A trace contribution to the mass current

Summary: Adding a trace contraction preserves coherent simplex frequencies and the common core, removes a high-momentum secular degeneracy, and gives a positive conserved norm for the constant far-field linearization; master selection and full spatial stability remain open.

The freedom left by the frequency calculation

The localized current action produces its simplex frequencies on an isotropic partner current carried by a coherent oscillator ray. Its Hilbert–Schmidt current norm is one inverse choice. The frequency calculation does not fix how the action curves away from that ray. That curvature matters for perturbations and coupling.

Let $r$ be the partner rank and let $R$ be any Hermitian retained current. Its matrix size can exceed $r$ because it also retains oscillator indices. Consider the invariant norm

$$\begin{gathered} F_q(R)=\sqrt{qr\operatorname{Tr}(R^2) +(1-q)[\operatorname{Tr}R]^2},\\ 0<q\leq1. \end{gathered} \tag{1}$$

This is positive for every nonzero Hermitian $R$ and smooth there, including when individual eigenvalues vanish. The entire zero current remains a nonsmooth point. At $q=1$, (1) is the original $\sqrt r\|R\|_{\mathrm{HS}}$.

Use the same complete doubled spinor, partner and oscillator fields as in the localized action. For each retained order $k$, the current $R_k$ sums the scalar-insertion currents of all grades $N\geq k$, tracing only their discarded oscillator spectators. Keep the same canonical and spatial Dirac terms and the same binding interaction. Replace only the mass potential by

$$U_{\mathrm{mass},q}=\omega_{0,d}\sum_{k\geq0}h_{k,d}F_q(R_k), \qquad h_{k,d}=\binom{k+d-1}{d-1}. \tag{2}$$

One $q$ applies to every grade at the given depth. There is no separate electron or muon mass coefficient in this replacement. The sums are taken on finite grade support, or on a convergent domain where their stated variations exist.

Complete variation and the unchanged frequency branch

Define a linear map on Hermitian matrices,

$$\mathcal L_q(X)=qrX+(1-q)\operatorname{Tr}(X)I. \tag{3}$$

The identity has the full retained matrix size. Differentiating the potential, rather than selecting its eigenvalue variations, gives

$$\begin{aligned} \nabla F_q(R)&=\frac{\mathcal L_q(R)}{F_q(R)},\\ D\nabla F_q(R)[X] &=\frac{\mathcal L_q(X)}{F_q(R)} -\frac{\mathcal L_q(R)\operatorname{Tr}[\mathcal L_q(R)X]} {F_q(R)^3}. \end{aligned} \tag{4}$$

These formulas retain every matrix direction. In the full field equation, each gradient is lifted back through the same current-splitting map that defined $R_k$. Its variation includes both the changed field and (4).

On the solved coherent core, $R_k=\rho I_r\otimes P_{z,k}$, with $P_{z,k}$ a unit-rank projector. Consequently

$$F_q(R_k)=r\rho,\qquad \nabla F_q(R_k)=q(I_r\otimes P_{z,k})+(1-q)I. \tag{5}$$

The occupied field lies in the range of this projector. Both terms in the second expression therefore act on it as their scalar coefficients. Every retained order supplies its original contribution to the frequency:

$$\boxed{\omega_N=\omega_{0,d}\sum_{k=0}^Nh_{k,d} =\omega_{0,d}\binom{N+d}{d}.} \tag{6}$$

The complete isolated mass source is unchanged. The lower-spinor sustaining equation and its smooth binding are also unchanged. Thus the same unit-norm core serves every occupied grade, with the same spatial size and the same fixed-point magnitude. The magnitude is not a separate mass variable. The isolated energy and canonical inertia remain $E_0=\hbar\omega_NQ$ and $M=\hbar\omega_NQ/c^2$.

The scalar insertion currents are invariant under the displayed spinor boosts. Since both the potential and its complete source agree on the boosted coherent branch, the localized action's boost and stress argument continues to apply. The common $Q=1$ condition and the identification of core gap with frequency unit remain independent inverse premises.

