Fixed-lock compatibility theorem: $q=1$ is the unique zero-drift branch of the selected frame

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 87: Fixed-lock compatibility theorem: $q=1$ is the unique zero-drift branch of the selected frame.

Provenance

Preserved from interaction notebook section 87, 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.

87. Fixed-lock compatibility theorem: $q=1$ is the unique zero-drift branch of the selected frame

Section 86 identified the $q\neq1$ mixer with the relative-frame generator

$$K_z=J_z^O-J_z^H.$$

This makes it possible to test the full time evolution against a fixed scattering-selected frame, rather than merely against the coarse invariant $s=0$.

The result is exact: within the integral family $(q,-q,-1,0)$, $q=1$ is the unique branch whose stationary Hamiltonian produces no drift of the selected frame link.

87.1 Use the singlet adapted to the selected diagonal lock

For the aligned frame link $Q_{\rm scat}=I$, choose the coupled basis

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

The last state is the one-dimensional invariant line of the selected diagonal $Spin(3)$:

$$|S\rangle=|0,0\rangle.$$

In the two-dimensional subspace

$$\operatorname{span}\{|1,0\rangle,|S\rangle\},$$

the general orbit generator is exactly

$$\boxed{ \widetilde G_q =\left(-\frac q2-\frac14\right)I -\frac{q-1}{2}K_z. }$$

Thus the entire nontrivial action on the selected singlet line is the relative-frame term.

87.2 Exact time evolution

Up to the common phase generated by the scalar term,

$$\boxed{ |S(t)\rangle =\cos\!\left(\frac{(q-1)t}{2}\right)|S\rangle +i\sin\!\left(\frac{(q-1)t}{2}\right)|1,0\rangle. }$$

Consequently

$$\boxed{ |\langle S|S(t)\rangle|^2 =\cos^2\!\left(\frac{(q-1)t}{2}\right). }$$

The instantaneous Fubini--Study velocity away from the selected singlet ray has squared norm

$$\boxed{ \left\|(I-|S\rangle\langle S|)\widetilde G_q|S\rangle\right\|^2 =\frac{(q-1)^2}{4}. }$$

Therefore

$$\boxed{ q=1 \quad\Longleftrightarrow\quad \text{zero projective drift away from the selected locked ray}. }$$

This is stronger than preservation of the whole maximally entangled orbit. The $q\neq1$ flow remains within the maximally entangled $s=0$ family; what it changes is the orientation of the frame link inside that family.

That distinction resolves an important subtlety: $s=0$ alone does not fix $Q$. The scattering construction does.

87.3 Exact frame-link motion

The relative spin rotation generated by $K_z$ induces opposite rotations of the observed and unobserved spin frames. On the associated real three-frames, the selected link therefore evolves as a relative $SO(3)$ rotation with angle magnitude

$$\boxed{\theta_{\rm rel}(t)=(q-1)t.}$$

In the aligned frame,

$$Q(t)=R_z((q-1)t)$$

up to the overall orientation convention.

For the scattering-frame functional already tested in sections 56--57,

$$V_{\rm frame}(Q) =\frac\kappa2\|I-Q\|_F^2,$$

this gives the exact mismatch energy

$$\boxed{ V_q(t) =2\kappa\left[1-\cos((q-1)t)\right] =4\kappa\sin^2\!\left(\frac{(q-1)t}{2}\right). }$$

Hence

$$\boxed{ V_q''(0)=2\kappa(q-1)^2. }$$

For positive lock stiffness,

$$q=1$$

is the unique integral branch with identically zero frame-mismatch cost.

87.4 What this does and does not prove

This closes a loophole left by section 86. We do not need to claim that relative-frame modes are impossible. They are legitimate excited motions. Instead:

$$\boxed{ q>1\text{ produces a periodic relative-frame path under the orbit-only generator}, }$$

whereas

$$\boxed{ q=1\text{ is the unique branch stationary at the scattering-selected frame minimum}. }$$

This orbit-only path need not solve the full Hamiltonian including the lock potential. It is a compatibility diagnostic, not a derived excited solution of that interacting system. Thus the integer family has the kinematic interpretation:

$$q-1$$

measures the winding rate of the relative-frame drift with respect to the primitive orbit parameter. It is not yet proven to be a quantized harmonic-oscillator occupation number, so that stronger interpretation should not be asserted.

87.5 Relation to the earlier minimum-norm result

Section 78 found independently that

$$\|\widetilde G_q\|^2=2q^2+\frac34$$

is uniquely minimized at $q=1$ over positive integral $q$.

The present result gives a more direct physical reason for selecting the same branch: the already-constructed scattering-frame potential assigns

$$V_q''(0)=2\kappa(q-1)^2$$

to the relative-frame component.

So there are now two distinct selectors with the same answer:

1. minimum invariant generator norm;
2. zero drift / zero mismatch of the scattering-selected frame.

The second is preferable because it refers to an interaction structure already developed in V2 rather than postulating a new variational functional over Hamiltonian circles.

87.6 Remaining dependency

The conclusion is as strong as the scattering-frame lock itself. The geometry and the exact evolution above are established, but current V2 has not yet derived the coefficient and precise $Q_{\rm scat}$ coupling from the master operator.

Therefore the present status is

$$\boxed{ \text{if the scattering-selected frame potential is the physical gross-lock term, then }q=1\text{ is dynamically forced for the stationary locked branch}. }$$

This substantially narrows the remaining proof obligation. One no longer needs an independent principle saying ``choose the smallest $q$.'' One needs to derive the already-motivated full-rank scattering-frame coupling from the projected master dynamics.

Related

• Research status — current proof boundaries and correction guidance.
• Relative-frame identification: the $q\neq1$ mixer is exactly a broken frame-rotation generator — preceding 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.