Localized oscillation and its inertial response

Summary: A reconstructed local current action has localized simplex-frequency solutions with common spatial profiles and an exact canonical boost inertia; its carrier, common normalization and full stability remain physical conditions to establish.

Frequency, a sustaining field and inertia in one action

Mass already has the equivalent frequency $\omega_m=mc^2/\hbar$. The dynamical question is whether a continuously oscillating, localized field can sustain itself and have the corresponding inertia under translation. The construction here answers that question for one explicit inverse action. It retains the simplex current hierarchy and adds local terms whose complete variations sustain a spatial core.

For its displayed branch, the exact results are

$$\boxed{\omega_N=\omega_{0,d}\binom{N+d}{d}},\qquad E_0=\hbar\omega_N Q,\qquad M_{ab}=\frac{\hbar\omega_N Q}{c^2}\delta_{ab}. \tag{1}$$

Here $Q$ is the dimensionless canonical field norm. One common $Q=1$ condition makes the inertial coefficient the mass-frequency value. No field magnitude is fitted separately for the electron and muon. Their spatial fixed point is the same, while their coefficient grades differ.

This is reconstructed dynamics. The original scalar quartic master term has not been shown to generate this action, its carrier or its weights. The oscillator coefficient grade has also not yet been identified with the actual spatial harmonics of a derived particle cavity. The existence and inertia calculation therefore closes a conditional field problem; it does not complete the physical IDWT mass-operator derivation.

Complete field components and local currents

Use ordinary flat physical $\mathbb R^d$ for this spatial witness, with even $d$ and $r=2^{d/2}$. The Hermitian $r\times r$ matrices obey

$$\{\alpha_a,\alpha_b\}=2\delta_{ab}I,\qquad \{\alpha_a,\beta\}=0,\qquad \beta^2=I. \tag{2}$$

Retain a doubled spinor and $r$ coefficient partners. At each oscillator grade the field has indices

$$\Psi_N\in\mathbb C^{2r}\otimes\mathbb C^r\otimes E_N, \qquad E_N=\operatorname{Sym}^N(\mathbf1\oplus\mathbb C^d). \tag{3}$$

These are components of the proposed same field, not additional particles or additional physical dimensions. All $d$ coordinates in the spatial equation are ordinary spatial directions. The solution imposes no external wall, compactification or preferred split of the six-dimensional sector. It does not establish the global geometry of the physical sector.

Define the principal and scalar-insertion matrices

$$\begin{aligned} A_a&=\sigma_x\otimes\alpha_a,& B_0&=\sigma_z\otimes I_r,\\ B_1&=\sigma_x\otimes\beta,& B_2&=I_2\otimes\beta. \end{aligned} \tag{4}$$

Each $B_j$ anticommutes with every $A_a$. The principal spatial operator is $-icA_a\partial_a$; its characteristic speed is $c$. Algebraic local nonlinearities do not change that principal symbol. This observation is not a global well-posedness theorem at nonsmooth current values.

For the moment suppress the oscillator index and regard $\Psi$ as a $2r\times r$ matrix. Define Hermitian partner currents

$$C=\Psi^\dagger B_0\Psi,\qquad J=\Psi^\dagger B_1\Psi,\qquad K=\Psi^\dagger B_2\Psi. \tag{5}$$

All complex, lower-spinor, partner and conjugate variations are retained. The signs of these matrices have no assigned antiparticle interpretation.

In the following equations set $\hbar=c=1$. The action is

$$S=\int dt\,d^dx\left[ \sum_N\frac i2\operatorname{Tr} (\Psi_N^\dagger\dot\Psi_N-\dot\Psi_N^\dagger\Psi_N) -\sum_N\operatorname{Tr}(\Psi_N^\dagger(-iA\cdot\nabla)\Psi_N) -U\right]. \tag{6}$$

The canonical norm is $Q=\int\operatorname{Tr}(\Psi^\dagger\Psi)$. It has not been identified with electric charge or derived as a quantized classical invariant.

The binding law and its complete variation

For one isolated coherent grade, use

$$U=\lambda_N\|C\|_{\mathrm{HS}} +m_c\operatorname{Tr}(J\beta_p)-g\operatorname{Tr}[Jf(K)], \qquad f(K)=\operatorname{sgn}(K)|K|^a. \tag{7}$$

The constant partner matrix $\beta_p$ is the same numerical Clifford involution as $\beta$, acting on the other matrix index. The common core coefficient $m_c$ has frequency units. It is independent of the occupied grade and is not an independently inserted mass for each particle.

