Tetrahedron vertex risk ======================== Under the uniform-simplex assumptions of :doc:`skeleton_risk_theory`, the tetrahedron vertex coefficient has a positive single sum involving only binomial coefficients and harmonic numbers. This eliminates the bivariate kernel convolution. The sum still has :math:`D` terms. An explicit certified degree recurrence is proved in :doc:`skeleton_risk_vertex_recurrence`; a bounded classical-special-function expression remains open. The next asymptotic term and a fully explicit error bound are proved in :doc:`skeleton_risk_vertex_asymptotics`. A binomial-transform diagonalization -------------------------------------- Put :math:`A_h=(-1)^h\binom{D-1}h\binom{D+1}h`, :math:`C_D=(D-1)!(D+1)!`, and :math:`W_{D,j}=\binom{D-1}j\binom{D+1}j`. As in :doc:`skeleton_risk_extensions`, define .. math:: \mathcal L_D(G)=\sum_{h,l=0}^{D-1} A_hA_lh!l!(2D-h-l)!\,[x^hy^l]G. .. prf:theorem:: Diagonalization by signed binomial transforms :label: thm-risk-binomial-gram-diagonal For every integer :math:`D\geq1` and arrays :math:`u,v` of length :math:`D`, set .. math:: \widehat u_j=\sum_{h=0}^j(-1)^h\binom jh u_h, \qquad \widehat v_j=\sum_{l=0}^j(-1)^l\binom jl v_l. Then .. math:: :label: risk-binomial-gram-diagonal \sum_{h,l=0}^{D-1}A_hA_lh!l!(2D-h-l)!u_hv_l =C_D\sum_{j=0}^{D-1}W_{D,j}\widehat u_j\widehat v_j. Consequently, if .. math:: \mathscr T(G)(x,y)= \frac{G(-x/(1-x),-y/(1-y))}{(1-x)(1-y)}, then .. math:: :label: risk-transformed-vertex-functional \mathcal L_D(G)=C_D\sum_{j=0}^{D-1} W_{D,j}[x^jy^j]\mathscr T(G). .. prf:proof:: Write :math:`n=D-1` and :math:`P_u(t)=\sum_{h=0}^n A_hh!u_ht^{D-h}`. The Laguerre expansion .. math:: L_k^{(2)}(t)=\sum_{r=0}^k (-1)^r\binom{k+2}{k-r}\frac{t^r}{r!} and binomial inversion give .. math:: P_u(t)=(-1)^n n!t\sum_{j=0}^n (-1)^j\binom{n+2}j\widehat u_jL_{n-j}^{(2)}(t). Equality of the coefficients of :math:`t^{r+1}` is exactly ordinary binomial inversion for :math:`u_{n-r}`. Integrate :math:`P_u(t)P_v(t)e^{-t}` on :math:`[0,\infty)`. The monomial moments are :math:`(2D-h-l)!`, while Laguerre orthogonality gives .. math:: \int_0^\infty t^2e^{-t}L_k^{(2)}(t)L_l^{(2)}(t)\,dt =\mathbf1_{k=l}\frac{(k+2)!}{k!}. Its surviving coefficient satisfies .. math:: n!^2\binom{n+2}j^2\frac{(n+2-j)!}{(n-j)!} =C_DW_{D,j}, proving :eq:`risk-binomial-gram-diagonal`. Apply the identity linearly to each coefficient of :math:`G`; the generating-function identity for signed binomial transforms is precisely :math:`\mathscr T`. All relevant arrays and polynomials are finite. The logarithmic formula for the outside kernel simplifies under this substitution. With :math:`f(t)=-\log(1-t)/t` and :math:`g(t)=(1-t)f(t)`, direct collection of logarithms gives .. math:: :label: risk-transformed-outside-kernel F\!\left(-\frac{x}{1-x},-\frac{y}{1-y}\right) =g(x)+g(y)-g(xy). Indeed, if :math:`X=-x/(1-x)`, :math:`Y=-y/(1-y)`, then :math:`1-X-Y=(1-xy)/((1-x)(1-y))`. Substitution into the numerator :math:`(1-X-Y)\log(1-X-Y)-(1-X)\log(1-X)-(1-Y)\log(1-Y)` of :math:`F(X,Y)` proves :eq:`risk-transformed-outside-kernel`. For every fixed vertex count, this transforms :math:`\mathcal L_D(F^{M-1})` into a weighted diagonal of the corresponding power of an additive kernel. No evaluation of all those diagonals is claimed. The positive single sum ------------------------- Write :math:`H_j=\sum_{r=1}^j1/r`, :math:`H_j^{(2)}=\sum_{r=1}^j1/r^2`, with both values zero at :math:`j=0`. .. prf:theorem:: Tetrahedron vertex coefficient at every degree :label: thm-risk-tetrahedron-vertex-single For every integer :math:`D\geq1`, .. math:: :label: risk-tetrahedron-vertex-single a_0(4,D)=\frac{96(D-1)!(D+1)!