A classical leading constant for eight vertices

Keep the uniform-simplex design, evaluation and noise assumptions of Theory of the skeleton risk. The finite construction of Finite MZV constructions for leading constants already handles every fixed constant index. This chapter reduces its seventh-index expression using only convergent identities of weight at most six.

Theorem 46 (A classical expression for c7)

\[\begin{split}c_7={}&211680\zeta(6)-189000\zeta(5)-75250\zeta(4) -18900\zeta(3)-3920\zeta(2)\\ &-78960\zeta(2)\zeta(3)+85680\zeta(2)\zeta(4) +50400\zeta(3)^2+5040\zeta(2)^3>0.\end{split}\]

Consequently, for eight vertices fixed,

\[a_0(8,D)\sim\frac{C_8}{D^9},\qquad C_8=10080c_7.\]

This is a leading-constant identity, not a finite-degree remainder, an all-degree comparison or a minimal classical-zeta basis.

The finite-MZV coefficient calculation

Use strict descending indices \(\zeta(k_1,\ldots,k_r)=\sum_{n_1>\cdots>n_r\geq1} n_1^{-k_1}\cdots n_r^{-k_r}\). The proved general construction gives

(90)\[\begin{split}c_7={}&-3920\zeta(2)-18900\zeta(2,1)-62020\zeta(2,1,1)\\ &-267960\zeta(2,1,1,1)+352800\zeta(2,1,1,1,1)\\ &+100800\bigl(\zeta(2,1,1,2)+\zeta(2,1,2,1) +\zeta(2,2,1,1)\bigr)\\ &-78960\bigl(\zeta(2,1,2)+\zeta(2,2,1)\bigr) -17640\zeta(2,2)+30240\zeta(2,2,2).\end{split}\]

For completeness, this is an exact rational specialization, not a coefficient fit. At \(n\geq1\), take

\[P_a=\sum_{l=0}^{a-1}(-1)^l a!(a-1)! E_{a-1-l}(1,1/2,\ldots,1/(a-1))H_{n-1}(1^l).\]

The initial harmonic kernels are the finite product

\[7e_6(n-1)=7\sum_{a=1}^{5}\binom6a \frac{P_aP_{6-a}}{(n)_a(n)_{6-a}}.\]

Expand each product of finite all-one words with the proved finite stuffle rule. For each rational kernel, the general partial-fraction identity \(f(n)=G(n+1)-G(n)+A/n^2\) and finite summation by parts reduce the remaining word depth. The terminal empty-word contribution is \(A\zeta(2)-G(1)\); a nonempty word \((s,v)\) contributes \(A\zeta(2,s,v)\) and the shorter kernel \(-G(n+1)/n^s\). These explicit rational operations give precisely the twelve coefficients in (90); their rational unit coefficient sums to zero at this index.

The general theorem proves these transformations before taking limits, including all finite upper boundaries \(G(N+1)H_N(w)\), their vanishing, and absolute convergence. Specializing its finite coefficient calculation to index seven does not require a second convergence assumption. The coefficient table is reproducible by leading_constant_expression(7); tests additionally retain every finite upper boundary and compare with the original elementary-symmetric coefficient route.

Reduction by convergent products and duality

The iterated-integral duality proved in A classical closed form for the codimension-four constant reverses a binary word and exchanges zero and one. In the present descending convention it gives

\[\begin{split}\zeta(2,1)&=\zeta(3),& \zeta(2,1,1)&=\zeta(4),& \zeta(2,1,1,1)&=\zeta(5),\\ \zeta(2,1,1,1,1)&=\zeta(6),& \zeta(2,1,2)&=\zeta(2,3),& \zeta(2,2,1)&=\zeta(3,2),\\ \zeta(2,1,1,2)&=\zeta(2,4),& \zeta(2,1,2,1)&=\zeta(3,3),& \zeta(2,2,1,1)&=\zeta(4,2).\end{split}\]

Each equality is directly checked by the corresponding finite binary word; the duality of its convergent integral supplies the equality of values. The earlier chapter also proves \(\zeta(2,2)=3\zeta(4)/4\).

