Abstract

Two proposed manuscript bridges are examined without identifying their distinct objects or domains. In a fully specified flat linearized-gravity setting, a compact tensor distribution admitting a smooth compact wave-shell representative has a smooth compact conserved representative on a vacuum solution space containing compact pure gauges if and only if the original distribution is conserved. The causal-propagator smoothness required by the conserved-current construction follows from the wave-shell relation once the bundles and operators have actually been identified. An explicit residual-gauge example shows that smooth compact wave-shell representability alone is insufficient.

For the proposed prepared-response-to-joint-graph bridge, we prove precise information-loss, reconstruction-descent, extension, and normal-jet results. Partial witness contraction is noninjective even on smooth compact kernels. Canonical-domain agreement does not establish global descent, and a bound on normal order does not by itself make supported ambiguities finite-dimensional.

These are completed mathematical results about bridge conditions. They neither identify the actual VII-B tensor with a conserved V-current nor construct the unresolved XV joint graph. The paper-specific conclusions are restricted to the supplied passages and registry descriptions.