Tetrahedron vertex risk

Under the uniform-simplex assumptions of Theory of the skeleton risk, the tetrahedron vertex coefficient has a positive single sum involving only binomial coefficients and harmonic numbers. This eliminates the bivariate kernel convolution. The sum still has \(D\) terms. An explicit certified degree recurrence is proved in Tetrahedron vertex degree recurrence; a bounded classical-special-function expression remains open. The next asymptotic term and a fully explicit error bound are proved in Quantitative tetrahedron vertex asymptotics.

A binomial-transform diagonalization

Put \(A_h=(-1)^h\binom{D-1}h\binom{D+1}h\), \(C_D=(D-1)!(D+1)!\), and \(W_{D,j}=\binom{D-1}j\binom{D+1}j\). As in Further evaluations of the skeleton risk, define

\[\mathcal L_D(G)=\sum_{h,l=0}^{D-1} A_hA_lh!l!(2D-h-l)!\,[x^hy^l]G.\]

Theorem 30 (Diagonalization by signed binomial transforms)

For every integer \(D\geq1\) and arrays \(u,v\) of length \(D\), set

\[\widehat u_j=\sum_{h=0}^j(-1)^h\binom jh u_h, \qquad \widehat v_j=\sum_{l=0}^j(-1)^l\binom jl v_l.\]

Then

(56)\[\sum_{h,l=0}^{D-1}A_hA_lh!l!(2D-h-l)!u_hv_l =C_D\sum_{j=0}^{D-1}W_{D,j}\widehat u_j\widehat v_j.\]

Consequently, if

\[\mathscr T(G)(x,y)= \frac{G(-x/(1-x),-y/(1-y))}{(1-x)(1-y)},\]

then

(57)\[\mathcal L_D(G)=C_D\sum_{j=0}^{D-1} W_{D,j}[x^jy^j]\mathscr T(G).\]

Proof. Write \(n=D-1\) and \(P_u(t)=\sum_{h=0}^n A_hh!u_ht^{D-h}\). The Laguerre expansion

\[L_k^{(2)}(t)=\sum_{r=0}^k (-1)^r\binom{k+2}{k-r}\frac{t^r}{r!}\]

and binomial inversion give

\[P_u(t)=(-1)^n n!t\sum_{j=0}^n (-1)^j\binom{n+2}j\widehat u_jL_{n-j}^{(2)}(t).\]

Equality of the coefficients of \(t^{r+1}\) is exactly ordinary binomial inversion for \(u_{n-r}\). Integrate \(P_u(t)P_v(t)e^{-t}\) on \([0,\infty)\). The monomial moments are \((2D-h-l)!\), while Laguerre orthogonality gives

\[\int_0^\infty t^2e^{-t}L_k^{(2)}(t)L_l^{(2)}(t)\,dt =\mathbf1_{k=l}\frac{(k+2)!}{k!}.\]

Its surviving coefficient satisfies

\[n!^2\binom{n+2}j^2\frac{(n+2-j)!}{(n-j)!} =C_DW_{D,j},\]

proving (56). Apply the identity linearly to each coefficient of \(G\); the generating-function identity for signed binomial transforms is precisely \(\mathscr T\). All relevant arrays and polynomials are finite.

The logarithmic formula for the outside kernel simplifies under this substitution. With \(f(t)=-\log(1-t)/t\) and \(g(t)=(1-t)f(t)\), direct collection of logarithms gives

(58)\[F\!\left(-\frac{x}{1-x},-\frac{y}{1-y}\right) =g(x)+g(y)-g(xy).\]

Indeed, if \(X=-x/(1-x)\), \(Y=-y/(1-y)\), then \(1-X-Y=(1-xy)/((1-x)(1-y))\). Substitution into the numerator \((1-X-Y)\log(1-X-Y)-(1-X)\log(1-X)-(1-Y)\log(1-Y)\) of \(F(X,Y)\) proves (58). For every fixed vertex count, this transforms \(\mathcal L_D(F^{M-1})\) into a weighted diagonal of the corresponding power of an additive kernel. No evaluation of all those diagonals is claimed.

The positive single sum

Write \(H_j=\sum_{r=1}^j1/r\), \(H_j^{(2)}=\sum_{r=1}^j1/r^2\), with both values zero at \(j=0\).

Theorem 31 (Tetrahedron vertex coefficient at every degree)

For every integer \(D\geq1\),

(59)\[a_0(4,D)=\frac{96(D-1)!(D+1)!}{(4)_{2D}} \sum_{j=0}^{D-1}\binom{D-1}j\binom{D+1}j G_j,\]

where

(60)\[\begin{split}G_j={}&\frac{H_j^2}{(j+1)(j+2)(j+3)} +\frac{H_j^{(2)}}{(j+1)(j+3)}\\ &-\frac{jH_j}{(j+1)(j+2)^2(j+3)} +\frac{2}{(j+1)^2(j+2)^2(j+3)}.\end{split}\]

Every weight \(G_j\) is strictly positive, including \(G_0=1/6\). All denominators are nonzero on the stated domain.

Proof. Set \(\phi_r(j)=[t^j]g(t)^r/(1-t)\). Elementary logarithmic coefficient identities give

\[\begin{split}\phi_1(j)&=\frac1{j+1},\\ \phi_2(j)&=\frac{2(1-H_j)}{(j+1)(j+2)},\\ \phi_3(j)&=\frac{6(H_j^2-H_j^{(2)}-3H_j+1)} {(j+1)(j+2)(j+3)}.\end{split}\]

For example, differentiate the finite product \(\binom\alpha{j+r}\) \(r\) times at \(\alpha=r-1\), using \(g(t)^r/(1-t)=(1-t)^{r-1}(-\log(1-t))^r/t^r\). The product has a simple zero there; the logarithmic derivatives of its remaining factors give the three formulas, including \(j=0\).

