Two-frequency fields in a readjusted nested pair

Summary: Both deformed nested Dirac profiles admit exact opposite-frequency components with independent polarizations, and the containing field can keep the lower field confined at finite frequency when its own mean scalar source vanishes.

Use the source of both occupied fields

The readjusted nested pair describes two projected channels of the one field on $Q_p\subset Q_D$, with $D=p+q$. The higher profile deforms because it shares $p$ ordinary macroscopic spatial directions with its lower partner. Those shared directions are equivalent in physical kind to its other $q$ directions; their grouping is relative to this pair.

The two-frequency construction can be extended to both of these deformed profiles. Its spatial algebra is stronger than a radial-density approximation, because it retains all four components of the anisotropic higher field.

Use the projected channel source and kinetic measures of the nested-pair model. With separately centered signed scalar moments, its common energy contains

$$\mathcal E_{\rm int}=\frac{g_0}{2}\operatorname{tr}S^2, \qquad S=IM_aI^T+M_b,\qquad A_a=g_0I^TSI,\quad A_b=g_0S. \tag{1}$$

Here $I$ is the physical spatial inclusion. Both source variations follow from this same energy. The projection is a stated assumption: a coherent unprojected field must also retain its transition-density source. The fields and the normal partner used below introduce no additional physical manifold or interaction particle.

The complete real anisotropic frame preserves the scalar source

At $x\in\mathbb R^p$ and $z\in\mathbb R^q$, set $C_x=\alpha(\widehat x)$ and $C_z=\alpha(\widehat z)$. For real radial functions $F_{ab}(|x|,|z|)$ define

$$P=F_{00}I+iF_{10}C_x+iF_{01}C_z+F_{11}C_xC_z, \qquad \phi_w=\frac{Pw}{\sqrt{\Omega_{p-1}\Omega_{q-1}}}, \quad \beta w=w,\quad \|w\|=1. \tag{2}$$

The full bound operator has scalar coefficient $m_b+A|x|^2+B|z|^2$. The four functions solve its product radial eigenproblem. In particular, $F_{11}$ is generally nonzero when $A\ne B$.

Choose an existing normal direction in a sector containing both objects, with Clifford matrix $Y$ anticommuting with all bound $\alpha_i$ and with $\beta$. The Clifford relations give

$$\boxed{P^\dagger P=\left(\sum_{a,b}F_{ab}^2\right)I, \qquad [P,\beta Y]=0.} \tag{3}$$

To check the first identity, terms linear in $C_x,C_z,C_xC_z$ cancel between the two products because the functions are real and $C_xC_z=-C_zC_x$. For the second, $\beta Y$ commutes with each bound Clifford matrix, since both $\beta$ and $Y$ anticommute with it. Consequently, for any two allowed rest polarizations,

$$\boxed{\phi_{w_1}^\dagger\beta Y\phi_{w_2}=0.} \tag{4}$$

Indeed the numerator reduces to $(\sum F_{ab}^2)w_1^\dagger\beta Yw_2$, which vanishes because $Y$ exchanges the $\beta$ eigenspaces. The diagonal scalar density is the same for every normalized $w$:

$$\sigma_0=\frac{F_{00}^2+F_{11}^2-F_{10}^2-F_{01}^2} {\Omega_{p-1}\Omega_{q-1}}. \tag{5}$$

These are pointwise spatial identities. A central lower field has the corresponding two-component identities. They require the stated real stationary profile class; arbitrary complex axial motion is not being assumed to stay within it.

An exact stationary-source family with two independent clocks

For each projected channel $i=a,b$, choose constants $u_i,v_i$ and two rest polarizations. Define

$$\boxed{\Psi_i(t)=u_i\phi_{i1}e^{-iM_it} +v_iY\phi_{i2}e^{iM_it},\qquad |u_i|^2+|v_i|^2=1, \quad s_i=|u_i|^2-|v_i|^2.} \tag{6}$$

Because $\{H_i,Y\}=0$, the normal partner has eigenfrequency $-M_i$. Its scalar density has the opposite sign, and (4) removes its interference with the positive component. Therefore

$$\Psi_i^\dagger\beta\Psi_i=s_i\sigma_{i0}, \qquad \int\Psi_i^\dagger\Psi_i=1. \tag{7}$$

Solve both profiles using these same weights. If $t_a$ is the lower scalar second moment per axis, and $t_x,t_z$ are the higher per-axis moments, their common source is

$$\boxed{A=g_0(s_at_a+s_bt_x),\qquad B=g_0s_bt_z.} \tag{8}$$

Each field in (6) solves its complete nonlinear projected channel equation with the unchanging source (8), uniformly in the selected normal coordinate. Its number currents can still oscillate. In particular,

$$\int j_{Y,i}\,dx_i =2\operatorname{Re}\left[ \bar u_i v_i\langle w_{i1},w_{i2}\rangle e^{2iM_it}\right]. \tag{9}$$

Both occupied polarizations and their relative phase remain visible in this output. Equation (9) is a number current of the specified field; its physical photon projection is not assigned here.

