Finite-degree errors for every fixed nonvertex q4 face size

Keep the uniform-simplex design, evaluation and noise assumptions of Theory of the skeleton risk. Put \(M=s+4\) for a fixed integer \(s\geq2\). The existing positive-diagonal theorem gives

\[a_{s-1}(s+4,D)\sim A_{s,4}D^{s-6},\qquad A_{s,4}=\binom{s+4}s^2\frac{(s)_4}{16(s-2)!}S,\qquad S=\sum_{j\geq0}\delta_4(j)=\frac{c_5}{5}>0.\]

This chapter adds an explicit finite-degree error. It uses a cutoff argument and does not assume the q3 discrete-derivative estimate for q4. The rate is deliberately conservative: \(O(D^{-1/2})\), rather than the \(O(\log D/D)\) proved separately at q3. These are coefficients of sample-size-asymptotic risk, not finite-sample risk.

Theorem 32 (A finite-D q4 error at every fixed source size)

Set \(Q=341944\), and define

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

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

(60)\[\left|\frac{a_{s-1}(s+4,D)}{A_{s,4}D^{s-6}}-1\right| \leq \frac{29Q}{8\sqrt D} +\frac{K_s(1+\log D)+B_s}{D}.\]

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

(61)\[\frac{29Q}{8\,2^{\lfloor k/2\rfloor}} +\frac{K_s(k+1)+B_s}{2^k}.\]

In particular, both the edge \(s=2\) and triangle \(s=3\) have relative error below one percent at every \(D\geq2^{54}\). This is a sufficient threshold, not the smallest useful degree. No new growing-\(s,D\) joint limit or q5 remainder is claimed.

A summable q4 diagonal bound

The already proved single-log diagonal estimate in (105), at \(q=4\), is

\[\left|\delta_4(j)-\frac{c_4}{(j+1)^2}\right| \leq341752\frac{1+\log(j+1)}{(j+1)^3},\qquad 0<c_4\leq192.\]

These constants are obtained by substituting \(q=4\) in (103); they are not fitted to data. Since \(1+\log x\leq x\) for \(x\geq1\), positivity of the transformed diagonals proves the global majorant

(62)\[0<d_j:=\delta_4(j)\leq\frac{Q}{(j+1)^2}\quad(j\geq0).\]

Direct rational coefficient extraction gives \(d_0=1,d_1=2,d_2=37/24\). Thus \(S>1+2+37/24=109/24>4\). The summability and identity \(S=c_5/5\) follow from Theorem 29.

Two exact inverse-factorial moments

Let \(0\leq L\leq D-s\), \(B=2D+s-L-1\), \(T=L+B=2D+s-1\), and

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

Vandermonde’s identity gives

(63)\[\begin{split}\mathbb E\frac1{(J+1)(J+2)} &=\frac{(T+1)(T+2)}{(L+1)(L+2)(B+1)(B+2)},\\ \mathbb E\frac{L-J}{(J+1)(J+2)} &=\frac{L(T+1)}{(L+1)(B+1)(B+2)},\\ \mathbb E(L-J)&=\frac{L^2}{T}.\end{split}\]

For the second identity, use \((L-j)\binom Lj=L\binom{L-1}j\) and \(\binom Bj/((j+1)(j+2))=\binom{B+2}{j+2}/((B+1)(B+2))\). At \(L=0\) both sides are zero. The first identity uses the same replacement without the deficit. The last is the hypergeometric mean. Because \(B\geq L\) and \(B+2\geq D+2s+1\),

\[\begin{split}\mathbb E\frac1{(J+1)^2} &\leq\frac8{(L+1)(L+2)},\\ \mathbb E\frac{L-J}{(J+1)(J+2)} &\leq\frac2{D+2s+1}.\end{split}\]

A cutoff averaging lemma

The following argument works for any nonnegative sequence satisfying (62); it does not use monotonicity or a difference bound for that sequence. On \(J=L\) the difference is zero, so the triangle inequality and \(\mathbf1_{J<L}\leq L-J\) give

\[\begin{split}|\mathbb E d_J-d_L| &\leq Q\mathbb E\frac{L-J}{(J+1)^2} +\frac{Q}{(L+1)^2}\mathbb E(L-J)\\ &\leq\frac{4Q}{D+2s+1}+\frac QT \leq\frac{9Q}{2D}.\end{split}\]

Alternatively, the preceding inverse-square moment and the bound for \(d_L\) give

\[|\mathbb E d_J-d_L| \leq\mathbb E d_J+d_L \leq\frac{10Q}{(L+1)(L+2)}.\]