The change appears in the full occupied-ray Hessian. Write $\phi=\sqrt\rho U$ and let $\mathcal B'_\phi$ be the complete real-linear spatial binding derivative. In the rotating frame,

$$\mathcal H'_q= \begin{pmatrix}\mathcal A_q&\mathcal B'_\phi\\ \mathcal B'_\phi&-2\omega_N I\end{pmatrix},\qquad \mathcal A_q(UX)=2q\omega_N U \left(\operatorname{Herm}X- \frac{\operatorname{Tr}\operatorname{Herm}X}{r}I\right). \tag{7}$$

All complex and conjugate directions remain. The change is a definite partner curvature in the same action, rather than a rescaling of a reported response by its particle mass.

Equal trace and exchange contractions

Two invariant quadratic expressions in a Hermitian current are $\operatorname{Tr}(R^2)$ and $(\operatorname{Tr}R)^2$. They are quartic in the original field. Let $\mathsf S$ exchange the two factors of the retained component space. Direct index contraction gives

$$\operatorname{Tr}[(I+\mathsf S)(R\otimes R)] =\operatorname{Tr}(R^2)+(\operatorname{Tr}R)^2. \tag{8}$$

Giving these two contractions equal coefficients and requiring (5) produces

$$F_*(R)=\sqrt{\frac r{r+1}} \sqrt{\operatorname{Tr}(R^2)+(\operatorname{Tr}R)^2}, \qquad q=\frac1{r+1}. \tag{9}$$

At depth six, $r=8$ and this comparison has $q=1/9$. At even depth ten, $r=32$ gives $q=1/33$. These ranks count field components, not extra spatial dimensions. Equation (8) is finite component algebra; it does not impose a statistical ensemble or a physical compact space.

Equal coefficients in (8) are a declared inverse choice. Invariance alone allows the whole family (1), and the master field has not been shown to select (8), its square root, or the shell weights. Thus (9) supplies a concrete comparison without tuning $q$ to the particle masses; it is not a forward derivation of a unique interaction.

The limiting wave problem and its exact reduction

Use natural units. In the decaying tail of the smooth common-gap core, the nonlinear binding coefficients vanish and the core gap $m_c$ remains. Freeze the limiting direction and the partner frame. A unitary change on the partner index takes the upper background matrix to $I_r$. The complex-linear spatial block then has the form

$$\mathcal D_p X=(\alpha\cdot p)X+m_c\beta X B, \qquad B^\dagger=B,\quad B^2=I, \qquad \mathcal D_p^2=(|p|^2+m_c^2)I. \tag{10}$$

The last equality follows from $\{\alpha_a,\beta\}=0$; no commuting assumption between independent matrix currents is used. Put $\varkappa=\sqrt{|p|^2+m_c^2}$. Multiplying each lower component by its corresponding $\mathcal D/\varkappa$ is unitary. Transposing the conjugate matrix component is another unitary change. The entire frozen generator then splits into one trace block $G_0(\varkappa)$ and $r^2-1$ traceless blocks

$$G_q(\varkappa)= \begin{pmatrix} q\omega&\varkappa&q\omega&0\\ \varkappa&-2\omega&0&0\\ -q\omega&0&-q\omega&\varkappa\\ 0&0&\varkappa&2\omega \end{pmatrix},\qquad \omega=\omega_N. \tag{11}$$

The evolution convention is $e^{-i\Omega t}$. Its squared frequencies are

$$\boxed{\Omega_\pm^2=\varkappa^2+2\omega^2 \ \pm\ 2\omega\sqrt{\omega^2+(1-q)\varkappa^2}.} \tag{12}$$

Both are positive for $0<q\leq1$ and $\varkappa>0$. The trace block has the real frequencies $\pm(\sqrt{\omega^2+\varkappa^2}\pm\omega)$. Both frequency signs have been retained. These statements concern the complete constant limiting operator, not only a radial restriction.

For numerical work, the slow root in (12) should be evaluated as

$$\Omega_-^2=\frac{\varkappa^2(\varkappa^2+4q\omega^2)} {\varkappa^2+2\omega^2+2\omega\sqrt{\omega^2+(1-q)\varkappa^2}}. \tag{13}$$

This avoids subtracting large nearly equal numbers. Direct diagonalization at the assigned muon coefficient can otherwise create spurious small imaginary frequencies. The unitary reduction and (13) determine the exact limiting operator's frequencies independently of that conditioning.

