The reconstructed localized-current action has an exact frequency and inertia relation. Its earlier spatial-tensor completion imposed an instantaneous identification between the coefficient frame and the field's timelike current. For polarized fields that identification produced a defective principal propagation matrix.

A dynamical frame supplies an explicit inverse completion of that kinetic construction. It preserves the localized frequency and boost solutions, while replacing the high-wave-number defect with a finite frame response. It does not derive the oscillator grades as actual spatial modes, choose their weights, or prove particle stability.

This is a different extension from the normal-band construction in A complete finite normal band. The two have not been combined into a single derived master action. The present calculation extends the maintained Localized oscillation and its inertial response and Spatial mode tensors carried by the field's rest frame.

Let the same-field frame respond

Use natural units c=hbar=1. Keep the complete grade fields Psi_N, their local current j^mu, and s=sqrt(j^mu j_mu)>0. Introduce a future unit timelike orientation u, with u^mu u_mu=1. It represents a retained collective frame degree of the same proposed field. Its realization in the master spinor is an additional inverse premise; it is not an exchanged particle or a Yukawa coupling.

The frame refers to all d ordinary spatial directions of a sector. Its components add no physical spatial dimensions. A constant frame does not mean that the particle's oscillating field is at rest internally.

Use the canonical boost B(u), its induced spatial rotation, and the spatial connection a_mu=(B^(-1) partial_mu B)spatial from the earlier carrier. The tensor connection Gamma(N,mu) acts on the full Sym^N(1 plus C^d) coefficient representation. The field derivative is

∇μΨN=∂μΨN−ΨNΓN,μ(u).(1) \nabla_\mu\Psi_N=\partial_\mu\Psi_N-\Psi_N\Gamma_{N,\mu}(u). \tag{1}

The crucial change is that u is varied independently. It is not replaced by j/s before varying the action.

An invariant local action

Retain the existing current potential U and its covariant kinetic term (1). Add

Lu=−Ifs2Gμν(u) ∂μuα∂νuα−κ(uμjμ−s),Gμν(u)=cf2ημν+(1−cf2)uμuν,If>0,κ>0,0<cf<1.(2) \begin{aligned} \mathcal L_u&=-\frac{I_f s}{2} G^{\mu\nu}(u)\,\partial_\mu u^\alpha\partial_\nu u_\alpha -\kappa(u_\mu j^\mu-s),\\ G^{\mu\nu}(u)&=c_f^2\eta^{\mu\nu} +(1-c_f^2)u^\mu u^\nu, \qquad I_f>0,\quad\kappa>0,\quad0<c_f<1. \end{aligned} \tag{2}

I_f is the common frame kinetic coefficient and kappa the common current-locking coefficient. They are not separately adjusted for different particle frequencies. In the frame u=e_0, the kinetic term has positive time coefficient and spatial wave speed c_f. The target metric is positive on variations tangent to u^2=1.

For future timelike j, u dot j is at least s. The second term therefore penalizes misalignment, with its minimum at u=j/s. Equation (2) gives that alignment a finite response time instead of imposing it identically. The factor s makes this an orientation term carried by the localized field density, rather than a prescribed nonzero background at infinity. The zero-current boundary still requires a separate nonlinear analysis.

Both terms are Lorentz scalars. The acoustic cone lies strictly inside the Dirac light cone. The frame's transformation induces the same unitary tensor rotation as before, so j still transforms as a vector. All added terms are invariant under each grade's local phase. The canonical grade norms Q_N therefore remain unchanged.

The existing mass-frequency solutions survive

For the localized matrix core, j=(rho_total,0) everywhere at finite position. Take u=e_0. The added frame kinetic term, connection term, alignment term and their first variations vanish on this branch. For the alignment term, variations are taken on the unit-timelike constraint; this also gives zero added stress on the aligned branch.

The same is true for its exact uniformly boosted field, with constant u equal to the boosted current direction. Covariance therefore retains

ωN=ω0,d(N+dd),E0=ωNQ,Mab=ωNQδab.(3) \boxed{\omega_N=\omega_{0,d}\binom{N+d}{d},\qquad E_0=\omega_NQ,\qquad M_{ab}=\omega_NQ\delta_{ab}.} \tag{3}

Restoring units gives M_ab=hbar omega_N Q delta_ab/c^2. These are the existing conditional solutions, with their existing weights, binding law and common Q=1 premise. The extension does not newly derive those inputs or replace frequency with a field magnitude.

More explicitly, a translation of the core has delta u=0. Its boost variation has delta u=e_a, constant in space and time. The existing spinor boost b_a=A_a Phi/2+i omega_N x_a Phi gives exactly the same current variation, delta(j/s)=e_a. Both new reciprocal locking sources therefore vanish. The frame adds zero to the translation–boost symplectic pairing, preserving the inertia calculation in (3).

