Localized polarization stationarity

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: With the same localized Gaussian envelope and magnetic kernel, uniform five-dimensional polarization has a nonzero spatial first variation whereas six-dimensional pure polarization is stationary.

Separately frozen dimensional control

The control protocol was written after the six-dimensional texture test and before evaluating this comparison. It is not retroactively part of that earlier protocol. Its fingerprint is e0ce516ebe9a85347dffa58e35df581060e2a0bca9076e18e781f9c531f7e362.

Keep the same closed-field magnetic kernel, fixed normalized unit-variance Gaussian envelope in each ambient dimension, and positive bare gradient energy. Use the fundamental Spin(5) spinor or one Spin(6) half-spinor, each in $\mathbb C^4$. Do not add the scalar-exchange vector channel. The first three spatial coordinates remain the visible three-plane in both.

This tests the uniform-polarization preparation, not every possible field solution in five dimensions and not the particle-depth assignment.

Exact first-variation distinction

In six dimensions $\|\mathcal P_kJ(z)\|^2=1$ for every pure ray and every nonzero wavevector. Differentiating at a uniform ray gives

$$\langle\mathcal P_kJ_0,\mathcal P_k\delta J\rangle=0.$$

Every spatial tangent variation is a sum of these constant spinor tangent matrices with scalar spatial coefficients. The identity makes the magnetic first variation vanish even for spatially varying coefficients, not only for a uniform rotation. The gradient term also has zero first variation at a uniform field. Thus the six-dimensional uniform preparation is stationary for the stated fixed-envelope model.

In five dimensions the identity is instead $\|\mathcal P_kJ\|^2=1-(n\cdot\widehat k)^2$. Its derivative is

$$\langle\mathcal P_kJ_0,\mathcal P_k\delta J\rangle =-(n_0\cdot\widehat k)(\delta n\cdot\widehat k).$$

It vanishes after an isotropic angular average for a uniform variation, but not for every spatially modulated variation. Choose $n_0=e_4$, $w=(e_0+e_4)/\sqrt2$, and $f(x)=\cos(qw\cdot x)$. These are zero-based coordinate labels. One unit spinor tangent gives $\delta n=2e_0$.

The Gaussian Fourier overlap contributes $e^{-q^2/4}$ and an angular tensor $T_d(q/2,w)=bI+(a-b)ww^T$. Its anisotropy has the exact expression

$$a_d(s)-b_d(s)=s^2\int_0^1(1-u)^{d/2}e^{-s^2u}\,du>0 \qquad(s\ne0).$$

Consequently that tangent has nonzero first variation

$$\frac{\delta E_5}{\kappa\int n(x)^2dx} =e^{-q^2/4}\left[a_5(q/2)-b_5(q/2)\right]>0.$$

Reversing the tangent lowers the energy. The bare positive gradient cost begins at second order in the perturbation, so no finite positive gradient coefficient restores exact stationarity at a nonzero magnetic coupling. This is a first-variation obstruction, not a threshold instability. The same argument applies to any uniform pure five-dimensional ray: its null axis can be rotated to $e_4$ and its ray tangents include changes of that axis. A textured five-dimensional equilibrium is not excluded.

Numerical verification under the frozen rules

The gradient norm of $E/(\kappa\int n^2)$ over the six real spinor tangents is:

All three five-dimensional preparations fail the $10^{-9}$ stationarity threshold; all three six-dimensional preparations pass this first-variation test. Direct nonlinear spinor-to-Fourier-field calculations at the two predeclared amplitudes reproduce the gradients within $2.8\times10^{-7}$, below the independent $10^{-6}$ agreement limit. Reversing the gradient gives explicit lower magnetic energies. The finite-amplitude witnesses do not include the gradient cost; the existence of a downhill direction for the complete specified energy follows from its nonzero first derivative.

localized_polarization_stationarity_checks.py performs 17 independent checks and records the model verdicts separately from diagnostic success. The complete report is dimensional-control-results.json.

What the comparison establishes

Five dimensions fail that stronger requirement in this magnetic model; six dimensions satisfy stationarity and the inherited weak-coupling texture test.

Six-dimensional stationarity is not unconditional stability: the separate frozen texture test found negative curvature in stronger-coupling cases. The kernel, envelope, leading-order approximation, omitted orbital/electric terms, and unidentified physical observable remain explicit premises. This is a conditional discriminator between specified preparations—not experimental proof that electrons have six spatial dimensions.

Related

• Frozen magnetic texture test — six-dimensional stability test, including its rejected strong-coupling preparations.
• Dimensional magnetic kernel comparison — supplies the null-axis and complex-structure identities used here.