Vertex coefficients at every fixed vertex count

This chapter keeps the uniform-simplex assumptions of Theory of the skeleton risk. It gives a constructive one-dimensional convolution formula at every vertex count and proves the leading vertex constant for each fixed \(M\geq4\). A quantified regime with both \(M\) and the degree growing is proved separately in A simultaneous vertex-count and degree regime.

For the separate dependence of the leading constant on the vertex count, The vertex leading constant at large vertex count gives its large-M prefactor, explicit remainder and certified intervals.

A rational univariate construction

Put \(g(t)=-(1-t)\log(1-t)/t\), with \(g(0)=1\), and set

\[\phi_r(j)=[t^j]\frac{g(t)^r}{1-t},\qquad e_r(j)=\sum_{a=1}^{r-1}\binom ra\phi_a(j)\phi_{r-a}(j).\]

Thus \(e_0=e_1=0\). Let \(b_k(j)=[t^j]g(t)^k\) and use ordinary sequence convolution. In particular, \(b_0\) is the unit sequence and \(b_k(j)=\phi_k(j)-\phi_k(j-1)\), where \(\phi_k(-1)=0\). Define

(82)\[\delta_q(j)=(-1)^{q+1}\phi_q(j) +\sum_{k=0}^{q-2}(-1)^k\binom qk(b_k*e_{q-k})(j).\]

An empty sum is zero. These sequences require only rational arithmetic. For example, writing \(E_l\) for an elementary symmetric polynomial,

(83)\[\phi_r(j)=\frac{r!(r-1)!j!}{(j+r)!} E_{r-1}\left(1,\frac12,\ldots,\frac1{r-1}, -1,-\frac12,\ldots,-\frac1j\right).\]

The positive list is empty at \(r=1\); the negative list is empty at \(j=0\). Adding \(-1/(j+1)\) updates the finite symmetric polynomial state, while the factorial ratio has a rational update.

Theorem 38 (Constructive vertex formula at every vertex count)

For integers \(M\geq2\), \(D\geq1\), let \(W_{D,j}=\binom{D-1}j\binom{D+1}j\) and \(Z_D=\binom{2D}{D-1}\). Then

(84)\[a_0(M,D)=\frac{M^2(M-1)!} {Z_D\prod_{h=1}^{M-1}(2D+h)} \sum_{j=0}^{D-1}W_{D,j}\delta_{M-1}(j).\]

Every \(\delta_q(j)\) is strictly positive. Formula (82) evaluates it by univariate convolutions, without constructing a bivariate outside kernel.

Proof. The transformed-kernel identity in Tetrahedron vertex risk gives

\[\delta_q(j)=[x^jy^j] \frac{(g(x)+g(y)-g(xy))^q}{(1-x)(1-y)}.\]

Expanding by powers of \(g(xy)\) gives \(\sum_{k=0}^q(-1)^k\binom qk(b_k*d_{q-k})(j)\), where \(d_r=2\phi_r+e_r\) for \(r\geq1\) and \(d_0=1\). Since \(b_k*\phi_{q-k}=\phi_q\), the edge terms combine to \((-1)^{q+1}\phi_q\). This proves (82). The previously proved binomial Gram diagonalization supplies (84); positivity is Proposition 6.

To prove (83), extract the coefficient of \(t^{j+r}\) after differentiating \((1-t)^\alpha\) exactly \(r\) times at \(\alpha=r-1\). The generalized binomial coefficient has one vanishing factor there. Expanding its remaining factors yields the displayed elementary symmetric polynomial, including both empty-list cases. Thus the evaluator does not require symbolic differentiation.

A positive integral for the leading constant

Theorem 39 (Vertex leading constant for every fixed M at least four)

For each fixed integer \(M\geq4\), put \(q=M-1\). Then

\[a_0(M,D)\sim\frac{C_M}{D^{M+1}},\qquad C_M=M^2(M-1)!\,2^{3-M}c_q>0,\]

where the following absolutely convergent expressions agree:

(85)\[c_q=q\sum_{j=0}^{\infty}e_{q-1}(j) =\frac{q}{2\pi}\int_{-\pi}^{\pi} \frac{\left(2\operatorname{Re}g(e^{it})\right)^{q-1}} {|1-e^{it}|^2}\,dt.\]

At \(M=4,5\), this recovers \(C_4=48\pi^2\) and \(C_5=7200(\zeta(2)-\zeta(3))\).

