Single-log vertex errors for every fixed M at least five

Keep the uniform-simplex assumptions of Theory of the skeleton risk. The general leading constant and its rational enclosures are proved in Vertex coefficients at every fixed vertex count. The explicit error in Explicit vertex errors at every fixed vertex count has logarithmic power \(M-2\). Finite first absolute moments remove this dependence from the logarithmic power, for every fixed \(M\geq5\). The existing critical \(M=4\) squared-log theorem and sharper five-vertex evaluation remain unchanged.

Explicit constants and theorem

Use the positive integers \(B_r,G_r,A_r,F_r\) of Explicit vertex errors at every fixed vertex count. Define, for integers \(p\geq0,\alpha\geq2\),

\[I_{p,\alpha} =\sum_{j=0}^p\frac{p!}{(p-j)!(\alpha-1)^{j+1}} =\int_1^\infty\frac{(1+\log x)^p}{x^\alpha}\,dx.\]

For \(k\geq2,r\geq3\), put

\[\begin{split}S_k&=G_k(1+I_{k-1,k}),\\ T_r&=A_r(1+I_{r-2,r-1}),\\ \widetilde U_r &=2^{r+1}F_r+(2+3^{r-1})A_r+18T_r.\end{split}\]

For \(k,r\geq2\), define

\[\widetilde V_{k,r} =2^{r+1}F_rS_k+5\,2^kG_kA_r.\]

Set \(q=M-1\geq4\) and

(103)\[\begin{split}J_q&=2qA_{q-1},\\ \widetilde K_q &=2B_q+A_q+q\widetilde U_{q-1} +\sum_{k=2}^{q-2}\binom qk\widetilde V_{k,q-k} +8(1+q^2+J_q),\\ \widetilde R_M &=\frac{M^2q!}{2^q} \left(96\widetilde K_q+(273+q(q+1))J_q\right).\end{split}\]

All these constants are positive computable rational numbers.

Theorem 52 (Single-log vertex error for every fixed M at least five)

For every pair of integers \(M\geq5,D\geq256\),

(104)\[\left|a_0(M,D)-\frac{C_M}{D^{M+1}}\right| \leq\widetilde R_M\frac{1+\log D}{D^{M+2}}.\]

This is an explicit finite-degree bound for each fixed \(M\). It does not assert uniform convergence when \(M\) grows with \(D\).

First absolute moments and elementary absorptions

Write \(L(x)=1+\log x\). Integration by parts proves the formula for \(I_{p,\alpha}\). When \(\alpha\geq p\), the integrand is decreasing on \(x\geq1\). Therefore a sum over positive integers is at most its first term plus that integral. The already proved coefficient bounds give

\[\begin{split}\sum_{l\geq0}l|b_k(l)|&\leq S_k\quad(k\geq2),\\ \sum_{m\geq0}(m+1)|e_r(m)|&\leq T_r\quad(r\geq3).\end{split}\]

The second displayed bound starts at \(r=3\): for \(r=2\), \(e_2(m)=2/(m+1)^2\) has a divergent first absolute moment. Similarly the first absolute moment of \(b_1\) diverges, but its partial sum through \(n\) is at most \(L(n+1)\). These two critical cases explain why \(M=4\) remains separate.

The following elementary inequalities hold on \(x\geq1\):

\[1\leq L(x)\leq x,\qquad L(x)^2/x\leq2.\]

For the second inequality set \(t=\log x\geq0\); the maximum of \((1+t)^2e^{-t}\) is \(4/e<2\) at \(t=1\). The spare inverse powers in the convolution bounds thus absorb every logarithm beyond the single critical factor.

A refined zero-sum split

For \(n\geq2\), put \(N=n+1\), \(H=\lfloor n/2\rfloor\) and \(L=L(N)\). Use the same exact zero-sum decomposition as in Explicit vertex errors at every fixed vertex count:

\[\begin{split}(b_k*e_r)(n) ={}&\sum_{l=0}^H b_k(l)(e_r(n-l)-e_r(n)) -e_r(n)\sum_{l>H}b_k(l)\\ &+\sum_{m=0}^{n-H-1}b_k(n-m)e_r(m).\end{split}\]

For \(k=1,r\geq3\), subtract \(-N^{-2}\sum_{m\geq0}e_r(m)\) from the last term. The four resulting error budgets are respectively

\[\begin{split}&2^{r+1}F_rL^{r-1}/N^{r+1},\qquad 2A_rL^{r-2}/N^{r+1},\\ &18T_r/N^3,\qquad 3^{r-1}A_rL^{r-2}/N^{r+1}.\end{split}\]

The first two use the first-difference estimate for \(e_r\), the exact identity \(b_1(l)=-1/(l(l+1))\), and its telescoping tail. The finite last-term difference uses the previously proved \(|b_1(n-m)+N^{-2}|\leq18(m+1)/N^3\) and the finite moment \(T_r\). The omitted tail starts at \(n-H\geq N/3\) and uses the previous logarithmic tail inequality. Since \(r\geq3\), the absorptions above prove

\[\left|(b_1*e_r)(n)+N^{-2}\sum_{m\geq0}e_r(m)\right| \leq\widetilde U_rL/N^3.\]

