The weighted-ball count operator admits an explicit extension in which both the wave and its material coefficients are varied. A common transformation law preserves the static count clocks, while the time-dependent action retains reciprocal wave/carrier coupling. Balanced component oscillations solve both equations exactly. Their ground branch has a complete positive energy bound, and simple spatial deformations produce mode-conversion selection rules tied to the same simplex counts.
This is an inverse constitutive candidate on a clamped active ball at d=2,3,4,5,6,10. The containing physical space remains ordinary shared flat space, currently modeled by R10. It is not a derivation of a freely localized particle in that containing space. Carrier and wave variables are proposed retained components of one field; their realization in the master field remains to be supplied. They are not mediator particles. Internal spatial profiles do not replace deterministic electron trajectories.
The reference wave operator is L0=-w^(-1) div[w(I-xx^T) grad], with w=sqrt(1-r^2), eigenvalues lambda_N=N(N+d+1), and complete degree-space dimension binomial(N+d-1,d-1). Its supplied positive polynomial P_d obeys P_d(lambda_N)=binomial(N+d,d)^2. All components use the same rule. The positive clamped elastic energy is
E_el(v)=1/2 int [2 epsilon(v):epsilon(v)+(div v)^2],
epsilon(v)=(Dv+Dv^T)/2,
sigma(v)=2 epsilon(v)+(div v)I,
so L_el=-Delta-2 grad div. These elastic coefficients are common material choices, not derived particle-specific numbers. Physical amplitudes remain free and are not fitted to observed masses.
Transform the Density and Stiffness Together
Let phi:B^d->B^d be a smooth orientation-preserving diffeomorphism equal to the identity on the boundary. Put F=Dphi, J=det F, and use y=phi(x) as a material label in the same d ordinary spatial coordinates. It does not add a physical dimension. Define
rho_phi(x)=J(x)w(phi(x)), w(y)=sqrt(1-|y|^2),
A_phi(x)=F^(-1)[I-phi(x)phi(x)^T]F^(-T),
L_phi=-rho_phi^(-1) div_x[rho_phi A_phi grad_x].
For a material function U(y), the chain rule and change of variables give
L_phi[U composed with phi]=(L_0 U) composed with phi,
int rho_phi |U composed with phi|^2 dx=int w|U|^2dy,
L_0=-w^(-1) div_y[w(I-yy^T)grad_y].
The quadratic gradient forms agree under the same substitution, proving the operator identity on their transported finite-energy domains. The entire polynomial P_d(L_phi) and its spectral domains transport with it. Every admissible static phi therefore has the same exact count frequencies. This uses the full Jacobian and inverse deformation gradient. Substitution only into w(phi) and I-phi phi^T is the loaded coefficient law and is not this isospectral material law.
A Complete Reduced Action on the Active Ball
Take a complex wave f, a positive common carrier inertia kappa, and the stated positive elastic strain form, with v=phi-x. The candidate is
S=1/2 int dt dx rho_phi [|f_t|^2-mu_d^2 f* P_d(L_phi)f]
+int dt [kappa/2 int |phi_t|^2 dx-E_el(phi-x)].
All spatial derivatives remain ordinary physical derivatives. For a fixed smooth carrier the wave potential is positive on its transported domain. The action has positive total energy
E=1/2 int rho_phi[|f_t|^2+mu_d^2 f*P_d(L_phi)f]dx
+kappa/2 int |phi_t|^2dx+E_el(phi-x).
No material convective derivative has replaced f_t in this action. That distinction matters for time-dependent coupling.
For any complete polynomial eigenmode L_0 u=lambda_N u,
phi=x, f=alpha u exp(-i omega_N t),
omega_N=mu_d binomial(N+d,d),
is an exact stationary-carrier solution of the joint equations below. The elastic first variation vanishes, and the wave carrier source also vanishes locally. This holds for every N and every maintained d using the same transformation rule. The scalar norm is not fixed. At these solutions, E=Q omega_N, Q=omega_N |alpha|^2 int w|u|^2, since the carrier's strain and velocity vanish.
