebra: with \(\sigma_2(c^0) = c^0_1\, c^0_2 + c^0_{12}\) (49) 🔗,
(46) 🔗: the intertwining (54) 🔗 at order two. The limit identity above is the partition Faà di Bruno formula (60) 🔗,
The lower-degree correction. The Möbius defect begins as the failure of the image parallelogram to close and ends as \(D(\phi;\, x;\, v_1, v_2)\), the degree-one summand of \(D_+(\phi;\, x;\, v_1 v_2)\); paired with \(D(f;\, y)\), it is the lower-degree term of the second-order chain rule. The Laplace exhibit shows how the inverse pushforward turns exactly these lower-degree terms into the Christoffel correction of a coordinate formula Finite stencils and the Laplace--Beltrami boundary 🔗.
Inverse pushforward
Pushing point operators through coordinate changes, as in the Laplace exhibit below, calls for the inverse of a pushforward. Individually, the formulas for inverse derivatives form a recursive hierarchy. The operator \(D_+(\phi;x)\) packages them into one triangular linear map. After its degree-preserving part is split off, inversion is a finite Neumann sum whose coefficients depend on the inverse first derivative and the higher derivatives of \(\phi\). For a linear map \(B:E\to F\), \(\SYM^r(B): \SYM^r(E) \to \SYM^r(F)\) denotes the induced map on symmetric powers, \(v_1 \cdots v_r \mapsto (B v_1) \cdots (B v_r)\), and \(\SYM(B) := \Vsum_r \SYM^r(B)\).
(109) Proposition (Inverse pushforward). Let \(\phi\) be \(C^k\) near \(x\) with invertible differential \(D(\phi;\, x): E \to F\). Split the pushforward by partition type,
where \(S\) collects the singleton partitions and \(N\) the partitions containing a block of size at least two. Then \(S^{-1} N\) is nilpotent, \(D_+(\phi;\, x)\) is invertible on \(ST_{\leq k}(X, x)\), and
By the Banach inverse function theorem, \(\phi\) has a \(C^k\) local inverse \(\psi\) near \(y=\phi(x)\), and \(D_+(\psi;\, y) = D_+(\phi;\, x)^{-1}\).
Proof. A partition of \([r]\) with a block of size at least two has at most \(r - 1\) blocks, so its term in (46) 🔗 lands in degree \(\leq r - 1\): \(N\) strictly lowers the degree and vanishes in degrees \(\leq 1\), while \(S\) is degreewise invertible with the stated inverse. Hence \(S^{-1} N\) strictly lowers the degree, \((S^{-1} N)^k = 0\) on \(ST_{\leq k}(X, x)\), and the finite geometric sum inverts \(1 + S^{-1} N\); then \(D_+(\phi;\, x) = S\, (1 + S^{-1} N)\) is invertible with the stated sum. No smallness enters. For the last claim, apply functoriality (58) 🔗 to \(\psi \circ \phi = \id\) and \(\phi \circ \psi = \id\).
In degree two, the degree-one component of the inverse formula gives \(D(\phi^{-1};\, y;\, w_1, w_2) = -A\, D(\phi;\, x;\, A w_1, A w_2)\), where \(A := D(\phi;\, x)^{-1}\).
(110) Remark (Inverse differencing). For comparison, let \(k \geq 1\). The cubical pushforward \(\Delta_+(\phi;\, x): CT_k(X, x) \to CT_k(Y, y)\) is bijective if and only if \(\phi\) is bijective. In that case
This follows directly from the conjugation in (11) 🔗. If \(d = \Delta_+(\phi;\, x)\, c\), the source vertices are recovered by \(\zeta_x(c)_T = \phi^{-1}(\zeta_y(d)_T)\).
(111) Remark. Expanded, the Neumann sum is the tree expansion of the classical inverse Faà di Bruno formula; in the Hopf-algebraic setting this unipotent inversion is performed by the antipode [FM2014].
Through the deformation. The Neumann-sum derivation uses only the invertibility of the differential and does not require constructing the local inverse \(\psi\) supplied by the Banach inverse function theorem. Exact functoriality (29) 🔗 applied to \(\psi\circ\phi=\id\) and \(\phi\circ\psi=\id\) gives, for \(t\in(0,1]\) and wherever the operators are defined,
zero-fiber functoriality (38) 🔗 gives the boundary identity
and under the symbol this is \(D_+(\psi;\, y) = D_+(\phi;\, x)^{-1}\). Thus finite-scale inversion, zero-fiber inversion, and the Neumann formula give the same operator. The Neumann formula can therefore be evaluated without constructing or invoking the local inverse map.
