A Responding Flat-Ball Carrier and Its Polarization Energy

The weighted-ball count action has a nonzero local material load when its coefficients are allowed to respond. One common clamped elastic rule now gives an explicit solution of that linear load, finite-load control of the weighted spectral domain, and a leading polarization-dependent effective energy. These are mathematical consequences under the declared constitutive and boundary assumptions. The containing space remains ordinary shared flat space; the calculation concerns the active d-dimensional pattern at d=2,3,4,5,6,10.

The underlying spatial operator is L_w=-w^(-1) div(w A grad), with w=sqrt(1-r^2), A=I-xx^T, and lambda_N=N(N+d+1). Its supplied positive polynomial P_d has P_d(lambda_N)=binomial(N+d,d)^2. The rotating mode f=alpha u exp(-i omega t) has omega^2=mu_d^2 P_d(lambda), weighted norm N0=int w|u|^2 and phase action Q=omega |alpha|^2 N0. Physical amplitude remains free. No amplitude is chosen to reproduce a measured mass.

The response operator is a separate elastic operator L_el=-Delta-2 grad div, with clamped v=0 at r=1 and form a_el(v,h)=int [2 epsilon(v):epsilon(h)+div(v)div(h)]. Both elastic constants equal one at every depth. They are stated material choices, not quantities derived from the count alone. The coefficient field is phi=x+v, inserted into w(phi), A(phi) with ordinary spatial derivatives. It is not an isospectral coordinate pullback.

For u=x1, put C=mu_d^2 P_d'(d+2)/2 and a=|alpha|^2. The complete variation of both the time term and spatial operator gives

J=C a j,
j=x[1-(d+3)x1^2]/sqrt(1-r^2)+2sqrt(1-r^2)x1 e1.

The physical leading displacement is C a times the unit-load solution. The boundary is clamped for the elastic response only; the wave uses its weighted finite-energy spectral domain, not a hard-wall Dirichlet domain. The singular source is in H-minus-one, as follows from its square-integrable stress representation below. Coercivity gives a unique weak response.

An Explicit Full First-Mode Solution

A smaller angular form is preserved by the differential operator:

v=x F(R)+e1 x1 G(R)+x x1^2 H(R).

For L=-Delta-2 grad div, direct differentiation gives

L v=x A(R)+x x1^2 B(R)+e1 x1 D(R),
A=-12R F''-6(d+2)F'-4G'-2H,
B=-8G''-12R H''-(6d+28)H',
D=-4R G''-(2d+12)G'-8R H'-4(d+3)H.

The full unit first-mode source is obtained by solving

A=1/sqrt(1-R),
B=-(d+3)/sqrt(1-R),
D=2sqrt(1-R),
F(1)=G(1)=H(1)=0,

with regularity at R=0. A regular solution of these equations is an exact solution of the full clamped elastic problem, and coercivity then supplies uniqueness. Angular closure alone is not an existence proof for the radial system. It identifies the exact remaining differential equations, and enables high-order radial Galerkin approximations without coordinate growth.

Write v=x F(R)+e1 x1 G(R)+x x1^2 H(R), R=r^2. Let

a_n=binomial(2n,n)/4^n,
g_n=-(2d^2+4dn+13d+6n+16)
      /[6(n+1)(n+2)(d+2n+4)(d+2n+6)],
h_n=(d^2+2dn+11d+6n+20)
      /[6(n+1)(d+2n+4)(d+2n+6)],
f_n=-(2dn-d+4n^2+14n+4)
      /[12(n+1)(n+2)(d+2n+4)(d+2n+6)].

Define the radial coefficients by

H(R)=H0+sum_(n>=0) a_n h_n R^(n+1),
G(R)=G0+G1 R+sum_(n>=0) a_n g_n R^(n+2),
F(R)=F0+F1 R+sum_(n>=0) a_n f_n R^(n+2),
H0=-sum a_n h_n,
G1=[-2-(4d+12)H0]/(2d+12),
G0=-G1-sum a_n g_n,
F1=-(1+4G1+2H0)/[6(d+2)],
F0=-F1-sum a_n f_n.

