Localized matrix sources, complete energy and translation response

The matrix-current action in A reciprocal matrix-current source with real uniform frequencies has a regular quartic source and a uniform branch with real Fourier frequencies. This note tests its nonconstant spatial realization, retaining the responding background. The physical target is one selected mass-frequency and its corresponding inertia. Frequency is never inferred from a fitted amplitude.

The construction remains conditional: its spatial carrier, binomial weights, background preparation and common physical normalization have not been derived from the master field. In particular, the uniform source is a proposed field background with its own dynamics, not an additional particle or an assumed Yukawa interaction.

Exact coupled radial equations

Use the same smooth binding interaction as the localized root action. For one coherent grade let D be its cumulative weight and M = D omega_0. All coefficients are common across grades. The target has four real stationary radial functions,

$$\Psi=\begin{pmatrix}fI+ih\alpha_{\hat x}\\uI+iv\alpha_{\hat x}\end{pmatrix}, \qquad \sigma_A=r(f^2+h^2-u^2-v^2), \tag{1}$$

where r is the partner rank and the physical radius is denoted s below. The per-copy background has components

$$B=\sqrt{b/r}\begin{pmatrix}(1+A/b)I\\i(B_1/b)\alpha_{\hat x}\end{pmatrix}, \qquad \omega_0=gb,\quad \mu=m_B+\omega_0. \tag{2}$$

B_1 here is a radial scalar, not the scalar-insertion matrix used in the binding action. Put epsilon = 1/b and define the residual total current

$$F=\sigma_A+2A+\epsilon(A^2-B_1^2). \tag{3}$$

The complete normal phase equation, at frequency mu, is

$$A'=-(2\mu+gF)B_1,\qquad B_1'=-\omega_0F(1+\epsilon A)-\frac{d-1}{s}B_1. \tag{4}$$

The target mass-source coefficient is M+(M/b)F. Its four equations retain the actual smooth binding source and every lower component. spatial_matrix_core.py solves them using U = 2Mu, V = 2Mv and omega = M+Lambda/(2M), which avoid cancellation between large phase frequencies. It fixes the target canonical norm at one and imposes regular origins and decaying asymptotic fields.

The vacuum target decay exponent satisfies

$$k_A^2=1-\Lambda-\frac{\Lambda^2}{4M^2}>0 \tag{5}$$

in common core-gap units. With R = K_{d/2}(k_A s)/K_{d/2-1}(k_A s), the full target tail relations are

$$k_A h=R\left[f-\left(1+\frac{\Lambda}{4M^2}\right)U\right], \qquad k_A V=R(-\Lambda f+U). \tag{6}$$

The normal tail is driven by the target density as well as its own homogeneous decay. Increasing the outer boundary verifies that the finite boundary condition does not determine the reported core.

Controlled leading spatial reduction

For large b, equation (4) yields

$$(-\nabla^2+\kappa^2)A=-\frac{\kappa^2}{2}\sigma_A, \qquad \kappa^2=4\mu\omega_0, \qquad F=\mathcal R\sigma_A+O(b^{-1}), \tag{7}$$

with R = (-Laplacian)(-Laplacian+kappa^2)^(-1). This is a derived static response of the declared same-field equations. It is not a separate interaction premise, and its static use does not replace the full causal time evolution.

Let B(phi) denote the original smooth spatial binding equation for the upper target. At large M, eliminating its lower component gives

$$\mathcal E_{\rm red}(\phi) =\|\mathcal B(\phi)\|^2 +\frac{\gamma}{2}\int\sigma_A\mathcal R\sigma_A\,d^dx, \qquad \gamma=\frac{2M^2}{b}, \tag{8}$$

$$\mathcal B'_\phi{}^\dagger\mathcal B(\phi) +\gamma(\mathcal R\sigma_A)\phi=\Lambda\phi, \qquad \|\phi\|^2=1. \tag{9}$$

This reduction is an expansion with stated small parameters. It does not substitute a static kernel into the exact calculation (4). For the depth-six control with gamma = 0.1, the refined coupled solve at M = 30 gives Lambda = -0.003356093237 and root mean squared radius 1.303860610. The independently refined reduced calculation gives Lambda = -0.003356078003. The difference approaches its remaining spatial discretization error as M increases.

These frequency and size changes occur under one common action. The core gap is unchanged, but equality of the resulting spatial radii has not been derived. The comparison that holds gamma fixed while changing M varies b and is an asymptotic control, not a particle comparison at one background preparation.

