A Matrix Scalar Current with the Same Count Backgrounds

The spatial persistence calculation exposes a response missed by the retained-pair analysis. The full spinor matrix supplies a concrete additional invariant to test without changing the fitted frequencies. Its stationary background equations agree with the preceding action, while its polarization response differs. This provides a common action-level comparison, without particle-specific amplitudes or inserted depth-five coefficients.

The Scalar Overlap Matrix

Retain the existing balanced rank-r spinor matrix Phi and its ordinary Dirac kinetic normalization. Define

$$M=\Phi\Phi^\dagger,\quad L=\Phi^\dagger\beta\Phi,\quad J=\operatorname{tr}L,\quad K_2=\operatorname{tr}L^2 =\operatorname{tr}(\beta M\beta M),$$

$$\boxed{n_S=\sqrt{rK_2-J^2} =\sqrt r\left\|L-\frac JrI\right\|_F.} \tag{1}$$

The matrix L records scalar overlaps among components of the same field. It adds no physical dimensions or mediator particles. For a local spinor Lorentz transformation S, S† beta S=beta, so L is unchanged by Phi -> S Phi. Both J and nS are therefore Lorentz scalars. A constant unitary change of field columns conjugates L and leaves (1) unchanged.

Assume the stated grading has r/2 positive and r/2 negative directions. Congruence cannot give L more than r/2 positive or negative eigenvalues. Write the sums of its positive eigenvalues and negative magnitudes as A and B. Cauchy–Schwarz gives

$$J^2=(A-B)^2\le A^2+B^2\le\frac r2K_2, \qquad \boxed{n_S\ge|J|.} \tag{2}$$

This proves the unit-interval ratio bound for arbitrary matrix fields, including rank-deficient ones. Nonzero fields with L=0 can still have nS=0. Smoothness and global continuation at that boundary require separate analysis; the ratio action below uses its open chart.

A Complete Local Source

Use the same count constitutive function f_d as the inverse benchmark, and compare the local potential

$$U_S=\frac{n_S}{R_d}f_d(J/n_S),\qquad x=J/n_S.$$

Vary the full matrix, including nS. In dimensionless radius units,

$$\nabla_\Phi n_S =\frac{r\beta M\beta-J\beta}{n_S}\Phi,$$

$$\boxed{G_S= \left[f_d'\beta+ \frac{f_d-xf_d'}{n_S}(r\beta M\beta-J\beta)\right]\Phi.} \tag{3}$$

Here the complex-gradient convention is delta U=2 Re Tr(G† delta Phi). Equation (3) includes all matrix-current derivatives. The scalar action requires its Hamiltonian gradient to transform contragrediently, G_S(S Phi)=S^(-dagger)G_S(Phi); the independently evaluated source obeys that law. Multiplying it by beta gives the corresponding spinor-covariant nonlinear term in the covariant Dirac equation. No preferred time vector was inserted into this potential.

The potential has amplitude degree two and its source degree one. Those facts do not establish positive unrestricted Dirac energy, localization or full well-posedness.

All Balanced Spatial Count Backgrounds Survive

For every actual balanced shell background of the previous calculation,

$$M=\frac\rho r(I+x\beta),\qquad J=\rho x,\qquad K_2=\frac{\rho^2}r(1+x^2).$$

Consequently nS=rho and

$$\frac{r\beta M\beta-J\beta}{n_S}=I. \tag{4}$$

The complete source in (3) equals the original source at every spatial point. This argument permits arbitrary nonzero local density and the actual spatially varying grading; it does not assume a constant field. Thus all fourteen maintained count backgrounds retain the same exact stationary field, frequency, potential value and complete energy E=omega Q. The original pair dynamics is also unchanged because every constant complex pair retains this balanced density form. Its restricted persistence result remains valid.

This preserves the known fitted frequencies; it does not derive them anew, select occupied indices or remove the matrix-carrier premise.

The Response That Changes

At a balanced background define B=f-xf' and

$$P=\frac{r\beta M\beta-J\beta}{n_S}=I.$$

Since delta nS=delta rho there, the remaining density-gradient variation is

$$\delta P=\frac{r\beta\delta M\beta-\delta J\beta-\delta\rho I}{\rho}. \tag{5}$$

The full source Hessian is

$$\delta G_S=(f'\beta+BI)\delta\Phi +\frac{2f''}{\rho}(\beta-xI)\Phi \operatorname{Re}\langle(\beta-xI)\Phi,\delta\Phi\rangle +B\delta P\Phi. \tag{6}$$

In an orthonormal Hermitian Clifford-matrix basis excluding I and beta, the last term includes every matrix moment. A basis matrix Gamma commuting with beta has a plus sign in beta Gamma beta; an anticommuting one has a minus sign. Thus (6) is not permission to retain only ordinary vector currents in a complete spatial stability claim.

Inside the exactly reducing rotational matrix algebra of the preceding note, only one extra moment is needed. With radial gamma, define the Hermitian tensor matrix C=-i gamma beta. A direct trace identity gives

$$\boxed{n_S^2=\rho^2-j_{\rm radial}^2-j_C^2.} \tag{7}$$

At depth two C is the remaining spatial Clifford vector, and nS is identically the original vector proper density. At higher depths it is an additional tensor polarization moment. The identity explains the extra response; it does not identify that moment with electric charge or color.

An Exact Persistence Constraint on the Natural Blend

Compare one common scalar-action blend

$$U_\kappa=(1-\kappa)U+\kappa U_S, \tag{8}$$

where U is the original vector-current potential. Every exact balanced count background survives for every kappa. On the ground-shell scalar harmonic blocks of the preceding note, Y is unchanged. At d>=3, X gains 2 kappa v0 S_L S_L^T; at d=2 the two actions are identical.

The first spatial harmonic is exactly reducing. Its squared-frequency polynomial again has a zero factor and a cubic. Rational characteristic polynomials and exact real-root inequalities certify the following at both common mixing preparations c=3/10 and c=3/5:

$$\boxed{\text{No }\kappa\in[0,1]\text{ removes the first-harmonic growth, at any occupied depth.}} \tag{9}$$

This excludes every positive blend as a repair of those ground controls, rather than only testing the two endpoints. The alternative still gives a useful invariant and exact backgrounds, but its extra tensor response does not by itself supply spatial persistence.

The report also retains exact allowed intervals for unrestricted real kappa in this single harmonic. Some intervals use negative weights in (8). Removing one harmonic's growth is not a full stability theorem; other spatial harmonics and the assigned excited shells remain necessary tests. These fixed preparations do not exclude every radius or every Lorentz-invariant action. There is no new sector preference or physical color-force identification.

Reproduction

matrix-current-completion.py checks full spinor-matrix action variations, arbitrary complex component fields, scaling, actual Lorentz boosts, and the source transformation. It tests all occupied Clifford ranks, every maintained actual spatial shell at two points, the tensor-current identity, and exact ground-block polynomials at both stated preparations. It independently solves the real-root inequalities including the entire convex coefficient interval. Its dated reports preserve the executed source hashes and numerical errors. The analytic inequality (2), background identity (4), and harmonic closure supply the general statements that finite samples alone cannot.

The unrestricted spatial repair analysis settles the real blend and its even-depth invariant extension. A distinct shell-preserving constitutive construction retains the exact clocks while changing the pair law between shells; its full persistence result requires the stated preferred-time density action.