The three sum definitions impose F(1)=G(1)=H(1)=0. All constants are explicit: partial fractions use the identity

sum_(n>=0) a_n/(n+A)=Beta(A,1/2), A>0.

The rational factors above have only distinct negative simple poles for the maintained depths. Thus their sums reduce to finite beta expressions. For example at d=2,

F0=71/1260, F1=-61/1260,
G0=-29/630, G1=19/126, H0=-139/630.

Verification of the Differential Equation

The coefficients of x x1^2 and e1 x1 at orders R^n and R^(n+1) give the two-by-two system

8(n+2) g_n+(12n+6d+28) h_n=(d+3)/(n+1),
(n+2)(4n+2d+16) g_n+(8n+4d+20) h_n=1/(n+1).

Its determinant is -12(n+2)(d+2n+4)(d+2n+6), so it is invertible. The coefficient of x then determines f_n using a_(n+1)/a_n=(2n+1)/(2n+2). The remaining lowest-order equations fix G1 and F1 as stated. Consequently the series solves all three radial equations for 0<=R<1. Its smoothness at the origin follows from being a power series in R multiplying ordinary coordinate polynomials.

Boundary Regularity and the Weak Solution

The elementary bound a_n<=1/sqrt(n+1) follows by induction from its exact ratio. Also h_n and f_n are O((n+1)^(-2)), while g_n is O((n+1)^(-3)). Hence F,G,H and their first derivatives converge absolutely and uniformly on [0,1]. Their second derivatives converge locally inside the ball, with the expected integrable boundary singularity.

The resulting v is C1 on the closed ball, vanishes on its boundary and belongs to H1_0. Its interior equation L v=j holds classically. Against compactly supported tests it therefore holds distributionally; both sides extend continuously to H1_0 using the elastic form and the L2 stress representation of j. Density of compact tests gives the full weak equation. Coercivity establishes that this is the unique full clamped unit-load solution. No finite Galerkin truncation is used in this existence argument.

Rigorous Boundary Admissibility and Truncation Bounds

Let F_m,G_m,H_m be the complete coefficients, including the constants. Because the functions vanish at R=1,

|v|/(1-R)<=K_d,
K_d=sum_(m>=1) m(|F_m|+|G_m|+|H_m|)<infinity.

Thus the physical displacement C a v preserves |x+C a v|<1 throughout the open ball whenever |C a| K_d<1/2. The coefficient sums are explicit beta sums: h_n>0, g_n<0, and f_n<0 for n>=1; only f_0 changes sign at d=4. This estimate applies to the full solution, unlike the earlier finite-polynomial bound.

For simple uniform truncation bounds set

Hc=(d^2+11d+20)/24,
Gc=(2d^2+13d+16)/24,
Fc=(2d+18+|4-d|)/48.

Then |a_n h_n|<=Hc(n+1)^(-5/2), |a_n g_n|<=Gc(n+1)^(-7/2), and |a_n f_n|<=Fc(n+1)^(-5/2). Omitting indices n>=M consequently gives

sup |v-v_M| <= (2/3)(Fc+Hc)M^(-3/2)
                       +(2/5)Gc M^(-5/2).

The load norm is bounded by integral |j|/Vol(B^d)<=(d+4) integral (1-R)^(-1/2)/Vol(B^d)+2. Multiplying this by the uniform tail estimate bounds the error in the computed compliance and in every first-shell load overlap. Numerical rounding must additionally be enclosed; an unrounded tail estimate alone does not certify a floating-point sum.

This solves the linear first-mode carrier problem at every maintained depth with one elastic rule. It does not establish the full coupled finite-amplitude eigenproblem, observed masses, or a second-mode solution.

Exact Shell Splitting

For each first-shell mode x_i, define

j_i=x[1-(d+3)x_i^2]/w+2w x_i e_i, w=sqrt(1-r^2),
u_i=L^(-1)j_i, N0=integral w x_i^2.

The loads belong to H-minus-one through their L2 symmetric stress representations. The clamped problem is coercive. Rotational covariance and reciprocity imply that the compliance matrix has diagonal entry c and off-diagonal entry h:

<j_i,L^(-1)j_i>=c, <j_i,L^(-1)j_j>=h for i!=j.

