Response of localized simplex-frequency fields

Summary: Solved localized branches have identical finite lower-grade probe energies and forces at common profile, while complete current Hessians and bounded spatial spectra distinguish that coupling identity from stability.

A common probe of actual localized targets

The localized current action has spatial solutions with $\omega_N=\omega_{0,d}D_N$ and common canonical norm and profile at a fixed depth. Its binding and mass terms belong to one action. This allows a finite probe comparison that retains both the probe source and the target's reciprocal force.

Write a target as $\Psi_N=e^{-i\omega_Nt}(\phi,0)^T\otimes\overline z^{\,N}$, where $\phi^\dagger\phi=\rho I_r$. A fixed test pattern occupies a distinct lower grade $M<N$ and may have arbitrary complex spinor, partner and oscillator coefficients. Its physical spatial profile is part of that field, rather than a prescribed overlap number.

The exact target mass current at every order accessible to the probe is

$$R_{N,k}(x)=\rho(x)I_r\otimes P_{z,k},\qquad 0\le k\le M. \tag{1}$$

It is independent of $N$. Define the local cross interaction by subtracting the two isolated energies from the combined action:

$$V_{N,M}(x)=U(\Psi_N,\Psi_M)-U(\Psi_N,0)-U(0,\Psi_M). \tag{2}$$

Every unshared mass-current order cancels. Every shared order contains the same target current (1). The binding currents trace all oscillator coefficients; the target's $J=0$ and $K=\phi^\dagger\beta\phi$ are also independent of $N$. Thus the entire cross interaction is the same for different targets with the same $\phi$ and $z$.

Where the complete source is differentiable, this proves

$$\boxed{ V_{N,M}=V_{N',M},\quad \frac{\delta V_{N,M}}{\delta\overline\Psi_M} =\frac{\delta V_{N',M}}{\delta\overline\Psi_M},\quad -\partial_XV_{N,M}=-\partial_XV_{N',M},\quad N,N'>M. } \tag{3}$$

Here $X$ translates the actual target profile. Integrating these identities gives the same cross energy and translation force whenever the integrals and their variations exist. No limit of vanishing probe amplitude is required. The independently normalized probe and target can both have $Q=1$.

The complete shape derivative on the target still contains the lifting maps $T_{Nk}^\dagger(\cdot\otimes I)T_{Nk}$ and depends on $N$. Equal force does not mean equal acceleration: its inertial coefficient is $\hbar\omega_N/c^2$. Subsequent shape changes can likewise differ. The statement therefore concerns an exact initial response and an energy functional, not identical encounter trajectories.

Independent tensor calculations use all seven coefficient slots at depth six, target grades two and three, and a grade-one Gaussian test pattern with unit norm and arbitrary full complex polarization. At three spatial points, the cross energies, full probe sources and reciprocal translation forces agree to numerical precision. The analytic identity (3) covers the assigned electron and muon grades for $M<12$ without materializing their enormous full coefficient arrays.

The test probe has not been identified as an observed particle solution. Oscillator grade is not physical spatial depth. Consequently (3) does not yet derive cross-depth coupling, photon vertices or neutrino suppression. The existing common-probe analysis also retains same-grade interference: two contributions to the same grade are one coherent sum, so treating them as two independent carriers is invalid.

The root law's noncommuting weak-field and vanishing-target limits still exist. Equation (3) avoids using that limit to infer a finite coupling, but it does not cure nonsmooth zero currents or fractional eigenvalue crossings during a general encounter.

The later two-core calculation prepares both objects as independently solved localized fields. It derives their full initial cross energy and reciprocal forces, and shows why aligned cores can have zero initial cross energy but nonzero shape response. It also finds a concrete cancellation of their fractional binding current and constructs smooth inverse bindings that retain the isolated frequency law. Those smooth actions require their own response checks; the spectra below continue to describe the original binding law.

