Reciprocal scalar response of excited spatial branches

Summary: A near-resonance derived from the common scalar action predicts spatial instabilities on nonlinearly dressed branches at the unchanged assigned indices, with converged full differential residuals and a direct constraint on the coupling response required by an inverse mass law.

The inverse interaction must support a responsive spatial field

The inverse scalar-law construction determines whether a proposed field can obey one real law $F(J)$ and how its derivative enters coupling. A stationary frequency alone does not settle the response of the resulting spatial state.

Here the same local scalar law is tested after allowing both spatial Dirac components to deform:

$$E_g[\psi]=\langle\psi,\mathscr D\psi\rangle+ \frac g2\int J^2\,d\nu,\qquad i\partial_\tau\psi=\mathscr D\psi+gJ\beta\psi,\qquad J=\psi^\dagger\beta\psi. \tag{1}$$

The action has the original first-order kinetic term and one common coefficient within each specified sector. The scalar source is a local current of the one field, with no mediator particle or independent coupling coefficient for each pair of modes.

The calculations include the unchanged six-dimensional indices $n=13,35$ and the ten-dimensional index $n=23$. Radii remain unknown and no dimensions are compactified.

Numerical values use radius one, unit spatial measure and unit canonical norm. The dimensionless coefficient is $g=R\kappa Q/\mathcal V$ when the physical local coefficient is $\kappa$, norm is $Q$ and volume is $\mathcal V$. Comparisons across different depths do not establish equality of their physical couplings. Frequencies and growth rates below are divided by $R$ when that radius is restored.

Continue the assigned mode through its full stationary equation

Let $\phi_n$ be a normalized zero-form spatial eigenfield and $\eta_n=\mathscr D\phi_n/\lambda_n$ its exact one-form partner. The positive free state is

$$\chi_n=\frac{\phi_n+\eta_n}{\sqrt2},\qquad \lambda_n=2\sqrt{n(n+2m-1)}. \tag{2}$$

The nonlinear continuation solves

$$(\mathscr D+gJ_{\psi_n}\beta)\psi_n=\omega_n\psi_n, \qquad \|\psi_n\|=1,\qquad \psi_n(0)=\chi_n. \tag{3}$$

Both even and odd profiles are expanded in the complete pole-centered paired spatial basis, with the cutoff enlarged independently. This radial symmetry class is preserved by the local equation. It supplies actual spatial fields, while nonradial polarizations remain additional stability directions.

At small coupling,

$$\omega_n(g)=\lambda_n+gA_n+O(g^2),\qquad A_n=\int J_n^2\,d\nu,\qquad J_n=\chi_n^\dagger\beta\chi_n. \tag{4}$$

The complete stationary source determines the complementary profile correction as in the nonlinear spatial reduction. The calculations below solve that changed field instead of holding $\chi_n$ fixed.

Keep both complex response blocks of the same action

Write $\psi=e^{-i\omega_n\tau}(\psi_n+h)$ and put $v=\beta\psi_n$. The full scalar variation gives

$$i\dot h=\mathcal A h+\mathcal B\bar h, \qquad \begin{aligned} \mathcal A&=\mathscr D-\omega_n+gJ_{\psi_n}\beta+g vv^\dagger,\\ \mathcal B&=g vv^{\mathsf T}. \end{aligned} \tag{5}$$

The last terms are pointwise spatial operators, followed by the full physical mode projection in a numerical calculation. They retain the ordinary and conjugate response of the same current. Dropping $\mathcal B$ would remove the instability mechanism tested here.

Derive the nearby reciprocal resonance

For the pole-centered representatives in (2), define $M_{kn}(x)=\chi_k^\dagger\beta\chi_n$. All spatial derivatives, densities and normalization factors are included in these fields. The relevant first-order coefficients are

$$\begin{aligned} a_\pm&=\int\left[J_nJ_{n\pm1}+|M_{n\pm1,n}|^2\right]d\nu-A_n,\\ b&=\int M_{n+1,n}M_{n-1,n}\,d\nu. \end{aligned} \tag{6}$$

The chosen real radial representatives make $b$ real. General phase conventions change its phase and leave $|b|$ and the following growth criterion invariant.