Complete quadratic equations at the core

Let y contain the real canonical spinor and partner variations of the occupied ray, and let v be the spatial variation of u. Define delta(j/s)=C(x)y. For an isotropic matrix core,

ρtotalC(x)C(x)T=2Id.(4) \rho_{\rm total} C(x)C(x)^T=2I_d. \tag{4}

Thus the coupling to canonically weighted frame fields is bounded even where the spatial profile is small. The full additional quadratic action is

Lextra(2)=Ifρtotal2(∣v˙∣2−cf2∣∇v∣2)−Nρtotal2vTMv˙−κρtotal2∣v−Cy∣2.(5) \mathcal L_{\rm extra}^{(2)}= \frac{I_f\rho_{\rm total}}2 (|\dot v|^2-c_f^2|\nabla v|^2) -\frac{N\rho_{\rm total}}2v^TM\dot v -\frac{\kappa\rho_{\rm total}}2|v-Cy|^2. \tag{5}

Here M_ab=2 Im(conj(z_a) z_b) is the spatial rotation moment of the occupied coherent ray. It is a real antisymmetric matrix, not the particle mass. All ordinary and conjugate spinor variations enter y. Coefficient directions orthogonal to the ray have zero first current variation and retain their earlier complete equations.

Writing J for the canonical real symplectic inverse and L_0 for the original full localized spinor linearization, (5) gives

y˙=L0y+κJρtotalCT(Cy−v),Ifρtotalv¨−Ifcf2∇ ⁣⋅(ρtotal∇v)+NρtotalMv˙+κρtotal(v−Cy)=0.(6) \begin{aligned} \dot y&=L_0y+\kappa J\rho_{\rm total}C^T(Cy-v),\\ I_f\rho_{\rm total}\ddot v &-I_fc_f^2\nabla\!\cdot(\rho_{\rm total}\nabla v) +N\rho_{\rm total}M\dot v +\kappa\rho_{\rm total}(v-Cy)=0. \end{aligned} \tag{6}

Both responses follow from the same quadratic action. Holding the frame fixed, or omitting the source in either line, gives a different calculation.

Recover the old connection only within its regime

At a constant core point, Fourier transformation gives

[κ+If(cf2∣k∣2−Ω2)−iΩNM]v=κCy.(7) \left[\kappa+I_f(c_f^2|k|^2-\Omega^2) -i\Omega NM\right]v=\kappa Cy. \tag{7}

When the frequency and wave number are small compared with the frame response scales, v approaches Cy. Its first frequency correction gives the old kinetic form Omega_0-N C^T M C. The latter is therefore a low-frequency reduction of (6), not its unrestricted principal equation. Taking instantaneous locking at every wave number is a nonuniform limit.

The spinor principal speeds of (6) are plus and minus one. The retained frame speeds are plus and minus c_f. Its principal matrix is diagonalizable; the gyroscopic term and locking terms have no spatial derivatives of y in this stationary background.

The affected spinor space is exactly the span of C^T, JC^T, AC^T and JAC^T, where A is the Clifford symbol in the tested spatial direction. Its dimension is 4d. Direct full-spinor embeddings verify this closure at depths two, four, six and ten. Every complementary spinor and partner direction has its original Clifford symbol. This is a complete decomposition of the affected principal equation, rather than a few-component polarization truncation.

Nonuniform frames retain separated principal cones

The repair is not restricted to a perfectly constant frame. On the smooth timelike-current domain, retain u and its first derivatives as independent first-order variables. The spinor principal operator stays the original Clifford operator: its new terms contain u and its retained derivatives algebraically. The frame equation contains first derivatives of the spinor through partial_mu s. Substituting the spinor time equation removes that mixed time derivative.

At a point choose the local frame u=e_0. Let A be the spinor symbol in one unit spatial direction, R the full real derivative of s with respect to y, and W_0,W_1 the background time and directional spatial derivatives of u. In frame variables (delta u_t,c_f delta u_1), the complete propagating principal block has the form

P=(A0BPf),Pf=(0−cfI−cfI0),B=−1s(W0RA+cf2W1R0).(7a) P=\begin{pmatrix}A&0\\B&P_f\end{pmatrix},\qquad P_f=\begin{pmatrix}0&-c_fI\\-c_fI&0\end{pmatrix},\qquad B=-\frac1s\begin{pmatrix}W_0RA+c_f^2W_1R\\0\end{pmatrix}. \tag{7a}

The connection's curvature terms have lower differential order here. Constraint and undifferentiated-frame variables have zero principal speed and do not overlap the Dirac speeds. Since A^2=I and P_f^2=c_f^2 I, the explicit matrix

