An exact flat-ball count clock and its conformal characterization

A scalar classical field on an ordinary spatial ball can have both the maintained cumulative mode count and the corresponding simplex frequency law. The construction below is inverse: the ball, its density and stiffness family, and the relation between its spatial clock and time evolution are specified. Requiring the exact infinite squared-count law then fixes the density exponent within that family. The conformal characterization supplies a second constraint and useful static and nonlinear identities.

The active depth is $d\in\{2,3,4,5,6,10\}$ in one infinite-dimensional ordinary spatial containing space. The ten-dimensional matter domain is currently modeled by $\mathbb R^{10}$; all remaining spatial directions are unbound for every particle. A depth classifies a pattern's dimensional involvement; it is not an independent physical sector manifold. The auxiliary spheres below are mathematical lifts with explicitly counted extra coordinates. They are not physical embeddings in that containing space and introduce no compactification or Kaluza--Klein tower. Spatial field profiles and spectral coefficients describe classical internal structure; they do not replace an electron's deterministic six-dimensional trajectory or define a position-probability law.

1. Carrier and material operator in ordinary coordinates

Let x belong to the unit ball B^d in ordinary R^d. The containing working physical space is flat R10; d labels the active pattern directions, not an independently existing sector. Set r^2=x dot x, E=x dot grad, and

$$w_a(x)=(1-r^2)^a,\qquad A(x)=I-xx^T,\qquad a>-1.$$

The density w_a and stiffness tensor w_a A are specified material coefficients of the trial pattern. They do not change the meaning of physical position or make the containing Euclidean metric curved. A has tangential eigenvalue 1 and radial eigenvalue 1-r^2. Define

$$L_a=-w_a^{-1}\nabla\cdot(w_a A\nabla) =-\Delta+E^2+(d+2a)E. \tag{1}$$

The differential identity follows from div(A grad)=Delta-E^2-dE and A grad log w_a=-2a x. This is a d-variable operator, not a sphere S^d embedded in d coordinates. Its actual boundary is S^(d-1).

For a=1/2, the outward polynomial flux is proportional to (1-r^2)^(3/2) partial_r f and vanishes. Thus for finite polynomials

$$\langle f,Lg\rangle_w=\int_{B^d}w\,\nabla f^*\cdot A\nabla g\,d^dx, \qquad L=L_{1/2},\quad w=\sqrt{1-r^2}. \tag{2}$$

This establishes symmetry and nonnegativity with the stated weighted kinetic inner product. Multiplying w by a constant changes neither L nor its frequencies; that constant is not a chosen physical field amplitude.

The standard ball polynomial equation and its separated basis are recorded in Piñar and Xu, Orthogonal polynomials and partial differential equations on the unit ball, equations (1.1) and (2.2). Here their mathematics is used as a spatial carrier; it supplies no IDWT material mechanism.

2. Spectrum and exact radial/angular count

Let V_N be polynomials of total degree N orthogonal to all lower degrees in L2(B^d,w_a dx). Equation (1) preserves total degree and has leading-degree action N(N+d+2a). Symmetry makes the lower-degree residual orthogonal to every lower-degree polynomial. The residual is itself lower degree, so it vanishes. Therefore

$$L_a f_N=N(N+d+2a)f_N,\qquad \dim V_N=\binom{N+d-1}{d-1}. \tag{3}$$

Finite polynomials are dense for the positive integrable ball weight. The diagonal spectral closure on this complete orthogonal basis defines the self-adjoint operator used here. Its domain has sum_N lambda_N^2 ||f_N||_w^2 finite. The squared wave action below has potential form domain $\mathcal D(T)$, specified explicitly below. An unweighted hard-wall Laplacian is not being substituted for this domain.

