An Exact Spatial Growth Test for Two Excited Count Waves

The simpler continuous count interaction retains the exact standing-wave frequencies and has a complete ground no-growth theorem. Its excited fields need a different test. Two assigned full-matrix zonal fields now have a full spatial growing linear mode under that action, even with the rotating-frame gyroscopic term retained.

The result concerns the strange-assigned depth-three field and the up-assigned depth-four field at the existing common diagnostic scales. It constrains this constitutive realization. It is not evidence against the sector count law, a result for every carrier, or an identification of observed decay. The separate locally affine envelope action has different spatial derivatives and retains its own persistence theorem.

1. Actual Field and Linearization

Use the ordinary round spatial sphere, radius units R=1, and the continuous count law. Write

$$h_l=l+d/2,\quad P_d(h_l)=\binom{l+d}{d},\quad c_l=\epsilon_dP'_d(h_l),\quad x_l=\sqrt{1-c_l^2},$$ $$\lambda_l=h_l/c_l,\quad m_l=h_lx_l/c_l,\quad \omega_l=\epsilon_dP_d(h_l),\quad \eta_l=\omega_l-\lambda_l. \tag{1}$$

The coefficient m_l is the source coefficient, while omega_l is the particle-frequency target. The action is

$$L_+=\|\dot\Phi\|^2-\|F(\Phi)\|^2,\qquad F=D\Phi+[f'(s/\rho)\beta+f-(s/\rho)f']\Phi.$$

All norms include the spatial integral. Let Gamma_0 be grading at the north pole, beta(X)=Gamma_X, and t=X_0. These Clifford matrices describe the spinor and its grading; they do not add spatial dimensions. Define

$$A_l=\frac{l+d-1}{d-1}C_l^{(d-1)/2}(t),\quad B_l=C_{l-1}^{(d+1)/2}(t),\quad p_l=A_l-tB_l,\quad g_l=\binom{l+d-1}{d-1},$$ $$Z_l=(p_lI+B_l\Gamma_0\beta)/\sqrt{g_l},\qquad \rho_l=Z_l^\dagger Z_l =\frac{A_l^2+(1-t^2)B_l^2}{g_l}>0. \tag{2}$$

The matrix norm uses normalized trace and normalized sphere measure. Thus the background is globally nodeless. Set

$$a_l^2=(1+c_l)/2,\quad b_l^2=(1-c_l)/2,\quad \phi_l=a_lZ_l+b_l\beta Z_l,\quad w_l=-b_lZ_l+a_l\beta Z_l.$$

It obeys F(phi_l)=omega_l phi_l. On real variations in the invariant space constructed below, the exact source derivative is

$$B=H+N,\qquad H=D+m_l\beta+\eta_l,\qquad Nu=q\,w_l\frac{\operatorname{Re}\operatorname{tr}(w_l^\dagger u)} {\rho_l},\quad q=2\lambda_l-\frac{2x_l^2}{\epsilon_dP''_d(h_l)}. \tag{3}$$

Consequently 0<=N<=qI, N squared=qN and N phi_l=0. Imaginary variations in this space have source derivative H. The homogeneous-gradient identity removes the apparent additional third-derivative contribution at the exact standing field. The complete rotating linear equations are

$$\ddot u+2\omega_l\dot v+A_ru=0,\qquad \ddot v-2\omega_l\dot u+A_iv=0,$$ $$A_r=B^2-\omega_l^2,\qquad A_i=H^2-\omega_l^2. \tag{4}$$

Negative stiffness alone is insufficient: the velocity terms in (4) are part of the test.

2. A Closed Spatial Space, Without a Mode Cutoff

Take the closure of real smooth four-function fields

$$f_0(t)I+f_1(t)\beta+f_2(t)\Gamma_0+ f_3(t)\Gamma_0\beta.$$

Their real pointwise trace Gram is

$$\begin{pmatrix}1&0&0&t\\0&1&t&0\\0&t&1&0\\t&0&0&1\end{pmatrix}. \tag{5}$$

It is positive definite for -1<t<1. The normalized coordinate measure is proportional to (1-t squared)^((d-2)/2)dt. The actual Dirac action, with h_0=d/2 and primes denoting t derivatives, is

$$\begin{aligned} (Df)_0&=h_0f_0-tf'_0+2(d-1)tf_3-f'_3,\\ (Df)_1&=-h_0f_1-tf'_1-f'_2,\\ (Df)_2&=f'_1+h_0f_2+tf'_2,\\ (Df)_3&=f'_0+(2-h_0)f_3+tf'_3. \end{aligned} \tag{6}$$

H and N preserve this real closed space; multiplication by i supplies its imaginary companion. At depths three and four all relevant traces in this four-function algebra are real. Thus (4) is the actual linearized field equation on an invariant subspace of the full matrix field problem. Growth here is full field growth, even though other angular perturbations have not been classified.