X=PfB+BA1−cf2satisfiesPfX−XA=−B.(7b) X=\frac{P_fB+BA}{1-c_f^2} \quad\hbox{satisfies}\quad P_fX-XA=-B. \tag{7b}

The invertible change with lower block X therefore separates P into the two Hermitian principal blocks A and P_f. Pulling back their positive norm gives a positive symmetrizer. This proves local strong hyperbolicity on the stated regular domain, with the strictly separated cones in (2). It does not extend the model through s=0 or prove global nonlinear regularity. Taking c_f=1 removes this separation and is not covered by the argument.

frame-principal-completion-20260914T042624_538609Z.json passes 46 checks. It retains all spinor and partner components in nonuniform frame jets, independently differentiates the actual timelike density, and verifies the mixed time elimination and the explicit symmetrizer at depths two, four, six and ten.

A spatial energy estimate without a mode cutoff

The full frame spatial operator is

Lρ=−ρtotal−1∇ ⁣⋅(ρtotal∇),⟨v,Lρv⟩ρ=∫ρtotal∣∇v∣2≥0.(8) L_\rho=-\rho_{\rm total}^{-1} \nabla\!\cdot(\rho_{\rm total}\nabla), \qquad \langle v,L_\rho v\rangle_\rho =\int\rho_{\rm total}|\nabla v|^2\ge0. \tag{8}

Its Friedrichs realization in the weighted space retains the complete spatial domain. On the analytic localized core, rho_total is proportional to (1+r^2)^(1-d). Multiplication by sqrt(rho_total) transfers it to

ρLρ1ρ=−Δ+(d−1)(r2−d)(1+r2)2.(9) \sqrt\rho L_\rho\frac1{\sqrt\rho} =-\Delta+\frac{(d-1)(r^2-d)}{(1+r^2)^2}. \tag{9}

The potential in (9) is bounded. Its negative central value does not contradict (8); the complete operator is nonnegative and has the normalized density-root zero mode.

Use the positive linear norm

E=∥y∥22+If∫ρ∣v˙∣2+Ifcf2∫ρ∣∇v∣2+κ∫ρ∣v∣2.(10) \mathcal E=\|y\|_2^2 +I_f\int\rho|\dot v|^2 +I_fc_f^2\int\rho|\nabla v|^2 +\kappa\int\rho|v|^2. \tag{10}

The antisymmetric gyroscopic term contributes zero to its derivative. Equation (4) bounds both reciprocal cross terms. The original spinor principal operator is norm-preserving; its localized potential Hessian is a bounded zeroth-order operator on the stated branch. If B_0 bounds its norm-growth contribution, a sufficient estimate is

E(t)1/2≤eCtE(0)1/2,C=B0+2κ+2(κ+κIf).(11) \mathcal E(t)^{1/2}\le e^{Ct}\mathcal E(0)^{1/2},\qquad C=B_0+2\kappa+\sqrt2\left(\sqrt\kappa+ \frac\kappa{\sqrt{I_f}}\right). \tag{11}

This estimate includes all spatial wavelengths and angular modes. In particular, it does not contain a factor proportional to wave number or to the gyroscopic degree N. The original potential bound B_0 can depend on the selected frequency. The estimate establishes a linear evolution bound, not spectral stability or bounded evolution for all time.

For I_f=kappa=1 and c_f=1/2, the frozen principal controls at time 2 pi approach amplification 6.4385 as wave number increases to 10000. The old instantaneous connection instead had amplification growing proportionally to wave number. Finite resonances and secular responses remain in the new lower-order equations.

Verification and physical limits

dynamical-frame-20260914T041717_300951Z.json passes 109 checks. It verifies the independent-frame Lorentz transformation, the complete kinetic action, the acoustic and locking terms, the full principal embedding, the low-frequency reduction, and the uniform propagation estimate. The ordinary eigenvalue solver is not used to infer stability near repeated roots; an exact triangular characteristic factorization controls those frequencies.

dynamical-frame-core-20260914T042117_674881Z.json passes 270 checks. Actual localized Cartesian fields verify every translation and boost current variation in six and ten dimensions. Complete moving fields give zero added action. Independent radial energy integrals and all-coordinate derivatives verify the weighted spatial operator in (8)-(9).

The retained orientation, its coefficients, the original current weights, and the physical realization of the coefficient modes remain inverse inputs. The earlier core's known unstable parameter controls remain relevant; adding frame dynamics does not erase them. A full nonlinear existence and stability theory, especially near zero or null current, is not supplied by (11). The result repairs a specific kinetic defect while preserving the conditional frequency–inertia branch.