Five-vertex risk comparison at every degree

Under the uniform-simplex design, evaluation and noise assumptions of Theory of the skeleton risk, the nonvertex boundary-lift bound from Boundary transfer and a codimension-three edge bound closes the five-vertex comparison. Put

\[R_{\mathrm{ind}}(D)=\left(\sum_{m=0}^4\sqrt{a_m(5,D)}\right)^2, \qquad R_{\mathrm{all}}(D)=\binom{D+4}4.\]

Inactive coefficients are zero. The common positive noise and output factors cancel. The conclusion concerns asymptotic statistical risk; it is not a finite-sample risk-ordering claim.

Theorem 67 (Complete five-vertex comparison)

For every integer \(D\geq1\), optimal inductive skeleton fitting on five vertices has strictly smaller asymptotic risk precisely when

\[D\in\{1,2\}\quad\text{or}\quad D\geq12.\]

All-at-once fitting strictly wins at \(3\leq D\leq11\). Equality never occurs. The finite prefix is computer-assisted.

Proof. We first prove a strict inductive win for every \(D\geq64\), then certify every smaller degree with exact rational radical bounds. We make no monotonicity or single-crossing assumption.

Vertex. The pointwise estimate in the proof of Theorem 66 writes, at \(n=j+1\),

\[\Psi_j=\frac{48(H_j^{(2)}-A_j)}{n(n+2)}+R_j,\qquad |R_j|\leq\frac{192H_j+229}{n^3}.\]

Use \(H_j\leq n\), \(H_j^{(2)}<2\) and \(A_j\geq0\) to obtain \(0<\Psi_j\leq517/n^2\leq517\). Vandermonde’s identity normalizes the positive weights in Theorem 64. Hence

\[a_0(5,D)\leq\frac{600\cdot517} {(2D+1)(2D+2)(2D+3)(2D+4)} <\frac{140^2}{D^4}.\]

This estimate is valid at every positive degree.

Edge. The all-facet inclusion-exclusion lift in Theorem 27 gives

\[a_1(5,D)<\frac{30000}{D^2}<\frac{174^2}{D^2} \qquad(D\geq2).\]

The inactive degree-one coefficient is zero.

Triangle face. In (26), put \(s=3\), \(j=L+1\) and \(N=2D+4\). The nonnegative bracket is at most \(H_{j+1}\) and \((N-j)(N-j+1)\geq D^2\). The multiplicity \(k+1\) is at most \(D\). Since

\[\sum_{j\geq1}\frac{H_{j+1}}{j(j+1)}=2,\]

summing the mode bound and multiplying by \(\binom53^2\) proves

\[a_2(5,D)\leq\frac{4800}{D}<\frac{70^2}{D}.\]

To verify the infinite sum, write \(H_{j+1}=H_j+1/(j+1)\). The first resulting sum is \(\zeta(2)\) by exchanging the positive sums; the second is \(2-\zeta(2)\) by partial fractions.

Facet. Reindex Theorem 13 with \(s=4\) and \(j=D-3-k\). For \(D\geq4\),

\[a_3(5,D)=50\sum_{j=1}^{D-3} \frac{(D-j-1)(D-j-2)}{j(2D+5-j)} \leq25\sum_{j=1}^{D-3}\frac{D-j}{j} \leq25D(1+\log D).\]

Indeed, \(2D+5-j\geq2(D-j)\) and both numerator factors are positive and at most \(D-j\). The function \(2t^{1/4}-1-\log t\) has its minimum at \(t=16\), where it equals \(3-4\log2>0\). Here \(\log2<3/4\) follows by integrating the strict secant bound for \(1/t\) on \([1,2]\). Therefore

\[\sqrt{a_3(5,D)}<\sqrt{50}\,D^{5/8} <\frac{15}2D^{5/8}.\]

Infinite tail. Set \(b=a_4(5,D)=\binom{D-1}4\) and \(X=\sum_{m=0}^3\sqrt{a_m(5,D)}\). For \(D\geq64\), \(D^{3/8}>19/4\), since \(64^3\cdot4^8>19^8\), and \(\sqrt D\geq8\). The four preceding bounds give

\[\begin{split}\frac XD &<\frac{15}2D^{-3/8}+174D^{-2} +70D^{-3/2}+140D^{-3}\\ &\leq Q=\frac{30}{19}+\frac{174}{64^2} +\frac{70}{64\cdot8}+\frac{140}{64^3} =\frac{2189881}{1245184}.\end{split}\]

Also \(\sqrt b\leq D^2/\sqrt{24}<5D^2/24\). Consequently

\[\frac{R_{\mathrm{ind}}-b}{D^3} =\frac{2\sqrt bX+X^2}{D^3} <\frac5{12}Q+\frac{Q^2}{64} =\frac{232531119030803}{297692773220352}<\frac56.\]

In contrast, direct polynomial expansion gives

\[R_{\mathrm{all}}-b =\binom{D+4}4-\binom{D-1}4 =\frac56D^3+\frac{25}6D>\frac56D^3.\]

Thus inductive fitting strictly wins for every \(D\geq64\).

Finite prefix. For each \(1\leq D\leq63\), evaluate the positive vertex sum, original scalar edge formula, single angular triangle-face sum, facet sum and top coefficient over the rationals. The integer square-root enclosures from Fixed-degree risk comparisons in every dimension separate every comparison at 32 fractional bits. The complete loss set is \(\{3,4,\ldots,11\}\), proving the theorem.

Reproduction

The optional checkout tool needs only the standard library by default:

$ python -m tools.risk_research.five_vertex_comparison --output .research/skeleton-risk/five-vertex-comparison.json

Pass --backend flint to use an already installed python-flint. Both backends clear the kernel denominators before exact integer convolution. The certificate includes all 63 finite coefficient rows, strict radical margins, the rational tail margin and hashes of all evaluation sources. Overlap, a possible equality, or changed sources raises an error and preserves any existing output file.