Common field response

Summary: Eliminating stable response modes of the same field gives a Gram correction to the direct coupling, with exact reciprocity and a proportional-profile criterion for rank one.

Direct coupling and reduction of one field

IDWT has one field $\Psi_\infty$ and no mediator particles. Its cross-sector interactions belong to that field's own action. Start with a specified background and independent real coordinates for its admissible variations in a finite region:

$$\delta\Psi_\infty=S_{\rm r}j+S_{\rm s}X.$$

Here $j$ describes retained variations and $X$ describes other variations of the same field. The maps must respect the actual spatial embeddings, constraints, and normalization. Real coordinates may describe a complex classical field or a specified mean-field energy; they add no physical dimensions. Independent variations do not imply separate physical fields.

At a stationary background, the energy's second variation has blocks

$$E^{(2)}=\frac12\langle j,H_{\rm rr}j\rangle +\langle X,H_{\rm sr}j\rangle+\frac12\langle X,LX\rangle, \qquad H_{\rm rs}=H_{\rm sr}^{*}.$$

Set $B=-H_{\rm sr}$. Thus both the direct block $H_{\rm rr}$ and the response source $Bj$ are derived from the same Hessian. A physical application must calculate these blocks; specifying them as a benchmark does not derive them from IDWT. Constraints and any zero modes must be handled before an inverse is taken.

For retained source currents $j_d$ on a finite collection of nested sectors, a source term $\sum_dB_dj_d$ is a further specified reduction of this one-field energy. Each map includes spatial identifications, measures, and polarization contractions. Its range is a real space $\mathcal H$ of response profiles of the same field. Nonlinear bilinear currents require the corresponding expansion and variations; they are not automatically identical to the linear coordinates above.

Assume a fixed positive self-adjoint stiffness $L\ge\ell_0 I>0$ on $\mathcal H$ and sources for which $L^{-1/2}B_dj_d$ exist. Work in a finite region with stated boundary conditions, or on a domain where these operator assumptions hold. An operator with zero modes requires a separate constraint, boundary, or limiting treatment. For this conditional response reduction, write

$$E[X,j]=E_{\rm direct}[j]+\frac12\langle X,LX\rangle -\left\langle X,\sum_dB_dj_d\right\rangle. \tag{1}$$

The currents and all maps use a specified common unit convention. Field normalization acts on every leg of a current as explained in Canonical coupling normalization.

Exact elimination and reciprocity

Set $S=\sum_dB_dj_d$. Completing the square gives

$$E=E_{\rm direct} +\frac12\|L^{1/2}(X-L^{-1}S)\|^2 -\frac12\langle S,L^{-1}S\rangle.$$

At fixed sources the unique minimizing response is $X_*=L^{-1}S$. The eliminated energy and its blocks are therefore

$$E_{\rm eff}=E_{\rm direct}-\frac12\sum_{a,b} \langle j_a,G_{ab}j_b\rangle,\qquad G_{ab}=B_a^*L^{-1}B_b,\qquad G_{ba}=G_{ab}^*. \tag{2}$$

The induced source potential is $-\sum_bG_{ab}j_b$, including the sign from (1). In the quadratic case the complete retained Hessian is $H_{\rm eff}=H_{\rm rr}-B^*L^{-1}B$. The Gram expression is a response correction, not the entire direct interaction. The full energy must still control stability as sources vary; positivity of the response stiffness alone does not do that. Different physical interaction channels can have different signs, constraints, derivative vertices, and tensor contractions.

Equations (1)–(2) give every pair in this correction from the same field-response operator and source maps. There is no independently adjustable coefficient for each pair in the correction once $B_d$ and $L$ have been specified. There can still be sector parameters inside $B_d$: a reduction in pairwise freedom is not their derivation.

Rank measures independent response profiles

Collect $B=(B_1,\ldots,B_N)$ and define $C=L^{-1/2}B$. Then

$$G=C^*C,\qquad \langle j,Gj\rangle=\|Cj\|^2\ge0,\qquad \ker G=\ker C. \tag{3}$$

On a finite retained source space, $\operatorname{rank}G=\operatorname{rank}C$. It is bounded by the number of independent response modes of the field to which the retained sources couple. One field has many spatial modes, so one field does not force this rank to be one.

For one fixed, normalized source profile $f_d$ in each sector, let $b_d=B_df_d$ and $v_d=L^{-1/2}b_d$. The pair numbers are

$$g_{ab}=\langle v_a,v_b\rangle,\qquad |g_{ab}|^2\le g_{aa}g_{bb}. \tag{4}$$

This is Cauchy–Schwarz in the response metric. If $g_{aa}=0$, then $v_a=0$ and $g_{ab}=0$ for every $b$ in this channel. Vanishing coupling to one particular other source only establishes orthogonality to that source's response, not absence from the whole channel.

When $g_{aa},g_{bb}>0$, equality in (4) holds precisely when $v_a,v_b$ are proportional. The complete nonzero matrix has rank one precisely when all nonzero response profiles lie on one common line:

$$v_d=\alpha_d v,\qquad g_{ab}=\alpha_a\alpha_b\|v\|^2. \tag{5}$$

For complex channel amplitudes, transpose becomes Hermitian adjoint, and the last product is $\overline{\alpha_a}\alpha_b\|v\|^2$. The rank and inequality statements are unchanged.

