A logarithmic finite-degree remainder for nonvertex q4 risks

Keep the uniform-simplex design, evaluation and noise assumptions of Theory of the skeleton risk. For each fixed integer \(s\geq2\), set \(M=s+4\) and use the existing positive leading constant

\[A_{s,4}=\binom{s+4}s^2\frac{(s)_4}{16(s-2)!}S, \qquad S=\sum_{j\geq0}\delta_4(j)=\frac{c_5}{5}>4.\]

The previous cutoff argument gives \(O(D^{-1/2})\) in Finite-degree errors for every fixed nonvertex q4 face size. Here a new, explicit q4 discrete derivative proves \(O(\log D/D)\). No q3 difference estimate is assumed for q4. The coefficients concern sample-size-asymptotic risk; they do not supply finite-sample risk guarantees.

Theorem 33 (A logarithmic q4 remainder at every fixed source size)

Put \(Q=5400\), \(V=179568\), and

\[K_s=V+\frac{Q(s-2)}4,\qquad B_s=2s+3+\frac Q4\left(2+(s-1)(s-2)\right).\]

For every pair of integers \(s\geq2,D\geq2(s-1)\),

(65)\[\left|\frac{a_{s-1}(s+4,D)}{A_{s,4}D^{s-6}}-1\right| \leq\frac{K_s(1+\log D)+B_s}{D}.\]

For every integer \(k\geq1\) with \(2^k\geq2(s-1)\), the relative error at every \(D\geq2^k\) is at most

(66)\[\frac{K_s(k+1)+B_s}{2^k}.\]

Both \(s=2\) and \(s=3\) therefore have relative error below one percent for every \(D\geq2^{30}\). The previous sufficient threshold was \(2^{54}\). Neither threshold is claimed to be the smallest useful degree. No growing-\(s,D\) joint limit or q5 remainder is asserted.

A convolution estimate including the first index

Write \(\Delta a_j=a_{j+1}-a_j\). Suppose two sequences satisfy

\[|a_j|\leq\frac A{(j+1)^2},\quad |\Delta a_j|\leq\frac U{(j+1)^3},\qquad |b_j|\leq\frac B{(j+1)^2},\quad |\Delta b_j|\leq\frac W{(j+1)^3}.\]

No positivity, monotonicity or zero total sum is required. For ordinary convolution \(c_n=\sum_{j=0}^na_jb_{n-j}\),

(67)\[|c_n|\leq\frac{16AB}{(n+1)^2},\qquad |\Delta c_n|\leq\frac{16(AW+BU+AB)}{(n+1)^3}.\]

Indeed, \(\sum_{j\geq0}(j+1)^{-2}\leq2\). Splitting the convolution at \(h=\lfloor n/2\rfloor\) leaves a denominator at least \((n+1)^2/4\) on each half, proving the first bound. An exact split of the difference, using the same \(h\) in both sums, is

\[\Delta c_n={}&\sum_{j=0}^ha_j\Delta b_{n-j} +\sum_{l=0}^{n-h-1}b_l\Delta a_{n-l} +a_{h+1}b_{n-h}.\]

The two derivative denominators are at least \((n+1)^3/8\). The last product is at most \(16AB/(n+1)^4\), hence also at most \(16AB/(n+1)^3\). This proves the second bound, including \(n=0\), where the second sum is empty.

Explicit coefficient norms for q4

Use the sequences \(\phi_r,b_r,e_r\) defined in Vertex coefficients at every fixed vertex count. Put \(x=j+1\) and let \(E_l^{(r)}(j)\) denote the elementary symmetric polynomial of order \(l\) in the signed reciprocal list in (83). With

\[p_r(j)=\frac{r!(r-1)!j!}{(j+r)!},\]

adding \(-1/(j+1)\) to that list gives the exact recurrence

\[\Delta\phi_r(j) =-\frac{r\phi_r(j)+p_r(j)E_{r-2}^{(r)}(j)}{j+r+1} \quad(r\geq2).\]

The absolute reciprocal sum is at most \(2+\log x\) for \(r=2\) and at most \(3+\log x\) for \(r=3,4\). Together with \(|E_l|\leq(\sum |u_i|)^l/l!\), this proves

\[\begin{split}|\phi_2(j)|&\leq\frac{2(2+\log x)}{x^2},& |\Delta\phi_2(j)|&\leq\frac{10+4\log x}{x^3},\\ |\phi_3(j)|&\leq\frac{6(3+\log x)^2}{x^3},& |\Delta\phi_3(j)|&\leq \frac{18(3+\log x)^2+12(3+\log x)}{x^4},\\ |\phi_4(j)|&\leq\frac{24(3+\log x)^3}{x^4},& |\Delta\phi_4(j)|&\leq \frac{96(3+\log x)^3+72(3+\log x)^2}{x^5}.\end{split}\]