An explicit basis has angular degree ell and radial index j:

$$f_{j,\ell,\nu}(x)=P_j^{(a,\ell+(d-2)/2)}(2r^2-1)H_{\ell,\nu}(x), \qquad N=2j+\ell, \tag{4}$$

where H is a homogeneous harmonic polynomial in the d physical coordinates. Its angular multiplicity is

$$h_d(\ell)=\binom{\ell+d-1}{d-1}-\binom{\ell+d-3}{d-1}.$$

Negative-upper binomial terms are zero. This convention handles d=2: h_2(0)=1 and h_2(ell)=2 for ell>0. The independent radial channels are essential:

$$\sum_{j=0}^{\lfloor N/2\rfloor}h_d(N-2j)=\binom{N+d-1}{d-1}, \quad D_N=\sum_{k=0}^N\dim V_k=\binom{N+d}{d}=S(N+1,d). \tag{5}$$

An angular shell or a single radial family does not have this count. Neither does the complete scalar spectrum of a full S^d, which has a different harmonic multiplicity and requires d+1 Euclidean embedding directions as an embedded shape.

With a=1/2, lambda_N=N(N+d+1). In the reduced radial coordinate s=r^2, the actual differential action on H_ell h(s) is H_ell times

$$-4s(1-s)h''+[(4\ell+2d+6)s-(4\ell+2d)]h' +\ell(\ell+d+1)h. \tag{6}$$

This expression, rather than integer arithmetic alone, was used in the exact frequency tests.

3. A local positive wave action turns the count into frequency

Put c=(d+1)/2. The identity

$$(N+j)(N+d+1-j)=\lambda_N+j(d+1-j)$$

pairs the desired rising factors. Define the positive polynomial

$$P_d(z)=\frac1{(d!)^2} \begin{cases} \displaystyle\prod_{j=1}^{d/2}[z+j(d+1-j)]^2,&d\text{ even},\\ \displaystyle(z+c^2)\prod_{j=1}^{(d-1)/2}[z+j(d+1-j)]^2,&d\text{ odd}. \end{cases} \tag{7}$$

Then

$$P_d(\lambda_N)=\binom{N+d}{d}^2. \tag{8}$$

For a real scalar field, take the explicit action

$$\mathcal A[f]=\frac12\int dt\int_{B^d}w \left[(\partial_t f)^2-\mu_d^2 fP_d(L)f\right]d^dx. \tag{9}$$

For multiple real field components, sum this same expression without changing the spatial count per component. Fully varying f on the stated domain gives

$$\partial_t^2 f+\mu_d^2P_d(L)f=0,\qquad \omega_N=\mu_d\binom{N+d}{d}=\mu_d S(n,d),\quad n=N+1. \tag{10}$$

This is the required count-to-frequency equation, not an identification of a degeneracy with frequency under the ordinary second-order wave law. That ordinary law would give omega proportional to sqrt(lambda_N), which is different.

Every coefficient b_k of P_d(z)=sum b_k z^k is positive. The potential term of (9) can be written as a sum of positive local derivative energies: even k=2m contributes b_k ||L^m f||_w^2, and odd k=2m+1 contributes b_k integral w grad(L^m f) dot A grad(L^m f). This gives a local energy density with at most d spatial derivatives, an equation of spatial order 2d, and nonnegative conserved complete energy within the prescribed medium.

The per-depth squared stiffnesses are:

At d=6, N=12 and 34 give 18564 and 3838380. At d=10, N=22 gives 64512240. Here $S(n,d)=\binom{n+d-1}{d}$ and $n=N+1$. These indices are input comparisons, not selected occupations. Because (10) is strictly increasing with N, the operator's own cumulative scalar count at omega_N is exactly omega_N/mu_d. The complete IDOS and frequency requirement holds together on this carrier.

For odd d, the positive first-time-order scalar frequency operator contains sqrt(L+c^2), and is nonlocal. The second-time-order action (9) is local at odd and even d. Depth five shares this parity feature with depth three; it supplies no five-specific suppression or cancellation.