Finite stencils and the Laplace--Beltrami boundary
The finite difference stencil and the coordinate expression of the Laplace--Beltrami operator are the two ends of one orbit of the deformation: the stencil transports exactly through a chart at every mesh width, and the differential operator is its boundary value under the symbol.
The finite object. The mesh-\(t\) Laplace stencil at \(y \in \IR^n\),
is the pairing of \(f\) against \(n\) affine cubes of order two; for \(n = 2\) it is the classical five-point stencil. Its differential counterpart is the Laplace probe \(L := \sum_a e_a e_a \in ST_2(\IR^n, y)\), representing \(\sum_a \del_a^2\).
Transport at finite scale. Let \(\phi: U \to \IR^n\) be a chart near \(x\): a \(C^2\) diffeomorphism onto an open set \(V \subseteq \IR^n\), with inverse \(\psi := \phi^{-1}\) and \(y := \phi(x)\). Transport the stencil cubes through the deformed pushforward of \(\psi\),
The legs of \(\lambda_t\, c^a(t)\) are the chart coordinates of the stencil points, \(\psi(y \pm t e_a) - x\), and its Möbius defect is the negative of their sum, since the geometric top vertex returns to \(\psi(y) = x\). The transported stencil is exact for every \(t\) with \(y \pm t e_a \in V\): by the discrete adjunction (18) 🔗 and exact functoriality (13) 🔗 applied to \(\phi \circ \psi = \id\),
The boundary. Boundary extension (35) 🔗 applies to \(\psi\): the transported cubes converge, \(c^a(t) \to c^{a,0} := Q^0_2(\psi;\, y;\, \Aff(e_a, -e_a))\), with legs \(\pm A e_a\) for \(A := D(\psi;\, y) = D(\phi;\, x)^{-1}\) and defect \(-D(\psi;\, y;\, e_a, e_a)\). The normalized stencil is a quotient jet along the moving cubes, \(t^{-2} L_t(f) = -\sum_a qj_2(\phi^* f;\, x;\, t;\, c^a(t))\), and for \(f\) that is \(C^2\) near \(y\) the uniform convergence on bounded cube sets (35) 🔗, with the zero jet identified in (90) 🔗, gives the boundary value
By the symbol intertwining (54) 🔗, with \(\sigma_2(\Aff(e_a, -e_a)) = -\, e_a e_a\),
the last equality by (109) 🔗: the exact curved stencil collapses onto the pairing with the inverse-pushforward image of the Laplace probe.
The chart expression. The probe \(\hat L = D_+(\phi;\, x)^{-1} L\) can be computed directly from the derivatives of \(\phi\) whenever the differential is invertible, without constructing the local inverse. Write \(D := D(\phi;\, x)\) and \(A := D^{-1}\); indices \(i, j, l\) refer to the chart basis and \(a\) to the Cartesian basis. With \(D_+(\phi;\, x) = S + N\) as in (109) 🔗 and \(S^{-1} = \SYM(A)\), the Neumann sum stops after one correction on the degree-two probe:
For the quadratic term, \(\SYM(A)\) is the algebra map induced by \(A\), so it takes the Cartesian sum of squares to
the coefficients forming the inverse metric. For the linear correction, \(N\) retains the one-block partition on a quadratic monomial, \(N(e_i e_j) = D(\phi;\, x;\, e_i, e_j)\); with \(\Gamma^l_{ij}\) defined by the first identity,
(112) Proposition (Laplace pushforward). Let \(\phi: U \to \IR^n\) be \(C^2\) near \(x\) with invertible differential \(D(\phi;\, x)\). With \(g^{ij}\) and \(\Gamma^l_{ij}\) defined above, let \(F\) be \(C^2\) near \(y = \phi(x)\) and \(G := F \circ \phi\). Then
Proof. The two displayed terms give \(\hat L = \sum_{i,j} g^{ij} e_i e_j - \sum_{i,j,l} g^{ij} \Gamma^l_{ij} e_l\) with \(D_+(\phi;\, x)\, \hat L = L\). By the smooth adjunction (59) 🔗, \(D(G;\, x;\, \hat L) = D(F;\, y;\, D_+(\phi;\, x)\, \hat L) = D(F;\, y;\, L)\), which is the display written out in coordinates.