Proof. Coefficient bounds. For each fixed \(r\geq1\), the elementary symmetric formula and \(|E_l(x)|\leq(\sum|x_i|)^l/l!\) give

(86)\[|\phi_r(j)|\leq B_r \frac{(1+\log(j+1))^{r-1}}{(j+1)^r}, \qquad B_r=r!r^{r-1}.\]

Indeed, the reciprocal sum is at most \(H_{r-1}+H_j\leq r+\log(j+1)\) and \(j!/(j+r)!\leq(j+1)^{-r}\). Differentiating the same binomial product at \(\alpha=r\) instead proves

\[b_r(j)=O_r\!\left(\frac{(1+\log(j+1))^{r-1}} {(j+1)^{r+1}}\right).\]

Consequently \(\sum_jb_r(j)=g(1)^r=0\) for \(r\geq1\): the coefficients are absolutely summable, so Abel’s limit is the ordinary sum. For \(r\geq2\), \(b_r\) also has an absolutely summable first absolute moment. For \(r=1\), \(b_1(j)=-1/(j(j+1))\) at positive indices, and its first absolute moment through \(n\) is \(O(1+\log n)\).

Products and differences of (86), using \(\phi_r(j+1)-\phi_r(j)=b_r(j+1)\), give, for \(r\geq2\),

\[\begin{split}e_r(j)&=O_r((1+\log(j+1))^{r-2}(j+1)^{-r}),\\ e_r(j+1)-e_r(j) &=O_r((1+\log(j+1))^{r-2}(j+1)^{-r-1}).\end{split}\]

In particular, \(e_r\) is absolutely summable.

The convolution limit. Fix \(k\geq1\), \(r\geq2\). Split \((b_k*e_r)(n)\) at \(l=\lfloor n/2\rfloor\) in the index of \(b_k(l)\). On the first part, subtract \(e_r(n)\sum_{l\geq0}b_k(l)=0\). The difference estimate bounds this part by

\[O_{k,r}\!\left(\frac{(1+\log n)^{r-2}}{n^{r+1}} \sum_{l\leq n/2}l|b_k(l)|\right) +|e_r(n)|\sum_{l>n/2}|b_k(l)|=o(n^{-2}).\]

For the second part set \(m=n-l<n/2\). The coefficients satisfy \(n^2b_1(n-m)\to-1\) and \(n^2b_k(n-m)\to0\) for \(k\geq2\), at every fixed \(m\). They are uniformly bounded over \(m<n/2\), by the coefficient bounds above. Absolute summability of \(e_r\) permits dominated convergence. Hence

\[n^2(b_1*e_r)(n)\longrightarrow-\sum_me_r(m),\qquad n^2(b_k*e_r)(n)\longrightarrow0\quad(k\geq2).\]

In (82), both \(\phi_q(n)\) and the \(k=0\) term \(e_q(n)\) are \(o(n^{-2})\) when \(q\geq3\). Its \(k=1\) term proves \(n^2\delta_q(n)\to c_q\) with the displayed series.

Boundary integrals and positivity. The sequences \(\phi_a\), \(a\geq1\), are square summable. Parseval, or finite polynomial Parseval followed by their Abel limits, gives

\[\sum_j\phi_a(j)\phi_b(j) =\frac1{2\pi}\int_{-\pi}^{\pi} \frac{g(e^{it})^a\overline{g(e^{it})}^{\,b}} {|1-e^{it}|^2}\,dt\quad(a,b\geq1).\]

All these boundary products are integrable: near zero they are bounded by a constant times \(|t|^{a+b-2}(1+|\log|t||)^{a+b}\). The analytic terms absent from \(e_r\) have zero integral. Indeed, on \(|z|=1\),

\[\frac{g(z)^r}{|1-z|^2}=-\frac{zg(z)^r}{(1-z)^2}.\]

The right side is analytic in the disk with zero constant coefficient. Its circle integrals vanish at every radius less than one. For radii at least one half its absolute value is dominated, uniformly near \(t=0\), by \(O_r((1+|\log|t||)^r)\) for \(r\geq2\); use \(|1-\rho e^{it}|\geq c|t|\) and the explicit formula for \(g\). Dominated convergence therefore proves that its boundary integral vanishes too. Summing the mixed Parseval identities gives (85).