4. What the inverse requirement does select

Keep the material family w_a=(1-r^2)^a and A=I-xx^T. Suppose the entire squared simplex law is a polynomial in L_a. Its degree must be d, since lambda_N is quadratic and the target product has degree 2d in N. Comparing the next highest coefficient gives

$$d(d+2a)=d(d+1),\qquad a=\frac12. \tag{11}$$

Thus the density exponent is unique inside this stated family if the exact infinite squared-count law and polynomial locality are required. Equations (7) and (11) do not derive that constitutive family, the occupied indices, or sector units. They are a concrete constraint produced by the inverse calculation, rather than freely choosing a new coefficient for each particle.

5. Physical measure, radius and boundary cost

For radius R and center X, replace x by y=(x-X)/R and use

$$L_R=-R^2w_R^{-1}\nabla_x\cdot[w_R(I-yy^T)\nabla_x]. \tag{12}$$

The eigenvalues of L_R remain dimensionless. The weighted volume is

$$\int w_R\,d^dx=R^d\pi^{d/2}\frac{\Gamma(3/2)}{\Gamma((d+3)/2)}.$$

The scale mu_d has dimensions of frequency and is independent of R unless a physical constitutive relation supplies one. Setting mu_d=c_wave/R is possible as an additional common material prescription, not a consequence of counting.

The weighted kinetic term can instead use ordinary Lebesgue measure by the actual field redefinition u=w_a^(1/2)f. Its spatial operator is

$$H_a=w_a^{1/2}L_aw_a^{-1/2} =-\nabla\cdot[(I-xx^T)\nabla]-ad+\frac{a^2r^2}{1-r^2}. \tag{13}$$

For a=1/2 the spatial profiles are u=(1-r^2)^(1/4)f_N. They vanish continuously at the boundary and have finite norm and the transformed action energy. Their ordinary first derivatives generally diverge too strongly for an unweighted uniform gradient energy. The singular potential and degenerate stiffness are indispensable here; this is not a regular compact-support solution of the previously written Dirac action. Under either field convention, the finite-mode boundary terms vanish under its stated action.

The elementary N=1 projector also shows why raw contact remains separate. With weight normalized to unit integral, mean x_i^2=1/(d+3), so

$$K_1(x,x)=1+(d+3)r^2,\qquad \int K_1w_{\rm normalized}\,dx=d+1.$$

The integrated trace is the count; local contact depends on position.

6. Exact surviving claim and missing physics

The surviving claim is an explicit positive local scalar wave action on a prescribed d-dimensional ball in ordinary flat coordinates, having the exact maintained cumulative mode count and exact simplex frequency law simultaneously for every N>=0 at all six maintained depths. It is an inverse spatial constitutive model. It improves the oscillator's squared-action spatial order from 4d to 2d, but does not derive the master law.

Its interior high-wave-number dispersion scales omega approximately |k|^d, so group speed is unbounded for d>1. Positivity and second order in time do not establish finite propagation speed. A low-frequency effective use would require a demonstrated regime and a causal completion; the exact infinite ladder cannot simply be asserted for a bounded-speed first-order parent.

The support radius, position, radial material coefficients, moving support and sustaining stress remain prescribed. Deriving them as responding components or bilinears of Psi_infinity is the next physical task. Their omitted dynamics may alter both normalization and spectrum. No mediator field or independent particles have been introduced.

The scalar construction supplies spatial shape degeneracies, not spin, vector polarization, electric charge or a photon zero. Its lowest frequency is mu_d at every d. A physical photon remains a separate control to realize. Arbitrary amplitudes are allowed and do not change (10), while total energy is quadratic in amplitude; no amplitude is fitted to a mass. Complete translation inertia and the mass-energy consistency check require a responding full action.

To prepare a finite field in R10 when d<10, a normalized transverse profile may multiply the ball mode, but its support and derivatives must enter a completed action. A law blind to those coordinates leaves an unlimited spectator degeneracy. Uniform unused directions instead give an infinite-volume norm. This construction therefore supplies the active d-direction carrier, not a complete finite-energy R10 particle. It does not replace deterministic electron center trajectories with scalar waves.