Real frequencies can still hide amplification

At large $\varkappa$, remove the common fast phase and compress (11) to one propagation direction. The remaining two-component matrix is

$$N_q=\frac\omega2 \begin{pmatrix}q-2&q\\-q&2-q\end{pmatrix},\qquad N_q^2=\omega^2(1-q)I. \tag{14}$$

For the original norm, $q=1$, this is nonzero but nilpotent. Its exact propagator is $I-itN_1$, with largest singular value

$$\boxed{\|I-itN_1\|_2 =\frac{\sqrt{4+(\omega t)^2}+|\omega t|}{2}.} \tag{15}$$

The full finite-wave-number frequencies are real, yet this response grows with elapsed time in the high-momentum limit. It prevents a bound uniform in both time and momentum for the original constant limiting evolution. It does not prove nonlinear instability of the localized core: the coefficient limit, spatially varying core and vacuum regularity are distinct questions.

For $0<q<1$, the positive matrix

$$H_q=\begin{pmatrix}1-q/2&-q/2\\-q/2&1-q/2\end{pmatrix} \tag{16}$$

satisfies $H_qN_q=N_q^\dagger H_q$. Its eigenvalues are $1$ and $1-q$, so the compressed response obeys

$$\|e^{-itN_q}\|_2\leq\frac1{\sqrt{1-q}}. \tag{17}$$

For (9) at depth six this bound is $3/\sqrt8=1.060660\ldots$. It is a bound for the compressed high-momentum response, not a bound of six percent on every disturbance of a particle.

A positive conserved norm for the complete frozen operator

There is also a full finite-momentum result. Let $\Sigma=\operatorname{diag}(1,1,-1,-1)$ and $\mu=\varkappa^2+2\omega^2$. Define

$$K_q(\varkappa)=\frac1{2\omega} \Sigma G_q(\mu I-G_q^2). \tag{18}$$

Direct multiplication gives $K_q^\dagger=K_q$ and $K_qG_q=G_q^\dagger K_q$. Its eigenvalues fall into two real quadratic pairs $\lambda^2-T_i\lambda+D_i=0$, where

$$\begin{aligned} T_1&=(2-q)\varkappa^2+2\omega^2,\\ D_1&=(1-q)\varkappa^4+\varkappa^2\omega^2,\\ T_2&=(2-q)\varkappa^2+2(q+1)\omega^2,\\ D_2&=(1-q)\varkappa^4+[1+4q(1-q)]\varkappa^2\omega^2 +4q\omega^4. \end{aligned} \tag{19}$$

For $0<q\leq1$ and $\varkappa>0$, each sum and product is positive. The Hermitian matrix $K_q$ is therefore positive. For $q<1$ and a nonzero gap, $\varkappa\geq m_c$, elementary bounds further give

$$\frac{1-q}{2-q}I\ \leq\ \frac{K_q(\varkappa)}{\varkappa^2} \ \leq\ \left[2-q+2(q+1)\frac{\omega^2}{m_c^2}\right]I. \tag{20}$$

Integrating this conserved Fourier norm proves a bound uniform in time and momentum for the constant limiting evolution. The bound is not uniform in $\omega/m_c$, nor does it apply automatically to the variable core coefficients. This positive perturbation norm is not the physical Hamiltonian. The regular fixed-norm family with energy unbounded below survives (1), since its current is proportional to $I_r$ and (5) still holds.

Reciprocal coupling retains the common-target identity

For two independently solved cores of distinct grades $N>M$ at one depth, let $\rho(x),\sigma(x)$ be their local partner densities and $s=|z^\dagger w|^2$ their coherent-ray overlap. Their initial root cross energy becomes

$$V_q(x)=\omega_{0,d}r\sum_{k=1}^Mh_{k,d} \left[\sqrt{(\rho+\sigma)^2-2q\rho\sigma(1-s^k)} -\rho-\sigma\right]. \tag{21}$$

It depends on both rays and remains independent of the larger grade. Every shared current and trace is identical when that target grade changes. The complete finite probe source and reciprocal initial translation forces therefore retain the common-target identity. The unchanged binding also retains its nonzero shape response for aligned coincident cores. Subsequent target deformation and encounter trajectories need not be identical.