Here h is a scalar, not a test field. The physical displacement generated by occupation of x1 is v=C a u1, with C>0 and a=|alpha|^2>0. Parity makes the first-shell perturbation matrix diagonal, and

delta_lambda_parallel=-C a c/N0,
delta_lambda_transverse=-C a h/N0.

The difference is exactly

delta_lambda_transverse-delta_lambda_parallel
  =C a (c-h)/N0
  =C a a(u1-u2,u1-u2)/(2N0)>0.                 (1)

The strict inequality follows because j1-j2 is not the zero distribution. Thus the exact full operator splits the occupied direction from every transverse direction. A finite-basis split is not needed as evidence for this theorem. The transverse shift's individual sign is not determined by (1). The loaded shift is negative because c>0.

An Explicit Lower Bound

Take the clamped test field z=(1-r^2)(x1 e1-x2 e2). Beta moments give

<j1-j2,z>=-4d N0/(d+5),
a(z,z)=8(d+6) Vol(B^d)/[(d+2)(d+4)].

For example the load moment uses integral w(x1^2-x2^2)^2=4N0/(d+5) and integral w^3(x1^2+x2^2)=6N0/(d+5). Elastic Cauchy--Schwarz consequently gives

c-h >= d^2(d+2)(d+4) N0^2
             /[(d+5)^2(d+6) Vol(B^d)] > 0.       (2)

This applies uniformly at depths 2,3,4,5,6,10. It supplies no exceptional depth-five cancellation. Both formulas use the full spatial source and the same boundary and constitutive rule. No particle-specific coefficient is introduced.

The matrix is the first-order spectral perturbation for the linear loaded carrier response. It does not establish a finite-amplitude coupled solution or the empirical particle spectrum. Coefficient admissibility follows separately from the full series bound above. It does show that the identity carrier's first-shell degeneracy cannot survive this nonzero mode-directed load at first order.

Complementary Stress Upper Bound

For j1 use S=w[1-(d+3)x1^2]I-[2(d+2)/3]w^3 e1 e1^T. Direct differentiation gives -div S=j1. Since S is L2, the exact compliance also obeys

c <= U0=1/2 integral [S:S-(tr S)^2/(d+2)].

Together with any conforming Galerkin lower bound cN, this yields the valid energy-norm error estimate ||u1-uN||_elastic^2<=U0-cN. Its quantitative sharpness is a separate calculation; it is not inferred from a finite set of test residuals.

Finite-Load Spectral Control

Boundary Control Gives Form Control

Let v be the full clamped first-mode response already constructed, and let

|v(x)| <= K_d(1-r^2),  phi(x)=x+epsilon v(x),
delta=|epsilon|K_d < 1/2,
w_phi=sqrt(1-|phi|^2), A_phi=I-phi phi^T.

Here epsilon=C|alpha|^2 for the physical first-mode load. The norm and gradient energy use the same deformed weight. The undeformed quantities are w=sqrt(1-r^2) and A=I-xx^T.

The boundary vanishing estimate gives the pointwise inequalities

(1-2delta)(1-r^2) <= 1-|phi|^2 <= (1+2delta)(1-r^2),
(1-2delta)A <= A_phi <= (1+2delta)A.

For the first inequality, expand |x+epsilon v|^2. The absolute change divided by 1-r^2 is at most 2delta r+delta^2(1-r^2), which is at most 2delta when 0<=r<=1 and delta<=1. For the matrix inequality conjugate the change by A^(-1/2): |A^(-1/2)x|=r/sqrt(1-r^2) and |A^(-1/2)epsilon v|<=delta sqrt(1-r^2). The same bound applies to its operator norm. This controls the radially degenerate stiffness, not merely the unweighted coefficient difference.

