Spatial mode tensors carried by the field's rest frame

Summary: A same-current rest-frame connection gives a covariant spatial tensor action and preserves zero-rotation-moment branches, while polarized principal modes expose a high-wave-number limitation of this completion.

A precise transformation law for spatial coefficients

The localized frequency and inertia action has an explicit spatial solution. Its initial boost calculation treats coefficient labels as spectators. If the oscillator coefficients instead represent ordinary spatial tensors, their transformations must compose consistently with the spinor boosts.

One local inverse completion uses the field's own current to define its rest frame. It introduces no independent frame field, additional particle or additional spatial dimension. It gives an exactly covariant action on a stated domain. It also produces a new kinetic response for complex polarized mode rays. That response has a high-wave-number limitation, derived below, so covariance alone does not complete the physical carrier.

Use natural units and the full spinor matrices $A_a$ and canonical grade fields of the localized action. Define

$$\begin{aligned} I_N^\mu&=\Psi_N^\dagger A^\mu\Psi_N, \qquad A^0=I,\quad A^a=A_a,\\ j^\mu&=\sum_N\operatorname{Tr}I_N^\mu,\qquad s=\sqrt{j^\mu j_\mu},\qquad u^\mu=j^\mu/s. \end{aligned} \tag{1}$$

The metric is $\eta=\operatorname{diag}(1,-1,\ldots,-1)$. This construction requires $j^0>0$ and $j^\mu j_\mu>0$. At a displayed localized core, $j=(\rho_{\mathrm{tot}},0)$ at every finite point, so the condition holds. Its zero-current and null-current boundaries remain outside this definition.

All spatial frames here concern the $d$ ordinary spatial directions of one sector. A particle's local zero-momentum frame does not stop its oscillation or split its physical dimensions into different kinds.

The subsequent retained-frame completion replaces the instantaneous frame response and repairs the stated principal defect. Its spatial source and mode-weight selection remain unresolved; this extension is distinct from the normal-carrier construction.

The induced spatial rotation and connection

For $u=(u^0,\mathbf u)$, the canonical boost sending $e_0$ to $u$ is

$$B(u)=\begin{pmatrix} u^0&\mathbf u^T\\ \mathbf u&I+\dfrac{\mathbf u\mathbf u^T}{1+u^0} \end{pmatrix}. \tag{2}$$

It satisfies $B^T\eta B=\eta$. Under a proper time-preserving Lorentz transformation $\Lambda$, put $u'=\Lambda u$. The matrix

$$B(u')^{-1}\Lambda B(u) =\begin{pmatrix}1&0\\0&R(\Lambda,u)\end{pmatrix} \tag{3}$$

defines an ordinary orthogonal spatial rotation. Matrix multiplication immediately proves the composition law

$$R(\Lambda_2\Lambda_1,u) =R(\Lambda_2,\Lambda_1u)R(\Lambda_1,u). \tag{4}$$

Thus spatial tensor rotations and boosts are parts of one transformation rule. A tensor is referred to its own current's spatial rest frame; it is not assigned an unrelated spectator boost.

Project the matrix one-form $B^{-1}dB$ onto its spatial block. Direct differentiation gives the real antisymmetric connection

$$\boxed{\mathfrak a_\mu= \frac{(\partial_\mu\mathbf u)\mathbf u^T -\mathbf u(\partial_\mu\mathbf u)^T}{1+u^0}.} \tag{5}$$

With the spacetime one-form transformation understood, it obeys $\mathfrak a'=R\mathfrak aR^{-1}-(dR)R^{-1}$. It is not generally flat. At rest its curvature evaluated on two velocity variations $v,w$ is $w v^T-v w^T$. A sequence of changes of velocity can therefore rotate the spatial coefficient frame.

Let $D_N(R)$ be the normalized symmetric-tensor representation on $E_N=\operatorname{Sym}^N(\mathbf1\oplus\mathbb C^d)$. The scalar summand is invariant and the spatial vector summand transforms by $R$. Its connection is

$$\Gamma_{N,\mu}=dD_N(0\oplus\mathfrak a_\mu). \tag{6}$$

The partner column labels of the localized matrix core remain spectators under both rotations and boosts. Only the oscillator spatial tensors use (6). This leaves the original spinor transformation on the field's row index intact. A transformation of the independent partner labels is not needed to prove (4).

In normalized occupation bases, a one-slot generator $G$ acts through $\sqrt{n_b(n_a+1)}G_{ab}$ between occupations differing by transfer of one index from $b$ to $a$, and through $\sum_an_aG_{aa}$ on the diagonal. This specifies (6) on every coefficient, not just a selected coherent ray.

