22 / C3-T: Pressure Diversification, Cone-Eigenvalue Non-Rigidity, and the Heredity Gap

C3-S drove margin uniformity all the way to cross-scale rigidity. This round turns back to examine two inferences that look natural but do not hold, honestly flagging the type-level gaps. The first is Half-Space Signature Non-Rigidity (Proposition 5.1): a fixed 5-dimensional mean-strain half-space (C3-P's $H_0$, or the common support matrix of C3-R/S) cannot pin down the sign of the middle eigenvalue $\lambda_2$ — the document gives an explicit two-matrix counterexample in which both matrices lie in the same half-space yet have opposite $\lambda_2$ signs. Only when the half-space narrows to a cone carrying an eigenvalue gap can Weyl's inequality (Theorem 8.1) lock down the sign. But even once the sign is locked, the Mean-to-Pointwise Middle-Eigenvalue No-Go (Proposition 11.1) proves that "$\lambda_2>0$ after averaging in time or space" does not imply "$\lambda_2>0$ pointwise" — yet another explicit counterexample (a construction that oscillates pointwise but averages positive). This is a direct reminder that the middle-eigenvalue criteria built up across C3-K/C3-L hold only at the pointwise level; averaged-level evidence cannot simply be substituted in. The second, more fundamental gap is the Hereditary Pressure-Poor Ancestry Gap (§18): C3-R/S proved only that "a pressure-poor witness core exists at every scale," but this does not imply "there exists a single causal ancestry ray that is pressure-poor at every generation, across all scales" — the document illustrates this with an explicit infinitely branching tree counterexample: every level has a witness node, but no single path hits a witness node at every level. The document explicitly names this gap the still-unproven Hereditary Pressure-Poor Ancestry Lemma and flags it as the genuine next open target for this line of research. The final result is an information-theoretic Mean-Motif/Operator-Fluctuation Separation No-Go (Theorem 24.1): a fixed mean-strain motif has only 5 scalar degrees of freedom, while the Miller operator-escape ratio is a functional on an infinite-dimensional kernel — 5 scalars cannot, information-theoretically, determine the behavior of an infinite-dimensional kernel. Finite mean geometry is inherently incapable of controlling infinite-dimensional operator escape — this is not a matter of computational difficulty, but a dimensional mismatch.