A classical closed form for the codimension-four constant

Keep the uniform-simplex assumptions of Theory of the skeleton risk. The general constant and fixed-codimension asymptotic are already proved in Vertex coefficients at every fixed vertex count and Positive diagonal formulas for nonvertex risk. This chapter evaluates the previously unevaluated constant \(c_5\) by a finite combination of ordinary Riemann zeta values. It does not change either asymptotic domain.

Theorem 44 (The fifth constant)

\[c_5=900\zeta(4)-600\zeta(3)-120\zeta(2)>0.\]

Consequently the six-vertex leading constant is

\[C_6=540c_5 =486000\zeta(4)-324000\zeta(3)-64800\zeta(2),\]

and, for every fixed integer \(s\geq2\),

\[a_{s-1}(s+4,D)\sim A_{s,4}D^{s-6},\qquad A_{s,4}=\binom{s+4}{s}^{\!2} \frac{s(s+1)(s+2)(s+3)}{80(s-2)!}\,c_5.\]

These are leading constants as the degree tends to infinity with \(s\) fixed, not a finite-degree or finite-sample error bound.

Proof. Write \(H=H_{n-1}\) and \(G=H_{n-1}^{(2)}\), including \(H_0=G_0=0\). The proved elementary-symmetric formula gives

\[\begin{split}\phi_1(n-1)&=1/n,\\ \phi_2(n-1)&=\frac{2(1-H)}{n(n+1)},\\ \phi_3(n-1)&=\frac{6(H^2-G-3H+1)}{n(n+1)(n+2)}.\end{split}\]

Thus, with \(t_n=e_4(n-1)/24\),

\[t_n=\frac{2(H^2-G-3H+1)}{n^2(n+1)(n+2)} +\frac{(H-1)^2}{n^2(n+1)^2}.\]

Define

\[U(n,H,G)=\frac{6n^2+9n+2}{2n^2(n+1)}H^2 -\frac{2n+3}{2n(n+1)}G+\frac{H}{n+1}.\]

Direct rational simplification, using \(H_n=H+1/n\) and \(H_n^{(2)}=G+1/n^2\), proves the finite identity

(88)\[t_n=U(n+1,H_n,H_n^{(2)})-U(n,H,G) +\frac{3H^2-G-5H-1}{n^2}.\]

All series here converge absolutely: use \(H_{n-1}\leq1+\log n\), \(G\leq2\) and the displayed rational denominators. The first boundary is \(U(1,0,0)=0\), and the other boundary tends to zero, since \(U(n,H,G)=O((1+\log n)^2/n)\). Telescoping is therefore valid without assigning values to any divergent harmonic series.

Use the descending-index convention \(\zeta(k_1,\ldots,k_r)=\sum_{n_1>\cdots>n_r\geq1} n_1^{-k_1}\cdots n_r^{-k_r}\). Expanding the finite square gives

\[\begin{split}\sum_{n\geq1}\frac{H_{n-1}^2}{n^2} &=2\zeta(2,1,1)+\zeta(2,2),\\ \sum_{n\geq1}\frac{H_{n-1}^{(2)}}{n^2}&=\zeta(2,2),\qquad \sum_{n\geq1}\frac{H_{n-1}}{n^2}=\zeta(2,1).\end{split}\]

The needed low-weight identities are \(\zeta(2,1)=\zeta(3)\), \(\zeta(2,1,1)=\zeta(4)\) and \(\zeta(2,2)=3\zeta(4)/4\). For completeness, the iterated integral of the word \(0^{k_1-1}1\cdots0^{k_r-1}1\) over \(1>t_1>\cdots>t_w>0\), with \(\omega_0=dt/t,\omega_1=dt/(1-t)\), equals the corresponding multiple zeta value, by expanding the geometric series and integrating. The substitution \(t_i\mapsto1-t_{w+1-i}\) reverses the word and interchanges 0 and 1. It sends 011 to 001 and 0111 to 0001, proving the first two identities. The diagonal/off-diagonal split of \(\zeta(2)^2\) gives \(2\zeta(2,2)+\zeta(4)\); Euler’s values \(\zeta(2)=\pi^2/6,\zeta(4)=\pi^4/90\) prove the third. These standard identities are also described in Henrik Bachmann’s Multiple Zeta Values, version 24, Corollary 2.6 and the discussion on printed page 65.

Since \(c_5=5\sum e_4(j)=120\sum t_n\), the telescope gives \(120(6\zeta(2,1,1)+2\zeta(2,2)-5\zeta(2,1)-\zeta(2))\), proving the claimed formula. Positivity follows from the already proved positive integral for \(c_5\); it is not inferred from floating-point cancellation. Substituting into the existing fixed-M and fixed-codimension theorems proves both consequences.

Exact rational evaluation of the finite expression

For integers \(k\geq2\), define

\[d_{m,k}=\sum_{j=0}^m\frac{(-1)^j\binom mj}{(j+1)^k} =\frac1{(k-1)!}\int_0^1(1-x)^m(-\log x)^{k-1}\,dx.\]

They satisfy \(0<d_{m,k}\leq1\). Summing the nonnegative geometric series inside the integral proves

\[\zeta(k)=\frac{1}{1-2^{1-k}} \sum_{m\geq0}2^{-m-1}d_{m,k}.\]

Indeed the integral of \(1/(1+x)\) equals \((1-2^{1-k})\zeta(k)\) by the even/odd split of the absolutely convergent ordinary zeta series. Therefore the sum of the first \(N\geq1\) terms is a rational lower bound, with upper error \(2^{-N}/(1-2^{1-k})\). No floating-point or unproved sign is used.

The evaluator uses the equivalent coefficient formula

\[d_{m,k}=\frac1{m+1}[z^{k-1}] \prod_{r=1}^{m+1}(1-z/r)^{-1}.\]

To prove it, sum the integral in powers of \(z\) near zero and evaluate \(\int_0^1x^{-z}(1-x)^m dx =m!/\prod_{r=1}^{m+1}(r-z)\), by integration by parts. Updating the complete homogeneous coefficients requires only a linear number of rational operations for fixed k. Rational integer bit lengths increase with N. This is an evaluator for a finite special-value formula; its requested numerical precision still requires increasing work. Negative coefficients in the c5 formula reverse the relevant endpoints. The positive-integral theorem permits clamping its lower bound to zero.

Use and scope

$ .venv/bin/python -m tools.risk_research.c5_closed_form --terms 64 --face-size 2 --output .research/skeleton-risk/c5.json
$ .venv/bin/python -m tools.risk_research.c5_closed_form --verify .research/skeleton-risk/c5.json

The record contains c5, C6 and A_(s,4) intervals, the exact finite zeta coefficients, and source/proof hashes. Verification restores saved terms and face size and compares the entire strictly typed JSON record. Hashes normalize CRLF to LF; raw-source guards still detect concurrent edits. A failed generation or verification preserves existing output. Outputs use a checked canonical parent and the original leaf name in a directory controlled by the caller. This is not a sandbox against another process replacing that canonical directory itself.

Tests check the finite telescope using exact arithmetic, compare the complete-homogeneous recurrence with the alternating binomial definition, and exercise saved options, record tampering and protected outputs. Frozen independent checks use the original general-vertex rational series and the original positive Arb integral, rather than another evaluation of the new zeta expression. These finite checks validate the implementation and normalizations; the identities above provide the proof. No c6 or arbitrary-q classical reduction, degree-error rate, finite-state recurrence, or simultaneous parameter limit is asserted.