Check the Carrier Equation, Not Only the Spectrum
There is also a direct check in laboratory coordinates. Let C_phi U=U composed with phi. Then P_d(L_phi)=C_phi P_d(L_0)C_phi^(-1). At the identity its variation is [v dot grad,P_d(L_0)]. In the on-mode action the density variation multiplies omega_N^2 |u|^2-mu_d^2 uP_d(L_0)u=0 pointwise. The remaining integral is proportional to int w u[v dot grad,P_d(L_0)]u=0 by the eigenvalue equation and weighted self-adjointness. This holds for arbitrary smooth clamped variations at every time, so it checks the local stationary carrier equation, not only an integrated force or dilation variation.
Define the material wave U(y,t)=f(phi^(-1)(y,t),t) and material velocity b(y,t)=phi_t(phi^(-1)(y,t),t). Then the very same physical action becomes
S_wave=1/2 int dt dy w[|D_t U|^2-mu_d^2 U*P_d(L_0)U],
D_t U=U_t+b dot grad_y U.
The b term follows from the ordinary time derivative f_t; it has not been added as an independent interaction. The wave equation is
partial_t(w D_t U)+div_y(w b D_t U)+w mu_d^2 P_d(L_0)U=0.
The material momentum conjugate to b is
m=w Re[(D_t U)* grad_y U].
For variation delta phi=eta composed with phi, delta b=eta_t+(b dot grad)eta-(eta dot grad)b. At fixed laboratory f, delta U=-eta dot grad U. On the wave equation, the latter contribution vanishes. Integration by parts therefore makes the wave's carrier source
-J {partial_t m+div_y(b tensor m)+(D_y b)^T m} composed with phi,
where the ith component of div_y(b tensor m) means partial_j(b_j m_i). Consequently the joint carrier equation, on the wave equation, is
kappa phi_tt-div_x sigma(phi-x)
=-J {m_t+partial_j(b_j m)+(D_y b)^T m} composed with phi.
For b=0 and U=alpha u exp(-i omega_N t), m is time independent, including when u is complex and carries a stationary spatial current. The source therefore vanishes pointwise, and phi=x satisfies its full interior Euler equation. This cancellation includes the determinant and inverse-gradient variations missing from the simpler substituted-coefficient candidate.
What Remains Coupled
General histories are not separated into independent wave and carrier actions: the material kinetic term contains b dot grad U. Expanding it produces a mixed current term and the nonnegative quadratic response
1/2 int w Re(partial_i U* partial_j U) b_i b_j.
Together with positive kappa, the full wave/carrier kinetic form is positive. The displayed term is an additional carrier-motion response; it is not by itself the complete particle's translational inertia or a proof of the required inertia-to-frequency ratio. Wave and carrier variations must be solved together when assessing that response.
The static count spectrum is protected here by a specified material transformation law. This offers an exact stationary support cancellation to compare with the genuine deformation shifts of the simpler material law. The mathematical difference is explicit and testable. Neither law has been selected empirically or derived from the master interaction.
Polarization Conditions for Exact Stationary Support
The Full Local Source
At the identity carrier, let every component have the same count frequency omega and write the real field as
Psi(x,t)=A(x) cos(omega t)+B(x) sin(omega t).
Every component of A and B belongs to the same complete spatial degree space. The wave equation is then satisfied exactly. The carrier momentum and force are
m_i=w partial_t Psi dot partial_i Psi,
F_i=-partial_t m_i.
Direct expansion gives
m_i=(omega w/2)[
(B dot partial_i B-A dot partial_i A) sin(2omega t)
+(B dot partial_i A+A dot partial_i B) cos(2omega t)
+B dot partial_i A-A dot partial_i B],
F_i=(omega^2 w/2) partial_i(|A|^2-|B|^2) cos(2omega t)
+omega^2 w partial_i(A dot B) sin(2omega t).
Thus the identity carrier is an exact joint solution if and only if
grad(|A|^2-|B|^2)=0, grad(A dot B)=0.
The two scalar invariants may be spatial constants; requiring them to be zero is sufficient but not necessary. This is a pointwise condition on the complete field, not a time-averaged cancellation. A stationary current in the final term of m is allowed.