Expand \((g(x)+g(y)-g(xy))^3\) and divide by \((1-x)(1-y)\). The pure powers contribute \(\phi_3(j)\). The terms \(g(x)^2g(y)\) and its symmetric counterpart contribute \(6\phi_1(j)\phi_2(j)\). The diagonal contributions of \(g(x)^2g(xy)\) and \(g(x)g(xy)^2\), together with their symmetric counterparts, cancel by ordinary convolution. Thus

\[[x^jy^j]\mathscr T(F^3) =\phi_3(j)+6\phi_1(j)\phi_2(j) -6\sum_{k=0}^j[t^k]g(t)\phi_1(j-k)^2.\]

Since \([t^0]g=1\) and \([t^k]g=-1/(k(k+1))\) for \(k\geq1\), the convolution equals \(1/(j+1)^2-B_j\), where

\[\begin{split}B_j={}&\sum_{r=0}^{j-1} \frac1{(r+1)(r+2)(j-r)^2}\\ ={}&\frac{H_j^{(2)}}{(j+1)(j+2)} +\frac{2(2j+3)H_j}{(j+1)^2(j+2)^2} +\frac{j}{(j+1)(j+2)^2}.\end{split}\]

The last equality follows by partial fractions and includes the empty sum \(B_0=0\). Substituting the three \(\phi_r\) formulas and simplifying gives \([x^jy^j]\mathscr T(F^3)=6G_j\). Apply (57) and the scalar risk identity \(a_0(4,D)=16\mathcal L_D(F^3)/(4)_{2D}\).

For positivity, group the first and third terms of \(G_j\) as

\[\frac{H_j(H_j-j/(j+2))}{(j+1)(j+2)(j+3)}.\]

At \(j\geq1\), \(H_j\geq1>j/(j+2)\); at zero the last rational term is positive. The second and last terms are nonnegative.

Large-degree behavior

Corollary 14 (Tetrahedron vertex asymptotic)

As \(D\to\infty\),

(61)\[a_0(4,D)=\frac{48\pi^2}{D^5} +O\!\left(\frac{\log^2D}{D^6}\right).\]

Proof. Vandermonde’s identity gives \(Z_D=\sum_jW_{D,j}=\binom{2D}{D-1}\). Let \(J\) have probabilities \(W_{D,j}/Z_D\). Since \(C_DZ_D=(2D)!\),

\[a_0(4,D)=\frac{576}{(2D+1)(2D+2)(2D+3)}\,\mathbb E[G_J].\]

The first two factorial moments, obtained from Vandermonde, give

\[\mu=\mathbb E[J]=\frac{D^2-1}{2D},\qquad \operatorname{Var}(J)=\frac{(D^2-1)^2}{4D^2(2D-1)}=O(D).\]

An elementary weight ratio controls the potentially small indices. For \(D\geq12\), put \(a=\lfloor D/4\rfloor\), \(b=\lfloor D/3\rfloor\). For \(0\leq j<b\),

\[\frac{W_{D,j+1}}{W_{D,j}} =\frac{(D-1-j)(D+1-j)}{(j+1)^2}\geq2.\]

The numerator is at least \(4D^2/9-1\), the denominator is at most \((D/3+1)^2\), and their ratio exceeds two on this range. Consequently

\[\mathbb P(J\leq a) \leq\sum_{j=0}^a2^{j-b}\frac{W_{D,b}}{Z_D} \leq2^{a+1-b}\leq4\,2^{-D/12}.\]

On \(j>a\), harmonic-number bounds in (60) give uniformly

\[G_j=\frac{\zeta(2)}{(j+1)(j+3)} +O(\log^2D/D^3).\]

Put \(r(t)=1/((t+1)(t+3))\). Between \(\mu\) and any \(j>a\), \(r''(t)=O(D^{-4})\). Taylor’s theorem, the mean-zero linear term, the variance bound, and the exponentially small omitted tail therefore give

\[\mathbb E[r(J)]=r(\mu)+O(D^{-3}) =4/D^2+O(D^{-3}).\]

Both \(G_j\) and \(r(j)\) are bounded by an absolute constant for all \(j\geq0\); the lower tail contributes exponentially little. Hence \(\mathbb E[G_J]=2\pi^2/(3D^2)+O(\log^2D/D^3)\). The outer prefactor is \(72/D^3+O(D^{-4})\), proving the result.

Verification and scope

tools/risk_research/vertex_single.py evaluates (59) with exact rational arithmetic and incremental binomial weights. Tests compare the pre-existing scalar sum through degree 20 and independent full Gram propagation through degree 6. After fixing the formula, the original integer-kernel implementation generated new held-out values at degrees 241, 247 and 260, beyond the previous exploration through degree 240. All three exact fractions agree; degree 260 is retained as an independent regression value. Finite checks verify the implementation; the all-degree claims use the proofs above.

The formula and its positivity avoid the severe alternating cancellation of the original scalar sum. Tetrahedron vertex degree recurrence supplies the explicit scalar degree recurrence, while Tetrahedron risk comparison at every degree classifies the optimal tetrahedron comparison at every degree. The nonlinear self-identification question remains separate, and the generic P-recursiveness theorem in Further evaluations of the skeleton risk continues to apply independently.