Full oscillator directions of the occupied grade

The current hierarchy supplies two contraction identities. If $j$ counts oscillator occupations outside the occupied coherent ray, they are

$$\begin{aligned} \sum_{k=0}^Nh_{k,d}\frac{\binom{N-j}{k}}{\binom Nk} &=\frac{D_N}{D_j},\\ \sum_{k=0}^Nh_{k,d}\frac{\binom kj}{\binom Nj} &=D_N\frac d{d+j},\qquad D_j=\binom{j+d}{d}. \end{aligned} \tag{4}$$

Unavailable binomials are zero. Retaining the new matrix partners gives the complete root-source derivative for $j\ge1$:

$$\delta\mathcal N_{\mathrm{mass},j} =\omega_N\left(\frac{B_0}{D_j}+\frac d{d+j}\Pi_u\right)\eta_j. \tag{5}$$

The binding currents have zero first variation in these orthogonal oscillator directions. Their derivative is complex-linear,

$$\delta\mathcal N_{\mathrm{bind},j} =m_cB_1\eta_j\beta_p-gB_1\eta_jf(K). \tag{6}$$

Each matrix insertion in (5) and (6) is Hermitian on the complete spinor and partner space. With the full self-adjoint spatial Dirac operator, these blocks conserve their linear perturbation norm. This is an exact statement for the stated transverse oscillator blocks, not for the occupied ray's spatial and partner response.

Independent full tensor variations at $d=6,N=3$ and $d=10,N=1$ verify (5), including every oscillator slot. They also verify the complete occupied block against the actual nonlinear gradient, with ordinary and conjugate variations both present.

An absent higher grade has additional previously zero root currents. Their leading directional source is generally nonlinear in the perturbation, so a single global linear Hessian cannot be assigned there without more regularity assumptions. The action does have an independent phase symmetry in every grade. Along a sufficiently regular solution with the required boundary decay, it conserves each grade's canonical norm. In the declared finite grade domain, a small absent-grade norm therefore stays small for as long as that solution exists. This conditional conservation statement does not establish existence or uniqueness at a zero current. This does not derive transitions between grades or mode selection.

Spatial and partner Hessian on the occupied ray

Use natural units and write $\phi=\sqrt\rho U$, with $U$ unitary. Define the full spatial binding operator

$$\mathcal B(\phi)=-i\alpha\cdot\nabla\phi +m_c\beta\phi\beta_p-g\beta\phi f(\phi^\dagger\beta\phi). \tag{7}$$

In the rotating frame, the complete ordinary/conjugate Hessian has block form

$$\mathcal H'\begin{pmatrix}\eta_u\\\eta_l\end{pmatrix} =\begin{pmatrix} \mathcal A\eta_u+\mathcal B'_\phi\eta_l\\ \mathcal B'_\phi\eta_u-2\omega_N\eta_l \end{pmatrix},\qquad \mathcal A(UX)=2\omega_N U\left(\operatorname{Herm}X -\frac{\operatorname{Tr}(\operatorname{Herm}X)}rI\right). \tag{8}$$

The real-linear derivative $\mathcal B'_\phi$ includes all spectral divided differences of the matrix function. A phase-only or positive-spinor truncation is not this Hessian.

For a radial matrix core $\phi=fI+ih\alpha_{\hat x}$, the complex radial subspace retains both upper and lower pairs. Conjugate away the radial measure, put $v=(f,h)^T$, $P=2vv^T/\rho-I$ and define

$$\begin{aligned} D_r&=\begin{pmatrix} 0&\partial_{r_x}+\dfrac{d-1}{2r_x}\\ -\partial_{r_x}+\dfrac{d-1}{2r_x}&0 \end{pmatrix},\\ L_+&=D_r+m_c\sigma_z-F(\rho)I-2F'(\rho)vv^T,\\ L_-&=D_r+m_c\sigma_z-F(\rho)P,\\ A&=2\omega_N(I-vv^T/\rho). \end{aligned} \tag{9}$$

Here $F(\rho)=g\rho^a$ for the power law. If upper perturbations are $x+iy$ and lower perturbations are $u+iw$, the full radial equations are

$$\begin{aligned} \dot x&=Ay+L_-w,&\dot y&=-L_+u,\\ \dot u&=L_-y-2\omega_Nw,&\dot w&=-L_+x+2\omega_Nu. \end{aligned} \tag{10}$$

A regular Laguerre basis and independent differential-equation matching test this same system. The latter propagates four regular origin solutions and four decaying solutions from infinity and compares their subspaces. Fixing the final subspace basis gauge is necessary before root finding; arbitrary eigenvector phases do not define a continuous matching determinant.

