A zonal differential bridge with exact kinetic pullback

Two actual height derivatives connect three occurrences of the same simplex spectral polynomial. The derivative maps change the canonical norm. Their induced metric, or equivalently their normalized polar map, must be retained to transfer a frequency equation. This result concerns zonal scalar functions and the positive linear branch after specified null modes are removed.

The weighted-ball clock is a prescribed material construction in $d$ ordinary spatial coordinates. Its $S^{d+2}$ representation is an auxiliary mathematical lift in $d+3$ Euclidean coordinates, not a physical sphere within $\mathbb R^d$ or $\mathbb R^{10}$. The original bending comparison instead uses a surface $S^{d-2}$ embedded in $\mathbb R^d$. No sphere dimension is silently equated with its Euclidean embedding dimension, and no compactification or Kaluza--Klein interpretation is used. Spectral inner products below are integrals of classical spatial functions, not an ontology of physical states moving in a Hilbert space.

1. Operators, dimensions, and what was already known

Use unit radius initially and write m=d−2. On a sphere S^p, the nonnegative scalar Laplacian restricted to functions of height t is

$$L_p=-(1-t^2)\partial_t^2+pt\partial_t, \qquad w_p(t)=(1-t^2)^{(p-2)/2},\quad -1<t<1.$$

Its degree-k eigenvalue is k(k+p−1). The pushforward of sphere area is |S^(p−1)|w_p dt. We first use the unnormalized one-variable measure w_p dt, making all geometric constants explicit when restoring area below.

The three order-d scalar spectral operators are

$$J_dY_\ell=\frac{\Gamma(\ell+d-1)}{\Gamma(\ell-1)}Y_\ell \quad\text{on } S^{d-2},$$ $$P_dY_k=\frac{\Gamma(k+d)}{\Gamma(k)}Y_k \quad\text{on } S^d, \qquad Q_dY_N=\frac{\Gamma(N+d+1)}{\Gamma(N+1)}Y_N \quad\text{on } S^{d+2}.$$

For J, define the multipliers at ell=0,1 to be zero using the finite rising product; for P the constant multiplier is zero. Q is strictly positive. The maintained particle label is n=ell−1=k=N+1. No assignment is changed.

For even d>=4, J is the actual conformal bending Hessian of the old surface construction. The relevant theorem concerns even-dimensional submanifolds, and its Hessian energy prefactor is 1/m. Takeuchi, Theorem 1.1 and section 6 supplies that geometric input. The bending construction already derives its simplex spectrum and first-order normal time law. The critical P construction and its conformal recurrence were already established in critical sphere construction, using Graham's sphere factorization. The weighted-ball construction proves that Q/d! restricted to SO(3)-invariant functions on an auxiliary S^(d+2) is the weighted-ball clock T.

For odd d=3,5, we define J by its spectral multipliers on the standard smooth-sphere domain. Its explicit form is pseudodifferential. The even-dimensional bending-energy theorem does not establish an odd-dimensional curvature energy with this Hessian. The target Q has the standard subcritical Gamma multiplier, as in Jin–Li–Xiong, equations (2)–(4); the source J lies outside that paper's stated subcritical parameter range. No analytical continuation of a formula is silently promoted to a proved physical bending energy.

The additional result is the differential dimension walk with its exact kinetic pullback and the precise relation to the earlier tensor map. The standard Gegenbauer derivative identity itself is external mathematics, not a new discovery.

2. The actual differential intertwining

Let A=partial_t, with source and target dimensions indicated by its argument. Direct differentiation gives

$$A L_p=(L_{p+2}+p)A. \tag{1}$$

Thus with D=A²,

$$\boxed{D L_m=(L_{m+4}+2m+2)D.} \tag{2}$$

These are identities of differential expressions on arbitrary smooth functions, not only checks at eigenvalues. For m>1 the zonal harmonics are Gegenbauer polynomials with parameter alpha=(m−1)/2. The usual derivative formula, DLMF 18.9.19, applied twice gives

$$D C_\ell^{(m-1)/2} =(m-1)(m+1) C_{\ell-2}^{(m+3)/2}. \tag{3}$$

The degree drops by two while the sphere dimension rises by four. Each of the two derivative steps changes the conformal input weight by one: the source has weight −1, the middle weight 0, and the target weight 1, using the conventions Omega^alpha f∘C. The output weights are d−1, d, d+1 respectively. This is a relation between their zonal differential realizations; it is not an intertwiner between the full conformal groups of differently dimensioned spheres.