Keep the background norm deficit in the energy

The background scalar-density change and canonical norm change differ:

$$\delta\sigma_B=2A+\epsilon(A^2-B_1^2),\qquad \delta q_B=2A+\epsilon(A^2+B_1^2). \tag{10}$$

For the first localized controls the integral of delta q_B is close to -1 per active copy. The full background-subtracted energy therefore contains a substantial change in the surrounding field's stored energy. It cannot be evaluated by retaining only the target frequency term.

Writing T_A and U_bind for the integrated target kinetic and binding energies, direct variation and integration by parts give

$$\begin{aligned} \mathcal F &:=E_{\rm excess}-\mu D\,\delta Q_B\\ &=T_A+U_{\rm bind}+M\int\sigma_A\,d^dx\\ &\quad+D\epsilon\int \left[(2\mu+2gF)B_1^2+\frac{\omega_0}{2}F^2\right]d^dx, \end{aligned} \tag{11}$$

with the actual finite-radius surface term retained in numerical checks. The identity uses integral T_B = -2 epsilon integral B_1 A' plus that surface term and the complete normal equation (4).

F is the excess of E-mu Q_normal at fixed normal phase. It is also the candidate large-volume energy at fixed total normal norm: compensating a finite deficit delta Q_B in a distant volume V changes its background energy by -mu D delta Q_B + O(V^(-1)). This interpretation requires the same boundary and ensemble limit; it is not an assertion that an arbitrary subtraction equals an isolated particle energy.

At M = 30 and gamma = 0.1, equation (11) gives approximately 30.00002388, while the selected target frequency is 29.9999440651. The corresponding inertial coefficient must still be computed from motion. At the assigned large M, direct subtraction of large energies loses accuracy. spatial_matrix_energy.py instead evaluates 2M(F-MQ_A) directly and checks it against an independent normal-equation identity. Those small corrections are the reliable quantities in the large-frequency reports.

First translation correction from the full response action

At fixed M and omega_0, expand in b^(-1) about the exact unperturbed localized core. Let sigma_0(x) be its scalar density. Its Lorentz-boosted profile has Fourier transform gamma_v^(-1) times the rest transform evaluated at the contracted momentum. The background is kept in its original frame. Both normal conjugate components and their full spatial kinetic action are retained when it responds.

For steady motion below the background response speed, the real quadratic action uses the complete multiplier

$$R(\Omega,p)= \frac{(\Omega^2-p^2)^2-4\mu^2\Omega^2} {(\Omega^2-p^2)^2-4\mu^2\Omega^2+\kappa^2p^2}. \tag{12}$$

The response action coefficient is -gD sigma R sigma/2. The full canonical field action independently verifies this factor, including the induced field's kinetic and potential contributions. This real steady coefficient must not be extended through radiative poles by silently using an instantaneous or acausal prescription.

Changing to rest-profile momentum k, and differentiating the action twice in a small velocity along direction i, gives

$$\delta E=\frac{gD}{2}\int\frac{d^dk}{(2\pi)^d} |\widetilde\sigma_0(k)|^2\frac{k^2}{k^2+\kappa^2}, \tag{13}$$

$$\boxed{\delta I_{ij}=\delta E\,\delta_{ij} +gD\int\frac{d^dk}{(2\pi)^d}|\widetilde\sigma_0(k)|^2 \frac{k_i k_j\kappa^2(4\mu^2+k^2)} {k^2(k^2+\kappa^2)^2}.} \tag{14}$$

The second term is positive. It is the background's extra response to relative motion. For a radial core, angular integration replaces k_i k_j by k^2 delta_ij/d. Equation (14) is a leading effective-action result. A finite-background moving-core solution and its stability are separate calculations.

For the depth-six smooth unit-norm core, with omega_0 = 1 and mu = 2, the leading coefficients are

The frequency coefficient follows from an independent continuum boundary problem for the full fixed-norm first variation, not a frozen profile overlap. The energy and inertia coefficients use independent position-space and Fourier calculations and direct differentiation of the moving action.

With the stated unit norm these corrections miss the required relation between selected mass-frequency and inertia. Their common leading factor M/b is also relevant: one must examine any proposed common normalization under the same action across grades, rather than declaring separate particle adjustments to be a derivation. No normalization remedy is established here. The positive matrix-current source supplies useful conditional spatial mathematics, but does not yet complete the physical mass operator.

The full polarized translation problem

The first-order response can also be tested without eliminating the normal field in advance. For translation in direction 1, a closed orthonormal angular matrix basis is

$$\begin{aligned} T_0&=\sqrt{d/r}\,\hat x_1 I,& T_1&=i\alpha_1/\sqrt r,\\ T_2&=\frac{id}{\sqrt{r(d-1)}} (\hat x_1\alpha_{\hat x}-\alpha_1/d),& T_3&=\sqrt{\frac d{r(d-1)}} (\alpha_1\alpha_{\hat x}-\hat x_1I). \end{aligned} \tag{15}$$

Their angular degrees are respectively 1, 0, 2 and 1. Together with both doubled spinor blocks and the target and collective normal field, they retain sixteen complex radial functions. All normal copies have identical driven responses in this sector, with collective canonical field equal to the per-copy field times sqrt(D). The other copy combinations and positive oscillator-charge sectors are not driven by an infinitesimal translation of this coherent core. This restriction therefore computes its translation response; it is not a stability classification of every perturbation.

