Quantitative five-vertex vertex asymptotics

Under the uniform-simplex assumptions of Theory of the skeleton risk, Five-vertex vertex risk gives a positive single sum and the leading constant of the vertex coefficient on five vertices. The following bound replaces its unspecified asymptotic remainder by an explicit one. Put \(c=\zeta(2)-\zeta(3)\), so \(0<c<1\).

Theorem 66 (Five-vertex leading term with an explicit remainder)

For every integer \(D\geq256\),

(128)\[\left|a_0(5,D)-\frac{7200c}{D^6}\right| \leq\frac{3000000(1+\log D)}{D^7}.\]

The constant is conservative: at the start of the domain the error bound can exceed the coefficient. The bound is \(O(\log D/D^7)\) and therefore gives a relative enclosure that shrinks to zero. It is a bound on the mathematical coefficient, in addition to any numerical rounding.

Proof. A pointwise bound. Use \(\Psi_j\) from (123), and put \(n=j+1\) and \(B_n=1+\log n\). Integral comparison gives

\[H_{n-1}\leq B_n,\qquad H_{n-1}^2\leq2n,\qquad H_{n-1}^{(2)}<2,\qquad H_{n-1}^{(3)}<5/4.\]

For the squared bound, the maximum of \((1+\log t)^2/t\) over \(t\geq1\) is \(4/e<2\). The last bound follows from \(\zeta(3)<1+1/8+\int_2^\infty t^{-3}\,dt=5/4\).

Separate the two terms containing the eventual constant:

\[\Psi_j=\frac{48(H_j^{(2)}-A_j)}{n(n+2)}+R_j.\]

The remaining cubic, quadratic, mixed, linear, second-harmonic, third-harmonic and rational terms have the following absolute bounds, in their order in (123):

\[\begin{split}|R_j|&\leq\frac{48H_j+120+96H_j+48H_j+48+60+1}{n^3}\\ &=\frac{192H_j+229}{n^3}.\end{split}\]

Here the second-harmonic contribution is the difference between its original coefficient and \(48/(n(n+2))\). The rational inequalities behind this bound hold throughout \(n\geq1\). To check them without an asymptotic expansion, write

\[\begin{split}Q&=(n+1)^2(n+2)^2(n+3),\\ P_2&=5n^3+7n^2-6n-12,\\ P_4&=2n^4+15n^3+44n^2+47n+24,\\ P_5&=2n^3+3n^2-2n-12.\end{split}\]

Then \(5Q\pm n^2P_2\) have nonnegative coefficients; \(2Q-nP_4=3n^4+18n^3+55n^2+56n+24\); and \(2(n+1)(n+2)(n+3)-n^2(2n+3)=9n^2+22n+12\). Similarly \(2(n+1)(n+2)^2(n+3)\pm nP_5\) have nonnegative coefficients. Finally \(Q-144n\), after substituting \(n=1+t\), has nonnegative coefficients. These respectively bound the quadratic, linear, mixed, second-harmonic correction and rational factors. All cleared denominators are positive.

For \(j\geq1\), the Euler-sum identity from Corollary 17 gives

\[0\leq H_j^{(2)}-A_j-c =\sum_{k=j+1}^{\infty}\frac{H_{k-1}-1}{k^2} \leq\zeta(3)-A_j \leq\frac{2+\log j}{j} \leq\frac{4B_n}{n}.\]

Consequently the additional error in replacing \(H_j^{(2)}-A_j\) by \(c\) is at most \(192B_n/n^3\). Since \(H_j\leq B_n\) and \(B_n\geq1\), \(192+229+192=613<625\) proves

(129)\[\left|\Psi_j-\frac{48c}{(j+1)(j+3)}\right| \leq\frac{625(1+\log(j+1))}{(j+1)^3}.\]

At \(j=0\), \(\Psi_0=1\) and \(0<c<1\), so the same bound follows from \(|1-16c|\leq15<625\). The function \((1+\log n)/n^3\) decreases for \(n\geq1\). Positivity of the transformed kernel therefore also gives the uniform bound \(0<\Psi_j\leq16+625=641\).

