Regular transport reconstructs a connection

3. Transport regularity and resolution

3.1 Smoothness at constant paths forces the first jet

Use the prepared trivialization to represent transports as matrices. Around a constant path, $C^1([0,1],U)$ is an open part of a Banach space. The differentiability in the next statement is in its $C^1$ norm and includes constant paths. Smoothness restricted to immersed paths is weaker.

Theorem 3.1 (Regular transport reconstruction). Suppose $T$ is multiplicative on successive subarcs, the identity on constants and invariant under orientation-preserving reparameterization. Suppose $T$ is continuously Fréchet differentiable on open $C^1$ neighborhoods of the constants, with $x\mapsto DT_{\mathrm{const}(x)}$ smooth as a map into bounded linear operators. Assume the corresponding joint higher differentiability for the parameterized variations and derivatives below. Then a smooth matrix-valued one-form $A$ generates the transport:

$$\frac{d}{ds}T_{\gamma|_{[0,s]}} =-A_{\gamma(s)}(\dot\gamma(s))T_{\gamma|_{[0,s]}}, \qquad T_{\gamma|_{[0,0]}}=I. \tag{7}$$

The first jet depends only on the current point and velocity and is linear in velocity. Conversely a smooth $A$ defines a smooth transport family of this kind. These constructions are inverse on the regular class.

Proof. At the constant path at $x$ set

$$A_x(v)=-DT_{\mathrm{const}(x)}[t\mapsto tv]. \tag{8}$$

The Fréchet derivative is linear on the tangent space, so this is linear in $v$. For a $C^2$ curve its actual short subarc satisfies

$$\gamma(s+\epsilon t)=\gamma(s)+\epsilon t\dot\gamma(s)+O(\epsilon^2) \qquad\text{in } C^1([0,1]). \tag{9}$$

Differentiability gives $T_{\gamma|_{[s,s+\epsilon]}}=I-\epsilon A_{\gamma(s)}(\dot\gamma(s))+o(\epsilon)$. Continuity of the derivative makes the remainder uniform on compact short-path families. Composition gives continuity of $s\mapsto T_{\gamma|_{[0,s]}}$ and the right derivative (7), whose right-hand side is continuous. The integral equation follows and gives the ordinary derivative. Compose across pieces for piecewise smooth paths.

Conversely solve the matrix integral equation of (7). Its Picard series converges uniformly on compact path images, with its $j$th iterated integral bounded by $M^j/j!$ for a suitable integrated generator bound. Uniqueness gives composition, reversal and reparameterization invariance. In a compact controlled path neighborhood, the Nemytskii map $\gamma\mapsto A(\gamma)\dot\gamma$ has the regularity of $A$. Differentiating the integral equation or its uniformly bounded series gives the required path-space smoothness. The generator is (8), and uniqueness gives the inverse identity for every family in the class.

This local sufficient condition realizes the established connection–transport correspondence Schreiber and Waldorf, 2009. It removes a separate first-jet postulate within the regular model. The smoothness remains raw-sector data.

Proposition 3.2 (Regularity obstruction). Composition, reversal and reparameterization invariance alone admit a nonlinear velocity law. In two coordinate directions take

$$F(v,w)=\frac{v^3}{v^2+w^2}\quad((v,w)\ne0),\qquad F(0,0)=0,\qquad T_\gamma=\exp\left[-\int F(\dot\gamma^1,\dot\gamma^2)\,ds\right]I. \tag{10}$$

The family has all three algebraic properties and is smooth on controlled immersed-path neighborhoods, but fails the required Fréchet differentiability at constants.

Proof. $F$ is odd and positively homogeneous of degree one, giving reversal and reparameterization invariance. Additivity of the integral gives composition. At a constant path, the straight variation $t\mapsto t(v,w)$ would have directional derivative $-F(v,w)I$. Since $F(1,1)=1/2$ and $F(1,0)+F(0,1)=1$, these values cannot come from a linear Fréchet derivative.

