The vertex leading constant at large vertex count
Keep the uniform-simplex assumptions and the positive leading constant \(C_M\) of Vertex coefficients at every fixed vertex count. This chapter concerns that constant as \(M\) grows. The degree limit used to define \(C_M\) remains a separate fixed-\(M\) theorem; no simultaneous \(M,D\) limit follows by substitution.
A quantified large-M theorem
Write \(q=M-1\), \(n=M-2\), \(L=\log2\) and \(a=L-1/2>0\). Define
Theorem 48 (Explicit large-vertex asymptotic constant)
For every integer \(M\geq1026\),
Consequently \(C_M\sim B_M\) as \(M\to\infty\). In particular, the relative error is at most one percent for every integer \(M\geq2502\). These constants and the starting vertex count are conservative.
Proof. The unique maximum and the exact normalization. Put \(h(\theta)=2\operatorname{Re}g(e^{i\theta})\) with the existing \(g(z)=-(1-z)\log(1-z)/z\). For \(0\leq x<\pi\), its explicit real form is
It extends continuously to zero at \(x=\pi\), is positive before that point, and strictly decreases. To check monotonicity, set \(c=\cos\theta\) in the already established positive integral:
The derivative of its integrand with respect to \(c\) is \(-(1+u)(1-u)^2/(1-2uc+u^2)^2<0\) for \(u<1\). This proves the unique maximum \(H=h(\pi)=4L\). Set \(f(x)=h(\pi-x)/H\), \(A(x)=\sec^2(x/2)\) and
Symmetry and \(|1-e^{i(\pi-x)}|^2=4\cos^2(x/2)\) give exactly
Local estimates with explicit constants. The elementary bounds \(2/3<L<7/10\) imply \(1/16<\alpha<1/12\). They follow, for example, by bounding the positive series \(L=2\sum_{j\geq0}3^{-2j-1}/(2j+1)\). For \(0\leq x\leq1/2\), direct differentiation gives
Its absolute value is at most one on this interval. Indeed, \(\cos x\geq7/8\), \(\cos(x/2)\geq31/32\), and \(\log(2\cos(x/2))\geq1/2\); the last inequality follows from \(e<3<(31/16)^2\). A lower bound is \(7/8+7/15-2>-1\). An upper bound is \(1/4+2(7/10)+1/2-2<1\). Taylor’s theorem, including the vanishing odd derivatives, yields
Monotonicity and this bound imply \(0\leq1-f(x)\leq(33/256)x^2\leq33/1024\). The inequality \(|\log(1-u)+u|\leq u^2/(2(1-u))\) therefore gives
For the second assertion one may use \(f(x)\leq1-(\alpha-1/256)x^2\) and \(\alpha-1/256>15/256>1/20\). Also \(1\leq A(x)\leq9/8\) and \(A(x)-1=\tan^2(x/2)\leq x^2/2\).
With \(G_n=\int_0^\infty e^{-\alpha nx^2}\,dx =\sqrt\pi/(2\sqrt{\alpha n})\), the mean-value inequality for exponentials gives
The Gaussian moments evaluate the ratio in the last line to at most
For an entirely rational check, use \(2\sqrt\alpha<3/5\), \(\sqrt{20}<9/2\); the resulting numerator is \(5373/256<21\).
The singular part is controlled, not discarded. Although \(A(x)\) is singular at \(x=\pi\), the exact second moment is finite. The known \(c_3=\pi^2\) gives
Thus \(I_n\) exists for all \(n\geq2\). Monotonicity and the local estimate give \(f(x)\leq f(1/2)\leq79/80\) for \(x\geq1/2\), so
Here \(G_n>3/\sqrt n\) follows from \(\alpha<1/12\) and \(\pi>3\). Separately, the elementary Gaussian-tail bound gives
For every real \(n\geq1024\), each of these two displayed right sides is at most \(1/n\). For the first, dominate \((79/80)^{n-2}\) by \(e^{-(n-2)/80}\) and differentiate the logarithm of \(n^{3/2}e^{-(n-2)/80}\); it decreases after \(n=120\). At \(n=1024\), \(e^{1022/80}>2^{18}>4\,1024^{3/2}\), using \(18\log2<18(7/10)<1022/80\). The second follows in the same way: \(n^{1/2}e^{-n/64}\) decreases after \(n=32\), and at 1024 its required bound follows from \(e^{16}>2^{16}>512/3\). Combining the three errors proves
The exact normalization makes this a relative bound with leading term \(\widetilde B_M=B_M\sqrt{q/n}\). Finally, \(1\leq\sqrt{q/n}=\sqrt{1+1/n}\leq1+1/(2n)\), and hence
This proves the theorem, including the prefactor and remainder.
Certified evaluation and independent integration
The standard-library evaluator
tools.risk_research.vertex_large_dimension saves exact rational
intervals for the normalized constant in
(95). It encloses \(\log2\) by its
positive series, \(\pi\) by Machin’s alternating arctangent formula,
and square roots by integer square roots. Removing the factorial and
exponential scale keeps reports short even at large \(M\).
The simple relative radius is \(25/(M-2)\) for \(M\geq1026\). For \(4\leq M<1026\), the same proof gives a weaker valid radius. Put
Then \((1+1/(2n))E_n+1/(2n)\) bounds \(|C_M/B_M-1|\); use \(e^{-1/64}<64/65\). The tool rounds \(\sqrt n\) outward and clips a negative lower constant endpoint to zero. The small-\(M\) fallback can be wide; the existing exact-series enclosure remains useful there.
Optional python-flint restores the factorial/exponential scale with Arb. A separate integration path directly evaluates the analytic interval of \(I_n\); it does not use the large-\(M\) error bound. At a cutoff \(0<b<\pi\), its omitted endpoint interval satisfies
The callback forwards the analytic flag to the logarithm, and both the cutoff kernel and finiteness are checked. This integration path covers the singular neighborhood by a proved tail, including \(n=2\). Exact rational-series constant intervals provide another independent comparison at moderate \(M\).
For example, from a source checkout:
$ .venv/bin/python -m tools.risk_research.vertex_large_dimension --vertices 2502 --relative-target 1/100 --output .research/skeleton-risk/constant2502.json
$ .venv/bin/python -m tools.risk_research.vertex_large_dimension --vertices 1026 --with-arb --integral-check --output .research/skeleton-risk/constant1026.json
The second command requires optional python-flint. Precision settings are restored after ball calculations. Saved ball strings are rounded outward; source changes or computation failures preserve an existing output file. No public sampling API or mandatory package dependency is added.
The theorem concerns \(C_M\) alone. A simultaneous growing-\(M,D\) vertex asymptotic requires separate control of the \(M\) dependence in the finite-degree remainder from Explicit vertex errors at every fixed vertex count.