Five-vertex vertex risk ======================= The additive-kernel transform from :doc:`skeleton_risk_vertex` also resolves the vertex coefficient on a simplex with five vertices. The result is one sum with four harmonic states, and its leading constant involves :math:`\zeta(2)-\zeta(3)`. A positivity argument for the transformed diagonals applies to every fixed vertex count. All statements use the uniform-simplex assumptions of :doc:`skeleton_risk_theory`. The sum has :math:`D` terms. It is not a bounded-length classical-special-function expression. :doc:`skeleton_risk_five_vertex_recurrence` gives an explicit certified order-seventeen degree recurrence with initial values and no exceptional positive integer indices. The broader construction and fixed-dimension leading constants are in :doc:`skeleton_risk_vertex_general`. The five-vertex leading term has an explicit remainder bound in :doc:`skeleton_risk_five_vertex_asymptotics`. Together with the nonvertex edge upper bound, this also completes the all-degree risk comparison in :doc:`skeleton_risk_five_vertex_comparison`. An exact harmonic weight ------------------------ Write :math:`H_j^{(r)}=\sum_{k=1}^j k^{-r}`, :math:`H_j=H_j^{(1)}`, and define the depth-two finite harmonic sum .. math:: A_j=\sum_{k=1}^j\frac{H_{k-1}}{k^2}. All four states vanish at zero. They update by :math:`H_j^{(r)}=H_{j-1}^{(r)}+j^{-r}` and :math:`A_j=A_{j-1}+H_{j-1}/j^2`; no inner summation is needed during evaluation. For :math:`n=j+1`, abbreviate :math:`H=H_j`, :math:`H_2=H_j^{(2)}`, :math:`H_3=H_j^{(3)}`, and :math:`A=A_j`. Define .. math:: :label: risk-five-vertex-diagonal \Psi_j={}&\frac{24H^3}{n(n+1)(n+2)(n+3)}\\ &-\frac{12(5n^3+7n^2-6n-12)H^2} {n^2(n+1)^2(n+2)^2(n+3)}\\ &+\frac{24(2n+3)HH_2}{n(n+1)(n+2)(n+3)}\\ &-\frac{24(2n^4+15n^3+44n^2+47n+24)H} {n^2(n+1)^2(n+2)^2(n+3)}\\ &+\frac{12(4n^4+22n^3+41n^2+26n+12)H_2} {n^2(n+1)(n+2)^2(n+3)}\\ &-\frac{48H_3}{n(n+1)(n+3)} -\frac{48A}{n(n+2)} +\frac{144}{n^2(n+1)^2(n+2)^2(n+3)}. .. prf:theorem:: Five-vertex vertex coefficient at every degree :label: thm-risk-five-vertex-single For every integer :math:`D\geq1`, let :math:`W_{D,j}=\binom{D-1}j\binom{D+1}j` and :math:`Z_D=\binom{2D}{D-1}`. Then .. math:: :label: risk-five-vertex-single a_0(5,D)=\frac{600} {Z_D(2D+1)(2D+2)(2D+3)(2D+4)} \sum_{j=0}^{D-1}W_{D,j}\Psi_j. Every :math:`\Psi_j` is strictly positive, with :math:`\Psi_0=1`. All denominators in the formula are nonzero on the stated domain. .. prf:proof:: Put :math:`g(t)=-(1-t)\log(1-t)/t` and :math:`\phi_r(j)=[t^j]g(t)^r/(1-t)`. Besides the three formulas in :doc:`skeleton_risk_vertex`, differentiating :math:`\binom\alpha{j+4}` four times at :math:`\alpha=3` gives .. math:: \phi_4(j)=\frac{24\left( -H^3+\tfrac{11}2H^2-6H+1 +(3H-\tfrac{11}2)H_2-2H_3\right)} {n(n+1)(n+2)(n+3)}. For sequences, let :math:`*` denote ordinary convolution and let :math:`g_k=[t^k]g(t)`. Expand the additive kernel to the fourth power. Terms in which one of :math:`g(x),g(y)` is absent combine by convolution. The result is .. math:: :label: risk-five-vertex-quartic-convolutions [x^jy^j]\frac{(g(x)+g(y)-g(xy))^4}{(1-x)(1-y)} =-\phi_4(j)+8\phi_1(j)\phi_3(j)+6\phi_2(j)^2 -24\bigl(g*(\phi_1\phi_2)\bigr)_j +12\bigl(g^2*\phi_1^2\bigr)_j. Here :math:`g^2` denotes the coefficient sequence of the square of the series, while the products of :math:`\phi` sequences are pointwise. In particular, the coefficient :math:`-24` includes both the :math:`g(x)^2g(y)g(xy)` term and its symmetric counterpart. The two remaining convolutions reduce to finite harmonic sums. Since .. math:: g_0=1,\quad g_k=-\frac1{k(k+1)},\qquad [t^0]g^2=1,\quad [t^k]g^2=\frac{4H_{k-1}-6}{k(k+1)(k+2)}\quad(k\geq1), their contribution, with :math:`n=j+1`, is .. math:: -24\bigl(g*(\phi_1\phi_2)\bigr)_j +12\bigl(g^2*\phi_1^2\bigr)_j =-\frac{48(1-H)}{n^2(n+1)}+\frac{12}{n^2} +\sum_{k=1}^{n-1}\bigl(R_H(k,n)H_{k-1}+R_0(k,n)\bigr), where .. math:: p(k,n)&=\frac1{k^2(k+1)(n-k)(n-k+1)},\\ q(k,n)&=\frac1{k(k+1)(k+2)(n-k)^2},\\ R_H(k,n)&=48(q(k,n)-p(k,n)),\\ R_0(k,n)&=48p(k,n)-72q(k,n). Partial fractions use only the seven denominators :math:`k,k^2,k+1,k+2,n-k,(n-k)^2,n-k+1`. The sums without :math:`H_{k-1}` are immediate harmonic numbers. For the remaining sums, put :math:`T=(H^2-H_2)/2`; their respective values are .. math:: \sum_{k=1}^{n-1}\frac{H_{k-1}}k&=T,\\ \sum_{k=1}^{n-1}\frac{H_{k-1}}{k^2}&=A,\\ \sum_{k=1}^{n-1}\frac{H_{k-1}}{k+1} &=T+H/n-1+1/n,\\ \sum_{k=1}^{n-1}\frac{H_{k-1}}{k+2} &=T+H\left(1/n+1/(n+1)\right)+1/(n(n+1)) -7/4+1/(2n)+3/(2(n+1)),\\ \sum_{k=1}^{n-1}\frac{H_{k-1}}{n-k}&=H^2-H_2,\\ \sum_{k=1}^{n-1}\frac{H_{k-1}}{(n-k)^2}&=A+HH_2-H_3,\\ \sum_{k=1}^{n-1}\frac{H_{k-1}}{n-k+1}&=H^2-H_2+2H/n-H. These identities follow by changing the order of the finite sums; for example, the penultimate sum is :math:`\sum_{i+r\leq n-1}1/(ir^2)`, and partial fractions at :math:`i+r=s` give :math:`2A+\sum_{s\leq n-1}H_{s-1}^{(2)}/s` :math:`{}=A+HH_2-H_3`. The formulas also hold when :math:`n=1` and the sums are empty. Substitution and collection give exactly :eq:`risk-five-vertex-diagonal`. The binomial-transform diagonalization now gives .. math:: a_0(5,D)=\frac{25(D-1)!(D+1)!}{(5)_{2D}} \sum_j W_{D,j}\Psi_j. Use :math:`(D-1)!(D+1)!Z_D=(2D)!` to obtain :eq:`risk-five-vertex-single`. Positivity follows from the general argument below. Positive diagonals for every vertex count ----------------------------------------- .. prf:proposition:: Outside-kernel powers remain positive after transformation :label: prop-risk-positive-transformed-kernel For every integer :math:`q\geq1` and :math:`j\geq0`, .. math:: [x^jy^j]\frac{(g(x)+g(y)-g(xy))^q}{(1-x)(1-y)}>0. .. prf:proof:: Let :math:`T_1,\ldots,T_q` be independent exponential random variables of mean one, and define the homogeneous polynomial .. math:: Q_h(T)=\sum_{\alpha_1+\cdots+\alpha_q=h} \prod_{r=1}^q\frac{T_r^{\alpha_r}}{(\alpha_r+1)!}. Since :math:`\mathbb E[T_r^k]=k!`, multiplication of the outside kernel :math:`F` gives .. math:: [x^hy^l]F(x,y)^q=\mathbb E[Q_h(T)Q_l(T)]. The signed binomial transform in each index therefore identifies the diagonal in the proposition with .. math:: \mathbb E\!