7. Auxiliary reduction with the correct dimension and measure

For this characterization write $\rho=w=\sqrt{1-|x|^2}$, $d\mu=\rho\,d^dx$, and define the positive spectral clock $T=\sqrt{P_d(L)}$, so $T|_{V_N}=D_N$. The notation $\mu$ for a spatial measure is distinct from the angular-frequency coefficient $\mu_d$.

Set m=d+2 and introduce an auxiliary mathematical sphere

$$S^m=\{(x,y)\in\mathbb R^d\times\mathbb R^3:|x|^2+|y|^2=1\}.$$

SO(3) rotates y while fixing x. For |x|<1 write y=rho u with u in S². The induced sphere metric, used only in this calculation, is

$$g_S=h+\rho^2g_{S^2},\qquad h_{ij}=\delta_{ij}+\frac{x_i x_j}{\rho^2},\qquad h^{-1}=A,\quad \det h=\rho^{-2}.$$

The fibre contributes rho² to volume while the base contributes rho^(−1), hence

$$d\mathrm{vol}_{S^m}=\rho\,d^dx\,d\Omega_2.$$

For an invariant function F(x,y)=f(x), the Laplace–Beltrami divergence formula therefore gives exactly

$$-\Delta_{S^m}F=(Lf)\circ\pi.$$

Thus f -> (4pi)^(−1/2)f∘pi is a unitary map from L²(B^d,mu) onto the SO(3)-invariant subspace of L²(S^m). The factor is only an integration convention, not a physical field amplitude. No KK expansion or fibre occupation is used. Invariant functions have no auxiliary angular dependence at all.

The volume of the ball weight is consequently |S^(d+2)|/(4pi), equivalently

$$\int_{B^d}\rho\,dx= \frac{\pi^{d/2}\Gamma(3/2)}{\Gamma((d+3)/2)}.$$

The same proof works with q auxiliary Cartesian coordinates and SO(q), q>=2: the sphere dimension is d+q−1 and the ball weight is rho^(q−2). Therefore a=1/2 in w=(1−r²)^a corresponds to q=3. This is an interpretation of the material exponent, not evidence that three additional physical directions exist.

8. Why the invariant multiplicity is exactly the ball multiplicity

A homogeneous SO(3)-invariant polynomial of total degree N is a sum of terms |y|^(2j) h_(N−2j)(x). The dimension of that invariant polynomial space is

$$I_N=\sum_{j=0}^{\lfloor N/2\rfloor}\binom{N-2j+d-1}{d-1}.$$

The ordinary harmonic/Fischer decomposition commutes with SO(3). Removing multiples of |x|²+|y|² times degree-N−2 polynomials leaves harmonic invariant dimension

$$I_N-I_{N-2}=\binom{N+d-1}{d-1}.$$

The full sphere harmonic multiplicity would instead be binomial(N+d+2,d+2)−binomial(N+d,d+2), so retaining the invariant subspace is essential. It is not permissible to count all sphere modes and call them d-dimensional ball modes.

A constructive lift verifies that this is an operator identity, not just count matching. Given any homogeneous h_N(x), the polynomial

$$H_N(x,y)=\sum_{j=0}^{\lfloor N/2\rfloor} \frac{(-1)^j|y|^{2j}\Delta_x^j h_N(x)} {4^j j!(3/2)_j}$$

is harmonic in the auxiliary ambient coordinates. The coefficient recursion follows from

$$\Delta_y |y|^{2j}=2j(2j+1)|y|^{2j-2}.$$

Restricting |y|²=1−|x|² gives a ball polynomial with L eigenvalue N(N+d+1). The checked examples use h_N=x_1^N and verify both the ambient harmonic equation and the actual mixed-coordinate ball differential equation.