For only one real field component, these two constants force A and B to be constant on the connected ball: (A^2+B^2)^2=(A^2-B^2)^2+4(AB)^2 is constant, and the pair (A,B) is restricted to a finite set unless both vanish. Continuity then fixes the pair. A nonconstant excited profile therefore needs at least two real components for this exact stationary carrier ansatz. The complex scalar action already supplies two; the real single-component version of the prescribed-medium action does not inherit the joint stationary family automatically.
Why the Complex Standing Family Works
For a real profile u, two real components with
A=(a u,0), B=(0,a u)
satisfy both conditions identically. They are precisely the circular component phase motion represented by a complex amplitude u exp(-i omega t), up to its sign convention. More generally a complex spatial profile has the real quadrature pair A=(Re u,Im u), B=(Im u,-Re u). Their norms agree and their dot product vanishes pointwise. Every exact stationary family already derived therefore survives this check.
An unequal component ellipse A=(a u,0), B=(0,b u) instead drives
F=omega^2(a^2-b^2) w u grad u cos(2omega t).
For an excited nonconstant real polynomial u this is nonzero if a^2!=b^2. A single real standing component is b=0 and drives the same second-harmonic source. The spatially constant ground profile has no such load for any ellipse. These statements hold at all six depths, with no special depth-five exception.
There is a useful distinction between component phase rotation and spatial rotation. A single real scalar profile
Psi=x1 cos(omega t)+x2 sin(omega t)
rotates its dipole in a physical two-plane, yet |A|^2-|B|^2=x1^2-x2^2 and A B=x1 x2 are not constants. It drives an oscillating carrier. Calling this spatial rotation circular does not make it the two-component stationary phase family. Conversely, the latter's complex notation does not derive physical spin or transverse photon polarization.
The consequence is a precise restriction on the exact stationary-support claim: it is an exact family of balanced component quadratures. Other polarizations require a responding carrier and may still admit periodic joint solutions. Their nonzero force does not establish that they fail to exist, decay, or lose the count frequency after self-consistent adjustment.
Complete Ground-Energy Control
An Exact Conserved-Energy Identity
Use the positive phase-action convention
Q=-Im int rho_phi f* f_t dx.
Q is the conserved phase action, not an identified electric charge.
For every admissible carrier, P_d(L_phi)>=I, by exact isospectral transport of the positive ball operator. The full energy satisfies
E-mu_d Q
=1/2 int rho_phi |f_t+i mu_d f|^2
+mu_d^2/2 int rho_phi f*[P_d(L_phi)-I]f
+kappa/2 int |phi_t|^2+E_el(phi-x) >=0.
The opposite sign gives E>=mu_d|Q|. Phase invariance and time-translation invariance conserve Q and E along any sufficiently regular joint solution with the specified fixed boundary. The identity includes all supporting energy and makes no claim of an unproved weak evolution theory.
Equality at Q>0 requires phi_t=0, zero elastic strain, f_t=-i mu_d f, and a spatially constant f. Clamped boundary data remove rigid carrier motions, so phi=x. The exact ground branch is consequently the global energy minimizer at its fixed phase action in this candidate's domain.
Quantitative Control from the Count Gap
Let W=int_B w dy=int_B rho_phi dx, which is independent of phi, and let fbar=W^(-1) int rho_phi f. The first count is d+1, hence the exact gap is
P_d(lambda_1)-P_d(0)=(d+1)^2-1=d(d+2).
For Delta=E-mu_d Q,
int rho_phi |f-fbar|^2 <= 2Delta/[mu_d^2 d(d+2)],
int rho_phi |f_t+i mu_d f|^2 <= 2Delta,
kappa int |phi_t|^2 <= 2Delta,
E_el(phi-x)<=Delta.
The clamped Korn and Poincare inequalities also control the carrier's H1 distance from the identity by its elastic energy. This uses the full spatial operator, not a truncated eigenbasis.
For fixed Q>0, put A=sqrt(Q/(mu_d W)). Writing n=int rho_phi |f|^2 and e=f_t+i mu_d f gives |mu_d n-Q|<=sqrt(n)||e||. Thus
|sqrt(n)-sqrt(Q/mu_d)|<=sqrt(2Delta)/mu_d,
inf_theta ||f-A exp(i theta)||_(rho_phi)
<=sqrt(2Delta)/mu_d [1+2/sqrt(d(d+2))].
