Complete form response to the current interaction

The local source in Spatial currents and orientation-dependent coupling drives modes outside the two initially occupied patterns. Resolving its full spatial form content shows that almost all of this forcing lies on low branches of the parent. The large intermediate gap therefore does not justify discarding it. Some of the additional patterns are exactly degenerate with the original frequency and require resonant evolution within the full mode family.

This calculation uses coincident spherical field features and a declared quartic interaction. It does not yet describe separated particle motion, derive the sustaining geometry, or identify observed interaction channels. Mass remains the selected particle frequency in equivalent units.

An invariant spatial class with all generated form components

On the unit S^m write

$$ds^2=d\theta^2+\sin^2\theta\,d\phi^2+ \cos^2\theta\,ds_{S^{m-2}}^2,\quad 0\le\theta\le\pi/2,\quad t=\sin^2\theta. \tag{1}$$

The two-plane patterns are invariant under O(m-1) acting on the remaining ambient directions. The corresponding basic forms have four components: scalar, one-form along theta, one-form along phi, and their exterior two-form. The Hodge operator and the specified local current source preserve this class. The pointwise source was also checked against every exterior component outside it. This is not a claim of stability against all nonsymmetric spatial perturbations.

Use the orthonormal components (f,a,b,c) multiplying (1,dtheta,sin(theta)dphi,dtheta wedge sin(theta)dphi), with azimuthal dependence exp(iq phi). Direct exterior differentiation and its weighted adjoint give

$$(d_S+\delta_S) \begin{pmatrix}f\\a\\b\\c\end{pmatrix} =\begin{pmatrix} -a'+[-\cot\theta+(m-2)\tan\theta]a-iq b/\sin\theta\\ f'+iq c/\sin\theta\\ iq f/\sin\theta-c'+(m-2)\tan\theta\,c\\ -iq a/\sin\theta+b'+\cot\theta\,b \end{pmatrix}. \tag{2}$$

Primes denote theta derivatives. The weight is sin(theta) cos(theta)^(m-2). In t, normalized angular integration uses k(1-t)^(k-1)dt with k=(m-1)/2.

Both complete harmonic families

Let q be any integer and j a nonnegative integer. A scalar seed is

$$f_{qj}=C_{qj}t^{|q|/2} P_j^{(k-1,|q|)}(2t-1)e^{iq\phi},\qquad L=|q|+2j,\quad \lambda_0=L(L+m-1). \tag{3}$$

The constant C normalizes its actual surface L2 norm. Together with d_S f/sqrt(lambda_0), it gives the scalar/exact-one-form pair. The constant scalar has no exact-one-form partner.

The second seed is a two-form with coefficient

$$c_{qj}=\widetilde C_{qj}t^{|q|/2}(1-t)^{1/2} P_j^{(k,|q|)}(2t-1)e^{iq\phi},\qquad L=|q|+1+2j,\quad \lambda_1=(L+1)(L+m-2). \tag{4}$$

Its partner is delta_S c/sqrt(lambda_1), a coexact one-form. The identities obtained by substituting the Jacobi differential equation in (2) establish both eigenvalue formulas. Jacobi orthogonality, and orthogonality of exact and coexact forms, give a complete orthonormal resolution in this basic spatial class. These formulas agree with the spherical form spectra of Ikeda and Taniguchi.

For B=d_S+delta_S+k Gamma, the two respective signed eigenvalues are

$$\beta_0=\pm(L+k),\qquad \beta_1=\pm\sqrt{(L+k)^2+m-2}. \tag{5}$$

The second family has a different sequence. Insert every signed beta into the same full first-order chain with links beta+j-(d-1)/2, j=0,...,d-1. Both low signs and all high chain modes remain present. The low eigenvalues are calculated with the stable scaled inverse; large eigenvalues use the ordinary Hermitian decomposition. Exact signed reversal is retained when evaluating degenerate frequencies.

Resolve the actual time-dependent source

At zeroth order take real spatial patterns A and B at their two positive parent frequencies, with coefficients sqrt(0.6) and sqrt(0.4). Factoring out exp(-i omega_a t), their relative phase advances at -(omega_b-omega_a)t. The complete cubic source has four Fourier orders

$$r=-1,0,1,2,\qquad \Omega_r=\omega_a+r(\omega_b-\omega_a). \tag{6}$$

With kappa from the preceding current note and real initial coefficients a,b, its spatial coefficients are

$$F_{-1}=-ig a^2b\,\kappa\cdot\gamma A,\quad F_1=ig a^2b\,\kappa\cdot\gamma A,\quad F_0=-ig ab^2\,\kappa\cdot\gamma B,\quad F_2=ig ab^2\,\kappa\cdot\gamma B. \tag{7}$$

The direct nonlinear source agrees with their sum at arbitrary relative phase. Each F is then projected onto (3)-(5) and every canonical chain eigenvector. Reconstruction in physical space and the complete 4-by-4 source Gram matrix check that this resolution has retained the forcing. The cubic source has finite angular degree; additional degrees and independent quadrature refinement verify the cutoff.

If f_mu,r is one of these coefficients, the first response in an empty mode with frequency nu_mu is exactly

$$\delta c_\mu(t)=\sum_r f_{\mu,r} \frac{e^{-i\Omega_rt}-e^{-i\nu_\mu t}}{\Omega_r-\nu_\mu}. \tag{8}$$

At equality of the frequencies the quotient is -it exp(-i nu_mu t). The implementation uses a sinc expression for that limit. Independent matrix-exponential evolution of the driven full chain verifies (8), including resonance. This is first order in g, with its usual limitation that the perturbation must remain small.

Low response and exact degeneracy

At eta=10,000, the fraction of summed source-channel norm carried by low chain modes is 0.9999974073 for depth-five degrees 11/16 and 0.9999846774 for depth-six degrees 14/36. The remainder is retained and its high-frequency response is bounded using its actual denominators. The small high response cannot account for the substantial low forcing outside the original pair.

For a reproducible weak-coupling comparison set g=epsilon omega_a and tau=omega_a t. The following are response norm squares divided by epsilon squared at tau=1. They include all low frequencies and both signed form polarizations.

These are conditional field responses, not empirical rates or a common physical g across depths. The degree-two rows are mathematical controls, not a photon assignment.

The r=0 and r=1 channels contain exact resonances in the original frequency families. For example, the degree-36 family also receives q=+8 and q=-8, j=14 patterns when it interacts with the degree-14 pattern. They have the same scalar degree and parent frequency, but a different spatial distribution. The opposite signed B polarization must also be kept. Such a response changes a spatial pattern without requiring a change of its leading particle frequency. It motivates the complete resonant-family calculation rather than a closed two-mode ansatz.

Verification and limits

The final spatial-form-spectrum-20260914T001621_669242Z.json report passes 78 checks, including independent finite differences of the full geometric operator and complete scalar/exact/coexact/two-form Gram matrices. The spatial-form-response-20260914T001627_066477Z.json report passes 41 checks, including the full source reconstruction, phase channels, canonical Parseval matrix, forced evolution and independent radial and angular refinement at the assigned degrees.

An earlier derivative-at-zero diagnostic used a time step that was not small compared with the fastest retained low branch in the high-ratio controls. The corrected check scales its time by that actual fastest frequency and uses a tighter error tolerance. Exact scalar beta values and signed reversal also prevent floating-point splitting of a known degeneracy. The original reports and source versions are retained.

The calculation does not establish nonlinear stability, an encounter cross section, ordinary electric neutrality, or a complete particle mass operator. It supplies the missing full spatial source and its first reciprocal response for the stated inverse action.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.