A logarithmic finite-degree remainder for nonvertex q4 risks ============================================================ Keep the uniform-simplex design, evaluation and noise assumptions of :doc:`skeleton_risk_theory`. For each fixed integer :math:`s\geq2`, set :math:`M=s+4` and use the existing positive leading constant .. math:: A_{s,4}=\binom{s+4}s^2\frac{(s)_4}{16(s-2)!}S, \qquad S=\sum_{j\geq0}\delta_4(j)=\frac{c_5}{5}>4. The previous cutoff argument gives :math:`O(D^{-1/2})` in :doc:`skeleton_risk_nonvertex_q4_error`. Here a new, explicit q4 discrete derivative proves :math:`O(\log D/D)`. No q3 difference estimate is assumed for q4. The coefficients concern sample-size-asymptotic risk; they do not supply finite-sample risk guarantees. .. prf:theorem:: A logarithmic q4 remainder at every fixed source size :label: thm-risk-nonvertex-q4-logarithmic-error Put :math:`Q=5400`, :math:`V=179568`, and .. math:: K_s=V+\frac{Q(s-2)}4,\qquad B_s=2s+3+\frac Q4\left(2+(s-1)(s-2)\right). For every pair of integers :math:`s\geq2,D\geq2(s-1)`, .. math:: :label: risk-nonvertex-q4-logarithmic-error \left|\frac{a_{s-1}(s+4,D)}{A_{s,4}D^{s-6}}-1\right| \leq\frac{K_s(1+\log D)+B_s}{D}. For every integer :math:`k\geq1` with :math:`2^k\geq2(s-1)`, the relative error at every :math:`D\geq2^k` is at most .. math:: :label: risk-nonvertex-q4-refined-dyadic \frac{K_s(k+1)+B_s}{2^k}. Both :math:`s=2` and :math:`s=3` therefore have relative error below one percent for every :math:`D\geq2^{30}`. The previous sufficient threshold was :math:`2^{54}`. Neither threshold is claimed to be the smallest useful degree. No growing-:math:`s,D` joint limit or q5 remainder is asserted. A convolution estimate including the first index ------------------------------------------------ Write :math:`\Delta a_j=a_{j+1}-a_j`. Suppose two sequences satisfy .. math:: |a_j|\leq\frac A{(j+1)^2},\quad |\Delta a_j|\leq\frac U{(j+1)^3},\qquad |b_j|\leq\frac B{(j+1)^2},\quad |\Delta b_j|\leq\frac W{(j+1)^3}. No positivity, monotonicity or zero total sum is required. For ordinary convolution :math:`c_n=\sum_{j=0}^na_jb_{n-j}`, .. math:: :label: risk-nonvertex-q4-convolution-norms |c_n|\leq\frac{16AB}{(n+1)^2},\qquad |\Delta c_n|\leq\frac{16(AW+BU+AB)}{(n+1)^3}. Indeed, :math:`\sum_{j\geq0}(j+1)^{-2}\leq2`. Splitting the convolution at :math:`h=\lfloor n/2\rfloor` leaves a denominator at least :math:`(n+1)^2/4` on each half, proving the first bound. An exact split of the difference, using the same :math:`h` in both sums, is .. math:: \Delta c_n={}&\sum_{j=0}^ha_j\Delta b_{n-j} +\sum_{l=0}^{n-h-1}b_l\Delta a_{n-l} +a_{h+1}b_{n-h}. The two derivative denominators are at least :math:`(n+1)^3/8`. The last product is at most :math:`16AB/(n+1)^4`, hence also at most :math:`16AB/(n+1)^3`. This proves the second bound, including :math:`n=0`, where the second sum is empty. Explicit coefficient norms for q4 --------------------------------- Use the sequences :math:`\phi_r,b_r,e_r` defined in :doc:`skeleton_risk_vertex_general`. Put :math:`x=j+1` and let :math:`E_l^{(r)}(j)` denote the elementary symmetric polynomial of order :math:`l` in the signed reciprocal list in :eq:`risk-general-phi-elementary`. With .. math:: p_r(j)=\frac{r!(r-1)!j!}{(j+r)!}, adding :math:`-1/(j+1)` to that list gives the exact recurrence .. math:: \Delta\phi_r(j) =-\frac{r\phi_r(j)+p_r(j)E_{r-2}^{(r)}(j)}{j+r+1} \quad(r\geq2). The absolute reciprocal sum is at most :math:`2+\log x` for :math:`r=2` and at most :math:`3+\log x` for :math:`r=3,4`. Together with :math:`|E_l|\leq(\sum |u_i|)^l/l!