Axial mixed tensors do not determine the orbit-generator spectrum

Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Research record, section 71: Axial mixed tensors do not determine the orbit-generator spectrum.

Provenance

Preserved from interaction notebook section 71, recorded 2026-09-05 and corrected 2026-09-06. Original sequence contents. This is an exploratory research record, not an independently established V2 result. Read the related later audits before using an intermediate conclusion.

71. Axial mixed tensors do not determine the orbit-generator spectrum

For a real mixed tensor $E:H\to O$, invariance under every simultaneous rotation $E\mapsto RER^T$ forces $E=\lambda I_3$. Invariance under axial rotations only gives

$$E=\begin{pmatrix}A&B&0\\-B&A&0\\0&0&C\end{pmatrix}.$$

Its symmetric part is $\operatorname{diag}(A,A,C)$, with $C/A$ unfixed when $A\ne0$.

These are statements about the mixed tensor in a specified physical split. The three real rotation-plane rates of an arbitrary six-dimensional generator are not generally the singular values of that mixed block. For the actual $G_q$ family, Section 82 finds a physical mixed block of rank one and beam-axis commutation for all $q$.

Thus axial symmetry alone neither derives the $1:1:3$ rotation-rate spectrum nor singles out $q=1$. The exact spinor weight calculations below and the conditional fixed-ray criterion are separate statements.

Related

• Research status — current proof boundaries and correction guidance.
• Beam-axis spinor decomposition: the orbit generator requires a bilinear lock term — following section in the research sequence.
• Leading-source orientation no-go: the written scalar and factorized $3\leftrightarrow6$ structures can select $s$ but cannot select the locked frame orientation — later audit limits or corrects this route.