Exact Spatial Disturbances Beyond the Inverse Count Pair

The inverse count action gives exact spatial fields and conserved energy with the supplied frequencies. Its retained pair persists under its own nonlinear dynamics. The broader spatial calculation below shows why that persistence cannot be transferred to arbitrary disturbances: a different, exactly reducing spatial harmonic has growing oscillations. This is a dynamical constraint on the fitted action, without requiring ignorance of measured masses.

The calculation retains the conditional ordinary sphere, its radius and connection, both Dirac signs, and the stipulated balanced matrix carrier. It supplies no replacement for literal deterministic electron trajectories. It does not identify this carrier or its unstable ground branches with an observed particle, nor derive a decay channel or lifetime.

An Actual Spatial Family

Write the spatial depth as d, set t=cos(theta), s=sqrt(1-t²), and use the positive Killing frame. Let gamma(n) be the radial Hermitian Clifford matrix and beta0 the grading at the pole. The physical parallel grading is

$$\beta(t)=(tI+i s\gamma(n))\beta_0.$$

Consider the matrix fields

$$\Phi=P(A,B)+P(C,D)\beta_0,\qquad P(A,B)=A(t)I-i sB(t)\gamma(n), \tag{1}$$

with arbitrary complex coefficient functions. Constants on the right label field columns; they are neither additional particles nor spatial directions. The exact dimensionless radial Dirac action on either right-column block is

$$\boxed{\mathscr D P(A,B)= P\left(\frac d2 A+dtB-(1-t^2)B',\ A'+(1-d/2)B\right).} \tag{2}$$

The terms involving d and t retain the sphere connection. The physical coefficient norm is the integral of |A|²+s²|B|²+|C|²+s²|D|², with the appropriate common matrix rank factor, against the normalized weight (1-t²)^((d-2)/2)dt. Smooth endpoint conditions follow from the original matrix fields.

The Killing-frame method and ordinary sphere spectrum are standard; see Christian Bär, The Dirac operator on space forms of positive curvature, section 2. The current Hessians and reducing blocks below are derived for the specified IDWT candidate, rather than inferred from that reference.

Define actual normalized positive shells by

$$\begin{gathered} A_l=\frac{l+d-1}{d-1}C_l^{(d-1)/2}(t),\qquad B_l=C_{l-1}^{(d+1)/2}(t),\qquad B_0=0,\\ g_l=\binom{l+d-1}{d-1},\qquad P_l=\frac{P(A_l,B_l)}{\sqrt{r g_l}},\qquad Q_l=\beta P_l\beta_0. \end{gathered} \tag{3}$$

Here r is the spinor rank. The scalar and vector functions of Q_l are tA_l-s²B_l and -A_l-tB_l. Equation (2) gives D P_l=(l+d/2)P_l and D Q_l=-(l+d/2)Q_l. The four collections P_l,Q_l,P_l beta0,Q_l beta0 form the complete rotational matrix family. Their integrated Gram matrix is the identity. Keeping only P_l and beta P_l omits half of this family even before varying its radial shapes.

The nonlinear source of U=n f(J/n), n=sqrt(rho²-|j|²), closes on (1). At d>=3 its vector current is radial. At d=2 an additional transverse current stays in the same matrix algebra. Independent projection with the full Clifford matrices verifies this distinction at all occupied depths; the extra channel is retained throughout.

Keep the Complete Complex Hessian

At the exact stationary shell l, put h=l+d/2, c=epsilon P_d'(h), x=sqrt(1-c²), a=sqrt((1+c)/2), b=sqrt((1-c)/2), and

$$\lambda=h/c,\qquad \omega R=\varepsilon P_d(h),\qquad \eta=1-\frac{x^2P_d'}{hP_d''},\qquad v_0=1-\frac{\omega R}{\lambda}. \tag{4}$$

Here P_d is the count polynomial defined in the linked inverse action; it is different from the spatial matrix P_l. For Phi=aP_l+b beta P_l, define T=(a beta-bI)P_l, V_a=alpha_a Phi and rho=Tr(Phi†Phi). The exact cancellation (beta-xI)Phi=cT avoids subtracting nearly equal large coefficients. The rotating source variation, in lambda units, is

$$\frac{\delta G-(\omega R)\delta\Phi}{\lambda} =(x\beta-I)\delta\Phi +\frac{2\eta}{\rho}T\operatorname{Re}\langle T,\delta\Phi\rangle +\frac{2v_0}{\rho}\sum_aV_a\operatorname{Re}\langle V_a,\delta\Phi\rangle. \tag{5}$$

The full spatial Dirac term acts in addition to (5). Both real and imaginary perturbations, and every current derivative, are included. The source identities follow from f-xf'=omega R-lambda and c²f''=lambda eta. The former retained pair remains an exact invariant subspace of this broader response.

