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.

Source and scope

Starting point: Gravity §3.10.1–2 and Appendix B §20a. The explicit rectangular identification and tensor generalization below are an adaptation derived during the import review, not a theorem quoted from V1.

Retain the second moments

Let x in R^d, y in R^e, and let B:R^e -> R^d be a specified identification. For unit-normalized densities define

KB(x,y)=g(xTBy)2,Mx=∫xxTρx(x) dx,My=∫yyTρy(y) dy. K_B(x,y)=g(x^{\mathsf T}By)^2,\qquad M_x=\int xx^{\mathsf T}\rho_x(x)\,dx,\quad M_y=\int yy^{\mathsf T}\rho_y(y)\,dy.

Then, exactly,

Vx(x)=gxTBMyBTx,Uxy=gtr⁡(MxBMyBT). V_x(x)=g x^{\mathsf T}B M_y B^{\mathsf T}x,\qquad U_{xy}=g\operatorname{tr}(M_x B M_y B^{\mathsf T}).

For fixed M_y and B, the coordinate Hessian of V_x is 2g B M_y B^T. This is not automatically the full coupled-field Hessian: the background moments can fluctuate too.

For equal dimensions and B=I, write M_x=aI+Q_x and M_y=bI+Q_y, with traceless Q_x,Q_y. Then

Uxy=g[dab+tr⁡(QxQy)]. U_{xy}=g\left[dab+\operatorname{tr}(Q_xQ_y)\right].

The historical isotropic reduction sets both Q tensors to zero. Retaining them gives an explicit orientation-sensitive term. Its sign and eigenvalue structure must determine the preferred orientation; alignment is not automatic. Even a preferred quadrupole-axis alignment need not determine a unique signed frame or the electron's complex-structure polarization.

Symmetry and missing inputs

Under independent orthogonal frame changes x->R x and y->S y, covariance requires B->R B S^T. Holding B fixed preserves only compatible transformations. An unspecified cross-sector dot product is therefore not invariant under arbitrary independent rotations. For complex coordinates, specify a Hermitian kernel such as |x-dagger B y|^2 and its representation separately.

Rank-one sector coefficients g_de=v_d v_e do not force the coordinate kernel or its projected matrix elements to be rank one.

Promising next calculation: derive B and background moments from the V2 field, retain their fluctuations, and project onto the actual polarization states. Do not identify this scalar coordinate Hessian with a spin-dependent lock operator without that projection and the corresponding kinetic form.