Primitive cocharacter norm test: the desired amplitude charge is the unique minimum-norm choice

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 78: Primitive cocharacter norm test: the desired amplitude charge is the unique minimum-norm choice.

Provenance

Preserved from interaction notebook section 78, 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.

78. Primitive cocharacter norm test: the desired amplitude charge is the unique minimum-norm choice

Section 77 showed that a primitive clock gap does not by itself force the signed amplitude charge to be

$$q=1.$$

The whole integer family

$$(q,-q,-1,0), \qquad q\in\mathbb Z_{>0},$$

is projectively primitive because the clock already contributes a unit weight difference.

There is, however, a canonical invariant discriminator: the norm of the traceless $SU(4)$ generator.

78.1 Traceless representative of the integer family

Start from the projective weights

$$\lambda(q)=(q,-q,-1,0).$$

Their mean is

$$\bar\lambda=-\frac14.$$

Subtracting the central component gives the traceless representative

$$\boxed{ \widetilde\lambda(q) =\left( q+\frac14, -q+\frac14, -\frac34, \frac14 \right). }$$

Using the Frobenius trace normalization on diagonal generators (equivalent to the Killing form up to an overall positive constant on $\mathfrak{su}(4)$), its squared norm is

$$\begin{aligned} \|\widetilde G(q)\|^2 &=\left(q+\frac14\right)^2 +\left(-q+\frac14\right)^2 +\left(-\frac34\right)^2 +\left(\frac14\right)^2\\ &=\boxed{2q^2+\frac34}. \end{aligned}$$

For positive integer $q$, this is strictly increasing with $q$.

Therefore

$$\boxed{q=1}$$

is the unique minimum-norm nonzero signed amplitude charge compatible with the primitive unit clock.

78.2 The minimum reproduces the existing electron generator exactly

At

$$q=1,$$

the traceless weights are

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

which are exactly the weights of

$$\widetilde G =\frac14\operatorname{diag}(5,-3,-3,1).$$

Thus the desired orbit generator is equivalently characterized as:

the unique minimum-norm traceless representative among primitive integral $U(1)$ generators having one signed amplitude pair and one primitive clock pair.

This is a substantially stronger characterization than simply choosing amplitude weights $\pm1$ by convention.

78.3 Relation to the existing minimum-curvature principle

The magnetic-flux module already proves that inside a fixed Chern class the harmonic Kähler curvature is the unique minimum-norm curvature representative:

$$\|F\|^2 =\|F_k^{\rm harm}\|^2+\|d\alpha\|^2 \geq\|F_k^{\rm harm}\|^2.$$

The present result is the representation-space analogue:

$$\boxed{ \text{primitive integral generator class} \quad\Longrightarrow\quad \text{minimum-norm cocharacter }q=1. }$$

If the projected master dynamics minimizes both the curvature norm and the norm of the induced primitive generator, then the amplitude charge magnitude is no longer arbitrary.

78.4 What is proved and what remains conditional

The mathematical minimization is exact:

$$\boxed{ \|\widetilde G(q)\|^2=2q^2+\frac34 }$$

and therefore $q=1$ is uniquely minimal for $q\ge1$.

What is not yet proved is that the physical projected electron connection must choose the minimum-norm primitive cocharacter among all allowed integral generators. The existing V2 flux module establishes minimum norm for the curvature representative within a fixed topological class, not automatically for the choice of Lie-algebra cocharacter itself.

So the new result supplies a clean variational closure candidate, but one action-level statement is still required:

$$\boxed{ \text{physical projected connection minimizes the induced generator norm over the allowed primitive cocharacters.} }$$

If that statement follows from the same quadratic connection energy already used for the harmonic-curvature argument, then

$$q=1$$

is derived rather than assumed.

78.5 Consequence if the minimum-norm principle is valid

Combining the results of sections 75--78 gives

$$\boxed{ \begin{aligned} &\text{primitive clock} &&\Rightarrow &&\Delta_{\rm clock}=1,\\ &\text{minimum-norm primitive signed amplitude cocharacter} &&\Rightarrow&& q=1,\\ &\text{signed amplitude pair} &&\Rightarrow&& \Delta_{\rm amp}=2,\\ &\text{locked-spin gap theorem} &&\Rightarrow&& \widetilde G=-\frac34I+\frac12J_z+\frac32J_z^2,\\ &\Lambda^2\mathbf4 &&\Rightarrow&& 1:1:3. \end{aligned} }$$

This is the cleanest non-circular route found so far to the dimensionless orbit-generator spectrum.

Related

• Research status — current proof boundaries and correction guidance.
• Extremal-doublet audit: the two-unit gap follows from a primitive signed amplitude charge, but that charge is still an input — preceding section in the research sequence.
• Rejected identification audit: an embedded-$U(1)$ curvature would select $q=1$, but it is not the Hopf connection — 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.