Explicit vertex errors at every fixed vertex count =================================================== Keep the uniform-simplex assumptions of :doc:`skeleton_risk_theory`. The leading constant :math:`C_M` and its exact rational enclosures are proved in :doc:`skeleton_risk_vertex_general`. Here the error of the finite-degree coefficient itself is bounded, for every fixed :math:`M\geq4`. For every :math:`M\geq5`, :doc:`skeleton_risk_vertex_refined_asymptotics` subsequently improves the logarithmic power to one, with explicit constants and the same degree start. The bounds here remain valid, including the critical :math:`M=4`. The constants are deliberately conservative. The sharper existing tetrahedron and five-vertex results remain useful. Computable constants and the error theorem ------------------------------------------ For integers :math:`r\geq1` define .. math:: B_r=r!r^{r-1},\qquad G_r=2^{r+1}r\,r!(r+1)^{r-1}. For :math:`r\geq2`, put .. math:: A_r&=\sum_{a=1}^{r-1}\binom ra B_aB_{r-a},\\ F_r&=2^{r-2}\sum_{a=1}^{r-1}\binom ra (G_aB_{r-a}+B_aG_{r-a}),\\ U_r&=2^{r+1}F_r+(20+3^{r-1})A_r,\\ V_{k,r}&=G_k\left(2^{r+2}F_r+5\,2^kA_r\right) \quad(k\geq2). Set :math:`q=M-1`, and define the following positive, explicit constants: .. math:: :label: risk-general-error-constants J_q&=2qA_{q-1},\\ K_q&=B_q+A_q+qU_{q-1} +\sum_{k=2}^{q-2}\binom qk V_{k,q-k} +8(1+q^2+J_q),\\ R_M&=\frac{M^2q!}{2^q} \left(96K_q+(273+q(q+1))J_q\right). An empty sum is zero. In particular these formulas apply at :math:`M=4`. .. prf:theorem:: Explicit vertex error for every fixed M at least four :label: thm-risk-general-vertex-explicit-error For all integers :math:`M\geq4` and :math:`D\geq256`, .. math:: :label: risk-general-vertex-explicit-error \left|a_0(M,D)-\frac{C_M}{D^{M+1}}\right| \leq R_M\frac{(1+\log D)^{M-2}}{D^{M+2}}. Thus the relative error tends to zero for each fixed :math:`M`. There is no claim that this convergence is uniform when :math:`M` grows with :math:`D`. The squared logarithm at :math:`M=4` is retained; the sharper single logarithm known at :math:`M=5` is not assumed at other vertex counts. Quantitative coefficient bounds -------------------------------- Use the sequences :math:`\phi_r,b_r,e_r,\delta_q` of :doc:`skeleton_risk_vertex_general`, and write :math:`L(x)=1+\log x`. The already proved elementary symmetric formula gives .. math:: |\phi_r(j)|\leq B_r L(j+1)^{r-1}(j+1)^{-r}. For :math:`j\geq1`, differentiating the binomial product at exponent :math:`r`, rather than :math:`r-1`, gives .. math:: |b_r(j)|=\frac{(r!)^2(j-1)!}{(j+r)!} \left|E_{r-1}\left(1,\frac12,\ldots,\frac1r, -1,-\frac12,\ldots,-\frac1{j-1}\right)\right|. The total absolute reciprocal sum is at most :math:`(r+1)L(j+1)`. Also :math:`j\geq(j+1)/2`, so .. math:: |b_r(j)|\leq G_r L(j+1)^{r-1}(j+1)^{-r-1}. The bound holds at :math:`j=0` too, since :math:`b_r(0)=1`. The series is absolutely summable and its sum is :math:`g(1)^r=0`. Using :math:`\phi_r(j+1)-\phi_r(j)=b_r(j+1)` and the product difference identity gives, for every :math:`j\geq0`, .. math:: :label: risk-general-cross-difference-bounds |e_r(j)|&\leq A_r L(j+1)^{r-2}(j+1)^{-r},\\ |e_r(j+1)-e_r(j)| &\leq F_r L(j+1)^{r-2}(j+1)^{-r-1}. Here :math:`L(j+2)\leq2L(j+1)` accounts for the factor :math:`2^{r-2}` in :math:`F_r`. We use a simple integral majorant repeatedly. For integers :math:`p\geq0,\alpha>p+1,T\geq1`, monotonicity and differentiation give .. math:: :label: risk-general-log-tail \sum_{m=T}^{\infty}\frac{L(m+1)^p}{(m+1)^\alpha} \leq\int_T^\infty\frac{L(x)^p}{x^\alpha}\,dx \leq\frac{L(T)^p}{(\alpha-1-p)T^{\alpha-1}}. Indeed the negative derivative of :math:`L(x)^p/x^{\alpha-1}` is at least :math:`(\alpha-1-p)L(x)^p/x^\alpha` on :math:`x\geq1`. In particular :math:`\sum_j|e_r(j)|\leq2A_r`, so :math:`0H}b_k(l)\\ &+\sum_{m=0}^{n-H-1}b_k(n-m)e_r(m). For :math:`k=1`, use the exact formula :math:`b_1(l)=-1/(l(l+1))` at positive indices. The first term is at most :math:`2^{r+1}F_r L^{r-1}/N^{r+1}` in absolute value, because :math:`\sum_{l=1}^H l|b_1(l)|\leq L`. The second is