Nested moment coupling

Summary: The dot-square kernel on centered isotropic nested subspaces gives a minimum-depth Gram matrix of full rank, despite factorized bare coefficients.

Applying the specified quadratic kernel

This calculation applies the existing cross-sector moment kernel to an arbitrary finite nested chain. It is a flat spatial benchmark, or a tangent-chart calculation with the flat kernel explicitly adopted. It does not replace the actual curved sector geometry or derive its global interaction law.

Use a finite containing spatial space $\mathbb R^D$, with isometric embeddings $E_d:\mathbb R^d\to\mathbb R^D$ whose images are nested. Set $P_d=E_dE_d^{\mathsf T}$. All these directions are ordinary spatial directions in the same containing manifold; the projectors express relations between subspaces, not an inherent split in any particle. Embedding a sector's moment tensor in this way is a coordinate identification inside the kernel. It does not assign a delta-sheet density to the particle or remove the lower-sector field's specified cylindrical continuation through the containing directions.

Let $\rho_d$ be a centered, normalized density with finite second moment

$$M_d=\int xx^{\mathsf T}\rho_d(x)\,dx.$$

Use the real positive bare factorization $g_{de}=v_dv_e$, $v_d>0$, and the kernel

$$K_{de}(x,y)=v_dv_e\left(x^{\mathsf T}E_d^{\mathsf T}E_e y\right)^2.$$

The common origin, embeddings, and profiles are fixed in this calculation. Allowing their motion requires the corresponding variations. Pairwise integration, using $(u^{\mathsf T}w)^2=\operatorname{tr}(uu^{\mathsf T}ww^{\mathsf T})$, gives

$$U_{de}=v_dv_e\operatorname{tr} \left(E_dM_dE_d^{\mathsf T}E_eM_eE_e^{\mathsf T}\right). \tag{1}$$

This result depends only on the second moments, so it holds for every centered profile satisfying the stated moment condition.

One tensor generates self and cross responses

Before imposing isotropy, the same kernel admits a compact expression. For the fixed embeddings and common origin used above, define

$$\mathcal M=\sum_d v_d E_dM_dE_d^{\mathsf T}.$$

Then the complete ordered-pair energy, including its symmetry factor, is

$$\boxed{E_K=\frac12\sum_{d,e}U_{de} =\frac12\operatorname{tr}(\mathcal M^2).}$$

Allow the densities to vary before imposing their normalization constraints. The induced moment variation gives

$$\delta E_K=\operatorname{tr}(\mathcal M\,\delta\mathcal M), \qquad \frac{\delta E_K}{\delta\rho_d(x)} =v_d x^{\mathsf T}E_d^{\mathsf T}\mathcal M E_dx.$$

Differentiating again with respect to $\rho_e(y)$ returns exactly $K_{de}(x,y)$. Thus the self-potential and reciprocal cross-response follow from the same tensor functional, with no independently assigned coefficient for each pair. The tensor contains the actual spatial second moments; replacing it by its trace is a further projection. Norm constraints add their usual scalar Lagrange multipliers.

This is an exact rewriting of the stated instantaneous density kernel. The coordinate origin, embeddings and physical currents must still be derived or varied as appropriate before it becomes a complete field action. geometric_channel_checks.py compares the tensor energy with independent two-position quadrature, checks its full density variation and common spatial rotation covariance, and retains all six geometric channels of the nested control.

Isotropic sectors give a minimum-depth matrix

Suppose each sector is isotropic in its own ordinary spatial directions: $M_d=s_dI_d$, $s_d>0$. Write $u_d=v_ds_d$. Nesting implies

$$P_dP_e=P_{\min(d,e)},\qquad \operatorname{tr}(P_dP_e)=\min(d,e).$$

Consequently

$$\boxed{U_{de}=u_du_e\,A_{de},\qquad A_{de}=\min(d,e).} \tag{2}$$

For amplitudes $J_d$ multiplying these fixed density profiles, the positive quadratic energy is

$$U(J)=\frac12\sum_{d,e}J_d u_d A_{de}u_e J_e =\frac12\left\|\sum_d u_dJ_dP_d\right\|_{\rm F}^2. \tag{3}$$

It has symmetric Hessian $G_{de}=u_du_e\min(d,e)$ and source response

$$\Sigma_d=u_d\sum_e u_e\min(d,e)J_e. \tag{4}$$

The sum in (4) retains the external depth $d$. In general it cannot be replaced by one common scalar $C$ in $\Sigma_d=u_dC$. This is a concrete operator-level test of factorization in the written kernel. It does not infer masses from its Hessian.

