Appendices C and D: reconstruction and scope

C. Analytic checks behind the reconstruction table

For the triangular matrix in §8, the diagonal entries solve $u'_{22}=-2tu_{22}$ and $u'_{11}=-tu_{11}$. Variation of constants gives

$$u_{12}(t)=-e^{-t^2/2}\int_0^t e^{-s^2/2}\,ds . \tag{71}$$

This yields (64) without a numerical differential-equation solver. The induced infinity norms satisfy $\|A\|\le2$ and $\|A'\|\le2$. Over length $h$ the fundamental solution has norm at most $e^{2h}$ and second derivative $(-A'+A^2)U$. Taylor's integral remainder bounds the Euler defect by $3h^2e^{2h}$. The other exact and Euler factors in the telescoping product contribute at most $e^{2(1-h)}$. Summing over $N$ equal intervals gives a global bound $3e^2/N$, already smaller than the declared $3e^{2.2}/N$ used in the saved comparison. No contractivity assumption is used.

For the oscillatory curvature control, take $A_1=(\sin(kx^0)/k)E_{01}$ and all other $A_\mu=0$. Since $E_{01}^2=0$, the exact path exponential is $I-E_{01}\int \sin(kx^0)\,dx^1/k$. The error from identity is $O(k^{-1})$ on bounded-length path families. Along $x^0=x^1=s$ the integral is $(1-\cos k)/k^2$, but $\partial_0(A_1)^0{}_1=\cos(kx^0)$ gives $R_{11}(0)=1$. This explicit limit distinguishes transport and curvature reconstruction, explaining the derivative norms in (16).

D. Scope of the recovery statements

There are three equivalences with different meanings.

1. Regular source transports and affine generators are off-shell inverse representations, including their allowed variations.
2. At fixed nondegenerate density, projective classes of generators and volume-restored transports are off-shell inverse representations.
3. The unrestricted affine action reduces to the metric Einstein–matter equations after its algebraic field equation eliminates nonprojective distortion. This is where the original torsion-free parent and the unrestricted quotient agree. Their off-shell connection dimensions, 40 and 60 respectively, exhibit the difference.

The results are local in the declared smooth sector, or apply to global histories with patching and boundary data obeying these contracts. A nontrivial tangent bundle requires compatible local frames and transport descent. The finite fan does not determine arbitrary global topology, singular extension or an all-sector Shadow equalizer. None of those enlargements is needed for the full local nonlinear Einstein–matter result.