Spatial Response of Proper-Current Count Charts

The geometric envelope gives exact nonlinear spatial persistence when its affine charts use the preferred-time density rho. Replacing rho with the Lorentz-scalar proper current n=sqrt(j_mu j^mu) preserves the stationary clock, but changes the spatial response. Neither flattening the scalar-ratio curvature nor reducing the carrier columns automatically removes that response.

This note compares the two carriers under the same proper-current chart at every occupied depth. Ground states are preparation controls, not the assigned excited electron, muon, tau or other particle states. Their growth does not negate the exact count solutions, observer transformation or local rotation-form theorems, and is not an observed lifetime.

1. The Complete Local Linearization

On one unique count chart, write

$$V=m s+\eta n,\qquad s=\operatorname{tr}\Phi^\dagger\beta\Phi, \quad n=\sqrt{\rho^2-|j|^2}.$$

At a stationary zero-current background Phi_0 with positive density, the full source is (m beta+eta I)Phi_0. Its linearization in the co-rotating frame is

$$i\partial_t\delta\Phi=\mathcal A\delta\Phi -\frac\eta\rho\sum_i\alpha_i\Phi_0\,\delta j_i, \quad \mathcal A=D_R+m\beta-\lambda I,$$ $$\delta j_i=2\operatorname{Re}\operatorname{tr} (\alpha_i\Phi_0)^\dagger\delta\Phi. \tag{1}$$

The scalar-ratio Hessian is zero, but the last term in (1) remains. Using the field's own proper current therefore does not give the preferred-time unitary equation. This is a derivative of the specified same-field action, with no inserted clock particle or independent mediator.

For the common ground control, take R=1, h=d/2, c=3/5, x=4/5 and

$$\epsilon_d=\frac{c}{H_d},\qquad \lambda=h/c,\quad m=h x/c,\quad \eta=\epsilon_d-\lambda. \tag{2}$$

The ground count is one. The condition c=epsilon P_d'(h) is satisfied at every depth under this shared recipe. These epsilons are ground controls, not the different sector scales used for all maintained excited assignments.

2. Exact Full-Matrix Ground Blocks

The earlier exactly reducing rotational harmonic family remains available on the square balanced carrier. Its real scalar-response factor is now zero, while the complete current block survives. Depth two also retains its actual transverse-vector contribution. Exact rational characteristic polynomials and real-root counts give:

The script tests L=1,2,3,5,10 and stores each polynomial. It removes zero factors before negative-real-root counting and counts nonreal roots separately. The first-harmonic zero generator is semisimple in this family at depths at least three, unlike the depth-two secular block. Semisimplicity of that zero does not remove the nonzero growth channels.

This excludes bounded linear response for these full-matrix ground preparations, rather than every radius, count action or particle state.

3. A Common Minimum-Carrier Spatial Test

The new minimum carrier has additional spinor identities. Its Hessian must be calculated directly rather than copied from the full matrix. A global Killing trivialization makes a small comparison possible. Let X be a unit vector in R^(d+1), Gamma_X=X_A Gamma_A and C the seed. In this representation the intrinsic operators are

$$\beta(X)=\Gamma_X,\qquad D=-\Gamma_X\Gamma\cdot\nabla_{S^d}+h. \tag{3}$$

This is the same ordinary d-dimensional sphere represented by its usual embedding coordinates, not a new physical sector at depth d+1. The ground field and its current-source vectors are

$$\Phi_0(X)=(a+b\Gamma_X)C,\qquad \alpha_A(X)=i(\Gamma_A\Gamma_X-X_AI). \tag{4}$$

The alpha_A are tangential: X_A alpha_A=0. The sum over these embedding components is therefore exactly the intrinsic current contraction in (1).

For each A use the finite polynomial fields

$$C,\quad\Gamma_XC,\quad X_AC,\quad X_A\Gamma_XC,\quad \Gamma_AC,\quad\Gamma_X\Gamma_AC, \tag{5}$$

their Clifford-volume multiples and their imaginary multiples. The free Dirac operator is closed on this family. For example,

$$D(X_AC)=(h+1)X_AC-\Gamma_X\Gamma_AC,$$ $$D(X_A\Gamma_XC)=-(h+1)X_A\Gamma_XC+\Gamma_AC. \tag{6}$$

The current source is evaluated from the actual seed and field in (4). It is not assumed to preserve (5). Fit its real operator on 28 sphere points, then verify closure independently on 12 new points. An eigenvalue is reported only if both residuals are below 1e-9. This is a finite spectral diagnostic, not a particle trajectory or nonlinear time simulation.

At depths five and six, carrier reduction changes the response and retains substantial growing eigenvalues in this tested ground family. It therefore gives no numerical support for removing the responding clock while keeping this particular preparation. At depth three, the separate full-matrix calculation above demonstrates why a favorable first-harmonic result does not settle larger spatial harmonics; its minimum-rank carrier still requires its own higher-harmonic test. Tiny floating eigenvalue splittings near zero, of order 1e-7, are not identified with physical growth. The displayed nonzero rates are far larger and occur in opposite growing/decaying oscillatory pairs.

The numerical closure at depths two through six is strong verification but is not yet an exact polynomial identity proof for the minimum-carrier family. Depth ten supplies a decisive leakage control: projecting its operator despite the failed closure would produce an unsupported spectrum. Its source couples to additional functions outside (5).

4. Reproduction and Consequences

An exact polynomial continuation now proves closure of a smaller eight-dimensional family at depths two, three, four, six and ten. Its depth-ten member is axial and remains closed despite the larger family leakage. At the stated ground preparation, every real scalar curvature retains growth at six and in this explicit depth-ten family. Depth five needs a larger coupled space. These results constrain the proper-current continuation, not the preferred-time nonlinear theorem or assigned excited particles.

proper-envelope-ground-response.py and its report pass 90 independent exact/numerical comparisons on the full matrix. minimal-carrier-ground-response.py and its report retain the closure residuals and all reported time-generator eigenvalues.

The test narrows a proposed simplification: replacing preferred-time density with the field's own proper current changes dynamics that must be retained. The existing preferred-time nonlinear theorem remains valid, as do the exact mass clocks and geometric forms. The responding same-field clock construction remains a distinct covariant option with its stated load and ground linear theorem. None of these comparisons establishes decay, assigned excited-state persistence, localization or new empirical preference for the occupied sector depths.