The complete nonlinear source is

$$\mathcal N(\Psi)= \lambda_N B_0\Psi\frac C{\|C\|_{\mathrm{HS}}} +m_c B_1\Psi\beta_p -g\left[B_1\Psi f(K)+B_2\Psi Df_K(J)\right]. \tag{8}$$

To define the last term, diagonalize $K=V\operatorname{diag}(k_i)V^\dagger$. Then

$$Df_K(J)=V\left[L_{ij}(V^\dagger JV)_{ij}\right]V^\dagger, \quad L_{ij}=\begin{cases} \dfrac{f(k_i)-f(k_j)}{k_i-k_j},&k_i\ne k_j,\\[4pt] a|k_i|^{a-1},&k_i=k_j\ne0. \end{cases} \tag{9}$$

This divided difference includes variations between positive and negative eigenspaces. Differentiating eigenvalues alone would omit physical field directions. A fractional law needs a separate definition at zero current eigenvalues; the root term likewise needs attention at $C=0$. The displayed profiles have nonzero $|k_i|$ at every finite spatial point, but arbitrary perturbations need not preserve that condition.

The binding term is a newly reconstructed interaction of the retained field components. It is not obtained by declaring a prescribed confining potential. Its nonconvex matrix structure differs from the action in the positive-current localization theorem, whose hypotheses and conclusion remain valid for that earlier action.

A localized spatial zero field gives an exact frequency branch

Write $\Psi=(\phi,\zeta)^T$. Suppose a spatial matrix $\phi$ satisfies

$$\begin{aligned} \phi^\dagger\phi=\phi\phi^\dagger&=\rho(x)I_r,\\ \mathcal B(\phi):=-i\alpha\cdot\nabla\phi +m_c\beta\phi\beta_p -g\beta\phi f(\phi^\dagger\beta\phi)&=0. \end{aligned} \tag{10}$$

Take $\zeta=0$. Its current $J$ vanishes, while $C=\rho I_r$. The upper source in (8) is $\lambda_N\phi/\sqrt r$. The lower component of the full field equation is exactly the second equation in (10). Consequently

$$\boxed{\Psi_N(t,x)=e^{-i\omega_Nt} \begin{pmatrix}\phi(x)\\0\end{pmatrix}\otimes\overline z^{\,N}, \qquad \omega_N=\lambda_N/\sqrt r,\qquad \|z\|=1} \tag{11}$$

solves the complete equation. No residual lower-spinor equation was discarded. The conjugate on $z$ is a matrix-current convention: ordinary column Gram matrices then give $P_{z,k}=|z^k\rangle\langle z^k|$.

For the full common action, split the oscillator indices with the isometries $T_{Nk}$ of the current hierarchy. Trace only the $N-k$ oscillator spectators, retaining the partner indices. Denote the resulting current by $R_k$ after summing all grades $N\ge k$. Set

$$U_{\mathrm{mass}}=\omega_{0,d}\sqrt r \sum_{k\ge0}\binom{k+d-1}{d-1}\|R_k\|_{\mathrm{HS}}. \tag{12}$$

The binding currents $J,K$ instead trace all oscillator coefficients and sum all grades before (7) is varied. These prescriptions define one action for self-interaction and reciprocal coupling.

On (11), $R_k=\rho I_r\otimes P_{z,k}$. Each mass-current order contributes $\omega_{0,d}\phi$ times its weight. Hockey-Stick gives

$$\sum_{k=0}^N\binom{k+d-1}{d-1}=\binom{N+d}{d}, \qquad \omega_N=\omega_{0,d}D_N. \tag{13}$$

The factor $\sqrt r$ in (12) is one stated normalization rule across depths: $\sqrt r\|\rho I_r\|_{\mathrm{HS}}=r\rho$ equals the canonical density. Without it, a common unnormalized coefficient would introduce an artificial rank factor between $r_6=8$ and $r_{10}=32$. The rule preserves the proposed shared charged-lepton frequency unit if $\omega_{0,6}=\omega_{0,10}$ is imposed. It does not derive that equality or remove the documented tau mass residual in the frequency-unit comparison.

The weights, root law, normalized partner trace and coefficient carrier remain inverse premises. Equation (13) proves their consequence, not their origin in the master field. The fixed assignments remain $N=12,34$ at depth six and $N=22$ at depth ten, giving $18564$, $3838380$ and $64512240$.

