Theorem: The specified VI–XVI-D coordinate comparison mismatches

Retain the slab, source, response sign, metric-sign transport, and ordinary coordinate-time data in Sections sec:native–sec:witness. For every $\lambda >0$, the direct comparison of VI's first coordinate coefficient on $h_{\lambda }$ with $X_{\mathrm{cand}}[R_{\lambda }]$ is a mismatch.

More precisely:

1. The two normalized flat forcings differ by the nonzero residual (eq:forcing-residual).

2. Their unique smooth zero-data coordinate coefficients are

   $$\begin{array}{rl}{w_{\lambda }}^{\mu }(t)& =-{{\int }_0}^t{V}_{\lambda }^{0\mu }(s)\mathrm{d}s,\\ {d_{\lambda }}^{\mu }(t):={w_{\lambda }}^{\mu }-{X_{\mathrm{cand}}}^{\mu }& =-{{\int }_0}^t(1-\sigma (s)){V_{\lambda }}^{0\mu }(s)\mathrm{d}s.\end{array}$$

   The time component ${d_{\lambda }}^0$ is nonzero somewhere in $(0,\epsilon )$.

3. A real linear combination of the witness coefficients and a compact real test detect that coordinate difference.

4. The response used as a first metric coefficient is not in VI's solution-jet domain, as shown by (eq:Einstein-witness).

This theorem excludes the stated direct first-order identification. It does not exclude a differently declared metric map, preparation, coupling-order comparison, or observable construction. It neither repairs nor contradicts XVI-D's conditional inherited-coordinate clause.

Proof. Item (i) is Lemma lem:B, with both operators normalized to ${\square }_{+}$.

The field in (eq:w-explicit) has zero position and velocity at $0$ and satisfies

Lemma lem:C identifies it with VI's smooth first coefficient. Equation (eq:Xcand) gives the other field; subtraction proves (eq:coordinate-difference).

If ${d_{\lambda }}^0$ vanished identically on $(0,\epsilon )$, then its derivative would give $(1-\sigma ){V_{\lambda }}^{00}=0$ there. On an interval with $0<\sigma <1$, this would force ${S_{\lambda }}^{00}=0$, contradicting (eq:S00), exactly as in Lemma lem:B. Thus item (ii) holds.

Choose $t_c\in (0,\epsilon )$ with ${d_{\lambda }}^0(t_c)\ne 0$ and a unit phase $\zeta$ such that $\mathrm{Re}(\zeta {d_{\lambda }}^0(t_c))>0$. A nonnegative compact test supported in a small neighborhood where that real part remains positive has a strictly positive pairing. Since the constructions are complex linear on the printed source coefficients, this pairing is the coordinate difference for the real metric coefficient $\mathrm{Re}(\zeta h_{\lambda })$. This proves item (iii).

Item (iv) is Proposition prop:jet-domain. None of these calculations uses an interaction switch $g$, a scalar-switch identity, a quantum coefficient, or a theorem from an unregistered sheet.