For the analytic critical core at $d=6,R=1,\omega=1$, both methods find a genuine growing mode $e^{\lambda t}$:

$$\lambda=0.983987427901+2.46986730435i. \tag{11}$$

Changing the origin cutoff, outer endpoint and integration tolerance leaves this value unchanged at the displayed accuracy. A smooth polynomial binding control, $F(\rho)=6\rho(1-\rho^2)$ with zero core gap, also has a localized core. At $\omega=1$ its independently matched growing mode is $\lambda=3.01769256948+3.72920233603i$. Thus neither exact localization nor smoothing the binding law alone proves stable particles.

Ordinary nonradial polarization multiplets

For a Hermitian Clifford $k$-form $T$, $T^2=I$, the radial background preserves the angular span

$$B_0=T,\quad B_1=i\alpha_{\hat x}T,\quad B_2=iT\alpha_{\hat x},\quad B_3=\alpha_{\hat x}T\alpha_{\hat x}. \tag{12}$$

These basis names are local to this section and do not denote the scalar insertions of the action. For $0<k<d$ let $s_k=(-1)^k(d-2k)$ and $t_k=s_k/d$. Angular integration gives diagonal Gram entries one and off-diagonal entries $G_{03}=G_{12}=t_k$ and their transposes. A normalized angular basis is

$$E_0=B_0,\quad E_+=\frac{B_1+B_2}{\sqrt{2(1+t_k)}},\quad E_-=\frac{B_1-B_2}{\sqrt{2(1-t_k)}},\quad E_3=\frac{B_3-t_kB_0}{\sqrt{1-t_k^2}}. \tag{13}$$

Its ordinary spherical-harmonic degrees are $0,1,1,2$. At $k=0,d$, only $B_0,B_1$ are independent. The spatial Dirac action on radial coefficients is exactly

$$\begin{aligned} D(aB_0)&=-a'B_1,\\ D(bB_1)&=[b'+(d-1)b/r_x]B_0,\\ D(cB_2)&=s_kcB_0/r_x+[c'-c/r_x]B_3,\\ D(eB_3)&=s_keB_1/r_x-[e'+de/r_x]B_2. \end{aligned} \tag{14}$$

Clifford multiplication and the complete matrix Hessian preserve this span. Independent arbitrary-normal matrix variations and Cartesian derivatives verify closure and (14) in six and ten ordinary dimensions. The $k=1$ sector includes translations. These are the constant-Clifford angular multiplets; higher angular representations remain untested.

Large frequency, canonical conditioning and a slow symmetry mode

The common-gap core uses $m_c=1$ and $a=1/6$. Its assigned electron and muon coefficients are $\omega=18564$ and $3838380$ if the stated inverse identification $m_c=\omega_{0,6}$ is made. Direct diagonalization then mixes very large and very small frequencies.

Write perturbations as $UX$, where $U=(fI+ih\alpha_{\hat x})/\sqrt\rho$. This is pointwise unitary and preserves the canonical norm. Its radial phase angle satisfies

$$\theta'=F(\rho)-m_c\frac{f^2-h^2}{\rho} -\frac{(d-1)fh}{r_x\rho}. \tag{15}$$

Transform the kinetic operator and the full Hessian together. In this frame the mass blocks are exact constant integer matrices times $\omega$. Forming the small binding Hessian before adding those blocks avoids subtracting nearly equal large numbers. A canonical squeeze $q\mapsto q/\sqrt\omega$, $p\mapsto\sqrt\omega p$ in stiff directions then improves conditioning without changing the symplectic form. The calculation retains raw unsymmetrized matrices and checks their Hermiticity independently.

The common-gap core has growing controls at $\omega=1$. Across every $k=0,\ldots,6$ in the tested bases, the $\omega=30$ and electron scans show no resolved growth. The electron's largest real parts are about $10^{-11}$ in core units, while the fast frequencies are about $37128$. Refinement changes radial basis size, length and quadrature together. These are bounded numerical observations, not a statement about the entire spatial spectrum.

