Spatial Rigidity of Pair-Preserving Current Repairs

The original spatial calculation found genuine ground-shell growth beyond the retained count pair. The matrix scalar current preserves that complete pair dynamics while adding another invariant response. This calculation settles the unrestricted real blend and identifies precisely what a broader repair must change. Ground shells are controls; the assigned electron, muon and tau occupy excited shells.

1. Exact Harmonic Comparison

Use the earlier exactly reducing ground harmonic matrices

$$X_L=H_L+2\eta T_LT_L^T+2\kappa v_0S_LS_L^T, \qquad Y_L=H_L+2v_0R_LR_L^T,$$

at depths at least three. At depth two the tensor is already an actual vector-current channel, so its coefficient is fixed rather than multiplied by $\kappa$. Here $\kappa$ is any real number, not just a positive blend. Squared disturbance frequencies are eigenvalues of $Y_LX_L$, in the normalized units of the preceding notes.

If all roots of its real characteristic polynomial are nonnegative real, its elementary symmetric coefficients have alternating signs and its discriminant is nonnegative. These are necessary conditions; for a quartic they are not alone sufficient. Their exact intersection gives:

Every exclusion quantifies over all real $\kappa$. No coefficient scan or cutoff convergence assumption enters it. The common mixing $c$ is a geometric preparation control; it is not a common sector $\epsilon$. It does not replace the assigned excited-state comparison.

2. A Larger Even-Depth Class Has the Same Obstruction

Consider an even occupied depth with the irreducible complex spatial Clifford carrier, a square full matrix field $\Phi$, and a local derivative-free potential. Impose the following additional conditions:

1. Right unitary column symmetry, so the potential depends on $M=\Phi\Phi^\dagger$.
2. Spatial Spin covariance and Lorentz spinor-boost invariance.
3. The repair preserves the complete original balanced pair law, not just the stationary values at the assigned shells.

The balanced densities are

$$M_0=\frac\rho r(I+x\beta),\qquad 0<x<1.$$

Let $K$ be the repair potential minus the original potential. Condition 3 says its gradient vanishes throughout this two-parameter family. Its Hessian therefore annihilates the scalar variations $I$ and $\beta$. Lorentz invariance transports a zero gradient to a zero gradient. An infinitesimal boost generated by $\alpha_a/2$ gives

$$\delta M_0=\tfrac12\{\alpha_a,M_0\} =\frac\rho r\alpha_a,$$

so the repair Hessian annihilates every $\alpha_a$ as well.

In even depth the full endomorphism algebra is the spatial Clifford algebra. Its Spin representation decomposes into exterior grades $\Lambda^p\mathbb R^d$. Only grades $1$ and $d-1$ can be vector copies: the exterior Casimir is $p(d-p)$, and equality with $d-1$ requires $(p-1)(p-d+1)=0$. These two vector copies are represented by $\alpha_a$ and $-i\alpha_a\beta$, with phases chosen Hermitian. Self-adjointness of the repair Hessian and its vanishing on the first copy leave just one real coefficient on the second copy.

The exact rotational matrix family has density variations only in $I,\beta,\alpha_{\rm rad},-i\alpha_{\rm rad}\beta$. Thus its entire repair Hessian is precisely the one tensor coefficient already varied by unrestricted $\kappa$. At depth two the two vector descriptions coincide, so the boost condition leaves no such freedom. At $c=3/5$ the exact exclusions above therefore cover this whole repair class at depths two, four, six and ten.

This is not a theorem about every same-field law. Odd-depth carriers have additional representation multiplicities. Derivative interactions, changed column symmetry, extra retained modes, a preferred time structure, or a change to the original pair dynamics fall outside the argument. Merely preserving the discrete stationary mass-frequency backgrounds is weaker than condition 3 and leaves a useful repair route open.

3. An Exact Three/Four Rotational Window

At the shared preparation

$$c=\frac3{10},\qquad\kappa=\frac{11}{10},$$

depths three and four have nonnegative real squared frequencies for every integer scalar harmonic $L\ge1$ in this reducing family. The proof is uniform in $L$, rather than an extrapolation from a numerical cutoff.

For $L\ge2$, form the monic quartic $p_L(z)$, its Newton sums $s_j$, and the four-by-four trace matrix

$$\mathcal H_L=(s_{i+j})_{i,j=0}^3.$$

All leading principal minors are strictly positive. The checker gives their exact rational expressions and substitutes $L=t+2$: numerator and denominator coefficients are nonnegative, with positive constant terms. This certifies positivity for every real $L\ge2$. Multiplication by $z$ in $\mathbb R[z]/(p_L)$ has companion matrix $C_L$. The independently checked identity

$$\mathcal H_LC_L=C_L^T\mathcal H_L$$

makes $C_L$ self-adjoint in a positive metric. Its roots are real. Strict alternating elementary coefficients exclude negative or zero roots. At $L=1$, the residual cubic has exactly three positive roots by exact root counting; the fourth root is zero.

The first-order eight-dimensional disturbance generator has nullities $1,2,2$ for its first three powers at that zero frequency. It therefore has a two-dimensional Jordan block: linear secular growth remains even though exponential spectral growth is absent. This is not a proof of bounded or nonlinear stability.

The window concerns ground rotational controls and an expressly changed action. It neither identifies color nor establishes a preference of the actual quark assignments. Other angular representations and excited particle states remain separate tests.

Reproducibility

The unrestricted repair checker and report preserve the exact polynomials, discriminants and inequality intersections at every occupied depth. The uniform checker and report preserve the all-harmonic trace-metric certificates and Jordan ranks. The invariant Hessian argument above is an analytical extension with its additional symmetry and pair-preservation assumptions explicitly stated.