Slow shape poles and their signed response

Summary: The two-frequency scalar shape response has poles whose frequency and residue decrease together, but their canonical sign and finite total response prevent identifying them from softness alone as stable weakly coupled particles.

Solve for the collective frequency instead of supplying it

The complete periodic response distinguishes the bound field's carrier from its collective source poles. In the radial rest examples, scalar quadrupole sidebands cancel and the remaining frequency equation can be solved directly. This gives a concrete test of whether a slow source also has a weak response.

Use the zero-bare-mass scalar model with one fixed $g_0$, unit number norm and scalar balance $s=|a|^2-|b|^2>0$. The bound profile dilates by $\ell=s^{1/5}$. Let $A(z)$ and $B(z)$ be the even and odd scalar functions of the full single-frequency spatial shape response, calculated at $s=1$. For a tensor eigenvector of the circulation action $T$ with eigenvalue $h$, the two-frequency bare response is

$$\boxed{\Pi_h(\omega;s)=A(z)+\frac{h}{s}B(z),\qquad z=\frac{\omega}{\ell},\qquad \chi_h=\frac{\Pi_h}{1-g_0\Pi_h}.} \tag{1}$$

The tensor labels describe spatial shape patterns over the existing macroscopic dimensions. They are not new physical directions. The $1/s$ factor comes from retaining both frequency components and their different scalar and canonical weights; it is not a chosen weak-coupling coefficient.

Derive the slow pole and its residue

Suppose the frozen source resolvents are analytic near $z=0$, and put

$$A(z)=A_0+O(z^2),\qquad B(z)=B_1z+O(z^3), \qquad D_0=1-g_0A_0. \tag{2}$$

For $hB_1\ne0$ and $D_0\ne0$, set $z=s\zeta$ in the denominator of (1). The implicit-function theorem gives the simple root

$$\boxed{\omega_h(s) =s^{6/5}\frac{D_0}{g_0hB_1} \left[1+O(s^2)\right].} \tag{3}$$

The source residue $R_h$ is defined by $\chi_h(\omega)=R_h/(\omega-\omega_h)+O(1)$. At the pole $\Pi_h=1/g_0$, so

$$\boxed{R_h=-\frac{1}{g_0^2\partial_\omega\Pi_h(\omega_h)} =-\frac{s^{6/5}}{g_0^2hB_1}\left[1+O(s^2)\right].} \tag{4}$$

Thus both the pole frequency and its source weight become small. Their signs also matter. The checked profiles have $A_0<0$, $B_1<0$ and $D_0>0$. The $h=+1$ member has negative frequency and positive residue; its conjugate $h=-1$ member has positive frequency and negative residue.

At one common $g_0=6.5$ and $s=0.001$, full radial response calculations give:

The same calculation supplies the slow poles at all three depths. It does not select a uniquely five-dimensional particle or an occupied mode.

Check the sign with the full canonical field realization

The scalar functions in (1) arise from both angular radial families and both signs of the Dirac spectrum. Let $L$ be their complete reference radial operator with its bound carrier subtracted, and let $v_h,v_{-h}$ be the respective spatial source vectors. The four-block realization is

$$\begin{gathered} L_4=\operatorname{diag}(\ell L,-\ell L,-\ell L,\ell L),\qquad \Sigma_4=\operatorname{diag}(I,I,-I,-I),\\ t=\ell^{-2}(a v_h,-b v_h,a v_{-h},-b v_{-h})^T,\\ \boxed{\mathcal B_4=L_4+g_0\Sigma_4tt^\dagger,\qquad \chi_h=t^\dagger(\omega-\mathcal B_4)^{-1}\Sigma_4t.} \end{gathered} \tag{5}$$

The amplitudes may be taken real in this single-helicity realization, with $a^2=(1+s)/2$ and $b^2=(1-s)/2$. The $\ell^{-2}$ factors are the dilated quadratic source moments. Expanding the four frozen blocks recovers (1), including its $1/s$ term.

The matrix obeys $\Sigma_4\mathcal B_4= (\Sigma_4\mathcal B_4)^\dagger$. For a simple real mode $\mathcal B_4v=\omega_hv$ with $v^\dagger\Sigma_4v\ne0$, its resolvent residue is

$$\boxed{R_h=\frac{|t^\dagger v|^2}{v^\dagger\Sigma_4v}.} \tag{6}$$

This independent normalization reproduces (4). Replacing the denominator by the positive Euclidean norm would erase the sign of the physical canonical form. A small source residue must be assessed with that form.

Interpret the work sign with the stability qualification

For a real source term $H_{\rm ext}=f(t)O$, let $f(t)=\operatorname{Re}(f_0e^{-i\omega t})$ and let a bounded particular output have amplitude $\chi f_0$. The mean supplied power is

$$\boxed{\langle\dot f\,\delta O\rangle =-\frac{\omega|f_0|^2}{2}\operatorname{Im}\chi.} \tag{7}$$

A simple real pole continued as $R/(\omega-\omega_0+i0)$ has positive absorptive weight at positive $\omega_0$ when $R>0$ and the opposite work sign when $R<0$. Equation (6) identifies this signed contribution without assigning an absolute physical energy to a Floquet frequency, which can be shifted by the background frequency.

An unstable full background has no generic settled periodic response. For it, (1) is a meromorphic source resolvent, and its individual real-pole weight does not establish stable absorption. The radial dynamics already transfer existing radial instabilities to these two-frequency fields. Stability and selection of the occupied field must therefore precede any particle interpretation.

A real canonical two-mode example checks the effect of opposite signs:

$$H=\frac12(p_a^2+\omega_a^2q_a^2) -\frac12(p_b^2+\omega_b^2q_b^2)+\varepsilon q_aq_b. \tag{8}$$

Both reciprocal equations follow from this same energy. Their frequencies satisfy

$$\boxed{(\omega^2-\omega_a^2)(\omega^2-\omega_b^2) +\varepsilon^2=0.} \tag{9}$$

At equal uncoupled frequencies a nonzero coupling gives growth, while the indefinite total energy remains conserved. This illustrates the normalization issue; it is not an added interaction field or a derived physical two-particle model.

A small pole weight does not make the full response small

At zero forcing frequency, $B(0)=0$, so

$$\boxed{\chi_h(0;s)=\frac{A_0}{1-g_0A_0},} \tag{10}$$

independent of $s$. At fixed nonzero $z=\omega/\ell$ with $hB(z)\ne0$, the $1/s$ term instead becomes large. Source feedback then gives

$$\boxed{\chi_h(\ell z;s)\longrightarrow-\frac1{g_0}.} \tag{11}$$

Neither limit is a weak total scalar response. The frequency of an incoming field, its selected spatial pattern, the pole residue and the full output map are separate parts of a coupling calculation, as explained in source frequencies.

Verification

slow_shape_pole_checks.py compares full four-block field resolvents with (1), locates the slow poles independently, and checks their residues using both source derivatives and the complete canonical eigenvectors. Increasing radial resolution verifies the quoted frequencies and signed residues. Both source components, both angular radial families and both signs of the Dirac spectrum are retained.

Direct real-time work verifies the convention in (7), and the complete canonical pair checks (9) and energy conservation during growth. Reports use timestamped filenames and verification remains active under Python optimization. The results establish the stated source spectrum and its limits, without a stable weak-particle identification.