On the polynomial core, and then the appropriate spectral domains,

$$\boxed{A J_d=P_d A,\qquad A P_d=Q_d A,\qquad D J_d=Q_d D.} \tag{4}$$

For even d, (4) can be derived by shifting the Laplacian factors with (1); exact differential checks apply the complete operators directly to arbitrary monomials. For odd d, (4) follows by spectral calculus and (3), with the square root fixed by its nonnegative eigenvalues. Finite polynomial expansions are cores, so this specifies the closure rather than assuming unrestricted equality between unbounded operators on every L² function.

The critical operator therefore is a concrete middle member of the bridge. A bare observation that the three spectra are rising products omits precisely these maps and their norms.

3. Measure, adjoint, boundary terms, and null modes

The source and target zonal measures satisfy w_(m+4)=(1−t²)²w_m. Two integrations by parts yield

$$D^\dagger g=w_m^{-1}\partial_t^2(w_{m+4}g)$$ $$=(1-t^2)^2g''-2(m+2)t(1-t^2)g' +(m+2)[(m+1)t^2-1]g. \tag{5}$$

The boundary expression is

$$[\overline{f'}w_{m+4}g-\bar f\,\partial_t(w_{m+4}g)]_{-1}^{1}.$$

It vanishes for smooth zonal functions and polynomials when m>=1: w_(m+4) has exponent (m+2)/2>=3/2, so it and its first derivative vanish at the endpoints. The initial core is the smooth zonal sphere core, not an arbitrary singular extension at the coordinate endpoints. No extra Dirichlet wall is introduced.

Direct differential multiplication now gives

$$\boxed{D^\dagger D=L_m(L_m-m)=:B,} \tag{6}$$ $$\boxed{DD^\dagger=(L_{m+4}+m+2)(L_{m+4}+2m+2)=:K.} \tag{7}$$

On source degree ell, B has eigenvalue

$$b_\ell=\ell(\ell-1)(\ell+m-1)(\ell+m).$$

It vanishes precisely at degrees 0 and 1 and is strictly positive thereafter. On target degree N,

$$b_{N+2}=(N+1)(N+2)(N+d-1)(N+d). \tag{8}$$

This proves the complete norm identity

$$\int_{-1}^{1}|f''|^2w_{m+4}dt =\langle f,Bf\rangle_{w_m}. \tag{9}$$

In particular, differentiating a mode does not preserve its canonical amplitude norm. Constant and linear displacement modes are lost, and cannot be inferred from the target's nonzero constant mode. The target constant is the image of the source degree-two deformation.

Define D as the closed operator from H_m=L²(w_m dt) to H_(m+4) specified by its polynomial spectral core. Its domain is the set with sum b_ell |f_ell|² finite, including its two-dimensional kernel. Its polar factor

$$\boxed{U=DB^{-1/2}=K^{-1/2}D} \tag{10}$$

is unitary from H_m orthogonal to span{1,t} onto H_(m+4). The inverse square root is zero on the removed kernel. Surjectivity follows because (3) reaches every target polynomial degree, a complete orthogonal basis. Consequently UJ_d=Q_dU with equal spectral domains. This unitary operator is nonlocal; raw D is the local derivative but changes the time metric.

At unit radius, for actual sphere area use sqrt(|S^(m−1)|/|S^(m+3)|)U. Both measures in (9) were stated explicitly, so the missing area constants cannot be mistaken for derived physical normalizations. At radius R, use the dimensionless height t and multiply each order-d operator by R^(−d); the dimensionless identities remain unchanged. The physical coefficients, radius powers in area, and uncalibrated field amplitudes remain in the action.

