Extremal-doublet audit: the two-unit gap follows from a primitive signed amplitude charge, but that charge is still an input

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 77: Extremal-doublet audit: the two-unit gap follows from a primitive signed amplitude charge, but that charge is still an input.

Provenance

Preserved from interaction notebook section 77, 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.

77. Extremal-doublet audit: the two-unit gap follows from a primitive signed amplitude charge, but that charge is still an input

Section 76 isolated the missing second condition as the two-unit separation associated with the amplitude branch. The magnetic/orbit construction clarifies exactly where that factor two comes from.

Let a two-state amplitude sector carry opposite charges under one $U(1)$ generator,

$$+q,\qquad -q.$$

Their separation is

$$\boxed{\Delta_{\rm amp}=2q.}$$

If the amplitude charge is primitive in the same group parameter used by the projective clock,

$$q=1,$$

then

$$\boxed{\Delta_{\rm amp}=2.}$$

At degree $L$, the extremal monomials carry

$$+Lq,\qquad -Lq,$$

so their separation is

$$\boxed{2Lq.}$$

For $q=1$, this reproduces the existing $\pm L$ extremal-weight construction.

77.1 What is and is not explained

The factor two itself is therefore elementary once a primitive signed doublet exists:

$$(+1)-(-1)=2.$$

The nontrivial issue is not the subtraction. It is whether the amplitude charge magnitude and the clock winding are normalized by the same physical $U(1)$ generator.

The current orbit theorem assumes one diagonal generator acts on both amplitude and clock coordinates and fixes the clock difference to one. Under those premises the relative normalization is built in.

What remains to be derived from the projected master connection is precisely that common-generator identification.

77.2 Magnetic curvature supplies the correct sign architecture

The existing magnetic coupling is

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

A rotation generator on an oriented two-plane naturally has opposite extremal eigenvalues. Therefore an actual projected curvature component can supply the required signed pair rather than a scalar level shift.

This is qualitatively different from the scalar self-contact term, which section 69 showed is projectively invisible.

77.3 Topology alone does not fix the amplitude magnitude

The line-bundle topology fixes integral weights once a primitive circle action is chosen. The primitive clock condition guarantees that the overall $U(1)$ action is not a multiple cover because one weight difference is one.

However, an amplitude pair

$$+q,-q$$

with integer $q>1$ is still compatible with the same primitive clock difference. The clock already makes the greatest common divisor of all projective weight differences equal to one.

Thus

$$\boxed{\text{primitive clock winding does not by itself force }q=1.}$$

An additional minimality or representation statement is required to choose the smallest nonzero signed amplitude charge.

77.4 Sharpened theorem

The missing dimensionless statement is now

$$\boxed{ \text{the projected electron connection realizes the amplitude pair in the minimal nonzero signed representation of the same }U(1)\text{ that supplies the primitive clock.} }$$

If that is proved, then

$$q=1,$$

and the two-unit amplitude separation follows automatically.

This is narrower than independently deriving either the factor two or the factor three.

Related

• Research status — current proof boundaries and correction guidance.
• Gap-origin audit: primitive winding and orientation reversal do not derive the two-unit branch — preceding section in the research sequence.
• Primitive cocharacter norm test: the desired amplitude charge is the unique minimum-norm choice — 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.