Thus a factorization $B_d=\alpha_d\widetilde B_d$ of bare coefficients is insufficient for (5); the $\widetilde B_df_d$ must generate the same spatial response up to scale. This is the exact additional condition needed by a common-anchor response. That note's subsequent conversion of source potentials to occupied mass poles still needs its kinetic and reference-count factors. Its displayed positive current-current energy also differs in sign from the negative term induced by (1). The shared rank criterion does not identify the two actions: an application must derive the interaction sign and the physical mass branch as well as the profile factorization.

These positivity and rank statements concern the static scalar response in (1). They are not universal inequalities for the sum of all gauge, gravitational, and matter interaction coefficients. A direct matter term can change the total quadratic form.

Spatial example across all depths

Choose a shared spatial chart $\mathbb R^s$, $s\le d$, and normalized sector densities

$$\rho_d(x,y)=p_{\sigma_d}(x)f_d(y),\quad \int f_d\,dy=1,\quad p_\sigma(x)=\frac{e^{-|x|^2/(2\sigma^2)}}{(2\pi\sigma^2)^{s/2}}.$$

Take source image $b_d=h_d p_{\sigma_d}$ and the local stiffness $L=\kappa I$, $\kappa>0$. This is a declared static contact benchmark, not a derived electromagnetic or weak response law. The exact overlap is

$$g_{ab}=\frac{h_ah_b}{\kappa} \frac{1}{[2\pi(\sigma_a^2+\sigma_b^2)]^{s/2}}, \qquad g_{aa}=\frac{h_a^2}{\kappa(4\pi\sigma_a^2)^{s/2}}. \tag{6}$$

For nonzero $h_a,h_b$, the magnitude of the normalized correlation is

$$\frac{|g_{ab}|}{\sqrt{g_{aa}g_{bb}}} =\left(\frac{2\sigma_a\sigma_b}{\sigma_a^2+\sigma_b^2}\right)^{s/2}. \tag{7}$$

The Gaussian integral can be evaluated one coordinate at a time. The inequality $2\sigma_a\sigma_b\le\sigma_a^2+\sigma_b^2$ proves (4) explicitly, with equality exactly at equal widths. Displaced centers multiply (7) by $\exp[-|x_a-x_b|^2/(2(\sigma_a^2+\sigma_b^2))]$.

For finitely many distinct positive widths, these centered Gaussians are linearly independent: along a radial line their exponentials have distinct decay rates, and the slowest-decaying term in any putative vanishing linear combination must have zero coefficient; repeat. Their Gram matrix is therefore strictly positive definite. A single scalar response component of the field can produce full rank among these preparations.

Equal widths with differing containing depths instead give (5), including for the assigned depths $3,4,5,6,10$ whenever the same shared marginal and vertex are chosen. Depth five is not singled out by this geometric contact model. A neutral sector's suppression has to enter through its derived channel map, not an arbitrary factor of five.

Retarded response and a controlled static limit

For linearized equations of the same field, let $D(\omega)$ be their operator in retained and remaining coordinates. Wherever the retarded inverse of $D_{\rm ss}$ exists, eliminating the remaining coordinates gives the exact block expression

$$D_{\rm eff}(\omega)=D_{\rm rr}(\omega) -D_{\rm rs}(\omega)D_{\rm ss}^{-1}(\omega)D_{\rm sr}(\omega).$$

The inverse is a response kernel for field evolution. This step assumes no exchanged particles. Both the blocks and their causal boundary conditions come from the original equations.

If that reduction yields harmonic response coordinates with positive inertia $M$, their equation is

$$M\ddot X+LX=S(t).$$

With Fourier convention $e^{-i\omega t}$ and retarded boundary conditions,

$$G_{ab}^{R}(\omega) =B_a^*\left[L-(\omega+i0)^2M\right]^{-1}B_b. \tag{8}$$

For real reciprocal coefficients its transpose reverses source indices at the same frequency. Above accessible resonances it need not be a positive static Gram matrix. An instantaneous interaction is the low-frequency approximation to this specified response, not proof of instantaneous physical signalling.

In a finite-mode model put $A=L^{-1/2}ML^{-1/2}$ and $C=L^{-1/2}B$. For $\epsilon=\omega^2\|A\|<1$, the Neumann series gives

$$G(\omega)=C^*(I-\omega^2A)^{-1}C,\qquad \|G(\omega)-G(0)\| \le \|C\|^2\frac{\epsilon}{1-\epsilon}. \tag{9}$$

This states when one static coupling matrix approximates the dynamic response. The field stiffness, inertia, and source maps must be derived before treating the resulting frequency dependence as a particle prediction.

Related and verification

• Generalized sector coupling — combines spatial reciprocity, normalization, and channel tests.
• Neutrino channel control — distinguishes a vanishing interaction term from suppressed field response.
• Cross-sector moment kernel — an existing example in which coordinate tensors obstruct automatic factorization.
• Gaussian contact profiles — derives the more general covariance and displacement overlap.

generalized_coupling_checks.py independently minimizes response energies, checks source gradients and response ranks, integrates the Gaussian profiles, and tests (9) against the full frequency-dependent inverse.