Locked-spin gap theorem: the orbit polynomial is uniquely fixed by the existing unit and two-unit projective gaps

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 75: Locked-spin gap theorem: the orbit polynomial is uniquely fixed by the existing unit and two-unit projective gaps.

Provenance

Preserved from interaction notebook section 75, 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.

75. Locked-spin gap theorem: the orbit polynomial is uniquely fixed by the existing unit and two-unit projective gaps

Section 74 rewrote the electron orbit generator as a quadratic polynomial of the total locked axial spin,

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

on the four states

$$m=1,0,-1,0.$$

The remaining question was whether the coefficients $b$ and $c$ can be derived from the already stated orbit premises rather than read backward from the desired generator.

They can, provided the two projective gap conditions already built into the amplitude/clock construction are accepted.

75.1 Translate the orbit premises into locked-spin gaps

The projectively invisible constant $a$ drops out of all level differences. Relative to the common $m=0$ level,

$$f(1)-f(0)=b+c,$$

and

$$f(-1)-f(0)=-b+c.$$

The existing orbit theorem contains two distinct relative-frequency requirements:

1. the primitive clock branch carries one unit of relative phase;
2. the opposite extremal amplitude branch differs from the common reference by two units.

In the locked-spin labeling these are

$$\boxed{f(-1)-f(0)=1}$$

and

$$\boxed{f(1)-f(0)=2.}$$

Therefore

$$-b+c=1,$$

$$b+c=2.$$

The solution is unique:

$$\boxed{b=\frac12,\qquad c=\frac32.}$$

Hence the lowest-degree axial polynomial is

$$\boxed{ f(J_z) =aI+\frac12J_z+\frac32J_z^2. }$$

This derives the ratio

$$\boxed{b:c=1:3}$$

from the one-unit and two-unit projective gap structure, without invoking $g_{66}$.

75.2 Traceless representative

On the four-state $(\mathbf2,\mathbf2)$ space the eigenvalues are

$$f(1),\quad f(0),\quad f(-1),\quad f(0).$$

Imposing tracelessness fixes only the central shift:

$$\boxed{a=-\frac34.}$$

Thus

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

with spectrum

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

up to ordering of the two $m=0$ states.

This is exactly the previously used traceless $SU(4)$ orbit generator.

75.3 Six-dimensional vector spectrum follows automatically

Passing to the real vector representation

$$\Lambda^2\mathbf4$$

adds the four spinor weights pairwise. The six weights are

$$\pm\frac12, \qquad \pm\frac12, \qquad \pm\frac32,$$

so the three positive real rotation-plane rates are

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

and therefore

$$\boxed{1:1:3.}$$

The vector-space factor three is thus no longer an unexplained extra coefficient once the locked-spin gap conditions are supplied.

75.4 What is actually new here

The earlier uniqueness theorem began with a chosen amplitude-plane/clock-plane decomposition of the four homogeneous coordinates and solved for the diagonal generator.

The present calculation gives an equivalent but more physical formulation after the $3+3$ frame lock and beam axis have been selected:

$$\boxed{ \text{locked }(\mathbf2,\mathbf2) +\text{axial }J_z +\text{unit clock gap} +\text{two-unit opposite gap} \Longrightarrow \widetilde G. }$$

So the generator can be reconstructed entirely in the diagonal-spin language.

75.5 What remains conditional

The primitive one-unit gap is already an explicit premise of the orbit theorem. The two-unit opposite gap is likewise equivalent to the existing amplitude/clock weight assignment. Therefore this section does not derive those spectral gaps from the projected master dynamics; it shows that once those two physical gap conditions hold, the polynomial and the $1{:}1{:}3$ generator are unique.

A weaker assumption set is insufficient. For example, imposing only

$$f(-1)-f(0)=1$$

leaves a one-parameter family of quadratic polynomials. The equality of the two $m=0$ states is automatic for any polynomial in $J_z$ and supplies no additional constraint.

Thus the remaining dynamical question has narrowed again:

$$\boxed{ \text{Why does the stationary projected electron interaction generate the relative axial gaps }(2,0,1,0)? }$$

If V2 can derive those gaps directly from the scattering-selected connection or projected master interaction, the full orbit generator, its $1{:}1{:}3$ real rotation spectrum, and its frame-lock compatibility follow without any appeal to the adopted value $g_{66}=1/4$.

Related

• Research status — current proof boundaries and correction guidance.
• Orbit generator as a polynomial of the total locked axial spin — preceding section in the research sequence.
• Gap-origin audit: primitive winding and orientation reversal do not derive the two-unit branch — 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.