Beam-axis symmetry leaves the integral charge free

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 81: Beam-axis symmetry leaves the integral charge free.

Provenance

Preserved from interaction notebook section 81, 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.

81. Beam-axis symmetry leaves the integral charge free

For the product-spin family $G_q\sim\operatorname{diag}(q,-q,-1,0)$, the physical diagonal beam generator is $J_z^{\rm diag}=\operatorname{diag}(1,0,0,-1)$. Consequently

$$[J_z^{\rm diag},G_q]=0$$

for every $q$. The induced $\Lambda^2$ actions also commute for every $q$.

Putting the three rotation rates of $G_q$ into a conveniently chosen purely mixed real block and then treating that split as the physical $O\oplus H$ split would mix incompatible basis identifications. It cannot be used to select $q=1$. Section 82 gives the physical branching; Sections 87--92 give the distinct, conditional fixed-lock criterion.

Related

• Research status — current proof boundaries and correction guidance.
• Physical $(3,1)\oplus(1,3)$ audit: beam symmetry leaves $q$ free; frame exchange would select $q=1$ — 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.