Completeness can be checked with polynomials. The even algebra f_0 of degree at most K and f_3 of degree at most K-1 has dimension 2K+1. It is D invariant, and has eigenbasis Z_k for 0<=k<=K and beta Z_k Gamma_0 for 0<=k<=K-1. The odd algebra f_1 of degree at most K and f_2 of degree at most K-1 has the corresponding basis beta Z_k and Z_k Gamma_0. Distinct signed Dirac eigenvalues and normalized trace establish orthonormality. Polynomial density then establishes completeness of all four infinite families.

Diagonalizing D+m_l beta gives eigenvalues

$$H_k^+=\omega_l+\frac{h_k^2-h_l^2}{\lambda_k+\lambda_l}, \quad H_k^-=\omega_l-\lambda_l-\lambda_k,\quad \lambda_k=\sqrt{h_k^2+m_l^2}, \tag{7}$$

each with two right-seed copies in this space. This rationalized positive branch avoids subtracting large nearly equal quantities.

3. Finite Projections Give Exact Lower Operators

Let C_j=C_j^((d-1)/2)(t) and choose actual field directions g_j=C_j w_l, 0<=j<=J. Their Gram matrix is

$$G_{ij}=\int\rho_l C_iC_j>0.$$

Every g_j is in the eigenvalue-q range of N. If T has columns g_j in the complete H eigenbasis, the orthogonal projection to their span gives

$$\boxed{N_{\rm low}=qTG^{-1}T^\dagger\le N.} \tag{8}$$

This is a lower bound on the full operator, not an assumption that the field contains finitely many modes. These g_j are polynomial Clifford fields. Their complete spectral expansions terminate; the implementation uses the conservative cutoff k<=l+J+2. All remaining source-projection coefficients vanish exactly. The diagonal H tail is retained analytically.

Polynomial integrals are rational because

$$\int t^{2r}=\prod_{j=1}^r\frac{2j-1}{d+2j-1},\qquad \int t^{2r+1}=0. \tag{9}$$

Only the explicit dressing square roots require interval arithmetic.

4. Bound the Number of Real Gap Directions

Split H into its positive- and negative-Dirac polarities. The following two sufficient inequalities hold for both certified backgrounds:

$$h_0^2-h_l^2+4\omega_l(\lambda_l-\omega_l)>0,$$ $$2\lambda_l-q-2\omega_l- \frac{h_l^2-h_0^2}{\lambda_l}>0. \tag{10}$$

The first places every positive-polarity H eigenvalue above -omega_l. The second places the entire negative block of H+N below -omega_l. It uses lambda_0>=lambda_l-(h_l squared-h_0 squared)/lambda_l and N<=qI, and is deliberately sufficient rather than necessary.

For z in [-omega_l,omega_l], the negative block of B-z is strictly negative. Its positive Schur complement is the supremum of the full quadratic form over negative-polarity variables. Therefore (8) gives

$$S_z(B)\ge S_z(H+N_{\rm low}). \tag{11}$$

At z=-omega_l it is strictly positive. As z increases, its derivative is strictly negative: it equals -I minus the squared negative-block resolvent contribution. Its zero crossings count B eigenvalues in the stated gap. At z=omega_l the selected phi_l is an exact decoupled zero and is removed.

Suppose the remaining finite Schur complement for N_low has r strictly negative eigenvalues and no zeros. Its untouched positive tail is strictly positive. By (11), the number of nonpositive directions of A_r on phi_l perpendicular is at most r. This includes possible additional zero directions of the true source: bounding only the strict negative index would not suffice near zero growth rate.

