Relative-frame identification: the $q\neq1$ mixer is exactly a broken frame-rotation generator

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 86: Relative-frame identification: the $q\neq1$ mixer is exactly a broken frame-rotation generator.

Provenance

Preserved from interaction notebook section 86, 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.

86. Relative-frame identification: the $q\neq1$ mixer is exactly a broken frame-rotation generator

Section 85 isolated the unwanted operator

$$V_z=|1,0\rangle\langle0,0|+|0,0\rangle\langle1,0|.$$

The next representation-theoretic calculation identifies it exactly.

Let the two spin factors carry generators

$$J_a^{O}=\frac12\sigma_a\otimes I, \qquad J_a^{H}=\frac12I\otimes\sigma_a.$$

Define common and relative rotations

$$J_a=J_a^O+J_a^H, \qquad K_a=J_a^O-J_a^H.$$

In the coupled basis

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

direct transformation gives

$$\boxed{K_z=V_z.}$$

Thus the precise operator that appears whenever $q\neq1$ is not merely an abstract singlet--triplet vector. It is the relative axial rotation of the observed and unobserved spin frames.

86.1 The three relative rotations are the singlet--triplet vector

For all three axes,

$$K_a=J_a^O-J_a^H$$

fails to commute with the locked-spin Casimir:

$$[J^2,K_a]\neq0.$$

By contrast the common rotations satisfy

$$[J^2,J_a]=0.$$

Therefore the two vector operator classes have a direct geometric meaning:

• $J_a$: common rotations of the already-locked frames;
• $K_a$: relative rotations that change their mutual identification.

The $q$ ambiguity found earlier contains

$$\boxed{-\frac{q-1}{2}K_z.}$$

Hence

$$\boxed{q\neq1\quad\Longrightarrow\quad\text{the stationary orbit generator contains a relative-frame rotation}.}$$

86.2 Real six-dimensional check

At the aligned locked point,

$$J_{\rm lock} =\begin{pmatrix}0&I\\-I&0\end{pmatrix}.$$

A common infinitesimal rotation has the real six-dimensional form

$$B_a^+ =\begin{pmatrix}R_a&0\\0&R_a\end{pmatrix},$$

while a relative rotation is

$$B_a^- =\begin{pmatrix}R_a&0\\0&-R_a\end{pmatrix}.$$

Exact matrix calculation gives

$$\boxed{[B_a^+,J_{\rm lock}]=0,}$$

but

$$\boxed{[B_a^-,J_{\rm lock}]\neq0.}$$

So the same common/relative distinction appears independently in the real vector representation and in the chiral four-spinor representation.

A common rotation belongs to the stabilizer of the selected frame link. A relative rotation moves the selected link orientation.

86.3 The scattering-lock Hessian gives the relative mode positive stiffness

The aligned quadratic lock operator derived earlier is

$$K_{\rm lock} =\kappa \begin{pmatrix} I&-I\\ -I&I \end{pmatrix}.$$

For every three-vector $v$,

$$K_{\rm lock} \binom{v}{v}=0,$$

whereas

$$\boxed{ K_{\rm lock} \binom{v}{-v} =2\kappa \binom{v}{-v}. }$$

Thus the lock itself separates the frame motions into:

$$\boxed{ \text{common mode: zero relative-lock cost}, \qquad \text{relative mode: stiffness }2\kappa. }$$

This is exactly the distinction between $J_a$ and $K_a$.

86.4 Consequence for the $q$ family

Section 85 gave

$$\widetilde G_q =\widetilde G_1 +(q-1)\left(-\frac12I+\frac12J_z+J_z^2-\frac12K_z\right).$$

The only part of this deformation that changes the locked irreducible sector is

$$-\frac{q-1}{2}K_z,$$

and $K_z$ is precisely a positive-stiffness relative-frame mode of the scattering lock.

Therefore the desired $q=1$ generator is uniquely the member of the integral family with no excitation of the relative frame coordinate.

Conditionally on the stationary gross orbit being evaluated at the selected scattering-frame minimum rather than with an independently excited relative-frame normal mode,

$$\boxed{q=1.}$$

This is substantially sharper than simply postulating singlet/triplet preservation: the forbidden mixer has now been identified with the very relative frame motion that the lock potential penalizes.

86.5 Remaining logical gap

One step still has to be justified before calling this unconditional.

A positive quadratic stiffness means the relative mode is energetically costly; it does not by itself prove that the stationary electron orbit has exactly zero excitation in that mode. A normal mode with nonzero frequency can in principle be excited.

The needed statement is therefore not

$$\text{relative rotations do not exist},$$

but rather

$$\boxed{ \text{the ground/stationary gross orbit is the zero-relative-mode solution of the selected frame-lock potential}. }$$

If the bound-state variational problem already defines the electron orbit as the minimum-energy stationary solution in the frame coordinate, this follows directly. If the orbit construction permits a persistent excited relative-frame mode, then $q\neq1$ remains possible as an excited branch rather than as the ground locked orbit.

86.6 New test

The next calculation should therefore compare the orbit-generator family with the normal-mode spectrum of the frame-lock Hessian. The decisive question is whether

$$q-1$$

can be interpreted quantitatively as a relative-frame excitation number. If so, $q=1$ is not an arbitrary choice at all: it is the locked ground branch, while $q>1$ are higher relative-frame branches.

Related

• Research status — current proof boundaries and correction guidance.
• Operator-class refinement: the unwanted $q\neq1$ piece is a singlet--triplet vector intertwiner — preceding section in the research sequence.
• Fixed-lock compatibility theorem: $q=1$ is the unique zero-drift branch of the selected frame — 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.