For \(x\geq1\), the elementary functions used on these right sides decrease after multiplication by the powers of \(x\) indicated below. For example, \((a+\log x)^b/x^c\) decreases whenever \(b/a\leq c\). Since \(\phi_1(j)=1/(j+1)\), it follows that

\[\begin{split}|\phi_1(j)|&\leq1/x,& |\Delta\phi_1(j)|&\leq1/x^2,\\ |\phi_2(j)|&\leq4/x,& |\Delta\phi_2(j)|&\leq10/x^2,\\ |\phi_3(j)|&\leq54/x,& |\Delta\phi_3(j)|&\leq198/x^2,\\ |\phi_4(j)|&\leq648/x^2,& |\Delta\phi_4(j)|&\leq3240/x^3.\end{split}\]

The last line retains stronger decay for the solitary \(\phi_4\) term. The exact identities

\[e_2=2\phi_1^2,\qquad e_3=6\phi_1\phi_2,\qquad e_4=8\phi_1\phi_3+6\phi_2^2\]

and the discrete product rule give the following absolute norms.

Table 14 Bounds valid at every integer j at least zero

Sequence a	A in \(|a_j|\leq A/x^2\)	U in \(|\Delta a_j|\leq U/x^3\)
\(e_2\)	2	4
\(e_3\)	24	84
\(e_4\)	528	2496
\(b_1\)	2	3
\(b_2\)	6	12

For the last two rows, \(b_1(0)=b_2(0)=1\) and, at \(j\geq1\),

\[\begin{split}b_1(j)&=-\frac1{j(j+1)},& \Delta b_1(j)&=\frac2{j(j+1)(j+2)},\\ b_2(j)&=\frac{2(2H_{j-1}-3)}{j(j+1)(j+2)},& \Delta b_2(j)&=\frac{2(11-6H_{j-1})}{j(j+1)(j+2)(j+3)}.\end{split}\]

At \(j=0\), their differences are \(-3/2\) and \(-2\). For positive indices use \(H_{j-1}\leq j-1\) and \((j+1)^2\leq j(j+3)\) for the \(b_2\) row. For \(b_1\), \(2(j+1)^2/(j(j+2))\leq8/3<3\). Thus the table has no omitted small-index exception. These are absolute bounds: the individual \(e_r\) terms need not be nonnegative.

The general transformed-diagonal identity specializes exactly to

\[\delta_4=-\phi_4+e_4-4(b_1*e_3)+6(b_2*e_2).\]

Applying (67) and the table gives

\[\begin{split}|\delta_4(j)|&\leq \frac{648+528+4(768)+6(192)}{(j+1)^2} =\frac{5400}{(j+1)^2},\\ |\Delta\delta_4(j)|&\leq \frac{3240+2496+4(4608)+6(960)}{(j+1)^3} =\frac{29928}{(j+1)^3}.\end{split}\]

The existing positive Gram-diagonal theorem supplies strict positivity. Since \((j+2)(j+3)\leq6(j+1)^2\), the second estimate also proves

(68)\[0<d_j:=\delta_4(j)\leq\frac Q{(j+1)^2},\qquad |d_{j+1}-d_j|\leq\frac V{(j+1)(j+2)(j+3)}.\]

The constants \(Q,V\) therefore come from explicit coefficient inequalities, not from a fit or a finite scan.

Averaging, angular weights and the infinite tail

Use the exact positive-diagonal averages from Positive diagonal formulas for nonvertex risk. Set \(n=D-s\), \(0\leq L\leq n\), \(B=2D+s-L-1\), \(T=L+B\), and

\[\mathbb P(J=j)=\frac{\binom Lj\binom Bj}{\binom TL}.\]

The inverse-factorial deficit identity is a distributional identity, independent of the q3 or q4 sequence. Vandermonde’s identity gives

\[\mathbb E\frac{L-J}{(J+1)(J+2)(J+3)} =\frac{L(T+1)(T+2)} {(L+1)(L+2)(B+1)(B+2)(B+3)} \leq\frac{4L}{(L+1)(L+2)(D+2s+2)}.\]

Indeed, replace \((L-j)\binom Lj=L\binom{L-1}j\) and \(\binom Bj/(j+1)_3=\binom{B+3}{j+3}/(B+1)_3\), then apply Vandermonde. At \(L=0\) both sides vanish. Since \(B\geq L\), the remaining upper bound uses \(T+1\leq2(B+1)\), \(T+2\leq2(B+2)\), and \(B+3\geq D+2s+2\).