(113) Remark (Classical identification). With the notation above, \((g^{ij}) = A A^{\mathsf T} = (D^{\mathsf T} D)^{-1}\): the inverse of the pullback metric. The coefficients \(\Gamma^l_{ij}\) are its Christoffel symbols, and the proposition is the coordinate expression of the Laplace--Beltrami operator [KobayashiNomizu1963].
(114) Corollary (Polar coordinates). Let \(I \subset \IR\) be an open interval of length less than \(2\pi\) and set \(U := (0, \infty) \times I\). For \(\phi(r, \theta) = (r \cos\theta,\, r \sin\theta)\) and \(f\) that is \(C^2\) near \(y\),
Proof. Direct differentiation gives
The two terms required by the general calculation are
The mixed entry \(D(\phi;\, x;\, e_r, e_\theta) = (-\sin\theta, \cos\theta)\) is nonzero, but \(g^{r\theta} = 0\), so it does not enter the contraction. The remaining diagonal term \(D(\phi;\, x;\, e_r, e_r)\) vanishes. Hence \(g^{rr} = 1\), \(g^{\theta\theta} = r^{-2}\), and \(\Gamma^r_{\theta\theta} = -r\). The inverse-pushforward formula gives the stated probe.
The exact polar stencil. For the polar chart, the transport identity above is an exact polar five-point stencil at every mesh width, and the collapse identifies its parts: the rescaled Möbius defects converge to \(-D(\psi;\, y;\, e_a, e_a)\) and supply, through the sign of the stencil, the Christoffel term \(\tfrac1r\, e_r\) of the polar Laplacian. In numerical practice the vertex equations \(\phi(q) = y \pm t e_a\) are solved by Newton iteration, whose linearized first step is the term \(S^{-1}\) of the Neumann sum (109) 🔗.
Polynomials as algebraic sections
For polynomial observables the deformation needs no limits: the section \(t \mapsto qj_k(p;\, x;\, t;\, c)\) of the family is polynomial in \(t\), and the boundary is reached algebraically. A map \(p: X \to \IR\) is a continuous polynomial of degree \(\leq d\) if, for some (equivalently, every) point \(x \in X\), \(p(x + w) = \sum_{s=0}^{d} A_s(w, \dots, w)\) with continuous symmetric \(s\)-linear forms \(A_s: E^s \to \IR\); such a \(p\) is smooth, and its Taylor expansion at every point is exact and terminates at degree \(d\).
If \(d=0\), then \(p\) is constant and every positive-order difference vanishes, so \(qj_k(p;\,x;\,t;\,c)=0\) for \(k\geq1\). Thus the nonconstant case below may be stated with \(d\geq1\).
The coefficients of the expansion below are the higher symbols \(\sigma_{k,m}\) of (55) 🔗.
(115) Proposition (Polynomial sections). Let \(p: X \to \IR\) be a continuous polynomial of degree \(\leq d\) for an integer \(d\geq1\), and let \(k \geq 1\).
-
For \(c \in CT_k(X, x)\) and \(t \in (0,1]\),
\[ qj_k(p;\, x;\, t;\, c) = \sum_{m=0}^{k(d-1)} t^m\, D(p;\, x;\, \sigma_{k,m}(c)). \]The quotient jet is polynomial in the deformation parameter and extends algebraically through \(t = 0\), with \(qj^0_k(p;\, x;\, c) = D(p;\, x;\, \sigma_k(c))\).
-
If \(d \leq k\), the section is constant on affine cubes: for all \(t \in (0,1]\),
\[ qj_k(p;\, x;\, t;\, \Aff(v_1, \dots, v_k)) = \Delta(p;\, x;\, v_1, \dots, v_k) = D(p;\, x;\, v_1, \dots, v_k). \]
Proof. 1) Every vertex of the rescaled cube is polynomial in \(t\), \(\zeta_x(\lambda_t\, c)_T = x + \sum_{A \in \KP_+(T)} t^{|A|} c_A\), so \(\Delta(p;\, x;\, \lambda_t\, c)\), an alternating sum of values of \(p\) at the vertices, is a polynomial in \(t\) of degree at most \(k d\). Curved collapse (57) 🔗 with \(M := k(d - 1)\) gives
and a polynomial in \(t\) of degree at most \(k d\) that is \(o(t^{k d})\) vanishes identically: the identity is exact for all \(t\). Dividing by \(t^k\) gives the display; the value at \(t = 0\) is the \(m = 0\) term, and \(\sigma_{k,0} = \sigma_k\).