`, this proves .. math:: |\phi_2(j)|&\leq\frac{2(2+\log x)}{x^2},& |\Delta\phi_2(j)|&\leq\frac{10+4\log x}{x^3},\\ |\phi_3(j)|&\leq\frac{6(3+\log x)^2}{x^3},& |\Delta\phi_3(j)|&\leq \frac{18(3+\log x)^2+12(3+\log x)}{x^4},\\ |\phi_4(j)|&\leq\frac{24(3+\log x)^3}{x^4},& |\Delta\phi_4(j)|&\leq \frac{96(3+\log x)^3+72(3+\log x)^2}{x^5}. For :math:`x\geq1`, the elementary functions used on these right sides decrease after multiplication by the powers of :math:`x` indicated below. For example, :math:`(a+\log x)^b/x^c` decreases whenever :math:`b/a\leq c`. Since :math:`\phi_1(j)=1/(j+1)`, it follows that .. math:: |\phi_1(j)|&\leq1/x,& |\Delta\phi_1(j)|&\leq1/x^2,\\ |\phi_2(j)|&\leq4/x,& |\Delta\phi_2(j)|&\leq10/x^2,\\ |\phi_3(j)|&\leq54/x,& |\Delta\phi_3(j)|&\leq198/x^2,\\ |\phi_4(j)|&\leq648/x^2,& |\Delta\phi_4(j)|&\leq3240/x^3. The last line retains stronger decay for the solitary :math:`\phi_4` term. The exact identities .. math:: e_2=2\phi_1^2,\qquad e_3=6\phi_1\phi_2,\qquad e_4=8\phi_1\phi_3+6\phi_2^2 and the discrete product rule give the following absolute norms. .. list-table:: Bounds valid at every integer j at least zero :header-rows: 1 * - Sequence a - A in :math:`|a_j|\leq A/x^2` - U in :math:`|\Delta a_j|\leq U/x^3` * - :math:`e_2` - 2 - 4 * - :math:`e_3` - 24 - 84 * - :math:`e_4` - 528 - 2496 * - :math:`b_1` - 2 - 3 * - :math:`b_2` - 6 - 12 For the last two rows, :math:`b_1(0)=b_2(0)=1` and, at :math:`j\geq1`, .. math:: b_1(j)&=-\frac1{j(j+1)},& \Delta b_1(j)&=\frac2{j(j+1)(j+2)},\\ b_2(j)&=\frac{2(2H_{j-1}-3)}{j(j+1)(j+2)},& \Delta b_2(j)&=\frac{2(11-6H_{j-1})}{j(j+1)(j+2)(j+3)}. At :math:`j=0`, their differences are :math:`-3/2` and :math:`-2`. For positive indices use :math:`H_{j-1}\leq j-1` and :math:`(j+1)^2\leq j(j+3)` for the :math:`b_2` row. For :math:`b_1`, :math:`2(j+1)^2/(j(j+2))\leq8/3<3`. Thus the table has no omitted small-index exception. These are absolute bounds: the individual :math:`e_r` terms need not be nonnegative. The general transformed-diagonal identity specializes exactly to .. math:: \delta_4=-\phi_4+e_4-4(b_1*e_3)+6(b_2*e_2). Applying :eq:`risk-nonvertex-q4-convolution-norms` and the table gives .. math:: |\delta_4(j)|&\leq \frac{648+528+4(768)+6(192)}{(j+1)^2} =\frac{5400}{(j+1)^2},\\ |\Delta\delta_4(j)|&\leq \frac{3240+2496+4(4608)+6(960)}{(j+1)^3} =\frac{29928}{(j+1)^3}. The existing positive Gram-diagonal theorem supplies strict positivity. Since :math:`(j+2)(j+3)\leq6(j+1)^2`, the second estimate also proves .. math:: :label: risk-nonvertex-q4-refined-diagonal 0n}d_j&\leq\frac Q{n+1}. The exact initial values :math:`d_0=1,d_1=2,d_2=37/24` imply :math:`S>109/24>4`. Hence the relative error is at most .. math:: :label: risk-nonvertex-q4-refined-unsimplified 1-P_s(D)+\frac14\left[ \frac{4V}{D+2s+2}\left(H_{n+1}+\frac2{n+2}-2\right) +\frac Q{n+1} +\frac QD\bigl((s-2)H_{n+1}+(s-1)(s-2)\bigr)\right]. At :math:`D\geq2(s-1)`, use :math:`(n+1)^{-1}\leq2/D`, :math:`H_{n+1}\leq1+\log D`, and :math:`1-P_s(D)\leq(2s+3)/D`. This proves :eq:`risk-nonvertex-q4-logarithmic-error`. Its right side decreases on :math:`D\geq1`, since its derivative is :math:`-(K_s\log D+B_s)/D^2<0`. Together with :math:`\log2<1`, this proves :eq:`risk-nonvertex-q4-refined-dyadic`. Its right side decreases to zero, giving a terminating rational threshold search for every positive rational target. At :math:`s=2,3,k=30`, the numerator :math:`K_s(k+1)+B_s` is respectively :math:`5569315` and :math:`5613867`, both strictly below :math:`2^{30}/100`. Certificates, scope and independent validation ---------------------------------------------- ``tools.risk_research.nonvertex_q4_refined`` evaluates the displayed bounds by exact fractions. It uses integer bit lengths to majorize harmonic numbers, and the smaller of :eq:`risk-nonvertex-q4-refined-unsimplified` and the available dyadic bound. It constructs no degree-sized risk or harmonic table. On the inactive prefix :math:`D