The vertex leading constant at large vertex count ================================================= Keep the uniform-simplex assumptions and the positive leading constant :math:`C_M` of :doc:`skeleton_risk_vertex_general`. This chapter concerns that constant as :math:`M` grows. The degree limit used to define :math:`C_M` remains a separate fixed-:math:`M` theorem; no simultaneous :math:`M,D` limit follows by substitution. A quantified large-M theorem ---------------------------- Write :math:`q=M-1`, :math:`n=M-2`, :math:`L=\log2` and :math:`a=L-1/2>0`. Define .. math:: B_M=\frac{M^2(M-1)!\,(2\log2)^{M-1}\sqrt{M-1}} {4\sqrt{\pi\log2\,(\log2-1/2)}}. .. prf:theorem:: Explicit large-vertex asymptotic constant :label: thm-risk-large-vertex-constant For every integer :math:`M\geq1026`, .. math:: :label: risk-large-vertex-error \left|\frac{C_M}{B_M}-1\right|\leq\frac{25}{M-2}. Consequently :math:`C_M\sim B_M` as :math:`M\to\infty`. In particular, the relative error is at most one percent for every integer :math:`M\geq2502`. These constants and the starting vertex count are conservative. .. prf:proof:: **The unique maximum and the exact normalization.** Put :math:`h(\theta)=2\operatorname{Re}g(e^{i\theta})` with the existing :math:`g(z)=-(1-z)\log(1-z)/z`. For :math:`0\leq x<\pi`, its explicit real form is .. math:: h(\pi-x)=4\cos^2(x/2)\log(2\cos(x/2))+x\sin x. It extends continuously to zero at :math:`x=\pi`, is positive before that point, and strictly decreases. To check monotonicity, set :math:`c=\cos\theta` in the already established positive integral: .. math:: \operatorname{Re}g(e^{i\theta}) =\int_0^1\frac{(1+u)(1-c)}{1-2uc+u^2}\,du. The derivative of its integrand with respect to :math:`c` is :math:`-(1+u)(1-u)^2/(1-2uc+u^2)^2<0` for :math:`u<1`. This proves the unique maximum :math:`H=h(\pi)=4L`. Set :math:`f(x)=h(\pi-x)/H`, :math:`A(x)=\sec^2(x/2)` and .. math:: I_n=\int_0^\pi f(x)^n A(x)\,dx,\qquad \alpha=\frac{a}{4L}. Symmetry and :math:`|1-e^{i(\pi-x)}|^2=4\cos^2(x/2)` give exactly .. math:: :label: risk-large-vertex-normalization \frac{C_M}{M^2q!\,(2L)^q}=\frac{q}{4\pi L}I_n. **Local estimates with explicit constants.** The elementary bounds :math:`2/3-1`. An upper bound is :math:`1/4+2(7/10)+1/2-2<1`. Taylor's theorem, including the vanishing odd derivatives, yields .. math:: |f(x)-1+\alpha x^2|\leq x^4/64. Monotonicity and this bound imply :math:`0\leq1-f(x)\leq(33/256)x^2\leq33/1024`. The inequality :math:`|\log(1-u)+u|\leq u^2/(2(1-u))` therefore gives .. math:: |\log f(x)+\alpha x^2|\leq x^4/32,\qquad \log f(x)\leq-x^2/20. For the second assertion one may use :math:`f(x)\leq1-(\alpha-1/256)x^2` and :math:`\alpha-1/256>15/256>1/20`. Also :math:`1\leq A(x)\leq9/8` and :math:`A(x)-1=\tan^2(x/2)\leq x^2/2`. With :math:`G_n=\int_0^\infty e^{-\alpha nx^2}\,dx =\sqrt\pi/(2\sqrt{\alpha n})`, the mean-value inequality for exponentials gives .. math:: \left|\int_0^{1/2}f(x)^nA(x)\,dx -\int_0^{1/2}e^{-\alpha nx^2}\,dx\right| &\leq\int_0^\infty \left(\frac{9n}{256}x^4+\frac12x^2\right)e^{-nx^2/20}\,dx\\ &\leq \frac{21}{n}G_n. The Gaussian moments evaluate the ratio in the last line to at most .. math:: \frac{2\sqrt\alpha}{n} \left(\frac{27\,20^{5/2}}{2048} +\frac{20^{3/2}}8\right)<\frac{21}{n}. For an entirely rational check, use :math:`2\sqrt\alpha<3/5`, :math:`\sqrt{20}<9/2`; the resulting numerator is :math:`5373/256<21`. **The singular part is controlled, not discarded.** Although :math:`A(x)` is singular at :math:`x=\pi`, the exact second moment is finite. The known :math:`c_3=\pi^2` gives .. math:: I_2=\frac{\pi^3}{12L^2}<12. Thus :math:`I_n` exists for all :math:`n\geq2`. Monotonicity and the local estimate give :math:`f(x)\leq f(1/2)\leq79/80` for :math:`x\geq1/2`, so .. math:: \frac1{G_n}\int_{1/2}^\pi f(x)^nA(x)\,dx \leq4\sqrt n\,(79/80)^{n-2}. Here :math:`G_n>3/\sqrt n` follows from :math:`\alpha<1/12` and :math:`\pi>3`. Separately, the elementary Gaussian-tail bound gives .. math:: \frac1{G_n}\int_{1/2}^\infty e^{-\alpha nx^2}\,dx \leq\frac{16}{3\sqrt n}e^{-n/64}. For every real :math:`n\geq1024`, each of these two displayed right sides is at most :math:`1/n`. For the first, dominate :math:`(79/80)^{n-2}` by :math:`e^{-(n-2)/80}` and differentiate the logarithm of :math:`n^{3/2}e^{-(n-2)/80}`; it decreases after :math:`n=120`. At :math:`n=1024`, :math:`e^{1022/80}>2^{18}>4\,1024^{3/2}`, using :math:`18\log2<18(7/10)<1022/80`. The second follows in the same way: :math:`n^{1/2}e^{-n/64}` decreases after :math:`n=32`, and at 1024 its required bound follows from :math:`e^{16}>2^{16}>512/3`. Combining the three errors proves .. math:: \left|I_n/G_n-1\right|\leq24/n\qquad(n\geq1024). The exact normalization makes this a relative bound with leading term :math:`\widetilde B_M=B_M\sqrt{q/n}`. Finally, :math:`1\leq\sqrt{q/n}=\sqrt{1+1/n}\leq1+1/(2n)`, and hence .. math:: |C_M/B_M-1|\leq(24/n)(1+1/(2n))+1/(2n)\leq25/n. 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 :eq:`risk-large-vertex-normalization`. It encloses :math:`\log2` by its positive series, :math:`\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 :math:`M`. The simple relative radius is :math:`25/(M-2)` for :math:`M\geq1026`. For :math:`4\leq M<1026`, the same proof gives a weaker valid radius. Put .. math:: E_n=\frac{21}{n}+4\sqrt n(79/80)^{n-2} +\frac{16}{3\sqrt n}(64/65)^n. Then :math:`(1+1/(2n))E_n+1/(2n)` bounds :math:`|C_M/B_M-1|`; use :math:`e^{-1/64}<64/65`. The tool rounds :math:`\sqrt n` outward and clips a negative lower constant endpoint to zero. The small-:math:`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 :math:`I_n`; it does not use the large-:math:`M` error bound. At a cutoff :math:`0