at most :math:`2A_rL^{r-2}/N^{r+1}`, since its tail equals :math:`-1/(H+1)`. For the last term, subtract :math:`-N^{-2}\sum_{m\geq0}e_r(m)`. On its finite range, .. math:: \left|b_1(n-m)+N^{-2}\right| \leq\frac{18(m+1)}{N^3}. This follows directly after clearing the positive denominator :math:`N^2(n-m)(n-m+1)`: use :math:`n-m\geq N/3` and :math:`n-m+1\geq N/2`. Also :math:`\sum_{m\leq n}(m+1)|e_r(m)|\leq A_rL^{r-1}`. For this last bound, take :math:`L(m+1)^{r-2}\leq L^{r-2}` outside the sum and use :math:`\sum_{m=0}^n(m+1)^{1-r}\leq H_N\leq L`. The missing tail starts at :math:`n-H\geq N/3`, and :eq:`risk-general-log-tail` bounds it by :math:`3^{r-1}A_rL^{r-2}/N^{r-1}`. Combining these bounds gives .. math:: \left|(b_1*e_r)(n)+N^{-2}\sum_{m\geq0}e_r(m)\right| \leq U_r L^{r-1}/N^3. For :math:`k\geq2`, the same first term uses :math:`\sum_{l=1}^H l|b_k(l)|\leq2G_kL^{k-1}`. The absolute tail in the second term is at most :math:`2^kG_kL^{k-1}/N^k`. In the last term use :math:`|b_k(n-m)|\leq2^{k+1}G_kL^{k-1}/N^{k+1}` and :math:`\sum_m|e_r(m)|\leq2A_r`. Since :math:`r,k\geq2`, this proves .. math:: |(b_k*e_r)(n)|\leq V_{k,r}L^{k+r-3}/N^3. Substitute these two bounds into :eq:`risk-general-transformed-diagonal`. The :math:`\phi_q` and :math:`e_q` terms are bounded by :math:`(B_q+A_q)L^{q-1}/N^3`. The added term :math:`8(1+q^2+J_q)` in :math:`K_q` also covers :math:`n=0,1`: here :math:`\delta_q(0)=1` and :math:`\delta_q(1)=(q^2-3q+4)/4`. Therefore, for every :math:`n\geq0`, .. math:: :label: risk-general-diagonal-explicit-error \left|\delta_q(n)-\frac{c_q}{(n+1)^2}\right| \leq K_q\frac{L(n+1)^{q-1}}{(n+1)^3}. Averaging and normalization ---------------------------- Use the distribution :math:`\mathbb P(J=j)=W_{D,j}/Z_D` from :doc:`skeleton_risk_vertex_general`. Its already proved moments and tail bound are, for :math:`D\geq12`, .. math:: \mathbb E[J-D/2]&=-1/(2D),\\ \mathbb E[(J-D/2)^2]&\leq D/4,\\ \mathbb P(J\leq\lfloor D/4\rfloor)&\leq\epsilon_D=4\,2^{-D/12}. For :math:`r(t)=(t+1)^{-2}`, the derivative bounds :math:`|r'(t)|\leq2/t^3`, :math:`|r''(t)|\leq6/t^4` and :math:`|r(t)-t^{-2}|\leq2/t^3` hold at :math:`t>0`. Taylor expansion at :math:`D/2` on the central event gives :math:`192/D^3` from the quadratic remainder, :math:`1/D^3` from the mean, and :math:`16/D^3` from the shift of :math:`r(D/2)`. The omitted Taylor polynomial is bounded by :math:`12/D^2`; the omitted true value is bounded by one. Consequently .. math:: \left|\mathbb E r(J)-4/D^2\right|\leq209/D^3+2\epsilon_D. On the central event, :math:`J+1>D/4`; on its complement use :math:`L(J+1)\leq L(D)` in :eq:`risk-general-diagonal-explicit-error`. Together with :math:`c_q\leq J_q`, this yields .. math:: \left|\mathbb E\delta_q(J)-4c_q/D^2\right| \leq L(D)^{q-1}\left( \frac{64K_q+209J_q}{D^3}+(K_q+2J_q)\epsilon_D\right). For :math:`D\geq256`, :math:`D^3\epsilon_D\leq32`, as in the five-vertex error proof: the expression decreases on that domain and its value at 256 is below 32. Finally write the exact prefactor as .. math:: a_0(M,D)&=\frac{M^2q!}{2^qD^q}\,Q_D\,\mathbb E\delta_q(J),\\ Q_D&=\prod_{h=1}^q(1+h/(2D))^{-1},\qquad 00` is the constant's certified lower endpoint, choose :math:`t\geq\max(8,2(M-2))` with .. math:: R_M(t+2)^{M-2}\leq\eta C_-2^t. Then all :math:`D\geq2^t` satisfy the target. Indeed, writing :math:`x=\log_2D`, the integer logarithm is at most :math:`x+2`, and :math:`(x+2)^{M-2}2^{-x}` decreases on that domain since :math:`\log2>1/2`. A zero lower constant endpoint requires more terms. The threshold can be very large: this documents the usefulness of the conservative bound separately from its correctness. Independent tests compare coefficient bounds and their differences with exact logarithmic series, test the zero-sum splits against exact rational convolutions, compare enclosures with the original scalar/Gram references and frozen independent six-vertex kernel values, and reparse low- and high-precision real balls. Such finite checks validate the implementation; the all-degree claims follow from the inequalities above.