2) On affine cubes the higher symbols are homogeneous of symmetric degree \(k + m\) (55) 🔗, and \(D(p;\, x;\, \cdot)\) vanishes above degree \(d \leq k\), so only the \(m = 0\) term \(\sigma_{k,0}(\Aff(v_\bullet)) = v_1 \cdots v_k\) remains: by 1) the section is the constant \(D(p;\, x;\, v_1 \cdots v_k)\). Its value at \(t = 1\) is the discrete pairing \(\Delta(p;\, x;\, v_1, \dots, v_k)\).
Newton to Taylor
Newton's formula writes vertex values as zeta transforms of face differences, exactly, for every observable at every scale (10) 🔗. Under the deformation each face difference has its affine collapse expansion (56) 🔗, and Möbius inversion reassembles the expansions into the multivariate Taylor formula: Taylor's theorem is the boundary value of Newton's.
(116) Proposition (Newton to Taylor). Let \(f: X \to F\) be \(C^n\) near \(x\) and \(v_1, \dots, v_k \in E\) with \(k \geq 1\). As \(t \to 0\),
Proof. Newton's formula (10) 🔗 at the rescaled legs reads
Expand each face by affine collapse (56) 🔗: the face \(T\) contributes exactly the Taylor monomials of full support on \(T\), so a pair of a face and a full-support multi-index on it is one multi-index \(\nu \in \IN_0^k\) with \(\set{i : \nu_i > 0} = T\), and the \(2^k\) remainders collect into \(o(t^n)\).
Newton's formula is exact at every \(t > 0\); Taylor's formula is its termwise boundary.
Order conditions and Richardson extrapolation
A finite-difference scheme approximates a differential by weighted observable values at scale \(t\). The plain difference quotient converges at first order. Numerical differentiation raises this order by choosing weights that cancel successive error coefficients. Here a scheme is a weighted family of affine probes. By affine collapse (56) 🔗, its deviation from the target differential is a power series in \(t\) whose \(t^m\)-coefficient pairs the derivatives of \(f\) against a fixed probe built from \(\sigma_{k,m}\). The order conditions are therefore linear equations on these coefficient probes. Combining the same scheme at different scales gives Richardson's deferred approach to the limit [Richardson1911] [RichardsonGaunt1927].
Fix a grade \(k\), weights \(a_\alpha \in \IR\), and legs \(v_\alpha = (v_{\alpha,1}, \dots, v_{\alpha,k})\) in \(E\), over a finite index set, and call
the associated scheme. For \(f\) that is \(C^{k+M}\) near \(x\), (56) 🔗 gives
with coefficient probes \(\xi_m \in ST_{k+m}(X, x)\).
(117) Proposition (Order conditions). Let \(\xi \in ST_k(X, x)\) and \(p \geq 1\). The following are equivalent:
-
\(S_t(f) = D(f;\, x;\, \xi) + O(t^p)\) as \(t \downarrow 0\), for every \(f\) that is \(C^{k+p}\) near \(x\);
-
\(\xi_0 = \xi\) and \(\xi_m = 0\) for \(1 \leq m \leq p - 1\).
Proof. If 2 holds, the expansion with \(M = p\) gives \(S_t(f) = D(f;\, x;\, \xi) + t^p D(f;\, x;\, \xi_p) + o(t^p)\). Conversely, evaluate on continuous polynomials \(q\): by (115) 🔗, \(S_t(q) = \sum_m t^m D(q;\, x;\, \xi_m)\) is a polynomial in \(t\), exactly, and a polynomial in \(t\) that is \(O(t^p)\) has no terms below \(t^p\); hence \(D(q;\, x;\, \xi_0 - \xi) = 0\) and \(D(q;\, x;\, \xi_m) = 0\) for \(m < p\), for every \(q\). By jet realization (84) 🔗 and the jet pairing (82) 🔗, every form \(\alpha \in ST^{\leq N}(X, x)\), \(N\) large, pairs to zero with \(\eta \in \set{\xi_0 - \xi,\, \xi_1, \dots, \xi_{p-1}}\). To show that such an \(\eta\) vanishes, choose a finite-dimensional subspace \(E_0\subseteq E\) containing all vector factors occurring in \(\eta\). By Hahn--Banach, a dual basis of \(E_0\) extends to continuous functionals on \(E\). Hence every symmetric form on \(E_0\) is the restriction of a continuous symmetric form on \(E\). The permanent pairing is nondegenerate on the finite-dimensional space \(\SYM_{\leq N}(E_0)\), so \(\eta=0\).