Exact rank, determinant, and pair correlation

Order distinct depths $0<d_1<\cdots<d_N\le D$, set $d_0=0$, and define $\Delta_i=d_i-d_{i-1}>0$. For the lower triangular matrix $T_{ij}=1$ if $j\le i$, and zero otherwise,

$$A=T\,\operatorname{diag}(\Delta_1,\ldots,\Delta_N)\,T^{\mathsf T}. \tag{5}$$

Indeed, the $(i,j)$ entry is the sum of the increments through $\min(i,j)$, namely $\min(d_i,d_j)$. Since $T$ is invertible,

$$A>0,\qquad \operatorname{rank}A=N,\qquad \det A=\prod_{i=1}^N\Delta_i. \tag{6}$$

For the working distinct depths $(2,3,4,5,6,10)$, the rank is six and the determinant is $2\cdot1\cdot1\cdot1\cdot1\cdot4=8$. The corresponding $G$ is also full rank when all $u_d\ne0$. Several profiles with the same depth and identical isotropic moments produce proportional columns instead; the theorem counts distinct projectors, not automatically distinct particles.

All the unknown positive $u_d$ cancel from the pair correlation:

$$\boxed{ \frac{G_{de}}{\sqrt{G_{dd}G_{ee}}} =\frac{\min(d,e)}{\sqrt{de}} =\sqrt{\frac{\min(d,e)}{\max(d,e)}}.} \tag{7}$$

Thus distinct depths do not saturate a rank-one response even when their bare coefficients satisfy $g_{de}^2=g_{dd}g_{ee}$. The squared correlation for depths three and four is $3/4$, not one, in this full moment model. These correlations describe the specified scalar kernel; they are not electric-charge ratios, scattering probabilities, or measured coupling constants.

Depth five has a nonzero row and no exceptional decoupling. Electric neutrality and low weak-interaction rates therefore cannot be obtained from this positive scalar moment matrix alone. The appropriate neutrino control needs derived channel maps.

Which projection would give rank one

The decomposition into the trace and traceless part in the containing space is

$$P_d=\frac{d}{D}I_D+\widetilde P_d,\qquad \operatorname{tr}\widetilde P_d=0.$$

Orthogonality in the Frobenius inner product gives

$$A_{de}=\frac{de}{D} +\operatorname{tr}(\widetilde P_d\widetilde P_e),\qquad \operatorname{tr}(\widetilde P_d\widetilde P_e)=\min(d,e)-\frac{de}{D}. \tag{8}$$

The first term has rank one. The second is a positive semidefinite Gram matrix and is generally nonzero. If the largest retained depth is $D$, its last row is zero and it has rank $N-1$: the remaining traceless projectors are independent, as follows from their successive spatial shells.

Keeping only the trace response would produce $G^{\rm trace}_{de}=u_du_e de/D$, with one common response. This is a definite additional projection of the interaction. It is not implied by each profile being isotropic on its own sector, because $P_d$ is not isotropic on the entire containing space when $d<D$.

Equation (8) identifies the extra physical condition a scalar common-anchor mechanism would need in this model: the actual interaction must select the trace channel, or otherwise make the normalized response profiles proportional. It must also explain what happens to the traceless response, preserve the kinetic normalization, and establish the relation to the complete occupied mass operator.

Verification and related

generalized_coupling_checks.py computes the double profile integral by exact second-moment cubature after arbitrary common spatial rotations, checks the Hessian by finite differences, verifies (5)–(8), and compares all six depth rows. No empirical parameters are fitted.

• Generalized sector coupling — the complete coupling construction and its physical scope.
• Common field response — gives the general Gram and proportional-profile criterion; its eliminated scalar energy has a different sign from (3).
• Rank-one common anchor — the source-cancellation proposal to which this moment calculation supplies an explicit test.