For \(k,r\geq2\), the three absolute error budgets are

\[\begin{split}&2^{r+1}F_rS_kL^{r-2}/N^{r+1},\\ &2^kG_kA_rL^{k+r-3}/N^{k+r},\\ &2^{k+2}G_kA_rL^{k-1}/N^{k+1}.\end{split}\]

Here the first term uses the finite first moment \(S_k\), the second uses the logarithmic tail of \(b_k\), and the last uses \(\sum_m|e_r(m)|\leq2A_r\) and \(n-m+1\geq N/2\). The ratios \((L/N)^{r-2}\) and \((L/N)^{k+r-3}\) are at most one; the last term leaves at most one factor \(L\). Consequently

\[|(b_k*e_r)(n)|\leq\widetilde V_{k,r}L/N^3.\]

In the transformed diagonal identity, the residual terms satisfy

\[|\phi_q(n)|\leq2B_qL/N^3,\qquad |e_q(n)|\leq A_qL/N^3\quad(q\geq4).\]

For the first inequality factor \(L^{q-2}/N^{q-3}=(L^2/N)(L/N)^{q-4}\leq2\). For the second use \((L/N)^{q-3}\leq1\). The \(k=1\) term supplies \(c_q=q\sum_me_{q-1}(m)\). The added \(8(1+q^2+J_q)\) covers \(n=0,1\) using \(\delta_q(0)=1\), \(\delta_q(1)=(q^2-3q+4)/4\) and \(0<c_q\leq J_q\). Thus for every \(n\geq0\),

(105)\[\left|\delta_q(n)-\frac{c_q}{(n+1)^2}\right| \leq\widetilde K_q\frac{L(n+1)}{(n+1)^3}.\]

Averaging and normalization

Apply the hypergeometric averaging proof of Explicit vertex errors at every fixed vertex count to (105). Only the diagonal error changes. Its logarithmic factor is now \(L(D)\). The central event, Taylor remainder and small-index tail give

\[\left|\mathbb E\delta_q(J)-4c_q/D^2\right| \leq L(D)\left( \frac{64\widetilde K_q+209J_q}{D^3} +(\widetilde K_q+2J_q)4\,2^{-D/12}\right).\]

For \(D\geq256\), \(D^3\,4\,2^{-D/12}\leq32\). Multiplying the exact prefactor, including its normalization error \(q(q+1)J_q/D^3\), gives (104) with precisely the constant (103).

Certified intervals and a six-vertex improvement

The checkout tool tools.risk_research.vertex_refined_asymptotic uses the exact integer \(1+\lceil\log_2D\rceil\) in place of \(1+\log D\). It reports the refined and previous analytic radii separately, uses their minimum, and combines it with the exact rational interval for \(C_M\). Below \(D=256\) it evaluates the exact univariate formula. Optional Arb output adds arithmetic rounding, restores ambient precision, and is checked again after serialization.

At \(M=6\), the explicit constants are

\[\widetilde K_5=107393368,\qquad J_5=5280,\qquad \widetilde R_6=1392034027680.\]

With the existing 128-term rational interval for \(C_6\), the new analytic error is at most one percent of \(C_6/D^7\) at every \(D\geq2^{38}\). The earlier sufficient threshold was \(2^{56}\). These are sufficient thresholds, with conservative constants, rather than exact smallest useful degrees.

In general, for a certified positive constant endpoint \(C_-\) and target \(\eta>0\), choose an integer \(t\geq8\) such that

\[\widetilde R_M(t+2)\leq\eta C_-2^t.\]

Then every \(D\geq2^t\) satisfies the target: writing \(x=\log_2D\), the integer logarithmic majorant is at most \(x+2\), and \((x+2)2^{-x}\) is decreasing for \(x\geq8\). This certifies the entire tail, including integer-logarithm jumps.

$ python -m tools.risk_research.vertex_refined_asymptotic --vertices 6 --degree 100000 --relative-target 1/100 --output .research/vertex6-refined.json
$ python -m tools.risk_research.vertex_refined_asymptotic --vertices 6 --degree 100000 --with-arb --bits 128 --output .research/vertex6-refined-ball.json

The JSON separates constant truncation, analytic error and optional arithmetic rounding. Source hashes use the explicit CRLF-to-LF policy; the concurrent-edit guard compares the original bytes and rejects even a newline-only change during calculation. Existing output is replaced atomically only after successful computation.

Independent checks cover exact logarithmic integrals, finite moments, both convolution parities, low-degree scalar and Gram references, the asymptotic boundary, native serialized balls and post-freeze holdouts. Finite checks support the implementation; the all-degree result follows from the inequalities above.