Linear Dirac-action variation does not fix the projector sigma-model coefficient

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 65: Linear Dirac-action variation does not fix the projector sigma-model coefficient.

Provenance

Preserved from interaction notebook section 65, 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.

65. Linear Dirac-action variation does not fix the projector sigma-model coefficient

Section 64 gave, under its separate-factor Clifford-commutation assumptions, the derivative identity

$$[D,P]=i\gamma^\mu\nabla_\mu P$$

and therefore a canonical positive quadratic norm whose Clifford trace is the Grassmannian sigma-model kinetic density.

The next question is whether the already written linear Dirac action automatically supplies that quadratic norm with fixed coefficient.

It does not.

65.1 Correct moving-subspace spectral test

Let

$$P(t)=R(t)P_0R(t)^T$$

and transport an orthonormal basis of the selected subspace with the same rotation,

$$U(t)=R(t)U_0.$$

The physically relevant compressed operator in the moving frame is

$$H_{\rm eff}(t)=U(t)^TDU(t).$$

For

$$D=\operatorname{diag}(d_1,d_2,d_3,d_4)$$

and simultaneous rotations $1\leftrightarrow3$, $2\leftrightarrow4$, direct calculation gives

$$H_{\rm eff}''(0) = \operatorname{diag} \left( 2(d_3-d_1), 2(d_4-d_2) \right).$$

Thus the second variation of the linear projected operator is controlled by ambient spectral gaps, not by the universal Grassmannian metric.

An earlier fixed-coordinate expectation test produced extra projector-normalization terms; transporting the selected basis removes that artifact and gives the correct gap formula above.

65.2 Scalar orientation-blind limit

If

$$D=mI,$$

then

$$H_{\rm eff}(t)=mI$$

for every projector orientation. Hence the linear scalar Dirac term gives exactly zero frame-lock stiffness.

This is consistent with the earlier result that the scalar self-contact/rest-frequency operator does not by itself select the $3+3$ orientation.

65.3 The positive commutator norm is distinct

For the same motion,

$$\operatorname{Tr}([D,\dot P]^\dagger[D,\dot P]) = 2\left[(d_3-d_1)^2+(d_4-d_2)^2\right].$$

This is a positive quadratic functional of the squared spectral gaps. It is not the same as the second variation of the linear compressed operator, which is linear in the gaps.

Therefore

$$\boxed{ \bar\Psi PDP\Psi \not\Rightarrow \operatorname{Tr}([D,P]^\dagger[D,P]) }$$

with a fixed universal coefficient.

65.4 Consequence for projector dynamics

The canonical candidate

$$S_P =\frac{\alpha}{2N_s} \operatorname{Tr}([D,P]^\dagger[D,P])$$

gives the Grassmannian kinetic metric only under the additional assumptions in section 64, but the current V2 action does not force this term merely from the unit coefficient of the linear Dirac kinetic term.

The overall coefficient

$$\boxed{\alpha}$$

therefore remains a genuine action-level quantity unless an additional master principle fixes it.

65.5 Status

The projector route survives, but one shortcut is ruled out:

$$\boxed{ \text{unit normalization of the linear Dirac action does not fix the projector sigma-model normalization.} }$$

What remains viable is either:

1. a genuine master-field quadratic commutator/projector term;
2. a symmetry or normalization principle fixing its coefficient;
3. or a nonperturbative relation tying the projector-gradient norm to an already derived V2 spectral ratio.

Related

• Research status — current proof boundaries and correction guidance.
• Dirac-projector commutator: a conditional sigma-model construction — preceding section in the research sequence.
• Orbit-generator spectrum versus the physical mixed block — following section in the research sequence.