Consequently, for every function in the common weighted form domain,

sqrt(1-2delta) ||u||_w^2 <= ||u||_(w_phi)^2
                                  <= sqrt(1+2delta) ||u||_w^2,
(1-2delta)^(3/2) a_0(u,u) <= a_phi(u,u)
                                  <= (1+2delta)^(3/2) a_0(u,u),

where a_phi=int w_phi grad(u)* A_phi grad(u). Both norms and forms are equivalent, with the same completed domain. Thus the deformed problem retains a positive self-adjoint weighted operator and compact form embedding. The latter follows from equivalence with the undeformed weighted polynomial operator, whose eigenvalues grow and whose individual multiplicities are finite.

Bounds on Every Ordered Eigenvalue

Apply the minimum-over-subspaces characterization to the bounded Rayleigh quotients. At every ordered index, including multiplicities,

a_delta lambda_j(0) <= lambda_j(phi) <= b_delta lambda_j(0),
a_delta=(1-2delta)^(3/2)/(1+2delta)^(1/2),
b_delta=(1+2delta)^(3/2)/(1-2delta)^(1/2).

This remains valid when a formerly degenerate shell splits. It does not require choosing or tracking an eigenvector within that shell. Constants remain the zero mode exactly. No finite-basis extrapolation is involved.

For the supplied increasing count polynomial P_d, define frequencies by omega_j(phi)=mu_d sqrt(P_d(lambda_j(phi))). Then

mu_d sqrt(P_d(a_delta lambda_j(0))) <= omega_j(phi)
                       <= mu_d sqrt(P_d(b_delta lambda_j(0))).

All d roots of P_d are negative in the maintained construction. Comparing each positive factor separately yields the simpler uniform bounds

a_delta^(d/2) omega_j(0) <= omega_j(phi)
                              <= b_delta^(d/2) omega_j(0).

These bound the whole family at once. At small delta their worst-case fractional width is 2d delta+O(delta^2); this is a conservative bound, not the actual mode-dependent shift. The exact response calculation supplies the latter for the first shell and already establishes nonuniform splitting.

The common same-law constants K_d are

The construction therefore retains a controlled spectral problem for a nonzero open interval of physical load at every maintained depth. It does not prove that actual particle loads lie in a desired numerical mass tolerance. No amplitude has been chosen to fit a mass, and the exact simplex clocks are the undeformed input spectrum rather than an invariant of arbitrary loaded coefficients.

Optional Geometric Map Bound

For v=xF+e1 x1 G+x x1^2 H, the same absolutely convergent coefficient sums bound ||Dv||_op by 5K_d. Hence |epsilon|5K_d<1 suffices to make phi an injective orientation-preserving map, with a Lipschitz inverse. The boundary is fixed, so its image is the same ball. This stronger geometric assertion is not needed for the form bounds above; the material coefficient field need not be reinterpreted as a physical coordinate map.

The First-Order Shift Is a Genuine Spectral Derivative

Let D be the fixed completed form domain with inner product a_0+b_0, where b_0 is the undeformed weighted norm. The ratios w_phi/w and A^(-1/2) A_phi A^(-1/2) are bounded analytic functions of epsilon in a neighborhood of zero. Their Taylor series converge uniformly as coefficient operators: the only square root is that of 1-2epsilon x.v/(1-r^2)-epsilon^2 |v|^2/(1-r^2), whose two coefficients are bounded. This gives bounded analytic forms a_epsilon,b_epsilon on D.

Represent a_epsilon+b_epsilon by S_epsilon and b_epsilon by B_epsilon in that fixed form inner product. Then S_0=I, S_epsilon is positive and invertible for small real epsilon, and B_epsilon is compact. Thus

K_epsilon=S_epsilon^(-1/2) B_epsilon S_epsilon^(-1/2)

is a bounded analytic compact self-adjoint family for real epsilon. Its nonzero eigenvalues are 1/(1+lambda_j(epsilon)). Analytic perturbation on each isolated finite-dimensional spectral cluster therefore gives the usual first-order matrix a'_0-lambda b'_0. This construction avoids assuming that a varying weight preserves a high-order differential domain.