The sphere's standard self-adjoint spectral closure restricts to precisely the polynomial-domain ball closure. In spectral notation,

$$\mathcal D(L)=\{f:\sum_N[N(N+d+1)]^2\|f_N\|_\mu^2<\infty\},$$ $$\mathcal D(T)=\{f:\sum_N D_N^2\|f_N\|_\mu^2<\infty\},\qquad \mathcal D(T^2)=\{f:\sum_N D_N^4\|f_N\|_\mu^2<\infty\}.$$

The potential form of the squared wave action is ||Tf||² and has form domain D(T). This fixes the realization without adding an unrelated hard-wall Dirichlet condition. Constants are in all three domains. Smooth invariant lifts have the appropriate regularity at the collapsing fibre; for polynomials their normal weighted flux rho³ partial_r f vanishes at the boundary.

The boundary-domain choice is substantive. If t=1−r, the leading radial expression is −2t partial_t²−3 partial_t. Its two zero-energy asymptotic solutions are a constant and t^(−1/2). Both are locally square integrable against t^(1/2)dt, but the singular branch has infinite quadratic L energy: t^(3/2)|partial_t f|² behaves as t^(−3/2). The regular invariant lift therefore agrees with the finite-energy/Friedrichs selection; interior differential coefficients alone do not authorize arbitrary boundary extensions. Constants can be approximated by smooth interior cutoffs with boundary energy tending to zero as the square root of their layer thickness.

9. Identification of the conformal clock and the local square

On S^m put

$$B=\sqrt{-\Delta_{S^m}+(m-1)^2/4},\qquad Q_{2s}=\frac{\Gamma(B+1/2+s)}{\Gamma(B+1/2-s)}.$$

Take m=d+2 and 2s=d. This is subcritical by two orders relative to the auxiliary sphere dimension. It is not the critical order-d operator on S^d. On degree N,

$$Q_d=\frac{\Gamma(N+d+1)}{\Gamma(N+1)}=d!D_N, \qquad T=\frac1{d!}Q_d\big|_{SO(3)}.$$

After restriction B=sqrt(L+(d+1)²/4). Pairing the rising factors gives

$$T=\frac1{d!} \begin{cases} \prod_{j=1}^{d/2}[L+j(d+1-j)],&d\text{ even},\\ \sqrt{L+(d+1)^2/4}\prod_{j=1}^{(d-1)/2}[L+j(d+1-j)],&d\text{ odd}. \end{cases}$$

Thus T² is exactly the polynomial P_d(L) of the ball wave construction above. The positive scalar clock T is local at even d and nonlocal at odd d. Its square is local at all tested depths. A local isotropic scalar operator cannot have a nonzero odd-order principal symbol proportional to |xi|^d; the square root is not removable by merely changing notation.

Depth five shares the nonlocal first-time-order scalar clock feature with depth three. No five-specific coupling suppression, axis protection or cancellation follows from it.

10. A direct ball symmetry and a conditional uniqueness theorem

The full auxiliary-sphere conformal law has input weight alpha=(m−d)/2=1 and output weight beta=(m+d)/2=d+1. There is a subgroup preserving the SO(3)-invariant functions. Its infinitesimal ball generators are

$$V_i=(\delta_{ij}-x_i x_j)\partial_j,\qquad \operatorname{div}_{\mu}V_i=-(d+2)x_i.$$

For a boost in the x_1 direction, with |a|<1,

$$C_a(x)_1=\frac{x_1+a}{1+a x_1},\quad C_a(x)_\perp=\frac{\sqrt{1-a^2}x_\perp}{1+a x_1},\quad \Omega_a=\frac{\sqrt{1-a^2}}{1+a x_1}.$$

These identities hold in the ordinary ball coordinates:

$$\rho(C_a x)=\Omega_a\rho(x),\qquad C_a^*d\mu=\Omega_a^{d+2}d\mu,$$ $$T(\Omega_a f\circ C_a)=\Omega_a^{d+1}(Tf)\circ C_a.$$

These are weighted material symmetries. The map need not be a conformal map of the physical Euclidean metric. The auxiliary metric h transforms conformally; none of that requires physical space to inherit h.

