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: Auxiliary model only: The positive rank-one scalar moment model has a unique symmetric ground-state covariance and a calculable mixed-source susceptibility.

Interpretation correction

This calculation concerns an auxiliary field or local spin model, not the actual electron orbit constructed in the 6D demo. Its stationary states, angular expectations, and stability results have not been mapped to that trajectory or its observer readout. They must not be used as a replacement for the existing orbit-shape derivation. See Actual orbit research correction.

Question and source boundary

Does retaining the unaveraged second moments of the historical quadratic kernel actually generate the missing oriented electron frame? A prescribed anisotropic density can give a directional potential; that does not establish that the coupled ground state creates the requisite anisotropy itself.

The source audit remains open: polarization module §13 specifies the target block P_d D_infinity P_e, not its computed matrix elements. Self-contact §8.6 separately records the unresolved coordinate-kernel to phase-ray bridge.

This note derives a result for the imported flat scalar functional, with a specified nested 3-in-6 embedding. Both field norms are one, both kinetic coefficients are set to one in common dimensionless coordinates, centroids are fixed at zero, and couplings v_3,v_6 are positive. No orbital angular momentum, external field, derivative vertex, or excited-state constraint is imposed. Cross energy is a product-density Hartree functional.

1. Exact reduction to moment matrices

Let P=(I_3,0), rho_d=|psi_d|^2, and M_d=integral xx^T rho_d(x) dx. Assume finite positive-definite second moments and sufficient decay for integration by parts. Use self couplings g_dd=v_d^2 and cross coupling g_36=v_3 v_6. The energy is

E=T3+T6+v322tr⁡M32+v622tr⁡M62+v3v6tr⁡(M3PM6PT). E=T_3+T_6+\frac{v_3^2}{2}\operatorname{tr}M_3^2 +\frac{v_6^2}{2}\operatorname{tr}M_6^2 +v_3v_6\operatorname{tr}(M_3 P M_6P^T).

There is an exact kinetic lower bound, not a Gaussian ansatz:

0≤∫∣∇ψ+12M−1xψ∣2dx=T−14tr⁡M−1. 0\le\int\left|\nabla\psi+\tfrac12 M^{-1}x\psi\right|^2dx =T-\tfrac14\operatorname{tr}M^{-1}.

The cross term follows from integration by parts and unit norm. Equality holds for the constant-phase Gaussian proportional to exp(-x^T M^-1 x/4), whose density covariance is M. Every positive-definite M is realized by such a normalized Gaussian. Therefore minimizing the field energy is exactly equivalent, within this model, to minimizing

F(M3,M6)=14∑d=3,6tr⁡Md−1+12tr⁡W2,W=v3PTM3P+v6M6. F(M_3,M_6)=\frac14\sum_{d=3,6}\operatorname{tr}M_d^{-1} +\frac12\operatorname{tr}W^2, \qquad W=v_3P^TM_3P+v_6M_6.

This square uses the mutually compatible fixed embedding and positive rank-one couplings. An arbitrary collection of pair identifications need not admit the same common-space square.

2. Strict convexity excludes a hidden broken-symmetry minimum

For symmetric variations H_d, the second variation is

δ2F=12∑dtr⁡(Md−1HdMd−1HdMd−1)+tr⁡(v3PTH3P+v6H6)2>0 \delta^2F=\frac12\sum_d\operatorname{tr} (M_d^{-1}H_dM_d^{-1}H_dM_d^{-1}) +\operatorname{tr}(v_3P^TH_3P+v_6H_6)^2>0

for any nonzero joint variation. In an eigenbasis of M, the first trace is a sum of positive multiples of H_ij^2. The functional is strictly convex on the product of positive-definite cones. It diverges at a singular covariance and as a covariance eigenvalue tends to infinity: positive self terms ensure coercivity. A unique minimizing covariance pair exists.

Independent rotations R in SO(3) of the observed coordinates and S in SO(3) of the unobserved coordinates preserve F under

M3↦RM3RT,M6↦diag⁡(R,S)M6diag⁡(R,S)T. M_3\mapsto RM_3R^T,\qquad M_6\mapsto\operatorname{diag}(R,S)M_6\operatorname{diag}(R,S)^T.