Two explicit sustaining profiles

An analytic witness uses $m_c=0$ and $a=1/(d-1)$:

$$\begin{aligned} \phi_R(x)&=A_R(1+|x|^2/R^2)^{-d/2}(I+i\alpha\cdot x/R),\\ A_R&=\left(\frac d{gR}\right)^{(d-1)/2},\qquad \rho_R=A_R^2(1+|x|^2/R^2)^{1-d}. \end{aligned} \tag{14}$$

Direct differentiation gives $-i\alpha\cdot\nabla\phi_R=d\phi_R/[R(1+|x|^2/R^2)]$. Writing $\phi_R=\sqrt\rho_R U$ with unitary $U$ also gives $\beta\phi_R f(\phi_R^\dagger\beta\phi_R)=\rho_R^a\phi_R$. These two identities prove (10), without needing a variational guess.

For $d>2$ its norm is finite, and for $d>4$ its second moment is finite:

$$Q=r\left(\frac dg\right)^{d-1}R \frac{\pi^{d/2}\Gamma(d/2-1)}{\Gamma(d-1)}, \qquad \langle |x|^2\rangle=\frac d{d-4}R^2. \tag{15}$$

At fixed $g$, this critical model has a scale family. A common $Q$ selects one $R$ for all grades; the classical action does not select $Q$ itself. This analytic core has a verified growing radial disturbance at $d=6$, $\omega R=1$, as detailed in the response calculation. Its exact existence must not be mistaken for stability.

A second witness has a common nonzero core gap and an exponential tail. In $m_c=g=1$ units choose $d=6$, $a=1/6$ and

$$\phi=f(r_x)I+ih(r_x)\alpha_{\hat x},\qquad \rho=f^2+h^2, \quad \begin{cases} f'=-(1+\rho^a)h,\\ h'=(\rho^a-1)f-5h/r_x. \end{cases} \tag{16}$$

Here $r_x=|x|$ is a radius, distinct from the representation rank $r$. Regularity requires $h(0)=0$. The decaying tail has $h/f\sim K_3(r_x)/K_2(r_x)$, where $K_\nu$ is the modified Bessel function. Shooting followed by boundary-value collocation supplies the nodeless solution

The physical field magnitude here solves the fixed-point equation. It does not define the particle mass. Dividing the entire common profile by $\sqrt Q$ and choosing the single common coefficient $g=Q^{1/6}$ produces the same size at $Q=1$ for every grade. The independent spatial virial and changes of outer boundary agree with this solution. Uniqueness and all excited spatial cores have not been classified.

If one additionally identifies $m_c=\omega_{0,6}$, restoring units gives

$$\boxed{L_{\mathrm{rms},6}=1.549216\ldots\frac c{\omega_{0,6}}.} \tag{17}$$

This supplies an explicit geometric conversion for the common cavity proposal in this particular action. Equality of the two coefficients is an inverse identification, not a derived master relation. This radius is not a measured charge radius; projection to the observed interaction form factor has not been calculated. With that identification, the electron and muon have dimensionless frequencies $18564$ and $3838380$ in core units. The stability control $\omega=1$ is therefore not the assigned electron.

The dimensionless binding extension derives the optional exponent rule $a=1/d$, verifies a general core virial, and solves nodeless profiles at depths five, six and ten. It supplies the corresponding geometric factors while keeping $m_c/\omega_{0,d}$ and the common canonical norm explicit as inverse conditions.

Full stress and a direct kinetic derivation of inertia

For the displayed radial cores in (11), both the spatial kinetic expectation and the binding energy vanish pointwise, because they connect the upper and lower blocks. Their variations are nonzero and are precisely what sustain (10). The complete symmetric stress has

$$T^{00}=\omega_N\rho_{\mathrm{tot}},\qquad T^{0a}=T^{ab}=0,\qquad \rho_{\mathrm{tot}}=\operatorname{Tr}(\phi^\dagger\phi). \tag{18}$$

For a constant boost velocity $v\hat n$, use the actual transformed field $S_v\Psi(t',x')$, with $S_v=\exp[\operatorname{artanh}(v)A_{\hat n}/2]$ in natural units. Since $S_v^\dagger B_jS_v=B_j$, every displayed scalar current is invariant. Direct substitution verifies the full boosted field equation. Its stress is

$$T^{\mu\nu}=\omega_N\rho_{\mathrm{tot}}(x')u^\mu u^\nu, \qquad E(v)=\gamma E_0,\qquad P(v)=\gamma E_0v. \tag{19}$$

Lorentz contraction in the spatial integral preserves $Q$. The field continues oscillating in the zero-total-momentum frame.