For the first shell, normalize the basis by sqrt(N0). The derivative matrix is exactly -<j_ij,v>/N0, including the derivative of the kinetic weight. For v=K_el j_11, its parallel and transverse eigenvalues are -c/N0 and -h/N0. The full-PDE signs already enclosed by the series computation therefore describe actual spectral branches near epsilon=0. The positive polynomial P_d of the deformed self-adjoint operator supplies the matching higher-order wave realization through spectral calculus. This does not claim that its high-order operator domain is independent of epsilon.

The external perturbation tools are described in Kato, Perturbation Theory for Linear Operators, Chapter VII, sections 3, 4 and 6 (especially Theorem 4.2 and the generalized eigenvalue construction). They apply to spatial integrals and operators here; they do not change the ordinary spatial interpretation of the field.

Polarization of the Complete First Shell

The Complete Quadratic Source

Let the occupied first-shell profile be u=z dot x, with a complex vector z. Define the real symmetric tensor T=Re(z* tensor z), rho=z* dot z, and sigma=|z dot z|^2. The same time-averaged carrier variation gives

j_T=x[tr T-(d+3)x.T.x]/sqrt(1-r^2)+2sqrt(1-r^2)T x.

Its physical source is C j_T, with C=mu_d^2 P_d'(d+2)/2. Thus no scalar amplitude factor is to be multiplied in again: it is already in T. Write K=(-Delta-2 grad div)^(-1) with the clamped boundary condition and define c=<j_E11,K j_E11>, h=<j_E11,K j_E22>. Rotational covariance and reciprocity determine the full bilinear response on symmetric tensors:

<j_T,K j_U>=h tr T tr U+(c-h)tr(TU).

To see this, orthogonally diagonalize a real symmetric T. Every diagonal source has self-overlap c and every distinct pair has overlap h. Polarization of that quadratic identity gives the displayed bilinear expression, without assuming simultaneous diagonalization of T and U.

Since tr T=rho and tr T^2=(rho^2+sigma)/2,

<j_T,K j_T>=(c+h)rho^2/2+(c-h)sigma/2.

The full-operator splitting theorem establishes c-h>0. The isotropic source also gives c+(d-1)h>0; this follows from positive elastic compliance and the nonzero source j_I. For d>=2 these imply c+h>0. In particular, the negative cross overlap h found numerically is not a negative self-energy.

Eliminate the Carrier at Fixed Phase Action

To the first nontrivial order, the carrier energy and coupling are

E_carrier(v)=a_el(v,v)/2-C<j_T,v>.

Its unique stationary minimum is v=C K j_T, and its contribution is -C^2<j_T,K j_T>/2. Let N0=int sqrt(1-r^2)x1^2 and omega^2=mu_d^2 P_d(d+2). Use canonical amplitudes a=sqrt(omega N0) z and total leading phase action Q=a* dot a. The corresponding leading effective Hamiltonian is

H_eff=omega Q-eta1 Q^2-eta2 |a dot a|^2,
eta1=C^2(c+h)/(4 omega^2 N0^2),
eta2=C^2(c-h)/(4 omega^2 N0^2)>0.

This is an expansion about zero load; its negative quartic term is not a globally bounded energy model and must not be extrapolated to large fields. The carrier is eliminated statically. A time-dependent application requires the carrier to respond adiabatically; finite carrier inertia is not included. For a linearly polarized branch, differentiating H_eff at fixed canonical phase action gives delta omega=-C^2 c Q/(omega^2 N0^2), agreeing with the earlier physical shift delta omega/omega=-C^2 rho c/(omega^2 N0).

A Conditional Polarization Preference and Conserved Geometry

At a given Q, 0<=|a dot a|<=Q. The lower quartic energy occurs at |a dot a|=Q, where a has a common phase times a real vector. The circular case has a dot a=0 and lies higher by eta2 Q^2. The preference follows from c-h>0 in every maintained dimension; it does not select depth five.

This is an energy comparison at fixed Q, not a relaxation or damping law. Writing a=p+i q, rotational invariance conserves

J_ij=2(p_i q_j-p_j q_i),
sum_(i<j) J_ij^2=Q^2-|a dot a|^2.

A nonzero angular momentum therefore prevents an isolated trajectory of this effective Hamiltonian from simply becoming linearly polarized. Calling the energy preference spontaneous alignment would omit that conservation law and the missing dissipative channel.