Averaging the rational leading term. Use the distribution

\[\mathbb P(J=j)= \frac{\binom{D-1}j\binom{D+1}j}{\binom{2D}{D-1}}, \qquad 0\leq j<D.\]

Put \(x=D/2\), \(X=J-x\), \(B=1+\log D\) and \(C=\{J>\lfloor D/4\rfloor\}\). The moments and the small-index bound in Quantitative tetrahedron vertex asymptotics give, for \(D\geq12\),

\[\begin{split}\mathbb E[X]&=-1/(2D),\\ \mathbb E[X^2]&= \frac{D^3-2D+2}{4D(2D-1)}\leq D/4,\\ \mathbb P(C^c)&\leq\epsilon_D=4\,2^{-D/12}.\end{split}\]

Write \(r(t)=1/((t+1)(t+3))\). For \(t>0\),

\[r(t)=\frac12\int_1^3(t+u)^{-2}\,du,\]

hence \(|r'(t)|\leq2/t^3\) and \(|r''(t)|\leq6/t^4\). Also \(|r(t)-t^{-2}|\leq4/t^3\). On the segment from \(x\) to any \(j\in C\), \(t>D/4\). First-order Taylor expansion thus has expected remainder at most \(192/D^3\). Its linear expectation has absolute value at most \(8/D^4\leq1/D^3\), while \(|r(x)-4/D^2|\leq32/D^3\). On the omitted range the absolute Taylor polynomial is bounded by \((4+8)/D^2\), since \(|X|\leq D/2\). It follows that

\[\left|\mathbb E[\mathbf1_Cr(J)]-4/D^2\right| \leq225/D^3+12\epsilon_D/D^2.\]

On \(C\), \(j+1>D/4\) and \(B_{j+1}\leq B\). Combining (129), the rational expectation bound and \(\Psi_j\leq641\) on the omitted range yields

\[\left|\mathbb E[\Psi_J]-\frac{192c}{D^2}\right| \leq\frac{40000B+10800}{D^3} +\frac{576\epsilon_D}{D^2}+641\epsilon_D.\]

For \(D\geq256\), both \(D^3\epsilon_D\) and \(D\epsilon_D\) decrease: their logarithmic derivatives are \(3/D-\log2/12\) and \(1/D-\log2/12\), respectively, and \(\log2>1/2\). At 256, \(\epsilon_D<2^{-19}\). Therefore \(D^3\epsilon_D<32\) and \(576D\epsilon_D<576/2048<1\). Since \(B\geq1\),

\[\left|\mathbb E[\Psi_J]-\frac{192c}{D^2}\right| \leq\frac{71313B}{D^3}.\]

Outer normalization. The exact single sum is

\[a_0(5,D)=\frac{75}{2D^4}q_D\mathbb E[\Psi_J],\qquad q_D=\prod_{r=1}^4(1+r/(2D))^{-1}.\]

Here \(0<q_D\leq1\) and \(1-q_D\leq\sum_{r=1}^4r/(2D)=5/D\). Thus the final error is at most

\[\left(\frac{75\cdot71313}{2}+36000\right)\frac{B}{D^7} =\frac{5420475}{2}\frac{B}{D^7} <3000000\frac{B}{D^7},\]

which proves (128).

Reproducible enclosures

tools/risk_research/vertex_five_asymptotic.py returns the exact rational majorant \(3000000(1+\lceil\log_2D\rceil)/D^7\). The integer bit count dominates the logarithm without a floating-point conversion. For this bound alone, no optional numerical dependency is needed:

python -m tools.risk_research.vertex_five_asymptotic --degree 100000 --bound-only --output .research/five-vertex-error.json

Without --bound-only, use the prepared Sage environment, which includes python-flint. Arb encloses \(\zeta(2)\) and \(\zeta(3)\), then adds both arithmetic rounding and the proved analytic error. The local precision context is restored. Reports record the evaluator hash and numerical version. Native tests also reparse the emitted JSON balls and compare them with the independent exact single sum. Finite checks validate the implementation; the all-degree guarantee is the inequality proof above.