There is also a direct canonical proof of the inertial coefficient. Let $\Phi=(\phi,0)^T$ and let $\mathcal H'$ be the full real-linear Hessian of the rotating equation. Differentiating the boosted solution at $v=0$ gives

$$\delta_{v_a}\Psi=e^{-i\omega_Nt}(b_a-t\partial_a\Phi),\qquad b_a=\frac12A_a\Phi+i\omega_Nx_a\Phi. \tag{20}$$

The full equations imply

$$\mathcal H'\partial_a\Phi=0,\qquad \mathcal H'b_a=-i\partial_a\Phi. \tag{21}$$

These identities also follow directly from (10). Local phase covariance of its binding source gives $\mathcal B'_\phi(ix_a\phi)=\alpha_a\phi$. Using $\{-i\alpha\cdot\nabla,\alpha_a\}=-2i\partial_a$ and $\{\beta,\alpha_a\}=0$ gives $\mathcal B'_\phi(\alpha_a\phi)=-2i\partial_a\phi$. The root-current Hessian annihilates the local scalar phase and the translation of an isotropic Gram matrix. Combining these terms proves the second identity in (21), retaining the lower component of $b_a$.

The canonical symplectic form from (6) is $\Omega(\eta,\xi)=-2\operatorname{Im}\int\operatorname{Tr}(\eta^\dagger\xi)$. The position variation is $-\partial_a\Phi$. Its pairing with the spinor part $A_b\Phi/2$ vanishes by upper/lower orthogonality. Its pairing with the phase part uses

$$\operatorname{Tr}[(\partial_a\phi)^\dagger\phi] =\tfrac12\partial_a\rho_{\mathrm{tot}},\qquad \int x_b\partial_a\rho_{\mathrm{tot}}=-\delta_{ab}Q. \tag{22}$$

The first equality holds for the displayed radial matrix cores; more general cores can contain a spatial trace phase current and need its additional terms retained. Hence

$$\boxed{\Omega(-\partial_a\Phi,b_b)=-\omega_NQ\delta_{ab}.} \tag{23}$$

Restoring the time-term coefficient $\hbar$ and the velocity factor $c^{-2}$ gives (1). This derives the center's inertia from the actual oscillating field's canonical kinetic term. It is not a redefinition of an amplitude as mass. It also differs from the two-translation surface pairing: the second variation here is a boost, including its frequency-dependent spatial phase.

A regular fixed-norm family has no lower energy bound

The energy of the localized solution is positive, but the full classical Hamiltonian in (6) is not bounded below at fixed norm. This distinction can be checked without approaching a singular current. For $0<s<1$, put

$$a_s=\sqrt{\frac{1+s}{2}},\qquad b_s=\sqrt{\frac{1-s}{2}},\qquad P_s=\frac1{\sqrt r}\begin{pmatrix}a_s I\\-b_s\alpha_1\end{pmatrix}. \tag{24}$$

Choose a smooth, strictly positive real envelope $h(x)$ with $\int h^2=1$, for example a normalized Gaussian. At a single coherent grade, consider the initial field

$$\Psi_{N,k}(x)=h(x)e^{ikx_1}P_s\otimes\overline z^{\,N}, \qquad \|z\|=1. \tag{25}$$

These are admissible smooth initial data for testing the Hamiltonian, not a proposed family of stationary particle masses. Their norm is one, and direct Clifford multiplication gives

$$C=\frac{s h^2}{r}I>0,\qquad J=0,\qquad K=\frac{s h^2}{r}\beta,\qquad j^0=h^2,\qquad j^a=-h^2\sqrt{1-s^2}\,\delta_{a1}. \tag{26}$$

Thus $K$ is invertible at every finite point and $j^\mu j_\mu=s^2h^4>0$. The binding energy vanishes because $J=0$, for both the original law and its smooth comparison. The mass-root energy integrates to $s\omega_N$. The envelope's integrated spatial derivative vanishes, while its phase contributes $-k\sqrt{1-s^2}$. Restoring units therefore gives exactly

$$\boxed{E[\Psi_{N,k}]=\hbar\left[s\omega_N-ck\sqrt{1-s^2}\right] \longrightarrow-\infty\quad(k\longrightarrow+\infty),\qquad Q=1.} \tag{27}$$

Positive mass current and a timelike field current therefore do not provide a lower energy bound. The unrestricted first-order spatial kinetic term already permits this family. The localized branch cannot be a global fixed-norm energy minimum, and such minimization cannot supply its stability proof. An unbounded Hamiltonian does not by itself prove that this particular branch is linearly or nonlinearly unstable. Any proposed physical admissibility condition must be stated, derived and shown to survive interactions; discarding field components or one frequency sign is not part of the present construction.

