Moment maps do not automatically select the frame-lock orbit

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 67: Moment maps do not automatically select the frame-lock orbit.

Provenance

Preserved from interaction notebook section 67, 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.

67. Moment maps do not automatically select the frame-lock orbit

For $G=\operatorname{diag}(1,-1,-1,0)$ and the coordinate family $Z(t)=(\cos t,\sin t,0,0)$, the exact moment map is $\mu_G=\cos(2t)$. Calling $\sin^2t$ the physical frame-lock invariant is not justified without specifying the product-spin identification.

In the product basis used in Sections 72--92, $Z(t)=|+\rangle\otimes(\cos t|+\rangle+\sin t|-\rangle)$ is a product state. Its concurrence is zero and its frame-lock invariant is $s=1$ for every $t$. Thus a maximum of $\mu_G$ on this curve does not prove selection of $s=0$.

The correct invariant fixed-lock test is $(I-P_*)G_qZ_*=0$ at a separately selected lock ray, as developed in Sections 87--92. It gives $q=1$ within the specified integer family and fixed-ray identification. It is a compatibility/stationarity condition, not a derivation of the locking potential or a proof of stability of the total Hamiltonian. The earlier comparison of two purported lock Hessians by a factor of four has no basis-independent physical interpretation.

Related

• Research status — current proof boundaries and correction guidance.
• Orbit-generator spectrum versus the physical mixed block — preceding section in the research sequence.
• Orbit-rate audit: three electron clocks are distinct until the stationary master generator identifies them — 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.