A conformal spatial operator with the simplex spectrum

Summary: The critical conformal scalar operator on a round even-dimensional sphere has exactly the normalized simplex spectrum, while its spinor realization, spatial geometry and master interaction remain additional requirements.

An exact geometric occurrence of the mass-count polynomial

There is a known natural differential operator whose normalized eigenvalues are exactly

$$S(n,d)=\binom{n+d-1}{d}. \tag{1}$$

For even $d$, it is the critical conformal scalar operator on a round $d$-sphere. The result follows from spatial conformal covariance and can also be verified by direct differentiation. It gives a distinct mathematical origin for the same sequence used in the mass-spectrum construction.

This calculation specifies a round physical spatial geometry $S_R^d$, with undetermined radius $R$. Its directions are ordinary macroscopic spatial directions; there is no compactification or hidden small radius. Spherical harmonic coefficients do not add dimensions.

The operator acts on scalar spatial functions. Its appearance in the master spinor interaction, and the identification of its harmonic degree with an occupied particle mode, must be derived separately.

Conformal covariance fixes the recurrence

Work first at unit radius. Let $P$ commute with round-sphere rotations, and let its scalar eigenvalue on the complete degree-$\ell$ harmonic space be $p_\ell$. Rotation covariance alone leaves those eigenvalues free. Impose the stronger critical conformal law

$$P(C^*f)=\Omega^d C^*(Pf),\qquad C^*g=\Omega^2g. \tag{2}$$

This is the scalar law with input conformal weight zero and output weight $-d$. Its infinitesimal form, for $\mathcal L_Vg=2\omega g$, is

$$[P,V]f=d\omega Pf. \tag{3}$$

Choose the height coordinate $t=X_{d+1}$ on the unit sphere and $V=\nabla t$. Since $\nabla^2t=-tg$, its conformal factor is $\omega=-t$.

For $\ell\ge1$, the complex harmonic $H_\ell=(X_1+iX_2)^\ell$ satisfies

$$VH_\ell=-\ell\,tH_\ell. \tag{4}$$

The ambient polynomial $tH_\ell$ is itself harmonic and homogeneous of degree $\ell+1$, because $H_\ell$ is independent of $t$. Applying (3) to (4) gives

$$-\ell(p_{\ell+1}-p_\ell)tH_\ell =-d\,p_\ell tH_\ell.$$

Thus

$$\boxed{\frac{p_{\ell+1}}{p_\ell}=\frac{\ell+d}{\ell}} \qquad(\ell\ge1,\ p_1\ne0). \tag{5}$$

Applying (3) to the constant function also gives $p_0=0$. Iteration yields

$$\boxed{ \frac{p_\ell}{p_1} =\prod_{j=1}^{\ell-1}\frac{j+d}{j} =\binom{\ell+d-1}{d}. } \tag{6}$$

The last expression is exactly the cumulative Hockey-Stick count in (1). Here it is an eigenvalue ratio forced by the specified conformal law, rather than a count being substituted for an eigenvalue. The chosen complex harmonic is a convenient way to derive the recurrence; rotation covariance gives the same value on every polarization of its harmonic space.

A local differential operator realizes that law

Let $\Delta=-\nabla^a\nabla_a$ be the nonnegative scalar Laplacian, and write $d=2h$. The critical GJMS operator has order $d$; critical means that its differential order equals the spatial dimension. On the round sphere it is

$$\boxed{ P_{d,R}= \prod_{j=1}^{h} \left[\Delta+\frac{(h-j)(h+j-1)}{R^2}\right]. } \tag{7}$$

The factors and the conformal transformation law are the standard Branson factorization, derived explicitly through stereographic projection in Graham, Proposition 1. The leading term in (7) is $\Delta^{d/2}$.