What remains physical input

The frequency and inertia identities do not select the shell weights. Replace the binomial weights in (12) by any positive sequence $w_k$, keeping one common action and the same sustaining core. The same contraction and canonical calculation give

$$\omega_N=\omega_{0,d}\sum_{k=0}^Nw_k,\qquad E_0=\hbar\omega_NQ,\qquad M_{ab}=\frac{\hbar\omega_NQ}{c^2}\delta_{ab}. \tag{28}$$

The spatial equation (10) does not contain these weights. This freedom tests what the inverse construction establishes; it does not propose different particle indices or a competing dimensional assignment. The Hockey-Stick identity is exact once its weights are supplied, but neither spatial existence nor the boost calculation derives them.

Likewise, $Q=1$ is a common physical normalization condition, not an arbitrary rescaling at fixed nonlinear coefficients. The coefficient $g$ in the unit-norm core was chosen once per depth to implement it. No quantization of $Q$ follows. The relation $m_c=\omega_{0,d}$ is a separate inverse identification. The relativistic kinetic structure in (6) is also supplied: the inertia calculation verifies its localized solution's response rather than deriving that kinetic structure.

The displayed boosts treat the oscillator and partner column labels as spectators. This makes the written local spinor action and its boosted solutions consistent. The later same-current rest-frame completion supplies an explicit Lorentz transformation law for the spatial tensors and varies its additional kinetic connection. The stationary frequencies and inertia survive, and zero-rotation-moment rays retain the complete linear response. Complex polarized rays expose a defective principal propagation matrix, however, so this completion does not yet settle a general physical carrier. Likewise, a list of coefficient grades is not a derivation of their underlying spatial harmonics, occupations or selection. Those are distinct carrier problems.

The master origin of the reconstructed terms, the common canonical norm, the relation of the core scale to $\omega_{0,d}$, and full spatial stability remain necessary for a physical particle interpretation. The complete response calculation gives finite normalized common-probe identities and bounded radial and angular tests. It does not yet identify observed electromagnetic or weak vertices, derive cross-depth coupling, or settle the neutrino control.

The solved-core pair calculation exhibits a specific fractional-current cancellation during an allowed pair preparation. Its smooth binding comparisons remove that particular variation singularity and still have localized depth-six solutions with the exact frequencies and full boosted stress. The smoothing scale is an additional common inverse coefficient; its spatial profiles and stability tests must be kept distinct from the original fractional model.

The trace-current comparison changes the current norm away from the occupied ray while preserving its complete isolated mass source, energy and spatial core. It removes a specific secular high-momentum response of the original norm and gives a positive conserved norm for the constant far-field linearization. This is a further inverse action choice. It does not select the shell weights, provide a physical energy floor, or resolve the polarized carrier's kinetic defect.

Verification

localized_current_inertia_checks.py checks independent variations of (7), the full six- and ten-dimensional Cartesian equations, complete boosted stress, exact current contractions, the common-gap fixed point and virial, both Hessian identities (21), and the entire six-direction tensor (23). It also checks finite normalized probe energies and reciprocal forces. Its preserved arrays contain the actual radial solution, current response and inertia tensors.

The executable action and core solvers are localized_current_model.py, localized_partner_tower.py and gapped_localized_core.py. The tolerances distinguish exact algebraic identities from numerical collocation and differentiation errors. Passing these checks establishes the stated model calculations, not the unresolved physical premises.

localized_action_audit_checks.py independently checks the complete potential's mixed second variation, arbitrary positive shell weights with every depth-six coefficient slot, and the exact currents and energy in (24)–(27). It also verifies the response calculation's Schur equation by direct block elimination with noncommuting matrices. The analytic critical profile has an exact existence proof; the common-gap and smooth profiles are refined numerical witnesses, not computer-assisted existence theorems.