The kinetic sequence is strictly concave. For a continuous argument $x>0$ and $c=2m-1$,

$$\frac{d^2}{dx^2}\left[2\sqrt{x(x+c)}\right] =-\frac{c^2}{2[x(x+c)]^{3/2}}<0. \tag{7}$$

Thus $\Delta_n=\lambda_{n+1}+\lambda_{n-1}-2\lambda_n<0$. The perturbation in $\chi_{n+1}$ lies close in frequency to the conjugate perturbation in $\chi_{n-1}$. Their leading response matrix is

$$K_{\rm near}= \begin{pmatrix} \lambda_{n+1}-\lambda_n+ga_+&gb\\ -g\bar b&\lambda_n-\lambda_{n-1}-ga_- \end{pmatrix}. \tag{8}$$

This is the first-order matrix in that near-resonant subspace. The full numerical response retains the other modes and the nonlinearly changed background; (8) does not replace them.

The two eigenvalues become nonreal when

$$\boxed{ \Gamma_{\rm lead}^2= g^2|b|^2-\frac14\left[\Delta_n+g(a_++a_-)\right]^2>0. } \tag{9}$$

The minus sign in the lower off-diagonal entry of (8) comes from the conjugate response equation. It permits exponential redistribution even though the underlying energy is real. It does not assign either frequency branch to an antiparticle.

In the three tested families, $a_++a_->2|b|>0$. Within the leading matrix, the instability interval is bounded by

$$g_{\rm low}=\frac{-\Delta_n}{a_++a_-+2|b|},\qquad g_{\rm high}=\frac{-\Delta_n}{a_++a_--2|b|}. \tag{10}$$

These are leading boundaries, not exact thresholds for the full nonlinear spatial equation. The latter also contains higher-order profile changes and other response channels.

These coefficients are spatial integrals of one interaction. Their different values are not separately fitted pair couplings.

Check the deformed field and the entire complex response

The stationary calculations use successively larger paired cutoffs $44,66,90$ at $n=13$; $110,150,190$ at $n=35$; and $78,118,158$ in the ten-dimensional comparison. All complex perturbations in each retained paired basis enter (5).

At twice the respective leading lower boundary, the converged positive growth rates are:

The final stationary residuals are below $2.4\times10^{-13}$. Independent differentiation of the actual curved complex response fields gives residuals below $5\times10^{-13}$. Both quantities converge with the enlarged spatial basis. A common complex phase rotation preserves the instability and the independently checked response equation.

The negative controls $g=-2g_{\rm low}$ and the positive controls $g=g_{\rm low}/2$ have no resolved radial exponential mode in the tested matrices. Tiny numerical splitting of the phase zero is kept below the diagnostic tolerance. Absence of a resolved radial instability is not a proof of full angular or nonlinear stability.

At one common six-dimensional coefficient $g=10^{-8}$, $n=13$ has no resolved radial growth, while $n=35$ has $\Gamma=0.000134195935$. Their stationary frequencies are respectively $30.594121102600$ and $74.833283773990$. The frequency ratio remains close to $2.446$ in this small-coupling control, rather than the assigned simplex target $3515/17$. The calculation therefore supplies neither the desired mass relationship nor a measured particle decay rate.

The energy-response calculation follows both branches as their common canonical norm changes under this same action. Both have positive energy curvature despite their different radial instabilities. The complete norm tangent also predicts their response to a common change in the interaction, providing an inverse constraint on mass energies as well as stationary frequencies.

The converged unstable perturbation is an allowed direction of the actual spatial field, including its complementary modes. The numerical evidence concerns these specified branches. It is not a theorem excluding every scalar law, every coupling strength or every spatial polarization.

Consequence for reconstructing the mass dynamics

In an inverse law $F(J)$, its values must support the complete stationary profiles. Its derivative controls the ordinary and conjugate feedback in (5), with $g$ replaced by the appropriate local $F'(J)$ in the source derivative. A law that gives a desired frequency can still produce an unacceptable spatial response.

This gives a concrete additional inverse condition: the same interaction must pass the full spatial source test, the common-law test and the required stability test. Changing the scalar response to repair an unstable branch also changes coupling; those adjustments cannot be treated as unrelated parameters. The local quartic law supplies a reproducible example of this constraint, while the physical particle fields remain unidentified.

Verification

excited_scalar_resonance_checks.py derives (6) from the full action Hessian and independently from curved spatial current integrals. It continues the normalized nonlinear fields, retains both complex response blocks, checks phase and norm identities, and enlarges every spatial cutoff.

The stationary and response residuals are evaluated through actual spatial derivatives on separate refined quadrature grids. The near-resonant response is checked directly, including the apparently stable controls, instead of selecting an arbitrary high-frequency numerical eigenvector from tiny roundoff growth. These calculations provide numerical convergence evidence; the finite matrix alone is not a continuum stability proof.