Uniform errors for growing nonvertex faces

Retain the uniform-simplex design, evaluation and noise assumptions of Theory of the skeleton risk. Write s for the number of vertices of the source face, q for its outside codimension and \(M=s+q\). The quantity below is a coefficient of sample-size-asymptotic risk; it is not a finite-sample risk bound. The positive leading constant \(A_{s,q}\) and explicit rational \(Q_q\) are those of Finite-degree errors at every fixed nonvertex codimension.

The earlier theorem keeps s,q fixed when describing its asymptotic scope. Its displayed finite-degree inequality, however, holds at every integer \(s,q\geq2\) whenever \(D\geq\max(4,2(s-1))\). All dependence on s is explicit. This chapter derives a uniform moving range from that inequality. No interchange of a fixed-s limit and a supremum is used.

Corollary 14 (A growing-face range at every fixed codimension)

Put \(L(D)=1+\log D\) and \(H(D)=\lfloor D^{1/4}\rfloor\). For every integer \(q\geq2\) and \(D\geq16\),

(73)\[\begin{split}\sup_{2\leq s\leq H(D)} \left|\frac{a_{s-1}(s+q,D)}{A_{s,q}D^{s-q-2}}-1\right| \leq\begin{cases} 88L(D)/\sqrt D,&q=2,\\ \bigl((39/2)Q_q+q(q+1)/4\bigr)/\sqrt D,&q\geq3. \end{cases}\end{split}\]

In particular this error tends uniformly to zero over the stated face range when q is fixed. The constants need not be sharp.

Proof. The elementary bound \(L(D)\leq2D^{1/4}\) holds for D>=1: set \(u=\log D\) and maximize \((1+u)e^{-u/4}\) on u>=0. Its maximum is \(4e^{-3/4}<2\), because \(e^{3/4}>1+3/4+(3/4)^2/2>2\). Set \(x=D^{1/4}\geq2\) and \(H=H(D)\leq x\). Then \(2(s-1)\leq2H\leq2x\leq x^4=D\). Thus the full range lies in the earlier theorem’s domain.

At q>=3 its numerator beyond the square-root term satisfies

\[2+(s-2)L(D)+(s-1)(s-2) \leq2+HL(D)+H^2\leq2+3\sqrt D\leq5\sqrt D.\]

Also \(H\leq\sqrt D\) and \(\sqrt D\geq1\) give

\[\frac{q(2s+q-1)}{4D} \leq\frac{q(q+1)}{4\sqrt D}.\]

Insert these inequalities into (71). The Q coefficient is \(29/2+5=39/2\).

At q2, \(Q_2=4\), and that same earlier upper bound is

\[\frac{58L}{\sqrt D}+\frac{s+1/2}D +\frac{8(L+1)}D +\frac{4(s-2)L^2}D +\frac{4(s-1)(s-2)L}D.\]

Its last four terms are bounded respectively by \((3/2)L/\sqrt D\), \(16L/\sqrt D\), \(8L/\sqrt D\), and \(4L/\sqrt D\). For the third use \(HL\leq2H D^{1/4}\leq2\sqrt D\); for the last use \(H^2\leq\sqrt D\). The sum of constants is \(58+3/2+16+8+4=175/2\leq88\). Each bound is independent of s, proving the supremum claim.

More general envelopes

The original explicit inequality also implies the following qualitative statements for integer envelopes \(h(D)\geq2\) eventually: at any fixed q>=3, \(h(D)=o(\sqrt D)\) suffices for uniform convergence over \(2\leq s\leq h(D)\); at q2, \(h(D)=o(\sqrt{D/\log D})\) suffices. Indeed the extra terms are bounded by fixed-q constants times \(h/D+hL/D+h^2/D\) in the first case, and \(h/D+L/D+hL^2/D+h^2L/D\) in the second. All tend to zero under the respective hypotheses; the earlier theorem’s domain also holds eventually. No effective threshold search for an arbitrary h is asserted. The evaluator below uses only the concrete fourth-root envelope. Growth of q together with D, an optimal face range, and minimal constants or starting degrees are outside this result.

Exact rational all-tail certificates

For \(k\geq4\), let \(\rho_k=2^{k/2}\) at even k and \(\rho_k=(4/3)2^{\lfloor k/2\rfloor}\) at odd k. The odd choice is a lower bound for \(\sqrt{2^k}\) because \((4/3)^2<2\). The preceding corollary gives

(74)\[\begin{split}\sup_{D\geq2^k}\ \sup_{2\leq s\leq\lfloor D^{1/4}\rfloor} \left|\frac{a_{s-1}(s+q,D)}{A_{s,q}D^{s-q-2}}-1\right| \leq E_q(k):=\begin{cases} 88(k+1)/\rho_k,&q=2,\\ \bigl((39/2)Q_q+q(q+1)/4\bigr)/\rho_k,&q\geq3. \end{cases}\end{split}\]

For q2, \(L(D)/\sqrt D\) decreases on D>=16, since its derivative has the sign of \(1-\log D\). At \(D=2^k\), \(L(2^k)\leq k+1\) uses \(\log2<1\). The q>=3 envelope decreases directly. In particular the face limit keeps growing at every future degree; it is not frozen at \(\lfloor (2^k)^{1/4}\rfloor\). Both rational envelopes decrease for k>=4 and tend to zero. For q2 the even-to-odd ratio is \(3(k+2)/(4(k+1))<1\), and the odd-to-even ratio is \(2(k+2)/(3(k+1))<1\). Consequently exact search terminates mathematically for every fixed q and every positive rational error target. It returns a sufficient dyadic threshold, not a minimal degree threshold.

The standard-library evaluator computes H with nested integer square roots, without a face-sized or degree-sized table. For a single degree it uses E at the containing dyadic block; this may be looser than the analytic bound. Its source-bound schema is skeleton-risk-nonvertex-growing-faces-v1. Each record stores its q,D, exact face limit, rational bound, optional target and tail, work limit, and normalized SHA256 hashes of the evaluator, proof and all dependencies of the earlier finite-degree theorem. The builder and verifier compare raw source bytes at both ends; the CLI also checks the saved input bytes during replay.

python -S -m tools.risk_research.nonvertex_growing_faces \
  --outside 5 --degree 18446744073709551635 \
  --relative-target 1/37 --work-limit 190000 --output /tmp/growing.json
python -S -m tools.risk_research.nonvertex_growing_faces \
  --verify /tmp/growing.json

The selected-input counter is \((q+1)^3+q+2+(2q+2)(b_D+b_\epsilon)\); it is checked before constructing Q or large dyadic powers. Here b_D is the supplied degree’s bit length (or the current trial exponent plus one), and b_epsilon is the maximum numerator/denominator bit length, or zero without a target. A record with a target stores the maximum counter over its requested degree and successful threshold search. An explicit limit up to 1,000,000 may be selected. Resource refusal raises an error and preserves an existing output file; it does not contradict mathematical stopping. The counter is not a bound on CPU, RSS, cumulative arithmetic work, or rational bit complexity. These records bound normalized degree error only. Leading-constant numerical error remains separate, and production APIs are unchanged.