Frozen magnetic texture test

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: A frozen-protocol test admits spatial polarization textures into the existing six-dimensional magnetic kernel and treats negative energy curvature as a failure of the uniform preparation.

Frozen question

This test holds its assumptions fixed while testing whether spatial textures destabilize the proposed state. The protocol fixes the kernel, envelope, inherited illustrative coefficients, perturbations, parameter scan, and rejection thresholds before the texture calculations. This is a model robustness test, not a blind experimental comparison.

The earlier magnetic calculation minimizes over fields at one spatially uniform spinor polarization. This test removes that uniformity restriction within a specified family, and includes the positive polarization-gradient cost. No force is added if the uniform preparation fails.

The physical electron trajectory remains a separate object: the field polarization varied here is not silently substituted for an actual orbit or interpreted as the electron's probability cloud.

Reference energy and perturbations

With $n(x)=(2\pi)^{-3}e^{-|x|^2/2}$ fixed, $z^\dagger z=1$, and $J_{ij}(z)=z^\dagger\Sigma_{ij}z$, use

$$E[z]=\tau\int n|\nabla z|^2\,dx -\frac\kappa2\int\|\mathcal P_k\widehat{nJ(z)}(k)\|^2\frac{dk}{(2\pi)^6}.$$

Here $\tau=1/(2m)$, $\kappa=(e/(2m))^2/Z_B$, and $\hbar=1$. The fixed benchmark $m=1.3,e=0.7,Z_B=1.8$ comes from the existing magnetic diagnostic; these are not inferred physical six-dimensional constants. The dimensionless strength is $\lambda=\kappa(4\pi)^{-3}/\tau$.

Test $z_\epsilon(x)=\cos(\epsilon f(x))z_0+ \sin(\epsilon f(x))v$, where $z_0=(1,0,0,0)$, $z_0^\dagger v=0$, $\|v\|=1$, and $f$ is a sine or cosine with the wave numbers fixed by the protocol. All six real polarization tangent directions are included. Additional fixed stress strengths test robustness without altering the inherited benchmark.

Results and their independent checks are recorded below, including rejected preparations. No finite scan is called full stability.

Analytic curvature, without fitting

Let $L_a$ be the six real tangent derivatives of $J$ at $z_0$. For a Gaussian Fourier weight shifted by $q e_0$, define

$$a(q)=\int_0^1 e^{-q^2u} \left[\tfrac12(1-u)^2+q^2(1-u)^3\right]du, \quad T(q)=\operatorname{diag}\left(a(q),\frac{1-a(q)}5 I_5\right).$$

This follows from a Laplace representation of $1/|k|^2$ in the original six-dimensional Gaussian integral; $a(0)=1/6$. With $G_{ab}(q)=\operatorname{tr}(L_a^TL_b[T(q)-I_6/6])$, the Hessian of $E/\tau$ in the prescribed texture family is

$$\mathcal H_f(q,\lambda)=q^2(1\mathord\pm e^{-2q^2})I_6 -\frac\lambda2(1\mathord\pm e^{-q^2})G(q),$$

where plus denotes sine and minus cosine. The magnetic part has four destabilizing and two restoring directions for $q>0$. The positive gradient term competes with it; positivity of the gradient coefficient does not automatically win at arbitrary magnetic strength.

Uniform polarization is stationary even against general spatial tangent variations in this fixed-envelope magnetic model. For every wavevector, $\langle\mathcal P_kJ_0,\mathcal P_kL_a\rangle=0$, obtained by differentiating the exact pure-spinor identity $\|\mathcal P_kJ(z)\|^2=1$. Thus a negative Hessian here is an actual energy instability of a stationary preparation, not merely failure to expand about a solution.

Recorded outcome

The protocol was written at approximately 2026-09-08 06:38:40 UTC, before the numerical scan. Its SHA-256 is 631f7e179bd1d7edaee66620167944fd5063adf03cc77c5605feb4a10bc31c03. The analytic perturbation method was selected beforehand; this is not being represented as an independent experimental preregistration.

The inherited benchmark gives $\lambda=5.27619253214\times10^{-5}$. Every one of its ten tested sine/cosine wavelength cases has positive curvature in all six polarization tangent directions. The smallest tested eigenvalue is approximately $0.00734391$. The scan alone does not prove positivity for arbitrary mixtures of wavelengths. The analytic supplement below establishes a sufficient all-texture second-variation bound, not a nonlinear global minimum.

Across the fixed benchmark plus four stress strengths, 23 of 50 cases are rejected for negative curvature. The smallest critical strength on the predeclared grid is approximately $5.69208$, for the sine at $q=0.5$. This is a sampled threshold, not a globally minimized critical value.

At $\lambda=32,q=1,\epsilon=0.005$, the independently computed energy changes $[E(z_\epsilon)-E(z_0)]/\tau$ are

• sine: $-6.14154273819\times10^{-5}$;
• cosine: $-2.41311245404\times10^{-5}$.

Increasing retained harmonics from 8 to 12, phase samples from 256 to 512, and tightening quadrature reproduces both energy changes to displayed precision. These are explicit lower-energy witnesses, not eigenvalues alone. The stronger claim of stability throughout positive coupling is therefore false for this specified model. No kernel or stabilizing term was changed after that result.