}{(4)_{2D}} \sum_{j=0}^{D-1}\binom{D-1}j\binom{D+1}j G_j, where .. math:: :label: risk-tetrahedron-vertex-weight G_j={}&\frac{H_j^2}{(j+1)(j+2)(j+3)} +\frac{H_j^{(2)}}{(j+1)(j+3)}\\ &-\frac{jH_j}{(j+1)(j+2)^2(j+3)} +\frac{2}{(j+1)^2(j+2)^2(j+3)}. Every weight :math:`G_j` is strictly positive, including :math:`G_0=1/6`. All denominators are nonzero on the stated domain. .. prf:proof:: Set :math:`\phi_r(j)=[t^j]g(t)^r/(1-t)`. Elementary logarithmic coefficient identities give .. math:: \phi_1(j)&=\frac1{j+1},\\ \phi_2(j)&=\frac{2(1-H_j)}{(j+1)(j+2)},\\ \phi_3(j)&=\frac{6(H_j^2-H_j^{(2)}-3H_j+1)} {(j+1)(j+2)(j+3)}. For example, differentiate the finite product :math:`\binom\alpha{j+r}` :math:`r` times at :math:`\alpha=r-1`, using :math:`g(t)^r/(1-t)=(1-t)^{r-1}(-\log(1-t))^r/t^r`. The product has a simple zero there; the logarithmic derivatives of its remaining factors give the three formulas, including :math:`j=0`. Expand :math:`(g(x)+g(y)-g(xy))^3` and divide by :math:`(1-x)(1-y)`. The pure powers contribute :math:`\phi_3(j)`. The terms :math:`g(x)^2g(y)` and its symmetric counterpart contribute :math:`6\phi_1(j)\phi_2(j)`. The diagonal contributions of :math:`g(x)^2g(xy)` and :math:`g(x)g(xy)^2`, together with their symmetric counterparts, cancel by ordinary convolution. Thus .. math:: [x^jy^j]\mathscr T(F^3) =\phi_3(j)+6\phi_1(j)\phi_2(j) -6\sum_{k=0}^j[t^k]g(t)\phi_1(j-k)^2. Since :math:`[t^0]g=1` and :math:`[t^k]g=-1/(k(k+1))` for :math:`k\geq1`, the convolution equals :math:`1/(j+1)^2-B_j`, where .. math:: B_j={}&\sum_{r=0}^{j-1} \frac1{(r+1)(r+2)(j-r)^2}\\ ={}&\frac{H_j^{(2)}}{(j+1)(j+2)} +\frac{2(2j+3)H_j}{(j+1)^2(j+2)^2} +\frac{j}{(j+1)(j+2)^2}. The last equality follows by partial fractions and includes the empty sum :math:`B_0=0`. Substituting the three :math:`\phi_r` formulas and simplifying gives :math:`[x^jy^j]\mathscr T(F^3)=6G_j`. Apply :eq:`risk-transformed-vertex-functional` and the scalar risk identity :math:`a_0(4,D)=16\mathcal L_D(F^3)/(4)_{2D}`. For positivity, group the first and third terms of :math:`G_j` as .. math:: \frac{H_j(H_j-j/(j+2))}{(j+1)(j+2)(j+3)}. At :math:`j\geq1`, :math:`H_j\geq1>j/(j+2)`; at zero the last rational term is positive. The second and last terms are nonnegative. Large-degree behavior ----------------------- .. prf:corollary:: Tetrahedron vertex asymptotic :label: cor-risk-tetrahedron-vertex-asymptotic As :math:`D\to\infty`, .. math:: :label: risk-tetrahedron-vertex-asymptotic a_0(4,D)=\frac{48\pi^2}{D^5} +O\!\left(\frac{\log^2D}{D^6}\right). .. prf:proof:: Vandermonde's identity gives :math:`Z_D=\sum_jW_{D,j}=\binom{2D}{D-1}`. Let :math:`J` have probabilities :math:`W_{D,j}/Z_D`. Since :math:`C_DZ_D=(2D)!`, .. math:: a_0(4,D)=\frac{576}{(2D+1)(2D+2)(2D+3)}\,\mathbb E[G_J]. The first two factorial moments, obtained from Vandermonde, give .. math:: \mu=\mathbb E[J]=\frac{D^2-1}{2D},\qquad \operatorname{Var}(J)=\frac{(D^2-1)^2}{4D^2(2D-1)}=O(D). An elementary weight ratio controls the potentially small indices. For :math:`D\geq12`, put :math:`a=\lfloor D/4\rfloor`, :math:`b=\lfloor D/3\rfloor`. For :math:`0\leq ja`, harmonic-number bounds in :eq:`risk-tetrahedron-vertex-weight` give uniformly .. math:: G_j=\frac{\zeta(2)}{(j+1)(j+3)} +O(\log^2D/D^3). Put :math:`r(t)=1/((t+1)(t+3))`. Between :math:`\mu` and any :math:`j>a`, :math:`r''(t)=O(D^{-4})`. Taylor's theorem, the mean-zero linear term, the variance bound, and the exponentially small omitted tail therefore give .. math:: \mathbb E[r(J)]=r(\mu)+O(D^{-3}) =4/D^2+O(D^{-3}). Both :math:`G_j` and :math:`r(j)` are bounded by an absolute constant for all :math:`j\geq0`; the lower tail contributes exponentially little. Hence :math:`\mathbb E[G_J]=2\pi^2/(3D^2)+O(\log^2D/D^3)`. The outer prefactor is :math:`72/D^3+O(D^{-4})`, proving the result. Verification and scope ------------------------ ``tools/risk_research/vertex_single.py`` evaluates :eq:`risk-tetrahedron-vertex-single` with exact rational arithmetic and incremental binomial weights. Tests compare the pre-existing scalar sum through degree 20 and independent full Gram propagation through degree 6. After fixing the formula, the original integer-kernel implementation generated new held-out values at degrees 241, 247 and 260, beyond the previous exploration through degree 240. All three exact fractions agree; degree 260 is retained as an independent regression value. Finite checks verify the implementation; the all-degree claims use the proofs above. The formula and its positivity avoid the severe alternating cancellation of the original scalar sum. :doc:`skeleton_risk_vertex_recurrence` supplies the explicit scalar degree recurrence, while :doc:`skeleton_risk_tetrahedron_comparison` classifies the optimal tetrahedron comparison at every degree. The nonlinear self-identification question remains separate, and the generic P-recursiveness theorem in :doc:`skeleton_risk_extensions` continues to apply independently.