Let H be the imaginary-quadrature Hessian of the complete stationary equations, and let t = -partial_1 Phi denote the translation vector in the real radial coefficients. The velocity response is i y, with

$$H y=t. \tag{16}$$

A direct boost of the uniform normal field does not decay at infinity. Instead boost the target and the normal deformation, subtracting that uniform field first. If y_kin denotes these boost coefficients divided by i, Lorentz covariance of the declared action gives the exact localized identity

$$H y_{\rm kin}=t-r_b,\qquad r_b=(H-H_\infty)y_{\rm background}. \tag{17}$$

The background boost includes its actual phase frequency mu. The localized residual r_b is evaluated from the full scalar currents and their variations; no spatial inverse kernel is inserted. The complete inertial coefficient is consequently

$$\begin{aligned} I={}&\omega Q_A+\frac{T_A}{d}\\ &+D\epsilon\int\left[ \mu(A^2+B_1^2) +\frac{A[B_1'+(d-1)B_1/s]-B_1A'}d\right]d^dx\\ &+2\langle t,H^{-1}r_b\rangle. \end{aligned} \tag{18}$$

This formula keeps both the kinematic boost and the responding field. The normal deformation norm in it is quadratic; the density deficit linear in A is not its norm. The boundary condition holds the far background in its original frame.

Self-adjointness and (17) provide a numerically better equivalent form,

$$\langle t,H^{-1}r_b\rangle =\langle y_{\rm kin},r_b\rangle +\langle r_b,H^{-1}r_b\rangle. \tag{19}$$

The target part of the first pairing vanishes pointwise: its boost leaves its scalar matrix current unchanged. Only the normal deformation contributes to that pairing. Equation (19) avoids magnifying a finite spatial projection error by the large assigned particle frequency. Independent full Cartesian checks verify (17) and the canonical pairing.

At M = 30 and gamma = 0.1, the refined result is 2M(I-MQ_A) = 0.00184342875 and 2M(I-omega Q_A) = 0.00519952198. The finite-gamma coefficient of M/b is 0.0184342875; it need not equal the first-order limit 0.0202832756 in the preceding table. The mixed and quadratic pairings agree within 2.9e-13 in the count-48 control. Increasing the radial basis to 64 gives the same resolved correction within the reported spatial accuracy.

The assigned branches in one background

A separate comparison fixes one background density for both depth-six assignments:

$$b=\frac{2(3838380)^2}{0.1},\qquad \omega_0=1,\qquad\mu=2,\qquad Q_A=1. \tag{20}$$

The same continuum-normalized binding coefficient and smoothing scale are used for both solutions. Units also identify the core gap with omega_0; that remains an inverse input. The resulting gamma differs between branches because gamma = 2M^2/b.

The radial equations have independently controlled boundary residuals and unit norms. The count-64 translation solves retain the full angular sector (15). At the large muon frequency, the mixed pairing is more sensitive to spatial projection error; the quadratic identity (19) is used for the reported correction. These finite spatial approximations are numerical evidence for the stated conditional model, not a proof of existence or stability.

The tiny relative frequency changes in (20) are consequences of the chosen very large background density. They are not an independently derived accuracy prediction. The full response still misses the required inertial relation at the stated normalization. Neither this source trial nor its approximately shared core size derives the spatial harmonic carrier or the binomial weights.

Reproducible checks

matrix_spatial_equations_checks.py passes 147 conditions by comparing the radial equations to complete Cartesian and oscillator matrix sources at depths two, four and six, including arbitrary complex binding variations. screened_core_checks.py passes 60 conditions with three spatial bases. The final spatial_matrix_core_checks.py report, 20260913T212442Z, passes 36 conditions, including refined full profiles and the small energy correction without cancellation of large quantities.

matrix_response_action_checks.py passes 18 independent full-action checks. moving_matrix_response.py passes 10 Fourier, position-space, normalization and direct moving-action checks. The continuum spatial_matrix_first_order.py passes four refinement and constraint checks; its final first-frequency slope changes by 3.3e-10 when both the outer boundary and the spatial tolerance are refined.

All source versions, reports and numerical arrays remain private research evidence. These checks establish the specified calculations and their accuracy, not the missing physical interpretation or a selected particle geometry.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.