Equation (21) is a same-depth initial interaction, not an observed gauge vertex or a cross-depth photon–neutrino calculation. The new norm changes its angular dependence while preserving the assigned masses.

Scope and verification

partner_current_metric_checks.py checks arbitrary indefinite and singular-eigenvalue currents, complete potential mixed second variations on the actual smooth core, full oscillator source variations, and the finite interaction of two solved cores. It verifies the complete frozen unitary reduction in depths four and six, the exact characteristic compression, direct matrix exponential gains, and the positive conserved form. The implementation is partner_current_metric.py, used by localized_partner_tower.py.

The higher angular calculation also tests the revised complete Hessian on the localized electron and muon branches. Its finite scans do not prove full spatial nonlinear stability. The physical mode carrier, shell weights, common normalization, core-frequency relation and master origin of the interaction remain open. The separate principal defect of the polarized rest-frame carrier is a kinetic issue and is unchanged by this potential replacement.

Alignment is an angular energy maximum for the prepared pair

Equation (21) vanishes when the two coherent rays align, $s=1$, although each isolated core keeps its nonzero mass frequency. That algebraic zero does not by itself provide a persistent weak encounter. Rotate one ray by a small physical polarization angle $\theta$, so $s=\cos^2\theta=1-\theta^2+O(\theta^4)$. For coincident equal-norm cores of distinct grades $N>M$, equation (21) gives

$$\frac{V_q(\theta)}{\omega_M} =-\frac{q}{2}\frac{\sum_{k=1}^{M}k h_{k,d}} {\binom{M+d}{d}}\theta^2+O(\theta^4) =-\frac{q dM}{2(d+1)}\theta^2+O(\theta^4), \qquad h_{k,d}=\binom{k+d-1}{d-1}. \tag{22}$$

The second equality uses the exact identity $\sum_{k=1}^{M}k h_{k,d}=d\binom{M+d}{d+1}$. For $q>0$ and $M>0$, the coefficient in (22) is negative. The maintained particle index is $n=M+1$; the depth-three index-one state has $M=0$ and no shared positive-order mass-current term. Along this prepared upper-field polarization path, the zero-cross-energy alignment is an angular maximum, and a small rotation lowers the energy. The full binding energy is zero on these upper-only initial data, though its complete field derivative can still drive lower-spinor shape response. Equation (22) is an energetic direction, not a derived collision trajectory.

The six-depth alignment check evaluates the exact sum and the closed form for all sixteen maintained indices, using $M=n-1$, then compares each coefficient with the finite-angle pair energy at $\theta=10^{-3}$. Its report contains the full table. With the conditional equal-trace/exchange choice $q=1/(r_d+1)$, examples of the coefficient of $\theta^2$ relative to the smaller core's mass frequency are $-0.75000$ at depth five, particle index ten; $-0.57143$ at depth six, particle index thirteen; and $-0.30303$ at depth ten, particle index twenty-three.

Thus this same-law mass current permits a vanishing initial pair energy with nonzero isolated masses. For every tested $M>0$, free alignment does not explain weak reactivity: the aligned preparation has a negative angular energy curvature. A stable suppression would need a derived selection or locking mechanism, or a different current geometry, in the same complete action. The pair formula concerns one depth and distinct grades; it does not calculate a neutrino's interaction with ordinary three-dimensional matter.

The sign cannot be reversed inside the globally positive quadratic-root family (1) by choosing $q<0$. Any nonzero traceless Hermitian test current $X$ would then have $F_q(X)^2=qr\operatorname{Tr}(X^2)<0$. At $q=0$ the same direction has zero norm and loses its polarization response. Hence every nondegenerate real norm of this form on all Hermitian currents requires $q>0$, which fixes the negative angular coefficient in (22). This is an obstruction for this invariant root family, not a claim about every possible same-field interaction or the complete spatial dynamics.