The canonical equation is

i dot a=(omega-2 eta1 Q)a-2 eta2 S a*, S=a dot a.

It gives dot Q=0 and dot S=-2i[omega-2(eta1+eta2)Q]S. Remove this common phase by a=exp(-i Omega t)b, Omega=omega-2(eta1+eta2)Q, and choose a fixed phase so S0=b dot b is real and nonnegative. With b=p+i q,

dot p=2 eta2(Q+S0)q,
dot q=-2 eta2(Q-S0)p.

The ellipse evolves at nu=2 eta2 sqrt(Q^2-S0^2), with its invariants fixed. This is bounded internal polarization motion in the stated reduced action. It is not a derivation of the literal electron's six-dimensional orbital precession, nor of photon polarization or physical spin quantization.

The full linear carrier solution therefore supplies more than one axis's frequency correction: it fixes the leading tensor response, polarization energy difference and conditional internal polarization dynamics under one constitutive law. The underlying count polynomial, elastic law and clamped boundary remain explicit model premises.

Higher Harmonic Modes

A Closed Elastic Block

Let P_L be a homogeneous harmonic polynomial of degree L and R=r^2. For L>=1, set v=x P_L F(R)+grad(P_L)G(R). Direct differentiation gives

div v=P_L[(d+L)F+2R F'+2L G'],
(-Delta-2grad div)v=x P_L A(R)+grad(P_L)B(R),
A=-12R F''-(6d+8L+12)F'-8L G'',
B=-2(d+L+1)F-4R F'-4R G''-(2d+8L-4)G'.

For L=0 the gradient term vanishes; only the first radial equation remains, with A=-12R F''-6(d+2)F'. The boundary condition is F(1)=G(1)=0 when both structures occur, together with regularity at the origin.

Write the source coefficients as A_n,B_n and unknown series coefficients as F_n,G_n. For n>=0 the exact two-by-two system is

[(n+1)(12n+6d+8L+12), 8L(n+2)(n+1)] [F_(n+1)] = [-A_n],
[2d+2L+6+4n,          (n+2)(4n+2d+8L)] [G_(n+2)] = [-B_(n+1)].

Its determinant is 12(n+1)(n+2)(2L+d+2n)(2L+d+2n+2)>0. After these coefficients are found, F0=-sum_(m>=1)F_m, G1=[-B0-2(d+L+1)F0]/(2d+8L-4), and G0=-G1-sum_(m>=2)G_m. These impose the boundary values and the lowest remaining equation. Convergence and source-specific tail bounds must still be checked; invertibility of this recurrence alone is not that proof.

Source from a Harmonic Occupied Mode

For a homogeneous harmonic mode u of degree ell, lambda=ell(ell+d+1), put Q=u^2. Its stripped source is

j=x[Delta(Q)/2-ell(2ell+d+1)Q]/sqrt(1-R)
    +ell sqrt(1-R) grad Q.

This follows directly from Delta(u^2)=2|grad u|^2 and x.grad u=ell u. Decompose Q into finitely many harmonic terms c R^s P_L with L+2s=2ell. One such term generates the radial load

A=c {[s(2L+d+2s-2)R^(s-1)-ell(2ell+d+1)R^s]/sqrt(1-R)
       +2ell s sqrt(1-R)R^(s-1)},
B=c ell sqrt(1-R)R^s.

Terms multiplied by s vanish when s=0. Both coefficients are polynomials times sqrt(1-R) or its inverse. Their Taylor coefficients are therefore finite shifts of the central-binomial sequence already used for the first mode. The same Navier block applies at every depth; no new elastic rule is needed for each occupied mode.

Complete Second-Mode Decomposition

For u=x1*x2, ell=2, define

H2=x1^2+x2^2-2R/d,
H4=x1^2*x2^2-R(x1^2+x2^2)/(d+4)+R^2/[(d+2)(d+4)].

Both are harmonic, and

u^2=H4+R H2/(d+4)+R^2/[d(d+2)].