3.2 The variations and curvature also descend

For compactly supported $\delta A$, differentiating (7) gives

$$\delta T_\gamma(1)=-\int_0^1 T_\gamma(1,s) \,\delta A_\mu(\gamma(s))\dot\gamma^\mu(s)\,T_\gamma(s,0)\,ds. \tag{11}$$

Its infinitesimal readout is $\delta A$. Conversely, differentiate the reconstructed transport equation in a jointly smooth one-parameter family. The tangent satisfies the inhomogeneous equation with zero initial value, whose unique solution is (11). Every regular tangent transport variation comes from its generator variation. Commuting the parameter derivative with the short-path derivative uses the stipulated joint second differentiability.

Compose the four transports around a positively oriented small coordinate rectangle. Expansion of the two-parameter integral equations gives

$$T_{\square_{\mu\nu}}=I-F_{\mu\nu}\Delta x^\mu\Delta x^\nu +O\left((\Delta x^\mu)^2\Delta x^\nu+\Delta x^\mu(\Delta x^\nu)^2\right). \tag{12}$$

The ordered products generate the commutator. For constant matrices the loop order is $e^{A_\nu t}e^{A_\mu s}e^{-A_\nu t}e^{-A_\mu s}$, whose mixed coefficient is $-[A_\mu,A_\nu]$. This fixes the sign. Endpoint derivatives and small-loop limits recover the differential and nonlinear connection operations.

3.3 Transport error and curvature error need different control

Proposition 3.3 (Finite-resolution transport). With bounded second path derivatives on the relevant neighborhood and a $C^2$ path $\gamma$, the subarc defect obeys

$$\left\|T_{\gamma|_{[s,s+h]}}-I+hA(\gamma(s))\dot\gamma(s)\right\|\le Ch^2 . \tag{13}$$

The first two derivatives of $T$ and $\gamma$ control $C$. If the generator along an interval of length $L$ has norm at most $M$, the left-endpoint product on a partition of mesh $h_{\max}$ satisfies

$$\left\|T_\gamma-\overset{\leftarrow}{\prod}_j [I-h_jA(\gamma(s_j))\dot\gamma(s_j)]\right\| \le CLh_{\max}e^{ML}. \tag{14}$$

Proof. Apply Taylor expansion to the Fréchet map using (9). This gives (13). Each exact subarc transport and Euler factor has norm at most $e^{Mh_j}$. Telescope the ordered products. The factors surrounding each local defect contribute at most $e^{ML}$ and $\sum_j h_j^2\le Lh_{\max}$.

With measured short-transport error at most $\epsilon$ and forward first-jet remainder at most $Kh^2$, the estimated generator satisfies

$$\|\widetilde A_h-A\|\le \epsilon/h+Kh. \tag{15}$$

At an admissible optimizing scale the bound is $2\sqrt{K\epsilon}$. With a controlled third derivative, a centered difference gives $\epsilon/h+K_3 h^2$ and optimized order $\epsilon^{2/3}$. These bounds price differentiation. They do not assume exact finite localization or an available instrument.

Curvature also needs spatial derivative control. Let $E=\widetilde A-A$, with $\max_\mu\|A_\mu\|\le M$, $\max_\mu\|E_\mu\|\le\delta_0$ and $\max_{\mu,\nu}\|\partial_\mu E_\nu\|\le\delta_1$. Then

$$\|F_{\mu\nu}(\widetilde A)-F_{\mu\nu}(A)\| \le2\delta_1+4M\delta_0+2\delta_0^2. \tag{16}$$

Expand the derivative and the three commutator differences and use submultiplicativity. Component Ricci errors introduce at most the additional contraction factor $n$. This is a local residual estimate; the nonlinear Einstein initial-value problem has a different stability question.

The derivative hypothesis cannot be dropped. In a flat chart choose only $\Gamma^0{}_{11}=\sin(kx^0)/k$ nonzero. It is torsion free and its generator tends uniformly to zero. Transports approach identity on bounded-length path families, but $R_{11}(0)=1$ for every $k$. Along $x^0=x^1=s$, $0\le s\le1$, the transport error is at most $2/k^2$ because the generator is a fixed nilpotent matrix times $\sin(ks)/k$. Small uniform transport errors therefore do not certify small gravitational residuals.