The latter estimate uses the constant projection and the triangle inequality; it is a sufficient bound, not an optimal constant. Together these inequalities give orbital energy stability of the ground branch over every interval on which the joint solution remains in the smooth diffeomorphism domain. They do not control the carrier gradient uniformly in high dimension or prove that the map stays invertible forever. The spatial norm is a classical field integral, not an electron-position probability.
Why the Excited Branch Needs Its Own Persistence Analysis
For an occupied clock omega_N>mu_d, completing the square instead produces the spatial form mu_d^2 P_d(L_phi)-omega_N^2. It has negative directions on lower shells. This does not by itself prove instability, since the linearized action has gyroscopic terms. It explains why the preceding ground-state minimization cannot be transferred to excited states, and why resonant transfer tests are relevant.
The ground theorem holds at every maintained depth. Of the fourteen supplied comparison labels, (d,N)=(3,0) is a ground assignment; this coincidence does not identify that mathematical branch with an observed particle. Other ground branches and physical occupied-state selection remain to be explained.
Internal Mode Conversion
These are interactions within the active ball, not a force derived between independently localized particles.
Distinct Natural Clocks Drive the Carrier
Choose real wave eigenfunctions u_n orthonormal under int w. For one carrier shape v with displacement z(t), put
A_nm=int w u_n v dot grad u_m,
U=sum_n q_n(t)u_n.
To leading order in carrier velocity the material kinetic action contains
z_t m_z, m_z=Re sum_(n,m) q_(n,t)* q_m A_nm.
Take two freely rotating modal coefficients q_n=a_n exp(-i omega_n t), q_m=a_m exp(-i omega_m t), retaining their separate natural frequencies. For Delta=omega_n-omega_m, their cross contribution is
m_z=-(omega_n A_nm-omega_m A_mn)
Im[a_n* a_m exp(i Delta t)],
-partial_t m_z=Delta(omega_n A_nm-omega_m A_mn)
Re[a_n* a_m exp(i Delta t)].
Thus the carrier is driven at the beat frequency, with a phase-sensitive source. A single stationary mode, or a stationary superposition within one degenerate shell, has no time-dependent carrier load. No equality of unlike clocks has been imposed. The leading carrier equation is
kappa M_v z_tt+K_v z=-partial_t m_z.
The higher-order wave-dependent carrier mass belongs to the full action; dropping it here is a stated small-wave approximation.
For canonical phase actions Q_n=omega_n |a_n|^2, the signed momentum coupling coefficient is
sqrt(Q_n Q_m)
[D_n A_nm-D_m A_mn]/sqrt(D_n D_m),
omega_n=mu_d D_n.
The explicit common mu_d cancels from this momentum coupling. Its time derivative restores the factor mu_d(D_n-D_m), and the physical response also depends on the elastic denominator K_v-kappa M_v Delta^2. Consequently the action distinguishes a common sector clock, count ratios, spatial overlaps and resonance. It does not establish a universal coupling that depends only on the fundamental or only on a particle's mass-frequency.
A Spatial Degree Selection Rule
Take a polynomial carrier shape v=(1-r^2)p(x), with each component of p of degree at most k. It is clamped at the sphere. Weighted integration by parts gives
div_w v=(1-r^2)div p-3 x dot p,
V=v dot grad, V*=-V-div_w v.
Both V and V* increase polynomial degree by at most k+1. Since the complete wave degree space V_N is orthogonal to all lower-degree polynomials,
A_NM=0 whenever |N-M|>k+1.
This is a rule for entire degree spaces, including their radial and angular multiplicities. It does not count one chosen angular family as the whole shell. If p has a definite homogeneous parity, the parity of N-M must also equal that of k+1 for a nonzero entry.
In particular, v=(1-r^2)e_i couples only neighboring degrees N and N+/-1. The radial shape v=x(1-r^2) couples only N,N+/-2. These zeros follow from the actual spatial coupling operator and the common ball measure.
