Frequency normalization requires the correct lock coordinate and kinetic metric

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 58: Frequency normalization requires the correct lock coordinate and kinetic metric.

Provenance

Preserved from interaction notebook section 58, 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.

58. Frequency normalization requires the correct lock coordinate and kinetic metric

The projective coordinate must be tied to the physical product-spin basis before its Fubini--Study metric can be used for a lock frequency. A unit-speed normal curve through a maximally entangled ray is

$$Z(t)=\bigl(\cos(\pi/4+t),0,0,\sin(\pi/4+t)\bigr).$$

It has concurrence $\mathcal C=|\cos2t|$, lock invariant $s=\sin^2(2t)$, and $g^{\rm FS}_{tt}=1$ in the variance normalization. The product-state curve $(\cos t,\sin t,0,0)$ does not have $s=\sin^2t$.

For the aligned trace potential from Section 57, near $t=0$,

$$V(t)=2\kappa(1-\sqrt{1-s(t)}) =2\kappa(1-\cos2t), \qquad V''(0)=8\kappa.$$

With an explicitly assumed quadratic kinetic term $\tfrac12R_6^2\dot t^2$, this normal-coordinate frequency is

$$\boxed{\omega_{\rm normal}^2=8\kappa/R_6^2.}$$

Using $u=2t$ instead gives $s=\sin^2u$, $g_{uu}=R_6^2/4$ and $V''(0)=2\kappa$, with exactly the same frequency. A coordinate Hessian without its matching kinetic metric is not a frequency prediction.

Even if a spatial Dirac eigenvalue fixes $R_6=\lambda_6/\omega_e$, a dimensionless coefficient remains. Define $\kappa=\widehat\kappa(\omega_eR_6)^2$ in this kinetic convention; then

$$\omega_{\rm normal}/\omega_e=\sqrt{8\widehat\kappa}.$$

Neither $\widehat\kappa$ nor this sigma-model kinetic normalization has been derived from the master action. Orientation modes on the locked orbit are distinct from this normal curve and require their own kinetic metric.

Related

• Research status — current proof boundaries and correction guidance.
• Variational derivation of the scattering-selected frame link — preceding section in the research sequence.
• Audit of existing V2 candidates for the dimensionless lock coefficient — following section in the research sequence.