Conditional theorem. Suppose an operator acts on every complete polynomial degree space V_N as a scalar t_N, has t_0=1, and obeys the displayed weighted conformal covariance on the polynomial core infinitesimally. Then it is T on that core and on the corresponding spectral closure.

Indeed the infinitesimal law is

$$[T,V_i]=-(d+1)x_iT+Tx_i.$$

For f_N in V_N, project to V_(N+1). The raising part of V_i f_N is −N times that of x_i f_N, since V_i=partial_i−x_iE and lower-degree terms cannot reach V_(N+1). At least one such raising part is nonzero. Hence

$$-N(t_{N+1}-t_N)=-(d+1)t_N+t_{N+1},$$ $$(N+1)t_{N+1}=(N+d+1)t_N.$$

Iteration yields t_N=binomial(N+d,d), including N=0. Exact differential checks also verify the full commutator, including lowering terms, on Gegenbauer polynomials at all six depths. Existence is supplied by the preceding sphere formula; the recurrence establishes uniqueness inside the specified class.

There are three limits to this theorem.

1. Rotation covariance on the ball alone does not require a scalar on all of V_N: its radial/angular pieces form several rotation representations. The spectral-function premise T=F(L), or a stronger auxiliary extension requirement, matters.
2. The ball subgroup covariance alone does not select the domain or operator. For example, multiplication by rho^(−d) satisfies the same finite covariance on smooth compactly supported interior functions. It does not preserve the polynomial spectral core and its constant input fails the required L² operator domain. This explicit alternative shows why formal interior covariance is insufficient.
3. Even the full normalized auxiliary conformal law does not establish that the physical same-field action has that symmetry. Adopting it would be a compact constitutive premise with predictive spectral consequences, not a completed material derivation.

The general q-coordinate construction makes the inverse selection transparent. Its conformal input weight is b=(q−1)/2 and its spectrum is Gamma(N+d+b)/Gamma(N+b). Its normalized recurrence is (N+d+b)/(N+b). Requiring the unshifted cumulative ball count gives b=1, hence q=3 and a=1/2. This agrees with the ball wave construction above's independent polynomial-locality coefficient comparison. Neither proof selects the constitutive family itself.

11. An explicit logarithmic static inverse in physical ball coordinates

A useful consequence of the auxiliary characterization is a closed ball kernel. Write dbar(mu)=dmu/volume(mu). With this normalization T1=1 remains unchanged. The normalized inverse on the auxiliary sphere has kernel

$$\frac{2d}{d+1}|X-Z|^{-2}.$$

The power is two because m−d=2. The prefactor is fixed by mapping the constant function to itself; the exact beta integral gives average |X−Z|^(−2)=(d+1)/(2d).

For ball points x,z define

$$a=1-x\cdot z,\qquad b=\rho(x)\rho(z).$$

Fixing one auxiliary fibre direction, the other has dot product t uniformly distributed on [−1,1], since the fibre is S². Since |X−Z|²=2(a−bt), elementary integration gives

$$\boxed{G_d(x,z)=\frac{d}{2(d+1)b} \log\frac{a+b}{a-b}},\qquad \boxed{T^{-1}f(x)=\int_{B^d}G_d(x,z)f(z)\,d\bar\mu(z)}.$$

For b=0 and a>0 use the continuous limit d/[(d+1)a]. For distinct interior points a>b>0. The diagonal logarithmic singularity is locally integrable. The kernel is positive and symmetric. Its normalization is verified independently by exact spherical beta moments, which reproduce inverse eigenvalues 1/D_N through N=12 at every tested depth. The derivation, not a truncation, establishes the complete operator identity.

This formula is a static response of the specified spatial law. No mediator field or particle is introduced. The static response of the actual squared-stiffness wave equation is T^(−2), with kernel obtained by composing G_d with itself; G_d alone must not be substituted for that response. A derivation from eliminating modes of Psi_infinity would still need their action and kinetic normalization.