Splitting the absolutely convergent products into greater, smaller and equal summation indices gives

\[\begin{split}\zeta(2,3)+\zeta(3,2)&=\zeta(2)\zeta(3)-\zeta(5),\\ \zeta(2,4)+\zeta(4,2)&=\zeta(2)\zeta(4)-\zeta(6),\\ 2\zeta(3,3)&=\zeta(3)^2-\zeta(6).\end{split}\]

For the triple product, partition the three indices at a finite cutoff into the six strict orderings, the three choices for either equal pair, and the all-equal case. Taking the absolutely convergent limit gives

\[\zeta(2)^3=6\zeta(2,2,2) +3\zeta(2,4)+3\zeta(4,2)+\zeta(6).\]

In particular, with \(T=\zeta(2,4)+\zeta(3,3)+\zeta(4,2)\),

\[\begin{split}T&=\zeta(2)\zeta(4)+\tfrac12\zeta(3)^2-\tfrac32\zeta(6),\\ \zeta(2,2,2)&= \frac{\zeta(2)^3-3\zeta(2)\zeta(4)+2\zeta(6)}6.\end{split}\]

Substituting into (90) proves the displayed classical expression by rational algebra. The positive original integral proves \(c_7>0\) independently of its signed representation. Finally the existing normalization gives \(C_8=8^2\,7!\,2^{-5}c_7=10080c_7\). No numerical integer-relation search, assumption of basis independence, or new even-zeta evaluation is used. The retained classical monomials need not be independent; reducing this bounded index does not initiate an automatic sequence of further individual constants.

Rational intervals and saved records

tools.risk_research.c7_closed_form uses the positive rational ordinary-zeta intervals proved in A classical closed form for the codimension-four constant. For the fixed orders two through six their Euler-transformed tail is at most \(2^{-N}/(1-2^{1-k})\). Positive factor intervals multiply endpointwise, including repeated factors in \(\zeta(3)^2\) and \(\zeta(2)^3\). A negative coefficient reverses the resulting endpoints. The original positive integral permits intersecting the final c7 interval with the nonnegative half-line.

For these fixed orders and monomials the rational-operation count is linear in N; integer lengths grow. This does not assert a uniform complexity bound for arbitrary zeta weight.

The mathematical tail bound holds for every integer \(N\geq1\). The evaluator separately accepts only \(1\leq N\leq4096\) (MAX_TERMS=4096). Direct evaluation, record generation and record verification reject larger cutoffs before invoking the ordinary-zeta helper. This operational cutoff is not a CPU-time or memory guarantee; it does not restrict the all-N mathematical identity.

$ python -m tools.risk_research.c7_closed_form --terms 64 --output .research/skeleton-risk/c7.json
$ python -m tools.risk_research.c7_closed_form --verify .research/skeleton-risk/c7.json

The independent record schema stores exact c7/C8 intervals, signed ordinary-zeta and monomial coefficients, descending MZV coefficients, terms and normalized source hashes. Reverification restores nondefault terms and compares the complete strictly typed record. The fingerprints bind imported research utilities, the finite construction and its proof, and the identities used here. Stable CRLF and LF sources share hashes; raw snapshots still reject any in-flight source change. Direct builders, verifiers and both CLI actions preserve that contract.

Duplicate or unknown fields, changed types, coefficients and sources are rejected. Protected evaluator/proof destinations cannot be replaced; failed generation and verification preserve existing output/input. The atomic writer fixes the canonical parent and original leaf in a caller-controlled directory. It is not a sandbox against another process replacing that directory inode.

After the formula and evaluator are frozen, fresh cutoff/precision holdouts compare with the original rational series and positive Arb integral. Index seven itself was already used in the previous phase, so it is not described as an unused index. These finite checks verify implementation and normalization; the convergent identities prove the formula. No finite-sample guarantee, finite-D remainder or arbitrary-index classical-zeta reduction follows from this chapter.