Telescoping the newly proved q4 derivative and summing over \(L\) therefore bounds the averaging error by

\[\begin{split}\sum_{L=0}^n|\mathbb E d_J-d_L| &\leq \frac{4V}{D+2s+2} \left(H_{n+1}+\frac2{n+2}-2\right)\\ &\leq\frac{4VH_{n+1}}{D+2s+2}.\end{split}\]

For active degrees \(D\geq s\), put

\[w_L=\prod_{r=2}^{s-1}\left(1-\frac{L+r}{D}\right),\qquad U_D=\sum_{L=0}^nw_L\mathbb E d_J,\qquad P_s(D)=\prod_{i=0}^3\left(1+\frac{s+i}{2D}\right)^{-1}.\]

The exact normalization is \(a_{s-1}(s+4,D)/(A_{s,4}D^{s-6})=P_s(D)U_D/S\). The product factors \(w_L\) lie in \([0,1]\), including the empty product at \(s=2\). The same finite-product calculation as in Finite-degree errors for every fixed nonvertex q4 face size, now with the proved smaller \(Q\), yields

\[\begin{split}\sum_{L=0}^n(1-w_L)d_L &\leq\frac QD\bigl((s-2)H_{n+1}+(s-1)(s-2)\bigr),\\ \sum_{j>n}d_j&\leq\frac Q{n+1}.\end{split}\]

The exact initial values \(d_0=1,d_1=2,d_2=37/24\) imply \(S>109/24>4\). Hence the relative error is at most

(69)\[1-P_s(D)+\frac14\left[ \frac{4V}{D+2s+2}\left(H_{n+1}+\frac2{n+2}-2\right) +\frac Q{n+1} +\frac QD\bigl((s-2)H_{n+1}+(s-1)(s-2)\bigr)\right].\]

At \(D\geq2(s-1)\), use \((n+1)^{-1}\leq2/D\), \(H_{n+1}\leq1+\log D\), and \(1-P_s(D)\leq(2s+3)/D\). This proves (65). Its right side decreases on \(D\geq1\), since its derivative is \(-(K_s\log D+B_s)/D^2<0\). Together with \(\log2<1\), this proves (66). Its right side decreases to zero, giving a terminating rational threshold search for every positive rational target. At \(s=2,3,k=30\), the numerator \(K_s(k+1)+B_s\) is respectively \(5569315\) and \(5613867\), both strictly below \(2^{30}/100\).

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 (69) and the available dyadic bound. It constructs no degree-sized risk or harmonic table. On the inactive prefix \(D<s\), the coefficient is exactly zero and its normalized relative error is one.

The leading-constant interval still uses the independent existing rational series for \(C_6=540c_5\). constant_terms changes that interval’s truncation width, while the new finite-D bound concerns the exact \(A_{s,4}\). The two uncertainties remain separate. No finite special-value reduction or new growing-source asymptotic is needed.

$ python -m tools.risk_research.nonvertex_q4_refined --face-size 2 --degree 1073741824 --constant-terms 64 --relative-target 1/100 --output .research/q4-refined-edge.json
$ python -m tools.risk_research.nonvertex_q4_refined --verify .research/q4-refined-edge.json

The new source-bound schema leaves existing saved-record schemas unchanged. Reverification restores the saved nondefault parameters and regenerates the complete record, rejecting changed types, duplicate or unknown fields, modified proof constants and conflicting overrides. Hashes normalize CRLF to LF; the concurrent-edit guard compares original bytes. Large decimal integers use the existing local chunked codec, without changing process-wide limits.

The CLI rejects destinations resolving to a bound evaluator, dependency or proof source. Its atomic writer uses the validated canonical parent and original leaf. The directory must be controlled by the caller; this is not a sandbox against concurrent replacement of the canonical directory inode itself. Failures preserve previous output contents.

Finite checks cover the exact coefficient and convolution identities, small-index norms, inverse-factorial moments, scalar/Gram coefficients, frozen original integer F4-kernel holdouts, saved options and failure preservation. Optional native checks independently integrate the original positive leading-constant integral. These finite checks validate code and normalization; the inequalities above prove the entire degree tail.