4. What the bridge transfers in a linear action

In one zonal normal pair write w=u−iv as in the maintained bending note. Suppress the constant sphere area factor by including it consistently in the one-variable integral. Its declared equation is

$$i\dot w=c_d J_dw,\qquad c_d=\kappa/(m\Gamma)$$

at unit radius. Applying U gives exactly

$$i\dot z=c_d Q_dz,\qquad z=Uw. \tag{11}$$

U preserves the quadratic norm, the normal-pair symplectic form, and the quadratic Hamiltonian. Thus this is a canonical equivalence of the restricted linear actions, not merely an equality of frequencies. The ball clock normalization is Q=d!T, so matching its frequency unit requires mu_d=c_d d!, with R^(−d) restored. Nothing in U determines kappa/Gamma or relates different depths.

The raw derivative a=Dw also solves i dot a=c_d Q_da, but its inherited symplectic form contains K^(−1):

$$\langle w_1,w_2\rangle_{w_m} =\langle a_1,K^{-1}a_2\rangle_{w_{m+4}},$$ $$H^{(2)}=\frac{\kappa}{2m} \langle a,K^{-1}Q_da\rangle_{w_{m+4}}. \tag{12}$$

Writing $a_u=Du$, $a_v=Dv$, the inherited unit-radius quadratic action is explicitly

$$\mathscr L_D^{(2)}= \frac\Gamma2\left[ \langle a_u,K^{-1}\dot a_v\rangle- \langle a_v,K^{-1}\dot a_u\rangle\right] -\frac\kappa{2m}\sum_{\sigma=u,v} \langle a_\sigma,K^{-1}Q_da_\sigma\rangle.$$

All inner products in this expression use $w_{m+4}dt$, with the same area convention absorbed in the coefficients as in the source action. $K$ is strictly positive on the entire target, so its inverse is well-defined. Varying both real coordinates gives $i\dot a=\kappa Q_da/(m\Gamma)$; the factor $K^{-1}$ cancels between the time and restoring terms, rather than disappearing from the action. Under $z=Uw$, the corresponding action has identity time metric and restoring operator $Q_d$. A second-order source action instead pulls back to time kinetic norm $\langle\dot a,K^{-1}\dot a\rangle$ and restoring form $\langle a,K^{-1}Q_d^2a\rangle$ when its original stiffness was $J_d^2$.

This yields a useful lower-order potential representation. For d>=4 even, K divides Q exactly, so the transformed potential is an operator of order d−4:

At depth four the bending potential becomes an L² norm of the differentiated deformation; the entire restoring frequency structure is carried jointly by that potential and the inverse fourth-order time metric. This is a legitimate action rewriting. Replacing K^(−1) by identity would change the frequencies and is not a canonical simplification.

The same unitary map transfers a stipulated second-order wave equation w_tt+c_d²J_d²w=0 to z_tt+c_d²Q_d²z=0. It cannot turn a first-order symplectic time law into ordinary second-order kinetic dynamics by spatial differentiation. Conversely a conventional kinetic action with potential J would still yield sqrt(J) frequencies under U.

For the ball, restrict to f(x)=g(x_1). Its operator L_ball=−Delta+E²+(d+1)E reduces exactly to L_(d+2) on g. Integrating the other ball coordinates gives

$$\int_{B^d}\sqrt{1-|x|^2}|g(x_1)|^2dx =c_d^{\rm vol}\int_{-1}^{1}(1-t^2)^{d/2}|g(t)|^2dt,$$ $$c_d^{\rm vol}=\frac{\pi^{(d-1)/2}\Gamma(3/2)}{\Gamma((d+2)/2)}.$$

This supplies a literal one-coordinate ball realization of (11), after a constant norm conversion. It is one mode per degree inside the complete ball shell, not the whole d-dimensional ball mode space. The auxiliary S^(d+2), including S12 at d10, remains mathematical; no physical extra sphere or compactification is asserted.