For an arbitrary positive integer \(H\), use the first bound on \(L<H\) and sum the second on \(L\geq H\). Extending the latter sum to infinity only increases its positive majorant. Even when \(H>D-s+1\) there are at most \(H\) terms in the first group. The telescoping reciprocal sum therefore yields

(64)\[\sum_{L=0}^{D-s}|\mathbb E d_J-d_L| \leq Q\left(\frac{9H}{2D}+\frac{10}{H+1}\right).\]

Choose \(H=\lfloor\sqrt D\rfloor\). Since \(H\leq\sqrt D<H+1\), this is at most \(29Q/(2\sqrt D)\).

Angular weights, tail and normalization

Write \(n=D-s\) and

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

For active degrees \(D\geq s\), all factors lie in \([0,1]\). The empty product at \(s=2\) is one. The same finite-product calculation used in Finite-degree errors for every nonvertex q3 face gives

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

Together with (64), these bound \(|U_D-S|\) by

\[E_s(D)=Q\left(\frac{9H}{2D}+\frac{10}{H+1} +\frac1{D-s+1} +\frac{(s-2)H_{D-s+1}+(s-1)(s-2)}D\right).\]

The exact coefficient formula, without an asymptotic substitution, is

\[\frac{a_{s-1}(s+4,D)}{A_{s,4}D^{s-6}} =P_s(D)\frac{U_D}{S},\qquad P_s(D)=\prod_{i=0}^3\left(1+\frac{s+i}{2D}\right)^{-1}.\]

Since \(S>4\) and \(0<P_s(D)\leq1\), its relative error is at most \(1-P_s(D)+E_s(D)/4\). This sharper unsimplified bound is valid at every active degree. For \(D\geq2(s-1)\), use \(1/(D-s+1)\leq2/D\), \(H_{D-s+1}\leq1+\log D\) and \(1-P_s(D)\leq(2s+3)/D\). This proves (60).

Both terms in that bound decrease on \(D\geq1\): the logarithmic term has derivative \(-(K_s\log D+B_s)/D^2<0\). Use \(\log2<1\) and \(2^{k/2}\geq2^{\lfloor k/2\rfloor}\) to obtain (61). Its right side decreases to zero, so it supplies a terminating exact search for every positive rational target. Direct rational substitution at \(k=54\) proves the stated one-percent sufficient threshold for \(s=2,3\).

Exact certificates and independent checks

tools.risk_research.nonvertex_q4_error uses isqrt(D) in the cutoff estimate and integer bit length to majorize harmonic numbers. Where applicable it takes the smaller of the unsimplified estimate and the all-tail dyadic estimate. The inactive prefix \(D<s\) is exactly zero, with normalized relative error one. It builds no table proportional to \(D\).

The positive constant interval uses the existing rational series for \(C_6=540c_5\), scaled by the proved nonvertex normalization. It does not require a special-value reduction of \(c_5\). constant_terms controls only that interval’s truncation error. The reported finite-D relative error concerns the exact \(A_{s,4}\), so those two error budgets must remain separate.

$ python -m tools.risk_research.nonvertex_q4_error --face-size 2 --degree 18014398509481984 --constant-terms 64 --relative-target 1/100 --output .research/q4-edge-error.json
$ python -m tools.risk_research.nonvertex_q4_error --verify .research/q4-edge-error.json

The saved record regenerates its complete contents, including nondefault constant terms and target. Scalar types, unknown or changed fields, duplicate keys and conflicting overrides are rejected. Source hashes normalize CRLF to LF; the concurrent-edit guard compares original bytes. The existing bounded decimal codec handles large rational and parameter integers locally without changing process-wide conversion limits. The new schema does not alter any existing saved-record schema.

The CLI rejects destinations resolving to the bound evaluator, dependency or proof sources. The atomic writer receives the validated canonical parent plus original leaf, so retargeting the original parent alias cannot redirect its write. Output directories must be controlled by the caller; this is not a sandbox against another process renaming the canonical directory itself. Failures preserve any previous destination.

Tests check both inverse moments, cutoff bounds for unrelated finite sequences, diagonal constants, all-tail behavior including odd dyadic exponents, endpoint handling, original scalar/Gram values and post-freeze original integer \(F^4\) kernel holdouts. Optional native checks use the original positive integral separately for the leading constant. Finite checks validate these implementations and normalizations; the displayed inequalities prove the infinite degree tail.