In general $M_a\ne M_b$. The full linearized pair has frequencies $2M_a$ and $2M_b$ in its coefficients, so it need not have a single period. The complete source response must then retain both frequency indices and the reciprocal cross curvature. The exact background construction alone does not establish stability. Reciprocal response with two spatial clocks derives the full pair equation, including deformation-induced scalar-center sidebands and both independent background frequencies.

The containing source changes the low-frequency limit

Set both bare scalar coefficients to zero and keep $g_0$ fixed. Write $\epsilon_d$ and $T_d$ for the frequency and total signed second moment of the unit-normalized central profile with quadratic coefficient one. Spatial dilation gives

$$M_d(A)=\epsilon_d A^{1/3},\qquad T_d(A)=T_d A^{-2/3},\qquad T_d=\frac{\epsilon_d}{3}. \tag{10}$$

For an isolated lower field, self-consistency gives $M_a\propto s_a^{1/5}$ as $s_a\to0$. In the pair, hold $s_b>0$ and instead let $s_a\to0$. The higher source becomes isotropic over all its $D$ directions, and (8) has the endpoint

$$\boxed{A_0=B_0= \left(\frac{g_0s_bT_D}{D}\right)^{3/5}>0.} \tag{11}$$

The lower channel still feels this higher source. Its limiting frequency and that of the containing field are

$$\boxed{M_{a,0}=\epsilon_p A_0^{1/3}>0,\qquad M_{b,0}=\epsilon_D A_0^{1/3}.} \tag{12}$$

As $s_a\downarrow0$, the positive- and negative-frequency weights become equal. The lower mean scalar density tends to zero while its total number norm stays one. The limiting profile remains localized in the containing potential and retains the finite clock in (12).

This is a limit of positive-balance stationary solutions. The complete centered scalar functional requires $Q_a=\int\sigma_a\ne0$ to define $X_a=\int x\sigma_a/Q_a$. At exactly zero balance that center is undefined. A limiting profile in a prescribed centered potential therefore does not by itself define a differentiable nonlinear field equation at $s_a=0$. No such endpoint evolution is asserted here.

Independent solves at one common $g_0=6.5$ and $s_b=0.3$ give:

These are frequencies of the declared model, not measured particle masses. They show why an isolated-field frequency scaling cannot be applied unchanged after the containing field has supplied part of the confinement. If both balances scale together, the common dilation law still gives $M_i\mapsto c^{1/5}M_i$.

A vanishing mean source retains reciprocal derivative coupling

On the stationary family, the full energy and its derivatives are

$$U=\frac56(s_aM_a+s_bM_b),\qquad \partial_{s_i}U=M_i,\qquad G_{ij}=\partial_{s_j}M_i=G_{ji}. \tag{13}$$

The readjusted scalar response derives these identities by varying both complete profiles. They apply here to the signed weights in (6); they do not supply the entire periodic field spectrum.

Equation (12) depends on $s_b^{1/5}$, giving the finite endpoint relation

$$\boxed{G_{ab}\longrightarrow\frac{M_{a,0}}{5s_b}>0.} \tag{14}$$

For $5\subset6$ at the table's common coefficient and higher balance, this limit is approximately $1.281502$. A zero lower mean scalar source therefore need not cancel its reciprocal response to changing the containing field. Mean source, induced response and normalized coupling to a specified incoming spatial mode are distinct calculations.

Other variations need not remain regular at the same limit. The lower scalar-center curvature is $-2A/Q_a$, with $Q_a=s_a\int\phi_a^\dagger\beta\phi_a$. Since $A\to A_0>0$ and the unit profile has a finite nonzero scalar integral, this curvature scales as $1/s_a$. This is a singular coefficient of the full moment response, not a normalized interaction rate. It is another reason to retain the center and the full field's kinetic form near zero balance.

Verification

nested_bifrequency_checks.py checks (3)–(7) using the complete containing Clifford matrices, independent rest polarizations and nonzero mixed upper components. Independent product radial quadrature reconstructs both spatial moments and the common source. Direct finite derivatives in every bound coordinate and time verify both nonlinear channel equations at their readjusted frequencies.

Decreasing the lower balance checks approach to (11)–(12), using an independently solved isotropic endpoint. Reciprocal source derivatives check (14) and the common scaling identity. The calculation uses the same coefficient at several nested depths, writes timestamped reports, and retains its checks under Python optimization.