5. Full nonzonal extension and its precise limitation

The covariant counterpart on S^m, for m>=2, is the traceless Hessian

$$\mathsf Hf=\nabla_a\nabla_bf-\frac{g_{ab}}m\nabla^2f.$$

The causal curvature parent, equations (3)–(5), already proves and checks

$$\mathsf H^\dagger\mathsf H=\frac{m-1}{m}L_m(L_m-m), \qquad (\nabla^*\nabla+2m)\mathsf H=\mathsf H L_m. \tag{13}$$

These hold for complete complex polarized harmonics, not just zonal functions. For a zonal f,

$$\mathsf Hf=f''\left[dt\otimes dt-\frac{1-t^2}{m}g\right],$$ $$|\mathsf Hf|^2=\frac{m-1}{m}(1-t^2)^2|f''|^2. \tag{14}$$

Equation (14) is the geometric reason that the dimension-walk norm produces the same fourth-order factor B as the full traceless Hessian. It connects the dimension-walk calculation to a previously established natural tensor structure.

On the scalar-derived tensor range, define L_T=nabla* nabla+2m (the round tracefree Lichnerowicz convention used here). If F_d is the source spectral function, then

$$\mathsf H J_d=F_d(L_T)\mathsf H.$$

Its normalized polar map is an isometry after removing scalar degrees 0,1. It maps every remaining scalar polarization into a constrained tensor field on the same S^m. The tensor range is not all symmetric traceless tensors. No map to complete scalar fields on S^(m+4), or to all ball fields, follows.

The obstruction is elementary and decisive. At N=0, the ball shell has dimension one. The source degree ell=2 has the following scalar multiplicities per normal direction:

For m>=2 the first column is m(m+3)/2; S1 has cosine and sine. Doubling real normal components on both sides does not cure the mismatch. An invertible spectral intertwiner must preserve these finite eigenspace dimensions, so no full unitary equivalence exists. Selecting a fixed height axis keeps one zonal member, makes (10) possible, and abandons complete physical rotation covariance as a scalar-to-scalar map. Allowing a different height axis for every pattern does not identify the full polarization spaces or their couplings.

6. The low-dimensional and odd-dimensional boundaries

At d=3, m=1, the naive Gegenbauer parameter is zero and C_ell^0 vanishes for ell>=1. Use the actual smooth reflection-even circle harmonics T_ell(t), where t=cos(theta). Then

$$D T_\ell=2\ell C_{\ell-2}^{2},\qquad J_3=\sqrt{L_1}(L_1-1).$$

All measure, adjoint, and unitary statements above remain valid on that reflection-even subspace, with w_1=(1−t²)^(−1/2). The full circle has an additional sine mode at each positive degree. Its traceless symmetric Hessian is identically zero in dimension one; equation (13) cannot be divided by m−1. Thus the zonal derivative walk survives while the natural rank-two tensor interpretation degenerates. This is a precise difference between d3 and d5, rather than an untested assertion that all odd depths behave identically.

At d=5,

$$J_5=\sqrt{L_3+1}\,L_3(L_3-3),$$

and the full traceless Hessian is nonzero. Both d3 and d5 have nonlocal positive scalar source and target clocks. Neither formula establishes a five-specific physical coupling suppression, fixed axis, or phase acceptance. Those require complete same-field dynamics.

At d=2 the putative source is S0, two points. Its area measure is atomic, not w_0 dt=(1−t²)^(−1)dt, which is nonintegrable. Harmonic homogeneous polynomials on its one-dimensional ambient line only have degrees zero and one. There is no bending harmonic tower at ell>=2 and no derivative D defined on arbitrary functions on two points. The formal m=0 differential expression is not this geometry and cannot be used as a physical continuation.

The target depth-two ball operator nevertheless remains valid: Q_2=L_4+2. The single derivative bridge from the actual critical sphere S2 survives:

$$\partial_t L_2=(L_4+2)\partial_t.$$

This offers a checked d2 middle-to-target identity; it does not fill the absent source bending construction or derive photon polarization.

7. Reusable physical equation and limits

A reduced normal-shape equation with the already declared bending time law can reuse (11) to calculate zonal count-clock evolution in ordinary ball coordinates, provided its canonical map and sphere/ball volume constants are retained. Equations (12) and the table give an alternative reduced action in a curvature-like coordinate with a lower-order potential and a nonlocal time metric. The full tensor representation (13) is available when all physical surface orientations and polarizations matter; it preserves the original harmonic representation rather than acquiring the ball multiplicities.