Pascal's Identity Becomes a Beat-Frequency Rule
For the neighboring-degree channel, the count clock gives
omega_N-omega_(N-1)
=mu_d [binomial(N+d,d)-binomial(N+d-1,d)]
=mu_d binomial(N+d-1,d-1).
The final binomial is exactly the scalar multiplicity of degree N. Thus a simple allowed carrier motion drives at the shell-count frequency, while the stationary pattern oscillates at the cumulative-count frequency. For a two-degree channel the beat is the sum of the two intervening shell multiplicities. More generally the difference of cumulative counts is the corresponding finite sum of shell counts.
This is a derived operator selection rule followed by a count identity. It is not an arithmetic recipe for adding particle indices or generating the observed families. Elastic resonance with these beats is a further condition; it has not been fitted to a lifetime or a measured interaction.
An Explicit Zonal Matrix Element
For an explicit example use Gegenbauer profiles u_N proportional to C_N^lambda(x1), lambda=(d+1)/2, and v=(1-r^2)e1. Integrating the unused coordinates gives
mean[1-r^2 at fixed x1]=3(1-x1^2)/(d+2).
Let h_N denote the unnormalized squared profile norm. The only nonzero matrix entries are
A_(M+1),M=-3M(M+1)/[2(d+2)(M+lambda)] sqrt(h_(M+1)/h_M),
A_(M-1),M=3(M+2lambda-1)(M+2lambda)
/[2(d+2)(M+lambda)] sqrt(h_(M-1)/h_M),
h_N/h_(N-1)=(N+2lambda-1)(N+lambda-1)/[N(N+lambda)].
The raising coefficient vanishes at M=0, because a constant has zero gradient. The reverse matrix element need not vanish. The asymmetry is required by the weighted divergence and is already retained in the physical source above. This zonal submatrix illustrates the full degree selection rule; it is not the complete angular response matrix.
A Common Excited-Branch Spectral Regime
For fixed real or complex spatial profiles, the complete linearized spectral analysis now establishes a sufficiently stiff common carrier regime with imaginary spectrum for all fourteen maintained labels at sufficiently small amplitude. The proof handles the full weighted domains, all negative-energy branches, and the exceptional depth-two wave collision. The stationary kernel is semisimple. This is distinct from the ground energy minimum and from nonlinear lifetime or material-map persistence. For complex current-carrying profiles the additional carrier gyroscopic form is derived and included in the same negative-index and collision argument.
The linked excited-branch theorem also establishes unique, uniformly bounded linear evolution in the complete form phase space. Its conserved positive norm includes stationary modes and supports explicit conditional forcing bounds. This does not control nonlinear material invertibility or determine physical decay widths. The separate finite-speed moving action still needs its own stability proof.
Verification and Remaining Physical Work
The material checker checks the complete nonlinear radial-map operator identity at symbolic depth, the full count-polynomial stationary variation at all six depths, and the ground-energy completed square. The arbitrary-map and arbitrary-variation proofs are the change-of-variables and commutator arguments above; sampled radial checks do not substitute for those proofs.
The polarization checker verifies the local source and distinguishes balanced component motion from a rotating scalar spatial dipole. The mode-selection checker checks 63 exact zonal matrix entries, eight norm ratios and forty Pascal beat identities per depth. The full degree-space selection rule follows from the operator and its weighted adjoint, not from the zonal sample alone.
The material map is restricted to smooth orientation-preserving ball diffeomorphisms with a clamped boundary. The energy bound controls the stated norms while a regular solution remains in that domain. It does not prove global evolution or continued invertibility. Excited branches do not inherit the ground energy minimum. The linked fixed-profile spectral theorem supplies a sufficient small-amplitude regime; nonlinear persistence remains open. The supplied high-order spatial dispersion also retains unbounded high-frequency group speed, so this is not a causal completion of the theory.
The ball's size, exterior stress, finite extension through unused spatial directions, complete translational inertia, sector frequency units and occupied-state selection remain open. The exact law is compatible with all maintained count assignments without a separate material coefficient for each particle, but it does not independently select those assignments or establish empirical spin, electric charge, photon polarization or a special depth-five coupling suppression.