12. Nonlinear equations that transfer, and those that change the clock

The same covariance weights make the stationary equation

$$Tu=\kappa |u|^d u$$

covariant under the stated ball group. For positive u its power is d+1. Its variational functional on real fields is

$$\mathcal F[u]=\frac12\int uTu\,d\bar\mu -\frac\kappa{d+2}\int |u|^{d+2}\,d\bar\mu.$$

Thus a cubic field equation is selected only for d=2. A cubic term at every depth would be an additional, different interaction. For odd d the sign-preserving real extension uses |u|^d u; the positive-field integer power must not silently change its signed behavior.

For kappa=1 a smooth positive family is

$$u_a(x)=\frac{\sqrt{1-|a|^2}}{1-a\cdot x},\qquad |a|<1, \qquad Tu_a=u_a^{d+1}.$$

It follows by applying covariance to T1=1. Every member is smooth on the closed ball for |a|<1 and lies in all spectral domains. At even d, direct differential checks verify two nonconstant rational examples by direct differentiation through the complete product T. The odd-depth result follows from the same exact conformal law, not from replacing its square root by a local operator.

These solutions do not supply the ball's supporting material: rho and A remain prescribed. They deform the field profile inside that material, and become concentrated toward the boundary as |a| approaches one. They neither select a particle radius nor select a simplex harmonic degree.

The sharp quadratic inequality restricts to this invariant family as

$$\left(\int |u|^{d+2}\,d\bar\mu\right)^{2/(d+2)} \leq\int uTu\,d\bar\mu.$$

Constants and the displayed conformal family saturate it. Hence they are constrained minimizers of this quotient. This is distinct from an unconstrained positive potential or full dynamical stability. At u=1 and kappa=1, the Hessian of F is

$$B=T-(d+1)I.$$

It has one negative constant direction, d zero degree-one directions, and strictly positive higher shells. The first three shell coefficients are −d, 0, d(d+1)/2. A fixed L^(d+2) constraint removes the constant tangent direction but leaves conformal neutral directions. These properties are shared systematically across all six depths.

There is an essential time-law distinction. With ordinary quadratic time kinetic energy and potential F, the linear frequency at u=0 is proportional to sqrt(D_N), not D_N. The original wave action uses T² and has the count clock, but T² is not a conformal intertwiner of the same type: the output weight of the first T does not match the input weight of the next T. It is not equal to the sphere's order-2d GJMS operator. Thus conformal covariance selects the spatial clock T only after the relation between the clock and time dynamics is stated.

There is also a short exact check of this failure. An order-2d conformal operator on S^(d+2) would have input and output weights alpha′=(2−d)/2 and beta′=(3d+2)/2. Applying its infinitesimal law to the constant would require alpha′(d+1)²−beta′=0 for T². The actual residual is −d³/2, nonzero at every tested depth.

For an explicit changed nonlinear candidate, the nonnegative functional

$$\mathcal E_{\mathrm{sq}}[u]=\frac12\|Tu-\kappa |u|^d u\|_\mu^2$$

has Hessian T² at zero and therefore retains the original linear count clock with ordinary quadratic time kinetics. On a positive field, its Euler derivative is

$$\big[T-\kappa(d+1)u^d\big]\big[Tu-\kappa u^{d+1}\big].$$

At its conformal stationary background u=1, kappa=1, the Hessian is [T−(d+1)]², so its perturbation frequencies are proportional to |D_N−(d+1)|, not D_N. It is local at even depth. At odd depth the linear T² term is local but the nonlinear cross term involving T remains nonlocal. This is a concrete transferable action with explicit changed properties; it is not proposed as already-derived self-support.

Likewise, if T acts on a real spinor bilinear J in an interaction eta/2 integral J T J, variation gives eta(TJ) beta psi, not T psi. Here T1=1, unlike the old critical operator's zero constant; constant-current contributions therefore survive. That is a useful difference to test in a future complete same-field reduction, but it does not turn the scalar result into the standard spinor mass operator.