Analytic supplement: an all-texture weak-coupling bound

This supplement was derived after the frozen scan, with the same energy. It is not an additional predeclared numerical outcome. Use the bivector inner product $\langle A,B\rangle=\frac12\operatorname{tr}(A^TB)$. For a horizontal tangent $u(x)\in\mathbb C^3$, write

$$z_\epsilon=(\sqrt{1-\epsilon^2|u|^2},\epsilon u),\qquad J(z_\epsilon)=J_0+\epsilon L(u)+\epsilon^2R(u)+O(\epsilon^3),$$

where $R(u)=J(uu^\dagger)-|u|^2J_0$ and $u$ is embedded orthogonally to $z_0$. Direct Clifford algebra gives $\|L(u)\|^2=8|u|^2$. Start with smooth bounded tangents; the resulting quadratic-form bounds extend to finite Gaussian-weighted $H^1$ tangents by density.

Let $A=\mathcal F^{-1}[\mathcal P_k(\widehat n J_0)]$ and $S(A)=\sum_{i<j}A_{ij}\Sigma_{ij}$. For each nonzero $k$, $\mathcal P_kJ_0$ is a unit simple bivector: $J_0^2=-I$ makes $\widehat k$ and $J_0\widehat k$ orthogonal unit vectors. Clifford multiplication therefore gives $S(\mathcal P_kJ_0)^2=I$ and operator norm one. Since the Gaussian $\widehat n$ is nonnegative,

$$\|S(A(x))\|_{\rm op}\leq \int\widehat n(k)\frac{dk}{(2\pi)^6}=n_{\max}=(2\pi)^{-3}.$$

The coefficient $\mathscr Q(u)$ of $\epsilon^2$ in the energy is one half of its second variation:

$$\mathscr Q(u)=\tau\int n|\nabla u|^2 -\frac\kappa2\|\mathcal P[nL(u)]\|_{L^2}^2 -\kappa\int n\langle A,R(u)\rangle.$$

The orthogonal-projection contraction and Parseval identity bound the second term below by $-4\kappa n_{\max}\int n|u|^2$. Also $\langle A,R(u)\rangle=u^\dagger S(A)u-|u|^2z_0^\dagger S(A)z_0$, so the last term is bounded below by $-2\kappa n_{\max}\int n|u|^2$.

Every spatially uniform ray is stationary against arbitrary textures, as shown above. Differentiating that stationarity along the uniform-ray family shows that constant tangents have zero mixed Hessian with every tangent. Thus $\mathscr Q(u)=\mathscr Q(u-\bar u)$, where $\bar u=\int nu$. For the unit-variance Gaussian, $\int n|u-\bar u|^2\leq\int n|\nabla u|^2$: in an orthonormal Hermite expansion the two sides are respectively $\sum_{|\alpha|\geq1}|c_\alpha|^2$ and $\sum_{|\alpha|\geq1}|\alpha||c_\alpha|^2$. Consequently,

$$\boxed{\mathscr Q(u)\geq \tau(1-48\lambda)\int n|\nabla u|^2.}$$

Here $n_{\max}/\int n^2=8$. Therefore $\lambda<1/48$ is a sufficient, non-sharp condition for positive second variation for every nonconstant polarization texture, modulo neutral global ray rotations. The inherited benchmark satisfies it, with margin $1-48\lambda=0.9974674275845725$. A spatial common-phase perturbation has no magnetic variation and adds only nonnegative gradient energy.

This is an all-texture infinitesimal result within the stated fixed-envelope energy. It does not prove a nonlinear global minimum, stability against envelope relaxation, or stability of the unreduced electron/nucleus system. The bound's sufficient threshold is not the actual transition threshold; the much larger sampled critical strengths remain separate results.

Verification and model boundary

frozen_magnetic_texture_checks.py checks the protocol fingerprint, first variations, all six-direction curvatures, and direct nonlinear Fourier energies for 108 selected tangent cases at three perturbation sizes each. It also verifies the shifted angular integral independently using Cartesian Hermite and radial Laguerre quadrature. The latter converges to within $6\times10^{-8}$ of the analytic reduction on the prescribed wavelength grid.

The initial report is preserved. A subsequent check corrected the implementation of the agreement gate to use the protocol's strict absolute-plus-relative rule, without a unit-size relative-error floor; all comparisons still passed. The verified result includes that strict gate and the independent angular integration.

The analytic-supplement report adds 80 independently sampled projected-source matrix-spectrum checks and the Gaussian normalization check, for 485 runtime conditions. The theorem rests on the algebra and inequalities above, not those finite samples. Both normal and optimized-Python executions retain the runtime checks. In-memory negative controls replacing the angular response by its isotropic value or reversing the Hessian's magnetic sign were rejected by the independent integrations; neither change was saved to the model.

The regression report records all 65 selected diagnostic scripts and source checks passing before the analytic supplement's extra checks were added. The two expensive cylindrical/fixed-Casimir continuation searches were explicitly excluded. After that addition, this script passed all 485 conditions in both normal and optimized Python. Diagnostic success includes correctly detecting rejected model preparations; it is not a count of successful physical predictions.

This remains a fixed-envelope, leading-order spin-magnetic reduction with a positive bare polarization-gradient cost. The minimal-coupling orbital $A$ terms, electric response, radiation, profile relaxation, nuclear dynamics, and higher-order Dirac terms are not included. Strong-coupling stress failures need not lie in a controlled nonrelativistic regime. Neither the inherited illustrative benchmark nor its field-envelope scale has been justified as a physical electron parameter set.

The concrete advance is a calculable stability boundary for the existing response kernel. It is not a rejection or confirmation of the electron's spatial depth, and it does not turn the polarization field into an orbit.

Related

• Localized polarization stationarity — separately frozen five-dimensional control for the same spatial-variation question.
• Magnetic bivector master bridge — supplies the unchanged field constraint, Pauli coefficient, and uniform-sector result being tested.
• Dimensional magnetic kernel comparison — identifies the uniform-polarization restriction and dimension-dependent response.