On a degree-$\ell$ scalar harmonic, $R^2\Delta=\ell(\ell+d-1)$. Each factor becomes

$$\ell(\ell+d-1)+(h-j)(h+j-1) =(\ell+h-j)(\ell+h+j-1). \tag{8}$$

The first factors cover $\ell,\ldots,\ell+h-1$ and the second cover $\ell+h,\ldots,\ell+d-1$. Therefore

$$\boxed{ P_{d,R}Y_{\ell,a} =\frac{\ell(\ell+1)\cdots(\ell+d-1)}{R^d}Y_{\ell,a} =\frac{d!}{R^d}S(\ell,d)Y_{\ell,a}. } \tag{9}$$

The label $a$ runs over the complete scalar harmonic multiplet. Its multiplicity is $\binom{\ell+d}{d}-\binom{\ell+d-2}{d}$, with the second term zero when $\ell<2$. That multiplicity is separate from the eigenvalue (9). The constant harmonic has eigenvalue zero.

For the two charged-lepton depths, the operators are particularly explicit:

$$\begin{aligned} P_{6,R} &=\Delta\left(\Delta+\frac4{R^2}\right) \left(\Delta+\frac6{R^2}\right),\\ P_{10,R} &=\Delta\left(\Delta+\frac8{R^2}\right) \left(\Delta+\frac{14}{R^2}\right) \left(\Delta+\frac{18}{R^2}\right) \left(\Delta+\frac{20}{R^2}\right). \end{aligned} \tag{10}$$

Define the dimensionless operator $\widehat P_d=R^dP_{d,R}/d!$, which normalizes the first positive harmonic eigenvalue to one. At the stated integer indices:

The degrees in the table have not been selected by (2) or (7). Identifying $\ell=n$ with a particle's occupied mode is an additional physical step. The harmonic degree $\ell$ is also different from the oscillator cutoff $N=n-1$. No change of particle dimensionalities or unrestricted mode search is being used.

At the level of scalar spatial operators, a trial family $M_d=\mu_d\widehat P_d$ has precisely the proposed eigenvalue sequence. It is one differential operator for all harmonic degrees, with a common scale in each dimension. Its cumulative harmonic compressions obey the ordinary consistency law. This is not the auxiliary rank-one contact family.

Normalizing by $d!$ is an explicit spectral convention. It does not derive the dimensional coefficient $\mu_d$, nor the charged-lepton relation $\mu_{10}=\mu_6$. A master reduction must fix those physical normalizations across sectors.

Verify it by differentiating spatial functions

In stereographic coordinates, put $s=|x|^2$ and $\Omega=2/(1+s)$. At unit radius, the critical conformal law gives

$$P_d f=\Omega^{-d}(-\Delta_{\mathbb R^d})^{d/2}f. \tag{11}$$

For a zonal harmonic,

$$f(s)=C_\ell^{(d-1)/2}\left(\frac{1-s}{1+s}\right),\qquad \Delta_{\mathbb R^d}f=4s f''+2d f', \tag{12}$$

where $C_\ell^\alpha$ is a Gegenbauer polynomial. Exact rational differentiation of (11)–(12) gives (9), including the assigned degrees.

A second family retains a complex angular phase. With the homogeneous Euclidean harmonic $H_\ell=(x_1+ix_2)^\ell$,

$$f(x)=2^\ell H_\ell(x)(1+s)^{-\ell},\qquad \Delta_{\mathbb R^d}(H_\ell h(s)) =H_\ell\left[4s h''+(2d+4\ell)h'\right]. \tag{13}$$

It gives the same eigenvalue independently of the zonal calculation. Thus the spectral identity is supported by both the conformal recurrence and the actual spatial differential action.

