Quantitative tetrahedron vertex asymptotics

Under the uniform-simplex assumptions of Theory of the skeleton risk, the positive single sum in Tetrahedron vertex risk gives a second asymptotic term and a completely explicit error bound. Write \(\gamma=\lim_n(H_n-\log n)\) for Euler’s constant and put \(L_D=\log(D/2)+\gamma\).

Theorem 39 (Two-term vertex expansion with an explicit remainder)

Define

(80)\[A_D=\frac{48\pi^2}{D^5} +\frac{576(L_D^2-L_D-1)-456\pi^2}{D^6}.\]

For every integer \(D\geq256\),

(81)\[\left|a_0(4,D)-A_D\right| \leq\frac{10^7(1+\log D)^2}{D^7}.\]

In particular the next-order coefficient contains a squared logarithm; the error is \(O(\log^2D/D^7)\). This result concerns the coefficient itself, rather than the error of a floating-point evaluation.

The constant is conservative. The inequality holds at the beginning of its domain even when its error bound is larger than the coefficient. It becomes an increasingly accurate relative enclosure as the degree grows.

Proof. Put \(z=\zeta(2)=\pi^2/6\), \(B=1+\log D\), \(x=D/2\). Use the distribution from the single-sum proof,

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

With \(G_j\) as in (60),

\[a_0(4,D)=\frac{72}{D^3}q_D\,\mathbb E[G_J], \qquad q_D=\frac1{(1+1/(2D))(1+1/D)(1+3/(2D))}.\]

Pointwise expansion. Integral comparison gives

\[0<H_j-\log j-\gamma<1/j,\qquad H_j^{(2)}=z-1/j+e_j,\quad 0\leq e_j\leq1/j^2.\]

The first follows by summing \(0<\log(1+1/k)-1/(k+1)<1/k-1/(k+1)\) from \(k=j\) to infinity. The second follows by bounding the tail of \(\sum k^{-2}\) between \(1/(j+1)\) and \(1/j\). Also \(0<\gamma<1\) and \(z<2\).

Write \(\ell_t=\log t+\gamma\) and define

\[u(t)=t^{-2},\qquad f(t)=\frac{\ell_t^2-\ell_t-4z-1}{t^3}.\]

For integers \(1\leq j<D\), direct rational expansions of the four terms of \(G_j\) give

(82)\[G_j=zu(j)+f(j)+R_j, \qquad |R_j|\leq160B^2/j^4.\]

Here are explicit bounds for that calculation. The errors in replacing the first denominator by \(j^{-3}\) and the third rational factor by \(j^{-3}\) are at most \(23/j^4\) and \(71/j^4\). For the second term,

\[\left|\frac1{(j+1)(j+3)}-\frac1{j^2}+\frac4{j^3}\right| =\frac{13j+12}{j^3(j+1)(j+3)}\leq25/j^4.\]

Thus the error before replacing \(H_j\) by \(\ell_j\) is at most \((23H_j^2+71H_j+25z+10)/j^4\). The replacement adds at most \((2B+1)/j^4\). Since \(H_j,\ell_j\leq B\), their total is at most \(157B^2/j^4\), which implies the stated bound.

Moments. Put \(X=J-x\), \(M_r=\mathbb E[X^r]\). Vandermonde’s identity gives

\[\mathbb E[(J)_{\underline r}] =\frac{(D-1)_{\underline r}(D+1)_{\underline r}} {(2D)_{\underline r}},\quad 1\leq r\leq4,\]

where the underline denotes a falling factorial. Converting these four moments to powers and centering at \(x=D/2\) gives

\[\begin{split}M_1&=-\frac1{2D},\\ M_2&=\frac{D^3-2D+2}{4D(2D-1)},\\ M_3&=-\frac{3D^2+2D-4}{8D(2D-1)},\\ M_4&=\frac{3D^5-4D^4-12D^3+20D^2+16D-24} {16D(2D-3)(2D-1)}.\end{split}\]

For \(D\geq12\), these imply

\[M_2\leq D/4,\qquad |M_2-D/8|\leq1/4, \qquad |M_3|\leq1,\qquad M_4\leq D^2.\]

These inequalities are rational: after clearing their positive denominators, substituting \(D=12+t\) leaves nonnegative polynomial coefficients. They therefore hold on the whole stated domain.

