Orbit generator as a polynomial of the total locked axial spin

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 74: Orbit generator as a polynomial of the total locked axial spin.

Provenance

Preserved from interaction notebook section 74, 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.

74. Orbit generator as a polynomial of the total locked axial spin

The corrected section 72 expressed the traceless orbit generator in the product-spin basis as

$$\widetilde G =\frac14\sigma_z\otimes I +\frac14I\otimes\sigma_z +\frac34\sigma_z\otimes\sigma_z.$$

This initially made the coefficient $3/4$ look like an independent two-frame bilinear that might have to be related to $3g_{66}$. There is a cleaner exact formulation.

Define the total axial generator of the diagonally locked spin frames,

$$\boxed{ J_z =\frac12\left(\sigma_z\otimes I+I\otimes\sigma_z\right). }$$

Since

$$J_z^2 =\frac12\left(I+\sigma_z\otimes\sigma_z\right),$$

the orbit generator is exactly

$$\boxed{ \widetilde G =-\frac34I +\frac12J_z +\frac32J_z^2. }$$

The scalar $-3I/4$ is projectively invisible, so the projective generator is equivalently

$$\boxed{ \widetilde G_{\rm proj} \sim \frac12J_z+\frac32J_z^2. }$$

74.1 Diagonal-spin basis

Under the locked subgroup,

$$(\mathbf2,\mathbf2) \downarrow_{Spin(3)_{\rm diag}} =\mathbf3\oplus\mathbf1.$$

Use the standard coupled basis

$$|1,1\rangle, \quad |1,0\rangle, \quad |1,-1\rangle, \quad |0,0\rangle.$$

Then

$$J_z =\operatorname{diag}(1,0,-1,0),$$

while the orbit generator becomes

$$\boxed{ \widetilde G =\operatorname{diag} \left( \frac54, -\frac34, \frac14, -\frac34 \right) }$$

up to the ordering of the two zero-$m$ states.

The two $m=0$ states — the triplet $|1,0\rangle$ and singlet $|0,0\rangle$ — have exactly the same orbit-generator eigenvalue. Therefore the generator does not need the total-spin Casimir $J^2$ to distinguish the $\mathbf3$ and $\mathbf1$ sectors.

Indeed, solving

$$\widetilde G =aI+bJ_z+cJ_z^2+dJ^2$$

forces

$$\boxed{d=0.}$$

So the orbit generator depends only on the axial quantum number $m$, not separately on whether the state lies in the singlet or triplet representation.

74.2 Why this is stronger than the $3g_{66}$ interpretation

The coefficient $3/4$ in the product basis is not an invariantly separate physical coupling. It is partly a consequence of rewriting the quadratic axial operator $J_z^2$ back into the uncoupled tensor-product basis.

Therefore the relation

$$\frac34=3g_{66}$$

must not be interpreted as evidence that three copies of the depth-six coupling are dynamically summed into an axial bilinear. Section 73 already showed that an actual isotropic three-axis sum produces the Heisenberg exchange operator instead.

The more invariant statement is

$$\boxed{ \widetilde G_{\rm proj} \sim \frac12J_z+\frac32J_z^2. }$$

This shifts the proof obligation away from deriving an isolated coefficient $3g_{66}$ and toward deriving a linear-plus-quadratic function of the total locked axial generator.

74.3 Representation-theoretic origin of the four weights

For

$$m\in\{1,0,-1,0\},$$

the polynomial

$$f(m) =-\frac34+\frac12m+\frac32m^2$$

gives

$$\boxed{ \left( \frac54,-\frac34,\frac14,-\frac34 \right), }$$

which is precisely the traceless orbit spectrum up to ordering.

Passing to

$$\Lambda^2\mathbf4$$

then produces the positive real rotation rates

$$\boxed{ \frac12,\frac12,\frac32. }$$

Thus the $1{:}1{:}3$ vector spectrum can be viewed as descending from a quadratic axial polynomial on the diagonally locked $(\mathbf2,\mathbf2)$ spinor.

74.4 What remains unexplained

This reformulation removes a misleading candidate coupling identification, but it does not yet derive the polynomial coefficients

$$\boxed{ \frac12,\qquad\frac32. }$$

The existing amplitude/clock uniqueness theorem fixes an equivalent generator once its five premises are assumed. What has not yet been shown is that the scattering-selected locked electron must realize precisely that polynomial of $J_z$.

The next useful test is therefore whether primitive projective closure, time-reversal/orientation reversal, and the required unit clock gap uniquely determine the lowest-degree polynomial

$$f(J_z)=a+bJ_z+cJ_z^2$$

on the four locked spin states. If those physical constraints force $b:c=1:3$, the factor three will have been derived from the locked-spin spectrum without invoking $g_{66}$ at all.

Related

• Research status — current proof boundaries and correction guidance.
• Three-axis contraction test: summing three depth-six channels does not yield the required axial bilinear — preceding section in the research sequence.
• Locked-spin gap theorem: the orbit polynomial is uniquely fixed by the existing unit and two-unit projective gaps — 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.