The classical moment conditions for difference schemes are thus the vanishing of the weighted higher-symbol coefficient probes. Parity eliminates half of the conditions: the probe \(\sigma_{k,m}(\Aff(v_\bullet))\) is homogeneous of degree \(k + m\), so a central scheme, one whose weighted family of legs is invariant under \(v \mapsto -v\), has \(\xi_m = (-1)^{k+m}\, \xi_m\): every coefficient with \(k + m\) odd vanishes identically, and the order conditions skip every other step.
For the central second difference, \(k = 2\), one cube with weight \(-1\) and legs \((e, -e)\),
the coefficient probes are \(\xi_0 = -\, e \cdot (-e) = e^2\), the odd \(\xi_m = 0\) by centrality, and
the three terms being \(\nu = (3,1), (1,3), (2,2)\): for \(f \in C^6\),
the classical second-order consistency with the \(\tfrac{1}{12}\)-law for the leading error.
Extrapolation. If the scheme has order \(p\) with first surviving coefficient \(\xi_p\), combining the scales \(t\) and \(t/2\) eliminates it: for \(f\) that is \(C^{k+p+1}\) near \(x\),
since the \(t^p\)-coefficients cancel while the constant term has coefficient \(2^p - 1\). Richardson extrapolation is coefficient elimination in the deformation parameter, and the eliminated coefficients are higher symbols. For the central second difference, parity removes the \(t^3\)-term as well: for \(f \in C^6\),
the classical fourth-order compact scheme.
Structural dictionary: \(D_+\) as \(L_\infty\)-morphism
Throughout this exhibit, \(\phi\) is smooth near \(x\) and all orders are taken: \(D_+(\phi; x): ST_*(X, x) \to ST_*(Y, y)\). The pushforward is always a coalgebra morphism (47) 🔗; it is an algebra morphism if and only if the Taylor expansion of \(\phi\) at \(x\) is affine, i.e. the higher derivatives at \(x\) vanish (118) 🔗. In general, the \(D^r(\phi; x)\) assemble into an \(L_\infty\)-morphism: smooth maps between affine spaces form a geometric family of \(L_\infty\)-morphisms in which the Taylor coefficients are the classical higher derivatives. The coderivations vanish, so the intertwining condition is automatic, and composition is the Faà di Bruno formula. The references used below describe the general formalism; we have not found this affine example stated explicitly in them.
(118) Proposition (Algebra behavior of the pushforward). Let \(\phi\) be smooth near \(x\). \(D_+(\phi; x)\) is an algebra morphism for the symmetric product if and only if \(D^r(\phi; x) = 0\) for all \(r \geq 2\); in that case
Proof. If \(D^r(\phi; x) = 0\) for \(r \geq 2\), only the partition into singletons contributes in (46) 🔗, and \(D_+(\phi;\, x;\, v_1 \cdots v_r) = \prod_i D(\phi;\, x;\, v_i)\) is multiplicative on monomials. Conversely, suppose \(D_+(\phi; x)\) is multiplicative and induct on \(r\): comparing
with \(D_+(\phi;\, x;\, v_1)\, D_+(\phi;\, x;\, v_2)\) forces \(D^2(\phi; x) = 0\) by comparing degrees; for the step, all partitions with a block of size \(2, \dots, r-1\) drop out by the induction hypothesis, and comparing with \(D_+(\phi;\, x;\, v_1) \cdots D_+(\phi;\, x;\, v_r)\) leaves \(D(\phi;\, x;\, v_1 \cdots v_r) = 0\).
For a general map, the higher derivatives \(D^r(\phi; x)\), \(r \geq 2\), are the correction terms by which \(D_+\) fails to be \(\SYM(D^1)\), the pushforward of the affine approximation of \(\phi\) at \(x\). We use the \(L_\infty\) conventions of [KDQ03], §4.1--4.3, and [LV2012], §10.1--10.2: an \(L_\infty\)-morphism is a morphism of Chevalley--Eilenberg coalgebras intertwining the coderivations ([CE48] classically; [LV2012], Proposition 10.1.20 and §10.2.2), determined by its Taylor coefficients ([KDQ03], §4.1--4.2) through the cofree property (123) 🔗 ([LV2012], Proposition 10.2.3). The deformation functor of an \(L_\infty\)-algebra \(\mathfrak g\) is defined over finite-dimensional nilpotent commutative algebras \(A\), non-unital: \(A\) plays the role of the maximal ideal, \(A = \mathfrak m_{A'}\) for the local Artinian algebra \(A' := \IR \oplus A\). It takes Maurer--Cartan solutions modulo gauge equivalence, defined homotopically for general \(\mathfrak g\) ([KDQ03], §4.5.2); for a differential graded Lie algebra, the case used below, gauge equivalence is the exponential action of the degree-zero part,
([KDQ03], §3.2). The references work without counit; the counital form used here has the same morphisms.