These results do not transfer the nonlinear bending energy or the ball's nonlinear stationary equation. For example,

$$\partial_t^2(f^3)=3f^2 f''+6f(f')^2$$

uses t as height, not time, and already shows why differentiating a local cubic does not preserve its form. A canonical transformed nonlinear functional is E[U^*z] with derivative U(delta E/delta w); it is generally spatially nonlocal. The discarded ell=0,1 modes may also be sourced nonlinearly, so their omission is a consistent linear restriction, not an automatic nonlinear truncation.

The surface-kinetic calculation remains binding: the complete normal-translation symplectic coefficient is a boundary current and cancels for the localized outer fields tested there. A mathematical unitary map cannot make that coefficient nonzero or supply the missing same-field support. Likewise, the older causal tensor parent has its own finite-gap corrections and full norm; the dimension walk does not remove them. No radius, occupied particle index, common cross-depth scale, complete inertia, or same-field interaction is selected here. Electron center trajectories remain separate from these classical deformation variables.

Radius, units and amplitudes

All differential maps above use the dimensionless height $t$. Time is the independent evolution variable in the dot notation; it is unrelated to that height. To restore a common physical radius, the order-$d$ multipliers become $J_{d,R}=R^{-d}J_d$, $P_{d,R}=R^{-d}P_d$ and $Q_{d,R}=R^{-d}Q_d$. Their eigenvalues have units $\text{length}^{-d}$. The source area measure is $R^m|S^{m-1}|w_mdt$; the target mathematical sphere area is $R^{m+4}|S^{m+3}|w_{m+4}dt$. A norm-preserving map between those two area conventions is therefore

$$U_R=R^{-2}\sqrt{\frac{|S^{m-1}|}{|S^{m+3}|}}\,U.$$

The $R^{-2}$ factor follows from their four-dimensional area difference. It is not needed when both one-variable measures are the dimensionless measures explicitly used in the derivation. The ordinary radius-$R$ ball norm of $g(x_1/R)$ instead has coefficient $R^d c_d^{\rm vol}$ in its one-variable integral. The norm-preserving source-sphere-to-ball map under these conventions is

$$U_{\Sigma\to B,R}=R^{-1} \sqrt{\frac{|S^{m-1}|}{c_d^{\rm vol}}}\,U, \qquad m=d-2.$$

These constants convert measures of different-dimensional classical fields; they do not identify their physical amplitudes. Any physical reduction must retain the actual field dimensions, area factors and kinetic coefficient. For the declared bending normal-pair action, $\kappa/(m\Gamma)$ has units $\text{length}^{d}/\text{time}$, and matching the dimensionless ball law requires

$$\mu_d=\frac{\kappa d!}{m\Gamma R^d}.$$

That is a conditional matching relation, not a derived common sector scale or a radius determination. With normalized one-variable measures the raw derivative kinetic metric is still $K^{-1}$; changing a constant integration convention never removes its mode dependence. Classical field amplitudes and the origin of a nonzero $\Gamma$ remain uncalibrated. The full localized outer-field cancellation in the surface-kinetic source cannot be repaired by a spatial isometry.

Verification and sources

The direct differential identities, weighted adjoint products and complete-basis polar decomposition establish the linear result. The existing exact checker records 676 finite conditions, including monomial operator applications, integrated adjoints, Gegenbauer and Chebyshev derivatives, odd square-root branches and the depth-two middle-to-target exception. That diagnostic count is not itself proof and is not a stability simulation. The portable checker has also been executed with the same 676 exact conditions and its own recorded source hash.

The Gegenbauer derivative is the standard identity DLMF 18.9.19. The actual even-surface bending Hessian is from Takeuchi, Theorem 1.1 and section 6. Sphere conformal factorization is from Graham, Proposition 1; target subcritical Gamma multipliers are from Jin, Li and Xiong, equations (2)--(4). The odd source $J_d$ is a declared spectral extension; that last reference does not prove an odd-dimensional bending energy. The full same-surface tensor identities are already in the causal curvature parent, and are not a new full nonzonal scalar-to-ball equivalence.

The portable zonal dimension-bridge checker preserves the original symbolic checks. Its dated result retains the exact executed portable-source hash.