Uniqueness forces the minimizer to be invariant under this entire group:

M3=aI3,M6=diag⁡(bI3,cI3),a,b,c>0. M_3=aI_3,\qquad M_6=\operatorname{diag}(bI_3,cI_3),\qquad a,b,c>0.

In particular its observed-unobserved covariance is zero. The model can distinguish the observed three-plane from its complement through b versus c, but cannot select an oriented isometry between those planes. This is stronger than only finding an isotropic stationary branch: the moment minimum is globally unique. Wavefunction phases remain arbitrary constants and are not fixed by this result.

For background on convex matrix optimization see Boyd and Vandenberghe, Convex Optimization. The specific second-variation proof and model conclusion above are derived here.

3. Closed-form self-consistent solution

Differentiation in the three invariant widths gives

14a2=v3(v3a+v6b),14b2=v6(v3a+v6b),14c2=v62c. \frac1{4a^2}=v_3(v_3a+v_6b),\qquad \frac1{4b^2}=v_6(v_3a+v_6b),\qquad \frac1{4c^2}=v_6^2c.

Writing s_0=v_3a+v_6b, their positive solution is

s0=(v3+v62)2/3,a=12v3s0,b=12v6s0,c=(4v62)−1/3. s_0=\left(\frac{\sqrt{v_3}+\sqrt{v_6}}2\right)^{2/3},\quad a=\frac1{2\sqrt{v_3s_0}},\quad b=\frac1{2\sqrt{v_6s_0}},\quad c=(4v_6^2)^{-1/3}.

Consequently b<c: the extra positive cross interaction tightens the observed coordinate profile. This is a dimensionless width effect in the specified model, not an electron radius prediction or a coupling-to-mass identification.

4. Mixed-source response is computable

Perturb M_6 by an off-diagonal block X and its transpose:

M6=(bI3XXTcI3). M_6=\begin{pmatrix}bI_3&X\\X^T&cI_3\end{pmatrix}.

The exact cross energy depends only on P M_6 P^T and is blind to X. The self interaction and kinetic term yield

F=F∗+α∥X∥F2+O(∥X∥4),α=b+c4b2c2+v62>0. F=F_*+\alpha\|X\|_F^2+O(\|X\|^4),\qquad \alpha=\frac{b+c}{4b^2c^2}+v_6^2>0.

Allowing the diagonal widths to relax does not change this quadratic coefficient: diagonal-mixed Hessian blocks vanish at X=0. Thus there is no spontaneous mixed covariance in this minimum.

As a response probe, add -h tr(E^T X), where E is an externally supplied mixed tensor and h is a small parameter. This term is not claimed to be present in the original action. Linear response is

X=h2αE+O(h3),Fmin⁡(h)=F∗−h24α∥E∥F2+O(h4). X=\frac{h}{2\alpha}E+O(h^3),\qquad F_{\min}(h)=F_*-\frac{h^2}{4\alpha}\|E\|_F^2+O(h^4).

The same rank-two source discussed in the Procrustes calculation would therefore induce rank-two mixed covariance at leading order. This provides a definite susceptibility for a supplied source, not a derivation of that source or its coupling to the spinor mixed block P_O J P_H.

5. What advances and what remains

The imported functional now has an explicit coupled ground-state solution, a positive full covariance Hessian, and a mixed response coefficient. Simply retaining second moments cannot supply the missing oriented lock in this positive, scalar, ground-state model. No physical time-dependent lock frequency is obtained from the covariance Hessian without a dynamical kinetic form.

This does not rule out driven or angular-momentum-carrying modes, nonfactorizing spinor vertices, correlated states beyond the product-density functional, different embedding dynamics, or other interaction terms. The useful next target is a derived mixed source or nonfactorizing transition from P_3 D_infinity P_6, not an assumed anisotropic ground-state density.

Verification

moment_kernel_checks.py checks stationarity, all 27 covariance-Hessian directions, random positive-definite comparisons, the mixed-block curvature against finite differences, the response equation, and Gaussian saturation at matrix level. Numerical checks supplement the analytic proof; they do not establish the source-to-V2 identification.