Degrees follow the shift convention: \(E[-1]\) is \(E\) placed in degree \(1\), so \(\mathfrak g^1 = E\) is the only nonzero component, and the shifted space \(\mathfrak g[1]\) entering \(CE_\bullet(\mathfrak g) = \SYM(\mathfrak g[1])\) is an ungraded copy of \(E\). In the standard ledger, \(\mathfrak g^0\) acts by gauge transformations and \(\mathfrak g^2\) contains obstruction classes; both vanish for \(E[-1]\), so the deformation problem below has trivial gauge action and no obstructions.
(119) Proposition (Affine spaces in the \(L_\infty\) dictionary). Let \(X\) and \(Y\) be affine spaces over \(E\) and \(F\), with base points \(x\) and \(y\), and let \(\mathfrak g := E[-1]\) and \(\mathfrak h := F[-1]\) be the abelian \(L_\infty\)-algebras: all brackets vanish.
-
The Chevalley--Eilenberg coalgebra is the symmetric bialgebra (120) 🔗,
\[ CE_\bullet(\mathfrak g) = \SYM(E) = ST_*(X, x), \qquad Q = 0. \] -
The exponential \(\xi \mapsto \exp(\xi)\) identifies \(\operatorname{Def}_{\mathfrak g}(A) = E \tensor A\) with the group-like elements of \(CE_\bullet(\mathfrak g) \tensor A'\) congruent to \(1\) modulo \(A\): \(\mathfrak g\) controls the deformations of the point \(x\) inside \(X\), the \(A\)-points \(x + \xi\) of its formal neighborhood.
Let \(\phi\) be smooth near \(x\) with \(\phi(x) = y\). Then:
-
Both coderivations vanish, so the intertwining conditions are empty, and \(D_+(\phi;\, x) = \exp_\star(D(\phi;\, x))\), the convolution exponential (122) 🔗 of the coalgebra lift (48) 🔗, is the \(L_\infty\)-morphism \(\mathfrak g \rightsquigarrow \mathfrak h\) whose Taylor coefficients are the higher derivatives,
\[ \SYM^r(E) \lra F, \qquad v_1 \cdots v_r \longmapsto D(\phi;\, x;\, v_1, \dots, v_r). \] -
The induced map on deformations is the Taylor expansion of \(\phi\) at \(x\),
\[ \operatorname{Def}_{\mathfrak g}(A) \lra \operatorname{Def}_{\mathfrak h}(A), \qquad \xi \longmapsto \sum_{r \geq 1} \frac{1}{r!}\, D^r(\phi;\, x)(\xi, \dots, \xi). \] -
The assignment is functorial: for \(\psi\) smooth near \(y\),
\[ D_+(\psi \circ \phi;\, x) = D_+(\psi;\, y) \circ D_+(\phi;\, x), \qquad D_+(\id_X;\, x) = \id, \]and the induced composition rule for Taylor coefficients is the partition Faà di Bruno formula.
Proof. 1) With all operations zero, the Chevalley--Eilenberg coalgebra is the cofree conilpotent cocommutative coalgebra on \(\mathfrak g[1] = E\) with zero coderivation. This is the symmetric bialgebra of the appendix.
2) The Maurer--Cartan equation is vacuous and the gauge group \(\exp(\mathfrak g^0 \tensor A)\) is trivial, so \(\operatorname{Def}_{\mathfrak g}(A) = \mathfrak g^1 \tensor A = E \tensor A\). The exponential lives in the unitalization: for \(\xi \in E \tensor A\), \(\exp(\xi) = \sum_r \xi^r / r! \in CE_\bullet(\mathfrak g) \tensor A'\) is a finite sum with constant term \(1\), as \(A\) is nilpotent, and is group-like: \(\Delta^{\times}(\xi) = \xi \tensor 1 + 1 \tensor \xi\) with commuting summands, so \(\Delta^{\times} \exp(\xi) = \exp(\xi) \tensor \exp(\xi)\) by the binomial formula (120) 🔗. Conversely, a group-like element \(g \equiv 1 \pmod{A}\) has the finite-sum logarithm \(\log g\), and \(\Delta^{\times} \log g = \log(g \tensor g) = \log g \tensor 1 + 1 \tensor \log g\) shows that \(\log g\) is primitive, an element of \(E \tensor A\); so \(\exp\) is a bijection onto these group-likes.