Vary the resulting action, including its frame

With matrix columns carrying the tensor representation, define

$$\Psi'_N(x')=S(\Lambda)\Psi_N(x)D_N(R)^{-1},\qquad \nabla_\mu\Psi_N=\partial_\mu\Psi_N-\Psi_N\Gamma_{N,\mu}. \tag{7}$$

Identity matrices on partner indices are implicit. The induced rotations are unitary, so tracing the column indices still gives $j'^\mu=\Lambda^\mu{}_{\nu}j^\nu$. Definition (1) is therefore consistent with its own nonlinear field transformation.

Replace the ordinary derivative in the canonical spinor action by (7), retaining the same current potential $U$. The complete extra term is

$$\boxed{\mathcal L_{\mathrm{frame}} =-i\sum_{N,\mu}\operatorname{Tr}(I_N^\mu\Gamma_{N,\mu}).} \tag{8}$$

The current hierarchy's partial traces commute with the tensor rotations, so the same root-current potential and binding potential remain invariant. The kinetic term is invariant only when the derivative of the field-dependent rotation in (7) is retained as well. Independent full first-derivative data in six and ten dimensions verify this equality for the complete action.

The connection is a local function of the fields and their first derivatives. It is linear in those derivatives, so this is a local first-order quasilinear action on its stated domain. Its coefficients must be varied with the fields. Holding $u$ fixed while varying $\Psi$ gives a different equation and would miss the effect below.

Each grade's local phase leaves $j$, $u$, $I_N$ and (8) unchanged. Only the original time and spatial terms contribute the phase-gradient change $-j_N^\mu\partial_\mu\theta_N$. The Noether current and conserved norm therefore remain the canonical $j_N$ and $Q_N$. The general field-space kinetic two-form nevertheless changes; a conserved norm is not the entire kinetic structure.

Which localized results survive exactly

At a static core, $u=e_0$ and $\partial u=0$. Around it, $\mathfrak a$ starts quadratically in variations of the spatial current velocity. Hence (8) and its first variation vanish at that field. The displayed localized solutions and their simplex frequencies remain exact stationary solutions of this completed action. The added kinetic term does not turn their amplitudes into masses.

For a coherent ray write $z=(z_0,z_1,\ldots,z_d)$, $\|z\|=1$, and define its spatial rotation moment

$$M_{ab}=2\operatorname{Im}(\overline z_a z_b),\qquad 1\le a,b\le d. \tag{9}$$

At the core, $I_N^0=\rho I_r\otimes P_{z,N}$ and $I_N^a=0$. The symmetric-tensor expectation is $\langle z^N,dD_N(G)z^N\rangle=N z^\dagger Gz$. Writing $v$ for the first variation of $\mathbf u$, the entire quadratic addition is consequently

$$\boxed{\mathcal L_{\mathrm{frame}}^{(2)} =-\frac{i\rho_{\mathrm{tot}}N}{2} z^\dagger\bigl(0\oplus[\dot v v^T-v\dot v^T]\bigr)z =-\frac{\rho_{\mathrm{tot}}N}{2}v^TM\dot v.} \tag{10}$$

A real coherent ray has $M=0$. More generally, zero spatial rotation moment is the precise condition; the scalar slack component can have a separate phase without producing $M$. Under this condition the new action starts at cubic order in arbitrary full field perturbations, including all spinor, partner, mode and derivative directions. Its complete linear response is therefore the one already tested in the localized response calculation.

For $M\ne0$, (10) must be retained. Independent variations of the actual action verify both the vanishing quadratic coefficient at a real ray and the nonzero coefficient at a complex polarized ray. An agreement of the stationary frequencies would not reveal this difference.

The exact translation/boost inertia is retained. At rest a translation changes the profile but not $u=e_0$, so (10) contributes zero to its pairing with a boost. The earlier coefficient $\hbar\omega_NQ/c^2$ is unchanged. Uniformly boosted stationary solutions also have constant $u$, making (8) zero on the solution. These statements do not establish the stability of the completed action.

A finite probe identity with the new kinetic term

For a target whose spatial rotation moment is zero, its contribution to (8) vanishes for every connection: the trace of each tensor rotation generator is zero on that ray. A fixed finite probe can change the common current $u$ and its derivatives, but their values remain the same for different larger target grades at the same target profile.

Thus the cross contribution of (8), its full probe variation and its target translation variation remain independent of target grade under the same distinct-grade conditions as the earlier coupling identity. The result uses the complete new action, rather than keeping its frame fixed during the probe variation.