Thus the full symmetric source requires one L=0 scalar block and the L=2 and L=4 two-function blocks: five radial functions. At d=2, H2 vanishes identically and that block is absent. This is a genuine low-depth boundary case, not a depth-five suppression. A one-axis trial basis cannot represent this full symmetry simply by adding radial powers.

Overlap Integrals Without Coordinate Expansion

For a harmonic polynomial define t_L as its squared mean over the unit sphere. Angular orthogonality makes different L blocks independent. In one block, the source/response overlap divided by the ball volume is

(d/2)t_L integral_0^1 R^(d/2+L-1)
  [R A F+L(A G+B F)+L(2L+d-2) B G/R] dR.

The gradient identity used here is mean_sphere |grad P_L|^2=L(2L+d-2)t_L. For the chosen second-mode polynomials,

t0=1,
t2=4(d-2)/[d^2(d+2)],
t4=4(d^2+4d+6)/[d(d+2)^2(d+4)^2(d+6)].

The mode's weighted norm divided by ball volume is Beta(d/2+2,3/2)/[2(d+2)]. The physical frequency correction still uses that mode's own C=mu_d^2 P_d'(lambda)/2 and amplitude squared a. Comparing shifts at common a differs from comparing them at common phase action omega a N0. Neither comparison can omit these factors.

Exact Positive Second-Mode Response

The clamped test displacement z=(1-R) grad H4 detects the complete second-mode source. Write a=d/2+4 and t4 for the angular norm above. Direct integration gives

<j,z>/Vol=-4d(d+2)t4 Beta(a,3/2),
a_el(z,z)/Vol=16d(d+14)t4/(d+8).

The first expression is strictly negative at every maintained depth, including d=2. The second is positive. Therefore the full compliance obeys

<j,L^(-1)j>/N0 >= <j,z>^2/[a_el(z,z)N0] > 0.

The exact lower bounds, in increasing maintained depth, are

Thus the earlier vanishing overlap with a purely radial trial does not make this mode mechanically unresponsive. Its degree-four angular component drives a nonzero response under the same elastic law. The longitudinal eigenvalue correction is negative for positive physical load C a, with its magnitude bounded below by C a times the displayed value. No independent amplitude or depth-specific coefficient has been chosen to obtain this sign.

Write W=sqrt(1-R), Q=u^2 and p=|grad u|^2-ell(2ell+d+1)Q. The source is -div(W p I)+W[grad p+ell grad Q], a derivative of a square-integrable tensor plus a square-integrable vector. It is therefore a bounded load on H1_0. Coercivity of the clamped elastic form gives existence and uniqueness of its weak response; this positivity argument does not depend on completing the radial series. It does not establish boundary admissibility of a finite physical load or the accuracy of a finite-basis approximation.

The checker check-harmonic-loaded-blocks.py verifies the common block operator by full coordinate polynomial differentiation at all six depths, the recurrence determinant, the harmonic decomposition, and the angular norms. A separate direct three-invariant moment calculation agrees with every trial load above.

This is a checked common reduction and a strict full second-mode response bound. It is not yet an explicit completed second-mode solution or a derived particle mass law.

Verification and Physical Scope

The exact recurrence and boundary argument give the full first-mode linear solution. The series checker uses Arb arithmetic and an analytic truncation bound at all six depths; it encloses negative parallel and positive transverse first-shell shifts. The harmonic checker verifies the full coordinate operator identities, angular moments and the strict second-mode lower bound. The polarization checker checks the tensor response identities, rotation covariance and all classical rotation charges at the same depths. The splitting-bound checker independently verifies the stress divergence and moment bounds.

The first-mode carrier now has an actual full response and a well-defined spectral perturbation, rather than only an integrated force cancellation. Its loaded shell splits. The leading reduced polarization action is adiabatic and small-load; it is not a globally bounded finite-amplitude completion. The full nonlinear carrier and wave equations, sustaining boundary stress, transverse localization in the containing space, physical scales and occupied modes remain to be derived. The parent high-order dispersion also retains its propagation limitation. These results do not identify an observed particle or its measured spin or charge, and the internal polarization motion does not replace an electron's literal orbit.