Scalar covariance is not automatically spinor covariance

The standard untwisted spinor Dirac spectrum on a round sphere is $\pm(k+d/2)/R$, with $k=0,1,\ldots$, as discussed in round-sphere Dirac counts. Its conformal differential powers have a different factorization: at unit radius they include

$$\mathscr D\prod_{j=1}^{m}(\mathscr D^2-j^2). \tag{16}$$

The spinorial intertwining family has multipliers proportional to

$$\frac{\Gamma(d/2+k+1/2-\lambda)} {\Gamma(d/2+k+1/2+\lambda)}. \tag{17}$$

For even dimension the intertwiner maps between the appropriate chiral bundles; it is not the scalar operator acting on independent component functions. These conventions and (16) are given in Clerc and Ørsted, Proposition 7.1 and Theorem 7.1.

At the parameter $\lambda=-d/2$, the magnitude of (17) becomes

$$\left(k+\frac12\right) \left(k+\frac32\right)\cdots \left(k+d-\frac12\right). \tag{18}$$

Its consecutive ratio is $(k+d+1/2)/(k+1/2)$, rather than (5) with integer $\ell$. An overall normalization does not remove this spectral shift. Consequently the successful scalar construction cannot simply be declared to be the standard untwisted spinor mass operator. A different physical twist or composite scalar source requires its own derivation.

An exact transfer through a complete Killing-spinor frame does produce a local spinor operator with the scalar spectrum. Its opposite Dirac signs carry adjacent scalar degrees, and its real scalar mass insertion changes the coefficient. This is a distinct operator construction, with its action and physical scope checked separately.

Acting on the scalar current changes the nonlinear equation

Another possible insertion would let $P_d$ act on the real scalar current $J=\psi^\dagger\beta\psi$. On a closed spatial manifold, consider the explicitly different interaction

$$\mathcal E_P(\psi)=\frac{\eta}{2}\int J P_d J\,d\mu. \tag{19}$$

Self-adjointness gives

$$\delta\mathcal E_P=\eta\int(P_dJ)\,\delta J\,d\mu,\qquad \frac{\delta\mathcal E_P}{\delta\psi^\dagger} =\eta(P_dJ)\beta\psi. \tag{20}$$

The variation is multiplication by $P_dJ$; it is not $P_d\psi$. Moreover,

$$P_d1=0,\qquad \int P_dJ\,d\mu=0. \tag{21}$$

Equation (19) contains derivatives of the current. It is an additional interaction, beyond the written algebraic scalar term tested by the local derivative calculation. Any claim that it arises by eliminating modes must identify those modes of the same master field and retain the induced kinetic normalization.

The complete response on a zero-current Killing background provides one exact spatial test of this proposed interaction. Its simplex coefficient enters collective frequencies; any nonzero derivative-current coefficient gives unstable high harmonics on that specified branch.

The exact result is therefore a geometric origin for the simplex scalar spectrum, with conformal covariance fixing the integer recurrence. The physical mass-operator problem still requires the actual spatial geometry, a scalar or spinor realization inherited from the master field, a selected harmonic mode, and a dynamical realization whose frequency matches the known mass-frequency and whose inertial response agrees with it. Converting mass to frequency units is already available; it is not a separate unresolved derivation.

The surface-bending construction gives a different geometric occurrence of the same sequence: it is the Hessian of an extrinsic curvature energy on a surface of dimension $d-2$ in ordinary $d$-dimensional space. Its complete normal pair and nonlinear curvature variation provide a separate dynamical test from the scalar operator here.

Verification

conformal_simplex_spectrum_checks.py checks the exact factorizations, the complete conformal commutator on zonal harmonics, the normalized simplex sequence, and independent stereographic differentiation of both zonal and complex angular fields. Exact rational arithmetic covers the assigned degrees without using a fitted spectral polynomial.

Those operator identities do not select a physical geometry or prove that the master action contains this conformal interaction.

A Distinct Flat-Ball Realization

The flat-ball count clock is the restricted subcritical order-d operator on an auxiliary sphere of dimension d+2, with a nonzero constant clock and its own polynomial multiplicities. The differential bridge passes through the critical operator here and derives its exact zonal kinetic normalization. The auxiliary sphere is a mathematical representation, not another physical sector manifold.