3) Both coderivations vanish, so the intertwining conditions are empty and an \(L_\infty\)-morphism is exactly a counital coalgebra morphism \(\SYM(E) \to \SYM(F)\). The coalgebra lift (48) 🔗 is the unique one with degree-one component \(D(\phi;\, x)\), and its components against the cofree structure (123) 🔗 are the higher derivatives.
4) A counital coalgebra morphism preserves group-like elements, so \(D_+(\phi;\, x)\, \exp(\xi) = \exp(\eta)\) for a unique \(\eta \in F \tensor A\), by 2) applied to \(\mathfrak h\). Compare degree-one components: \(\exp(\eta)\) has component \(\eta\), and \(D_+(\phi;\, x;\, \xi^r)\) has the one-block component \(D(\phi;\, x;\, \xi^r) = D^r(\phi;\, x)(\xi, \dots, \xi)\) (46) 🔗; hence \(\eta = \sum_{r \geq 1} \tfrac{1}{r!}\, D^r(\phi;\, x)(\xi, \dots, \xi)\).
5) is smooth functoriality (58) 🔗 and the partition Faà di Bruno formula (60) 🔗.
Kontsevich's gloss for the coalgebra morphism of a map of formal pointed manifolds, "the pushforward on distributions supported at zero" ([KDQ03], §4.1), is realized analytically by (75) 🔗.
Appendix: The symmetric bialgebra
This appendix collects the coalgebra background behind the coalgebra lift (48) 🔗. Throughout, \(V\) is a real vector space. The permanent pairing lives in (61) 🔗, the product-coproduct adjunction in (63) 🔗.
(120) Definition (Symmetric bialgebra). Let \(\SYM(V) = \Vsum_{r \geq 0} \SYM^r(V)\) be the symmetric algebra with its degree filtration \(F_k := \Vsum_{r \leq k} \SYM^r(V)\).
- The coproduct \(\Delta^{\times}: \SYM(V) \to \SYM(V) \tensor \SYM(V)\) is the unique algebra morphism with \(\Delta^{\times}(v) = v \tensor 1 + 1 \tensor v\) for \(v \in V\); the counit \(\varepsilon\) is the degree-zero projection. On monomials, \(\Delta^{\times}(v_1 \cdots v_r) = \sum_{I \sqcup J = [r]} v^I \tensor v^J\) with \(v^\emptyset := 1\), matching (44) 🔗.
- The reduced coproduct of an element \(a\) of positive degree is \(\uDelta(a) := \Delta^{\times}(a) - a \tensor 1 - 1 \tensor a\).
- A counital coalgebra \(C\) with coaugmentation \(1_C\) is conilpotent if for every \(c \in \ker \varepsilon_C\) there is an \(m\) with \(\uDelta^{(m)}(c) = 0\).
(121) Lemma (Conilpotence). The coproduct preserves the degree filtration, so each \(\SYM_{\leq k}(V)\) is a subcoalgebra; the reduced coproduct is strictly degree-decreasing, \(\uDelta(F_k) \subseteq F_{k-1} \tensor F_{k-1}\), so \(\SYM(V)\) and every \(\SYM_{\leq k}(V)\) are conilpotent.
Proof. In the monomial formula both tensor factors have degree \(\leq r\), and in the reduced part both are proper sub-monomials, of degree \(\leq r - 1\).
(122) Definition (Convolution). For a counital cocommutative coalgebra \(C\) and a commutative unital algebra \(A\), the convolution product on \(\Hom(C, A)\) is
associative and commutative with unit \(1_\star := \eta_A \circ \varepsilon_C\).
(123) Proposition (Cofree property; unique coalgebra lift). Let \(C\) be a coaugmented, counital, conilpotent cocommutative coalgebra and \(f: C \to V\) linear with \(f(1_C) = 0\), viewed as a map into \(\SYM(V)\) of pure degree one. Then the convolution exponential
is a finite sum on every element of \(C\), and it is the unique counital coalgebra morphism \(C \to \SYM(V)\) with degree-one component \(f\), that is, with \(\pi_1 \circ \exp_\star(f) = f\).
