Orbit-generator spectrum versus the physical mixed block

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 66: Orbit-generator spectrum versus the physical mixed block.

Provenance

Preserved from interaction notebook section 66, 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.

66. Orbit-generator spectrum versus the physical mixed block

The traceless generator $\widetilde G=\operatorname{diag}(5,-3,-3,1)/4$ has positive $\Lambda^2\mathbf4$ rotation rates $(1/2,1/2,3/2)$.

Pairing one real axis from each rotation plane with its quadrature partner puts this generator in a full-rank mixed block for that chosen split. It does not show that the block is full rank in the independently selected physical $O\oplus H=(\mathbf3,\mathbf1)\oplus(\mathbf1,\mathbf3)$ split.

In the physical product-spin branching calculated in Section 82, the mixed block has rank one, including at $q=1$. The orbit generator's spectrum alone therefore does not supply an oriented three-frame lock. A physical source must be evaluated in the physical splitting; changing the splitting to maximize mixed rank changes the claim being tested.

Related

• Research status — current proof boundaries and correction guidance.
• Linear Dirac-action variation does not fix the projector sigma-model coefficient — preceding section in the research sequence.
• Moment maps do not automatically select the frame-lock orbit — 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.