The diagnostic uses a unit-norm Gaussian probe with a spatially varying matrix polarization, retaining constant and linear spatial coefficient terms, their phases and a moving center. The added cross term is nonzero. Targets of grades two and three give equal values and equal arbitrary first-derivative probe variations at three points. This is a conditional functional comparison, not a solved encounter or an observed vertex.

The polarized principal symbol has a defective part

The nonzero coefficient in (10) permits an exact high-wave-number test. At one isotropic core point normalize $\rho_{\mathrm{tot}}=1$ and $\phi=I_r/\sqrt r$. Use real canonical coordinates $y=(\sqrt2\operatorname{Re}\eta,\sqrt2\operatorname{Im}\eta)$ for all spinor and partner components of the occupied oscillator ray. The velocity variation is $v=Cy$, with

$$C_a=\sqrt2\bigl(\operatorname{Re}\operatorname{vec}t_a, \operatorname{Im}\operatorname{vec}t_a\bigr),\qquad t_a=\begin{pmatrix}0\\\alpha_a\phi\end{pmatrix}. \tag{11}$$

The canonical and additional kinetic matrices are

$$\Omega_0=\begin{pmatrix}0&-I\\I&0\end{pmatrix},\qquad \Omega_1=-NC^TMC. \tag{12}$$

Actual first derivatives of (8) in two independent lower-spinor directions verify this added two-form. It is not inferred just from the frequency spectrum. Since $C\Omega_0^{-1}C^T=0$, the matrix $Z=\Omega_0\Omega_1$ satisfies $Z^2=0$. The kinetic form remains invertible: $\Omega^{-1}\Omega_0=I+Z$.

Let $A$ be the real form of the original Clifford principal matrix in one ordinary spatial direction. It obeys $A^2=I$ and exchanges upper and lower spinor blocks. The new principal evolution matrix is

$$\boxed{P=(I+Z)A.} \tag{13}$$

The correction $Z$ acts only in the lower block, giving $ZAZ=0$. Consequently

$$K=P^2-I=Z+AZA,\qquad K^2=0. \tag{14}$$

All characteristic roots remain $+1$ or $-1$, but for a nonzero rotation moment the matrix need not be diagonalizable. Decompose it exactly as

$$P=P_s+P_n,\qquad P_s=P(I-K/2),\qquad P_n=PK/2, \quad P_s^2=I,\quad P_n^2=0,\quad [P_s,P_n]=0. \tag{15}$$

The Fourier propagator of this frozen principal equation is

$$e^{-iktP} =(\cos kt\,I-i\sin kt\,P_s)(I-iktP_n). \tag{16}$$

The last factor grows with wave number when $P_n\ne0$. For the explicit ray $z_1=1/\sqrt2$, $z_2=i/\sqrt2$, degree $N=2$ and $d=6$, the largest amplification at time $2\pi$ is approximately $12.645$, $125.672$ and $1256.638$ at integer wave numbers $1$, $10$ and $100$. Every complete spinor and partner component is included; the orthogonal oscillator directions have the original principal symbol.

This proves failure of a wave-number-independent canonical norm estimate for that principal equation. It does not demonstrate superluminal characteristic speeds, and it is not a numerical claim of a particle lifetime. A lower-order mass term does not repair diagonalizability of the principal matrix. The simple induced connection therefore does not supply the general strong-hyperbolicity control required for the physical carrier. At $M=0$, $Z=0$ and the complete original Clifford symbol is recovered, but that special branch is not a proof of nonlinear regularity in a neighborhood containing polarized fields.

The remaining carrier requirement

This construction supplies an actual transformation law, its local action, its reciprocal kinetic response and a precise propagation restriction. A more complete kinetic reduction must address the polarized defect while retaining the mass-frequency branch and its normalization. Additional terms or constraints would need their own same-field origin and variations.

It also remains necessary to identify the coefficient grades with physical spatial modes and explain their selection. Rest-frame covariance is not that identification. The derivative-carrier calculation already shows why simply declaring the coefficients to be normalized derivatives can introduce a nonlocal response. Neither a representation dimension nor a boost composition law bypasses that dynamical test.

Verification

rest_current_carrier_checks.py checks 60 conditions: boost-frame composition and curvature, complete tensor generators, the full field-dependent action in six and ten dimensions, phase current, full first-derivative variations, a finite textured probe, the actual kinetic two-form and the exact polarized principal propagator. The growing principal control is an expected mathematical result, not a test silently adjusted to pass stability. The action and representation calculations are in rest_current_carrier.py.