Proof. Finiteness. Decompose each term of the iterated coproduct \(\Delta_C^{(m-1)}(c)\) by which tensor factors are the coaugmentation \(1_C\): since \(f(1_C) = 0\), only the terms with all \(m\) factors in \(\ker \varepsilon_C\) remain after applying \(f^{\tensor m}\), and they constitute exactly the iterated reduced coproduct \(\uDelta^{(m-1)}(c)\). For \(c \in \ker \varepsilon_C\), conilpotence makes \(\uDelta^{(m-1)}(c)\) vanish for \(m\) large; hence \(f^{\star m}(c) = 0\), and \(f^{\star m}(1_C) = f(1_C)^m = 0\) for \(m \geq 1\).
Morphism. The value \(f(c) \in V\) is primitive, so \(\Delta^{\times} \circ f = (L + R)(f)\) with \(L(f)(c) := f(c) \tensor 1\) and \(R(f)(c) := 1 \tensor f(c)\), and \(L(f)\), \(R(f)\) commute under the convolution of \(\Hom(C, \SYM(V) \tensor \SYM(V))\). Exponentiating, \(\Delta^{\times} \circ \exp_\star(f) = \exp_\star(L(f)) \star \exp_\star(R(f)) = (\exp_\star(f) \tensor \exp_\star(f)) \circ \Delta_C\); counitality holds since all positive-degree terms have \(\varepsilon = 0\).
Uniqueness. Let \(g, h\) be counital coalgebra morphisms with the same degree-one component. Counitality gives \(\pi_0 g = \varepsilon_C = \pi_0 h\). For \(r \geq 2\), taking the components of \(\Delta^{\times} \circ g = (g \tensor g) \circ \Delta_C\) in \(\SYM^i \tensor \SYM^j\) with \(i + j = r\), \(i, j \geq 1\),
depends only on the components of degree \(< r\), and \(\uDelta\) is injective on \(\SYM^r(V)\) for \(r \geq 2\): on a monomial, the multiplication returns \(\mu\, \Delta^{\times}(v_1 \cdots v_r) = \sum_{I \sqcup J = [r]} v^I\, v^J = 2^r\, v_1 \cdots v_r\), and removing the two extreme terms \(I = [r]\) and \(J = [r]\) leaves \(\mu \circ \uDelta = (2^r - 2)\, \id\), invertible for \(r \geq 2\). Induct on \(r\).
(124) Corollary (Coalgebra lift, truncated form). Let \(f: ST_{\leq k}(X, x) \to F\) be linear with \(f(1) = 0\). There is a unique counital coalgebra morphism \(f^+: ST_{\leq k}(X, x) \to \SYM(F)\) with degree-one component \(f\), \(\pi_1 \circ f^+ = f\); it preserves the degree filtration, hence corestricts to \(\SYM_{\leq k}(F)\), and on monomials it is the partition sum
If \(Y\) is an affine space modeled on \(F\) and \(y \in Y\), the identification \(\SYM_{\leq k}(F) = ST_{\leq k}(Y, y)\) reads \(f^+\) as a map of symmetric tangent spaces; in particular the symmetric pushforward (46) 🔗 is the unique coalgebra morphism with degree-one component the differential \(D(\phi; x)\).
Proof. \(ST_{\leq k}(X, x) = \SYM_{\leq k}(T_x X)\) is a conilpotent subcoalgebra (121) 🔗, so (123) 🔗 applies with \(f^+ = \exp_\star(f)\). On a monomial, the iterated coproduct is the sum over ordered decompositions \([r] = I_1 \sqcup \dots \sqcup I_m\) into possibly empty blocks; \(f\) kills the empty blocks, and \(\frac{1}{m!}\) converts ordered partitions into unordered ones, giving the partition sum. Each term has degree \(|\pi| \leq r\), whence the filtration bound.
(125) Remark (Literature). Over an algebraically closed field of characteristic zero, the cofree cocommutative coalgebra over \(V\) decomposes as \(\bigoplus_{P \in V} \SYM_P(V)\), and the partition formula for the unique lift of a linear map is Theorem 2.22 of [Murfet2015]; Clift and Murfet use it in differential linear logic [CM2020]. The convolution-exponential proof above works in the truncated, single-basepoint coalgebra over \(\IR\).
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