Material backreaction and physical normalization

Prescribed-coefficient eigenfunctions remain exact under the action above. Promoting those coefficients to responding field components requires their full local Euler equation. A useful interior diagnostic already exposes the missing load. For a carrier $\phi=x$, take $w(\phi)=\sqrt{1-|\phi|^2}$ and $A(\phi)=I-\phi\phi^T$, with $u=x_1$ and $\lambda=d+2$. Varying the time term, density and the complete polynomial stiffness gives the local carrier source

$$J_a=\frac{\mu_d^2P_d'(d+2)}2 \left[\frac{x_a[1-(d+3)x_1^2]}{w} +2w\delta_{a1}x_1\right].$$

At $x_1=0$, $x_2=r\in(0,1)$, this has $J_2=\mu_d^2P_d'(d+2)r/(2\sqrt{1-r^2})\ne0$. Cancellation after integration against constant carrier translations therefore does not establish pointwise stationarity. No same-field support claim follows from the exact polynomial spectrum.

Here $\omega$ and $\mu_d$ are angular frequencies; a cyclic-frequency convention changes both by $2\pi$. The dimensionless $T$ and $P_d(L)=T^2$ do not fix a length. An order-$d$ dimensionful conformal operator at radius $R$ would be $R^{-d}d!T$; matching it to the dimensionless ball clock requires its dimensional coefficient explicitly. The count-to-frequency law does not independently select $R$, $\mu_d$, or their relation. Overall classical wave amplitude remains uncalibrated: scaling a solution changes its quadratic energy while preserving this linear frequency. Mode normalization and normalized integration measure do not calibrate that physical amplitude.

The common action on shared R10 is a separate inverse realization with responding global moments, different spatial profiles and its own localization theorem. Its complete center dynamics and bounded localization must not be transferred to this prescribed-ball action. Conversely this ball's conformal characterization does not establish the other action's local material or propagation law.

The responding carrier continuation solves the full linear first-mode clamped elastic response at all six maintained depths. Its boundary bounds preserve the weighted spectral domain for small finite loads. The same response determines first-shell splitting and a leading polarization-dependent energy at fixed phase action. The nonlinear coupled carrier remains open, and its spectral shifts do not preserve the exact undeformed ladder automatically.

A distinct complete material pullback has exact joint stationary wave/carrier solutions with the count clocks. The Jacobian and inverse deformation gradient are transformed together; this is a changed constitutive law. Balanced component quadratures cancel the local stationary carrier load, while general motion remains coupled. The complete ground-energy bound and internal mode-conversion rules follow under the clamped active-ball assumptions. Free exterior support remains open.

Verification and sources

The analytic polynomial-domain argument, factor pairing, invariant lift, conformal recurrence and fibre-integrated kernel establish the stated identities. Finite exact diagnostics accompany those proofs: the ball checker records 1,254 checks and the conformal checker 713. These counts describe tested conditions, not a proof or physical validation by themselves. The executed scripts and results are preserved separately; the portable versions have also been executed with matching exact check counts and their own recorded source hashes.

The ball polynomial equation and separated basis use Piñar and Xu, equations (1.1) and (2.2). The even-order sphere factorization uses Graham, Proposition 1. General-order sphere multipliers, the Riesz inverse and the conformal nonlinear exponent use Jin, Li and Xiong, equations (1)--(7). These sources supply mathematics, not IDWT physical premises. The zonal dimension bridge explains the exact relation to the earlier bending and critical-sphere operators, including the kinetic metric and unequal complete multiplicities.

Portable calculation sources are the ball differential checker and conformal characterization checker. Their dated results retain the exact executed portable-source hashes.

The IDOS energy and coherence continuation derives the exact filled-ball trace, distinguishes it from coherent local energy, and tests conservative equal-energy occupations with their supporting dynamics. Its additional modal and scale actions retain their own stated assumptions.