The quadratic normalization

For any fixed norm on $Z$, injectivity implies a finite constant $C_Z$ with

Neither the subspace membership $r\in Z$ nor injectivity follows from having printed a finite table.

Proof. The exact assertion is $\ker P=\{0\}$. For the quantitative assertion, the continuous function $r\mapsto {\Vert P(r)\Vert }_{\infty }$ has a positive minimum on the unit sphere of a nonzero finite-dimensional $Z$. Its reciprocal gives $C_Z$. The zero space is immediate.

Let $O\subset V$ be any nonempty open set and ${\ell }_1,\dots ,{\ell }_N$ any finite family of linear evaluations on smooth compactly supported symmetric tensors. There is a nonzero

such that

No assertion of a zero trace is required.

Proof. For $\psi \in {C_c}^{\infty }(O)$, define

It is symmetric and divergence free. If $S_{\psi }=0$, taking its trace gives $-3\square \psi =0$. A spacetime-compact homogeneous wave is zero by Cauchy uniqueness, so $\psi =0$.

Choose $N+1$ nonzero bump functions with mutually disjoint compact supports in $O$. Their tensors $S_{{\psi }_a}$ are nonzero and linearly independent. The $N$ evaluations on their span have a nonzero kernel. A nonzero tensor in that kernel has all the required properties.

This establishes the finite-probe limitation locally. It does not append a new stress source to the action, construct an Einstein solution, or verify OpenAI Family 282.

The independent non-implication from record conservation to Einstein dynamics is the existing I-B :not-source and X-A-ST :noeinstein boundary CheungIB,CheungXAST. No third geometric counterexample is needed here.

complete Enter report interface

finite supplied catalogue

Before execution, freeze all evaluation configurations, tests, units, source data, and approximation enclosures. A possible catalogue consists of the following entries.