Exact Ground-Shell Harmonic Blocks

For the ground shell l=0, each scalar harmonic degree L>=1 has the exact four-complex-dimensional reducing basis

$$(P_L,\ Q_L\beta_0,\ Q_{L-1},\ P_{L-1}\beta_0). \tag{6}$$

This is an exact spatial subspace, rather than a finite approximation to an infinite coupled block. Set

$$u^2=\frac{L+d-1}{2L+d-1},\qquad v^2=\frac{L}{2L+d-1},\qquad h_0=d/2,$$

and let beta4 exchange entries 1/2 and 3/4. In this orthonormal basis,

$$\begin{gathered} H=\frac1\lambda\operatorname{diag} (h_0+L,-h_0-L,-h_0-L+1,h_0+L-1)+x\beta_4-I,\\ T_L=(-bu,au,-bv,av)^T,\\ R_L=(av,-bv,-au,bu)^T,\qquad S_L=(bv,-av,-bu,au)^T,\\ X=H+2\eta T_LT_L^T+\mathbf1_{d=2}\,2v_0S_LS_L^T,\qquad Y=H+2v_0R_LR_L^T. \end{gathered} \tag{7}$$

The indicator in (7) records the proved Clifford trace distinction; it is not an inserted particle selection. For real perturbation components p,q, dot p=lambda Yq and dot q=-lambda Xp. Squared frequencies in lambda units are eigenvalues z of YX. A nonreal z gives a genuine exponentially growing oscillation of this actual spatial field.

For the count polynomial at the ground shell,

$$P_d=1,\quad P_d'=H_d,\quad P_d''=H_d^2-H_d^{(2)},\quad \varepsilon=c/H_d,\quad v_0=1-\frac{c^2}{h_0H_d}. \tag{8}$$

Take the common mixing c=3/5, x=4/5. This is one shared preparation; epsilon varies with depth according to (8), so it is not the earlier shared epsilon=2^-32 radius comparison. At L=1 the characteristic polynomial has one zero factor and a cubic with strictly negative exact discriminant at every occupied depth. An independent Sturm count gives exactly one real cubic root and no negative real cubic roots. The other two squared frequencies are nonreal.

For example the exact depth-six polynomial at c=3/5 is

$$z\left(z^3-\frac{257132}{177625}z^2 +\frac{11566259}{22203125}z -\frac{957732}{11328125}\right). \tag{9}$$

Its cubic discriminant is -1058711215813006922641212/15553795685123443603515625 < 0. The complete rational certificates are in the reproduction report. The zero frequency is retained; it does not remove the nonzero growing pair.

Exact real-root isolation of the discriminant as a polynomial in c extends the result. At depths 4,5,6,10 there is no discriminant zero in 0<c<1, so this L=1 instability holds throughout the entire open chart. At depths 2 and 3 it holds below the first positive discriminant boundary, isolated respectively by the rational intervals (59029/86305,111927/163646) and (139556/147047,78711/82936). It therefore also covers the small ground mixings from epsilon=2^-32.

These ground shells are controls for electron, muon and tau, whose assigned shells are excited. The maintained down assignment does have l=0; its realization in this particular matrix/sphere action is covered when its c lies below the depth-three boundary. A free-quark lifetime or the behavior of a confined composite does not follow from that statement.

Assigned Excited Shells Still Need Convergence

All fourteen maintained assignments have been projected into the broader rotational family using one epsilon per depth and no amplitude fitting. The explicit diagnostic prescription is epsilon_d=0.3/max P_d'(h_assigned), with the maximum over maintained assignments in that depth. It probes radii while keeping all frequencies of the same fitted constitutive construction; it is not a derived radius law. One run uses cutoff l+10 and another l+24 with finer quadrature.

Both projections show growing candidates, but several maxima move substantially. For the electron they move from approximately 8.217 to 19.785 in inverse-radius units. For the tau they move from 14.521 to 31.801. These are unconverged finite-projection diagnostics, not certified growth rates or electron-instability theorems. Increasing the basis can expose additional channels as well as change a previously leading root. The exact ground blocks do not have this truncation qualification.

A separately labeled comparison replaces n by rho. That changes the action and removes the vector-current Hessian; it is not permission to freeze a derivative of the original action. Its finite projections reduce many growth candidates, but do not establish persistence of the assigned excited states. The comparison selects a preferred time direction and does not supply a Lorentz-invariant replacement by itself.

Reproduction and Scope

inverse-current-radial.py checks exact radial differential identities, actual full Clifford projections, Gram normalization, the stationary equation, both complete real Hessians, and the old pair's invariance. On the ground shell its quadrature-projected matrices agree with the independently derived exact blocks (7). Its dated reports retain separate cutoff and action controls.

inverse-current-ground-proof.py derives the rational characteristic polynomials, discriminants and Sturm certificates without numerical eigenvalue signs. Independent normalized matrices agree with their coefficients.

inverse-current-radial-variation.py tests independent nonlinear action and source variations with arbitrary complex matrix coefficients at 120-digit precision. Its 112 evaluations cover every maintained assignment, two spatial points and both complete actions. Action and Hessian discrepancies are below 3.4e-65 and 6.8e-66, respectively. Exact reports retain the executed source hashes and errors.

The new result narrows a concrete inverse mechanism: fitting the clocks and proving pair persistence do not establish spatial persistence. The exact ground instability does not exclude IDWT's count law or its sector assignments. It does not establish an observed decay, a localized particle or a preference for the proposed depths. No new supporting claim for the sector-case document follows from this calculation alone.

The matrix scalar-current continuation adds a Lorentz-invariant polarization response while retaining every balanced count background. Exact first-harmonic inequalities show that its natural positive blends do not remove growth at the two stated ground preparations.

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.