Read the printed-source corrections

The supplied historical copy contains expressions that cannot all hold with their surrounding equations. This register preserves the literal forms and the separately derived consistent replacements.

A. Printed-source qualifications and correction register

This appendix concerns the supplied copy of [5], checked against its page images and the publisher-hosted PDF. Printed page numbers, not PDF offsets, are used. The correction decisions follow direct algebra or explicit witnesses. No assertion is made about authorship of a typographical error, priority of its detection, editorial history, or later editions. The source’s valid equations are retained, and the inconsistent branch is identified rather than silently replaced.

The rank statement on page 784

After discussing completely antisymmetric tensors, the copy prints “Symmetrische Tensoren” in the sentence denying existence above rank four. Taken literally for symmetric tensors, it is false: for a nonzero covector $a$, $a^{\otimes5}$ is a nonzero symmetric tensor. The valid statement in the surrounding discussion is that a completely antisymmetric tensor above rank four vanishes because every component repeats an index. Proposition 4.1 retains the arbitrary-rank symmetric tensor algebra and the correct exterior-algebra cutoff. This is a qualification of the printed word, not a modification of the transformation formulas Einstein (14)–Einstein (15a).

The trace-shift footnote on page 804

The footnote considers $B_{\mu\nu}+\lambda g_{\mu\nu}\operatorname{tr}_qB$ and states that setting it to zero returns $B=0$. In four dimensions its trace is $(1+4\lambda)\operatorname{tr}_qB=0$, so the reduction requires $\lambda\ne-1/4$. At the exceptional value, (7.12) gives a nonzero $B=g/3$ satisfying the trace-free equation. The reconstruction retains the selected vacuum law and does not use the footnote as an unrestricted uniqueness theorem.

The unnumbered variation sign on page 805

The line immediately before Einstein (48) prints a negative coefficient of $\Gamma_{\!\mathrm E}^\alpha{}_{\mu\nu}\delta(\partial_\alpha q^{\mu\nu})$, whereas numbered Einstein (48) gives the positive coefficient. Theorem 7.1 proves

$$\delta H=-\operatorname{tr}(C_\mu C_\nu)\delta q^{\mu\nu} +\Gamma_{\!\mathrm E}^\alpha{}_{\mu\nu}\delta(\partial_\alpha q^{\mu\nu}). \tag{A.1}$$

It agrees with numbered Einstein (48) and the ensuing field equation. The final publication audit also differentiates $H$ directly for $g=\operatorname{diag}(-1,-1,-1,N^2-\varepsilon N^4K)$, $N=1+x^2$, $K=1+x+x^2$. At $\varepsilon=0$, $\delta q^{44}=K$ and $\delta H=-2(N')^2K-NN'K'$, which rejects the opposite derivative sign without using a vacuum or unit-volume restriction.

The gravitational exchange line on page 809

Immediately before Einstein (57), the copy prints

$$\partial_\alpha t^\alpha{}_{\sigma} +\tfrac12\partial_\sigma q^{\mu\nu}T_{\mu\nu}=0.$$

Numbered Einstein (57) uses that plus sign for the matter array $T$. Taken together with total conservation, the two plus signs would force all exchange to vanish. The action and field equation instead give

$$\partial_\alpha t^\alpha{}_{\sigma}-e_\sigma=0, \qquad \partial_\alpha T^\alpha{}_{\sigma}+e_\sigma=0. \tag{A.2}$$

The same-realization dust witness (10.22) has nonzero $e_4$ in (10.23). The printed gravitational residual is $16r^{-5/3}/(81\kappa)$, whereas both corrected relations and the full coupled field equation vanish. Numbered Einstein (57) and Einstein (57a) are retained unchanged.

The mass coefficient on page 819 and its propagation

The supplied Einstein (70a) prints $\alpha=\kappa M/(8\pi)$. Yet the preceding Einstein (68a) and the displayed $g_{44}=1-\alpha/r$ imply

$$\phi=-\frac{\kappa M}{8\pi r},\qquad g_{44}=1+2\phi=1-\frac{\kappa M}{4\pi r},\qquad \alpha=\frac{\kappa M}{4\pi}. \tag{A.3}$$

The latter agrees with the Newtonian force and general clock-period expression. The same $M$ and asymptotic time normalization are used throughout. The mass expression $\kappa M/(4\pi b)$ printed in Einstein (74) propagates this half-normalization; its parameter expression $2\alpha/b$ is correct. This is one inconsistency carried into a later expression, not two independent physical errors.

The coordinate light-speed sign on page 822

The copy prints a positive first-order correction to $\gamma=\sqrt{-g_{44}/g_{22}}$. In the stated ray direction, the displayed metric gives

$$(1-\alpha/r)-(1+\alpha y^2/r^3)\gamma^2=0, \qquad \gamma=1-\frac\alpha{2r}\left(1+\frac{y^2}{r^2}\right)+O(\alpha^2). \tag{A.4}$$

The printed positive choice leaves a nonzero first-order residual and reverses the inward-positive derivative integral. The reconstruction uses the root fixed by the null condition and retains the correct parameter bending result, with the mass correction above.

Statements requiring hypotheses rather than typographical correction

Several source statements are retained with explicit logical scope, not labelled transcription errors. The coordinate condition beside Einstein (47a) does not specify a constrained variational problem; the unrestricted-variation interpretation and the failure of early trace restriction are established in Section 7. The discussion of four material equations and the fluid variable count does not prove general initial-value well-posedness. The free-charge prose on p. 814 does not follow from the displayed Maxwell equations alone; the actual algebraic conservation conclusion uses $\xi=0$. Local flatness and local potential statements are not global topology theorems.

The original optical integral concerns asymptotic endpoints. The finite-endpoint term in Theorem 16.1 qualifies an extension of that operation; it does not invalidate the asymptotic result. Its finite output is a coordinate-direction change until observer frames are specified. Finally, Einstein (75) is a stated result with external dependencies. The exact spherical and orbit calculation here is identified as a completion of that dependency, not falsely attributed to intervening derivations in the 1916 article itself.

Consolidation changes relative to the checkpoints

The final audit introduced no alteration to the six retained scientific checkers. Their complete result objects reproduce. The publication adds a separately written forty-group audit and tightens two descriptions: finite optical bending is explicitly chart-dependent at finite endpoints, and the bound-orbit domain is accompanied by the positive factorization of $\mathcal E^2$. It also supplies the radial comparison one-forms explicitly, separates all theorem assumptions in a public register, and expands the equation correspondence to every numbered and lettered target. These are consolidation proofs and scope clarifications, not a claim that replay alone validates every theorem.