Central range and tail. Let \(C=\{J>\lfloor D/4\rfloor\}\) and \(\epsilon_D=4\,2^{-D/12}\). The weight-ratio argument in Corollary 14 proves \(\mathbb P(C^c)\leq\epsilon_D\) for \(D\geq12\). Along the segment from \(x\) to any \(j\in C\), \(t>D/4\) and \(0<\ell_t\leq B\). Restricted expectations below sum only over \(C\); neither \(u(0)\) nor \(f(0)\) is evaluated.

Taylor expansion of \(u\) through degree three uses \(u^{(4)}(t)=120/t^6\). Its expectation on the central range, with the moment bounds above, satisfies

\[\left|\mathbb E[\mathbf1_Cu(J)] -\frac4{D^2}-\frac6{D^3}\right| \leq\frac{20511}{D^4}+\frac{40\epsilon_D}{D^2}.\]

Indeed the linear, quadratic-correction, cubic and fourth-derivative errors are bounded by \(8,12,11,20480\) times \(D^{-4}\). The absolute value of the cubic Taylor polynomial on the omitted tail is at most \((4+8+12+16)/D^2=40/D^2\).

For \(f\), differentiation gives

\[\begin{split}f'(t)&=\frac{-3\ell_t^2+5\ell_t+12z+2}{t^4},\\ f''(t)&=\frac{12\ell_t^2-26\ell_t-48z-3}{t^5}.\end{split}\]

The numerator bounds are \(34B^2\) and \(137B^2\). First-order Taylor expansion with a second-derivative remainder yields

\[\left|\mathbb E[\mathbf1_Cf(J)] -\frac{8(L_D^2-L_D-4z-1)}{D^3}\right| \leq\frac{17559B^2}{D^4} +\frac{360B^2\epsilon_D}{D^3}.\]

Its linear expectation is at most \(272B^2/D^5\leq23B^2/D^4\); the quadratic remainder is at most \(17536B^2/D^4\). On the tail the absolute linear Taylor polynomial is at most \((88+272)B^2/D^3\).

The positive single-sum weight also satisfies \(0<G_j\leq1\) for every \(j\geq0\), as shown in the tetrahedron comparison. Combine this tail bound with (82) and \(j^{-4}<256/D^4\) on \(C\). Since \(40960+2\cdot20511+17559=99541<100000\),

\[\begin{split}\mathbb E[G_J]={}&\frac{4z}{D^2} +\frac{8(L_D^2-L_D-1)-26z}{D^3}+\mathcal R_D,\\ |\mathcal R_D|\leq{}&\frac{100000B^2}{D^4} +\epsilon_D\left(1+\frac{80}{D^2}+\frac{360B^2}{D^3}\right).\end{split}\]

Outer normalization. Direct multiplication gives \(0<q_D\leq1\) and \(|q_D-1+3/D|\leq7/D^2\) for \(D\geq12\). Thus the coefficient of \(D^{-6}\) is

\[72\left(8(L_D^2-L_D-1)-26z-12z\right) =576(L_D^2-L_D-1)-456\pi^2.\]

Put \(T_D=8(L_D^2-L_D-1)-26z\). The remaining error before multiplication by \(72/D^3\) is bounded by

\[\frac{7\cdot4z+(3+7/D)|T_D|}{D^4} \leq\frac{360B^2}{D^4},\]

using \(4z<8\), \(|T_D|\leq76B^2\) and \(3+7/D<4\). Thus the additional normalization error is at most \(25920B^2/D^7\). The remaining tail, divided by \(B^2/D^7\), is at most

\[72\epsilon_D(D^4+80D^2+360D)<10^6,\qquad D\geq256.\]

Each summand decreases there, because \(\log2>1/2\). At \(D=256\), replace \(\epsilon_D\) by \(2^{-19}\); the resulting rational expression is less than \(10^6\). Finally \(7200000+25920+1000000<10^7\), proving (81).

Reproducible enclosures

tools/risk_research/vertex_asymptotic.py returns the rational upper bound \(10^7(1+\lceil\log_2D\rceil)^2/D^7\), which dominates (81) without evaluating a rounded logarithm. The optional real-ball evaluator uses python-flint/Arb for \(\pi\), \(\gamma\) and the logarithm, and includes both arithmetic and analytic error in its returned coefficient enclosure. Its precision context is restored.

For an error bound using only the project Python and standard library:

python -m tools.risk_research.vertex_asymptotic --degree 10000 --bound-only --output .research/vertex-error.json

For a real-ball enclosure, run the same command without --bound-only in the prepared Sage environment, which already contains python-flint. Finite exact comparisons and moment checks detect implementation errors; the all-degree guarantee follows from the inequalities above.