One muon computation produces an apparent real growth rate near $5.398\times10^{-4}$ in the $k=6$ symmetry sector. An exact Schur elimination diagnoses it. In this sector the real and imaginary Hessians have the finite projected blocks

$$H_R=\begin{pmatrix}2\omega I&L_R\\L_R&-2\omega I\end{pmatrix},\qquad H_I=\begin{pmatrix}0&L_I\\L_I&-2\omega I\end{pmatrix}. \tag{16}$$

Put $z=\lambda^2$, $A(z)=(zI+L_IL_R)/(4\omega^2)$. Eliminating the fast block gives the exact slow matching matrix

$$\boxed{\mathcal F(z)=zI+L_I^2 -L_I[I+A(z)]^{-1}A(z)(L_I-L_R).} \tag{17}$$

This form avoids cancellation of terms of order $\omega^2$. Evaluating it in the raw $L_I$ eigenbasis further resolves its smallest mode without forcing either Hessian to be symmetric numerically. The apparent growing value fails (17): its smallest matching singular value is approximately $2.927\times10^{-7}$, instead of zero.

The elimination does not assume that $L_I$ and $L_R$ commute. Where $I+A(z)$ is invertible, it also gives the exact determinant identity

$$\det(zI+H_IH_R)=(4\omega^2)^p\det[I+A(z)]\det\mathcal F(z), \tag{18}$$

where $p$ is the size of $L_I$. Independent noncommuting matrix tests check both the direct Schur complement and this characteristic determinant.

The actual slow squared growth is negative. With 26, 40 and 55 radial basis functions, its oscillation frequencies are respectively $8.99508\times10^{-4}$, $3.59022\times10^{-5}$ and $1.16970\times10^{-6}$. They approach the exact phase-symmetry zero mode, rather than a growing disturbance. Thus a small residual relative to the enormous original generator is insufficient to identify this mode; the well-conditioned reduced equation is the decisive control.

The other tested muon multiplets have real parts at floating-point accuracy. Neither this scan nor (17) proves stability for all angular momenta, radial wavelengths or finite perturbations. The action's nonsmooth currents also remain a separate evolution question.

The full classical Hamiltonian also has no lower bound at fixed unit norm, even for positive mass current and strictly timelike field current; the localized-action calculation gives an explicit smooth family. This precludes a global energy-minimum argument. It does not turn the unresolved spectral questions into a proved instability.

What the combined result establishes

The same reconstructed action now supports a localized spatial fixed point, the selected simplex carrier frequencies, their exact canonical inertia, and a finite common-probe identity. The pair comparison retains both polarizations and reciprocal translation variation. Its continuing limitations concern the actual physical carrier, observed interaction channels, full encounter evolution and complete spatial stability.

In particular, the oscillator labels still need to arise as modes of the same physical field. Their tensor interpretation and complete spacetime transformation law cannot be inferred from the dimension count alone. The inverse action is a concrete target for that derivation, not evidence that its premises are already realized by particles.

The later rest-frame carrier calculation provides an explicit covariant tensor connection from the field's own current. Its additional kinetic term preserves these linear calculations on zero-rotation-moment rays and preserves the finite common-probe identity for such targets. For complex polarized rays it changes the kinetic form and produces a defective principal response. Those new terms cannot be omitted when assessing that completed action's general stability.

The later higher angular response gives the complete polynomial angular block classification at even depth and tests additional electron and muon blocks. Its finite spectra retain their stated radial and angular limits. The trace-current comparison also identifies a secular high-momentum response that real frequencies alone do not exclude. Adding a positive trace contraction removes that specific degeneracy and yields a positive conserved norm for the complete constant far-field operator. Its localized frequency and common-probe identities survive, while its changed Hessian requires its own tests.

Verification

localized_current_response_checks.py checks the full angular source and spatial derivatives, independent direct radial matching, complete asymptotic spinor/partner symbols, raw canonical structure, finite angular spectra and the exact slow Schur equation. It preserves the complete finite eigenvalue arrays, representative eigenvectors and projected generators, including the misleading unconditioned slow result rather than silently deleting it.

The response implementations are localized_clifford_response.py, localized_radial_galerkin.py, localized_radial_evans.py and localized_full_asymptotic.py. The smooth binding control is implemented in polynomial_localized_core.py. The finite probe and inertia calculation uses localized_current_inertia_checks.py.

localized_action_audit_checks.py independently checks (17) and (18) by block Gaussian elimination and a full characteristic determinant, and compares arbitrary complex mixed second variations of the actual potential with the complete Hessian.