5. A Gyroscopic Pole Argument Forces Growth

Seek an actual solution (u,v) exp(sigma t), sigma>0. Away from the finite negative spectrum poles of A_i, eliminate v to obtain the self-adjoint compact-resolvent operator

$$K_\sigma=A_r+\sigma^2+ 4\omega_l^2\sigma^2(A_i+\sigma^2)^{-1}. \tag{12}$$

The selected phi_l plane decouples and cannot grow. On its orthogonal complement, as sigma tends to zero from above, K_sigma converges in bounded perturbation norm to A_r+4 omega_l squared P_ker(A_i). It therefore has at most r negative directions for sufficiently small sigma. Compact resolvent and the nonpositive-index bound handle zero eigenvalues in this statement.

The first inequality (10) makes exactly the k<l positive-polarity branches of A_i negative. Their total multiplicity here is m=2l. At each pole, the resolvent term changes from negative infinity to positive infinity on its finite-dimensional residue space. If the complementary Schur block is invertible, the negative index of K_sigma drops by that residue multiplicity.

If the complementary block has a kernel at a pole, there is already a growing solution of the original two equations (4). To see this, take u in that complementary kernel, set its pole projection to zero, solve for v off the pole, and choose v's pole component to satisfy the remaining projected first equation. The second equation at the pole requires exactly that zero projection of u.

Finally K_sigma is strictly positive for sigma>omega_l, since both A_r and A_i are bounded below by -omega_l squared. If no growing solution existed, the negative index could only fall by a total m while starting at most r and ending at zero. This is impossible when

$$\boxed{r<m.} \tag{13}$$

Thus (13) proves at least one actual growing spatial mode with 0<sigma<=omega_l. No precise number or growth rate is inferred from the index deficit alone.

This is a direct pole and inertia argument for this particular pencil. General quadratic-pencil instability index methods provide related mathematics; see Bronski, Johnson and Kapitula, An Instability Index Theory for Quadratic Pencils and Applications. No general theorem from that paper is being applied without checking its kernel assumptions.

6. Outward Interval Certificates

Use one diagnostic scale per occupied depth, epsilon_d=(3/10)/max_assigned P'_d(h_l). These are not calibrated physical radii. Every maintained assignment is checked against (10); only the two following cases receive the full source certificate.

For strange, c=3/10, lambda=15, omega=18/37 and q=3383/225. For up, c=171/2646490, lambda=10585960/171, omega=18/264649 and q=2136192347630711/26926712505. The report retains the exact gap margins and source Gram matrices.

The checker uses 85-digit outward intervals. A floating eigensolver only proposes a congruence matrix. Every proposed entry is then converted to its exact binary rational; an interval Gram bound proves invertibility, and strict signed row diagonal dominance proves the exact inertia by a nonsingular homotopy to its diagonal. The rational congruence matrix and every interval margin are saved for reproduction. The result does not rely on taking floating eigenvalue signs as proof.

continuous-count-source-index.py and its report pass 164 conditions. The full conclusion additionally uses sections 2 through 5 above.

7. Finite Spatial Calculations and Their Limits

The common curvature continuation retains all count clocks and proves sufficient growth intervals, rather than testing only the full continuous curvature. It also connects the same linearized operators to the explicit analytic forward count-energy correction.

The companion continuous-count-excited-spatial.py also compares all fourteen assignments with six ground controls. Its real stiffness uses the actual projected B squared,

$$P B^2P=H_P^2+H_PN_P+N_PH_P+qN_P, \tag{14}$$

rather than squaring the projected B. This follows from N squared=qN and from the H-invariant spectral projection. The gyroscopic pencil is retained. Larger finite calculations show growing roots in the displayed excited cases, while the ground controls show none.

Those numerical roots alone are not a full spatial proof. With all shells through l+64, the strange rate is approximately sigma/omega=0.59573454 and the unprojected residual divided by omega squared is about 0.0002013. At the same diagnostic cutoff, up, the first neutrino comparison, electron and tau have much larger normalized full residuals: approximately 30.75, 27.36, 250.62 and 3.007. These calculations therefore cannot establish their full growth rates or full-field stability. The up existence result instead uses the independent exact lower-operator certificate.

The last finite report passes 1,895 implementation and algebra checks; earlier cutoffs, source versions and the initial trace-projection failure remain preserved. These checks assess the stated finite calculations. They do not certify all-mode convergence.

The two rigorous existence results concern full-matrix zonal fields, which differ from the maintained minimum-column particle candidates. They identify a concrete persistence issue for the continuous law. They supply no new depth-five preference, observed lifetime or outgoing decay channel, and do not change the exact count frequencies.

The exact source executed for the final index report is retained separately from the later curvature-generalized current script. Its SHA-256 matches the report.