Gap-origin audit: primitive winding and orientation reversal do not derive the two-unit branch

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 76: Gap-origin audit: primitive winding and orientation reversal do not derive the two-unit branch.

Provenance

Preserved from interaction notebook section 76, 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.

76. Gap-origin audit: primitive winding and orientation reversal do not derive the two-unit branch

Section 75 showed that the desired orbit generator follows uniquely from two relative axial gaps,

$$f(-1)-f(0)=1, \qquad f(1)-f(0)=2,$$

for a quadratic polynomial

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

The next question is whether the second equation follows from the first together with orientation reversal or another elementary symmetry of the locked beam geometry.

It does not.

76.1 Primitive winding fixes only one equation

The primitive clock condition supplies

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

For

$$f(m)=a+bm+cm^2,$$

this gives

$$-b+c=1,$$

or

$$\boxed{b=c-1.}$$

Thus a one-parameter family remains. Primitive winding alone does not determine the orbit generator.

76.2 Orientation reversal does not supply the missing target equation

There are two obvious ways to impose a reversal condition on the axial spectrum.

If reversal is taken to make the two extremal gaps antisymmetric about the $m=0$ level,

$$f(1)-f(0) =-[f(-1)-f(0)],$$

then together with the unit gap one obtains

$$\boxed{b=-1,\qquad c=0.}$$

This is a purely linear axial generator, not the required one.

If reversal is instead taken to make the spectrum even,

$$f(1)=f(-1),$$

then one obtains

$$\boxed{b=0,\qquad c=1.}$$

which is also not the required generator.

Therefore neither elementary reversal prescription produces

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

76.3 The two-unit branch is genuinely additional information

The target generator requires

$$f(1)-f(0)=2.$$

In the currently written orbit construction, that two-unit separation comes from the extremal amplitude weights

$$+L \qquad\text{and}\qquad -L$$

relative to the chosen clock/reference sector. It is therefore already encoded in the amplitude-plane premise of the earlier uniqueness theorem.

The locked-spin reformulation does not eliminate that premise; it translates it into a spectral condition on $J_z$.

76.4 Consequence

The derivation chain is now accurately separated into two levels:

$$\boxed{ \text{primitive clock winding} \Longrightarrow \text{one unit gap}, }$$

but

$$\boxed{ \text{two-unit opposite gap} \not\Leftarrow \text{primitive winding + reversal symmetry alone}. }$$

So the factor-three orbit spectrum is not yet derived from the beam lock alone.

The remaining physical input is precisely the origin of the extremal amplitude splitting. A genuine V2 closure must show why the stationary projected electron operator has two axial extremal states separated by two primitive clock quanta.

76.5 Sharpened next target

The most promising place to look is the magnetic orbital coupling already written as

$$H_{\rm mag} =-\frac{q}{2m}\sum_{A<B}F_{AB}L^{AB}.$$

A single rotation generator naturally has opposite extremal eigenvalues. If the primitive projective clock fixes their unit normalization, an actual projected curvature tensor might supply the required $+1$ and $-1$ amplitude weights directly, making their separation two rather than postulating it.

The next test should therefore determine whether a beam-compatible full-dimensional curvature with the V2 photon/electron geometry can produce a normalized axial orbital generator with eigenvalues $+1$ and $-1$ while simultaneously retaining the primitive clock gap.

Related

• Research status — current proof boundaries and correction guidance.
• Locked-spin gap theorem: the orbit polynomial is uniquely fixed by the existing unit and two-unit projective gaps — preceding section in the research sequence.
• Extremal-doublet audit: the two-unit gap follows from a primitive signed amplitude charge, but that charge is still an input — 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.