\left[ \left(\sum_{h=0}^j(-1)^h\binom jh Q_h(T)\right)^2\right]. The polynomial inside the square is nonzero: its highest homogeneous part is :math:`(-1)^jQ_j`, including the case :math:`j=0`. Exponential density is strictly positive on the positive orthant, so the expectation is strictly positive. This proves positivity without requiring each term of a collected harmonic expression to be positive. The leading asymptotic constant ------------------------------- .. prf:corollary:: Five-vertex vertex asymptotic :label: cor-risk-five-vertex-asymptotic As :math:`D\to\infty`, .. math:: :label: risk-five-vertex-asymptotic a_0(5,D)=\frac{7200(\zeta(2)-\zeta(3))}{D^6} +O\!\left(\frac{\log D}{D^7}\right). .. prf:proof:: The Euler sum :math:`A_j` tends to :math:`\zeta(3)`. Indeed, monotone convergence and the power series for the logarithm give .. math:: \sum_{k=1}^\infty\frac{H_{k-1}}{k^2} =\frac12\int_0^1\frac{\log^2(1-t)}t\,dt =\frac12\int_0^1\frac{\log^2t}{1-t}\,dt =\zeta(3). Moreover, for :math:`j\geq1`, .. math:: 0\leq\zeta(2)-H_j^{(2)}\leq1/j,\qquad 0\leq\zeta(3)-A_j\leq(2+\log j)/j. The second inequality follows from :math:`H_{k-1}\leq1+\log k` and comparison with the decreasing function :math:`(1+\log x)/x^2`. The coefficient of :math:`H_2` in :eq:`risk-five-vertex-diagonal` differs from :math:`48/(n(n+2))` by .. math:: -\frac{12(2n^3+3n^2-2n-12)} {n^2(n+1)(n+2)^2(n+3)}=O(n^{-3}). All the other terms apart from :math:`-48A/(n(n+2))` are :math:`O(\log n/n^3)`; the cubic harmonic term is :math:`O(\log^3n/n^4)=O(\log n/n^3)`. Consequently .. math:: \Psi_j=\frac{48(\zeta(2)-\zeta(3))}{(j+1)(j+3)} +O\!\left(\frac{\log(j+2)}{(j+1)^3}\right). Let :math:`J` have probabilities :math:`W_{D,j}/Z_D`. The mean, variance and small-index bound proved in :doc:`skeleton_risk_vertex` apply unchanged: .. math:: \mathbb E[J]=\frac{D^2-1}{2D},\qquad \operatorname{Var}(J)=O(D),\qquad \mathbb P(J\leq\lfloor D/4\rfloor)\leq4\,2^{-D/12} \quad(D\geq12). On the complementary event, the remainder is uniformly :math:`O(\log D/D^3)`. The sequence :math:`\Psi_j` is bounded, so the omitted event contributes an exponentially small amount. Taylor expansion of :math:`1/((x+1)(x+3))` about the mean, with second derivative :math:`O(D^{-4})` and variance :math:`O(D)`, gives .. math:: \mathbb E[\Psi_J] =\frac{192(\zeta(2)-\zeta(3))}{D^2} +O(\log D/D^3). Finally, :eq:`risk-five-vertex-single` is :math:`600\mathbb E[\Psi_J]/\prod_{r=1}^4(2D+r)`. Expanding this rational factor proves the assertion. Reproduction and limits ----------------------- ``tools.risk_research.vertex_five.five_vertex_coefficient`` evaluates :eq:`risk-five-vertex-single` using exact rational arithmetic, four harmonic states, a binomial weight and an accumulator. The number of rational states is fixed, while their bit lengths increase with :math:`D`. Floating-point evaluation of the ungrouped weight can lose precision through cancellation; the exact evaluator avoids that loss. The tests compare the new expression with the original scalar kernel sum at degrees 1 through 20, independent Gram propagation at degrees 1 through 5, and direct signed binomial transforms of :math:`F^4`. After the candidate was frozen, a separate integer-polynomial :math:`F^4` computation at degrees 193, 227 and 251 agreed exactly. The committed holdout file records the candidate and oracle source hashes. These checks support the implementation; the algebraic and asymptotic arguments above establish the mathematical identities.