Finally, for \(z=e^{it}\ne1\),

\[g(z)=\int_0^1\frac{1-z}{1-uz}\,du,\qquad \operatorname{Re}\frac{1-z}{1-uz} =\frac{(1+u)(1-\cos t)}{|1-uz|^2}>0.\]

The integral for \(c_q\) is thus strictly positive. Its absolute convergence also follows from the preceding boundary bound.

The weighted vertex limit. Under probabilities \(W_{D,j}/Z_D\), the random variable \(J\) satisfies \(J/D\to1/2\) in probability and \(\mathbb P(J\leq\lfloor D/4\rfloor)\leq4\,2^{-D/12}\), as proved in Tetrahedron vertex risk. The diagonal limit implies a bound \(|\delta_q(j)|\leq K_q/(j+1)^2\). On the complementary central event, \(D^2\delta_q(J)\) is bounded and converges in probability to \(4c_q\); its expectations therefore converge. The small-index event contributes at most \(4K_qD^2 2^{-D/12}\to0\). Multiplying the outer prefactor in (84) proves the theorem.

For the known constants, \(e_2(j)=2/(j+1)^2\) gives \(c_3=6\zeta(2)\). At \(M=5\), the same limit agrees with Corollary 17, giving \(c_4=48(\zeta(2)-\zeta(3))\).

The critical triangle case is separate: \(q=2\) leaves \(\delta_2=-\phi_2+e_2\), whose first term has a logarithm at order \(j^{-2}\). The known triangle formula gives \(a_0(3,D)\sim36\log D/D^4\), rather than a finite constant at that normalization.

Certified constant enclosures

The infinite constant does not require unverified numerical integration. For \(r\geq2\), put

\[A_r=\sum_{a=1}^{r-1}\binom ra B_aB_{r-a},\qquad m=r-2,\qquad L_T=1+\lceil\log_2T\rceil.\]

For every integer \(T\geq1\), (86) and the decreasing function \((1+\log x)^m/x^r\) give

(87)\[\left|c_q-q\sum_{j=0}^{T-1}e_{q-1}(j)\right| \leq\frac{qA_r}{T^{r-1}} \sum_{k=0}^{m}\frac{m!L_T^{m-k}} {(m-k)!(r-1)^{k+1}},\qquad r=q-1.\]

To check the integral exactly, substitute \(u=\log x\) in \(\int_T^\infty(1+\log x)^m x^{-r}\,dx\) and integrate by parts \(m\) times. Replacing \(1+\log T\) by the integer \(L_T\) increases every positive term. The tool computes this rational bound and the partial sum exactly, then scales by \(M^2(M-1)!2^{3-M}\). The known positivity permits clipping a negative lower bound to zero. Increasing \(T\) converges to the true constant for each fixed \(M\); no claim of uniform cost in \(M\) is made.

From a source checkout, for example:

$ .venv/bin/python -m tools.risk_research.vertex_general --vertices 6 --constant-terms 128 --output .research/skeleton-risk/vertex6-constant.json
$ .venv/bin/python -m tools.risk_research.vertex_general --vertices 6 --degree 32 --output .research/skeleton-risk/vertex6-exact.json

vertex_general.vertex_coefficient implements the all-degree formula; leading_constant_enclosure returns two exact fractions. These are checkout-only research tools, with no additional runtime dependency or public sampling API. This chapter certifies the constant’s truncation error. Explicit vertex errors at every fixed vertex count separately proves an explicit finite-degree error for every fixed \(M\geq4\) and \(D\geq256\), combining the constant interval with an analytic remainder. The sharper five-vertex finite-degree error remains a separate result.

Independent implementation checks compare the elementary rows with logarithmic-series expansion, the convolutions with the original bivariate kernel and signed transform, and the coefficient with the original scalar and rational Gram oracles. After freezing the candidate, a separate integer-polynomial computation of \(F^5\) at degrees 67, 79 and 97 agreed exactly; the committed artifact records both source hashes. Native Arb checks place the independently known constants at M4/M5 inside the exact rational intervals. These finite checks support the implementation; the general claims use the proofs above.