Six-vertex risk comparison at every degree

Under the uniform-simplex design, evaluation and noise assumptions of Theory of the skeleton risk, put

\[R_{\rm ind}(D)=\left(\sum_{m=0}^5\sqrt{a_m(6,D)}\right)^2, \qquad R_{\rm all}(D)=\binom{D+5}5.\]

Inactive coefficients are zero. Common positive noise and output factors cancel. These are asymptotic statistical risks, not finite-sample risks.

Theorem 49 (Complete six-vertex comparison)

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

\[D\in\{1,2\}\quad\text{or}\quad D\geq17.\]

All-at-once fitting strictly wins at \(3\leq D\leq16\). There are no ties. The finite prefix is computer-assisted.

Proof. We prove the entire tail \(D\geq128\) analytically, and certify every smaller degree over exact rational square-root intervals. No monotonicity or single-crossing assumption is made.

Vertex. Write \(G(x,y)=g(x)+g(y)-g(xy)\), where \(g(t)=-(1-t)\log(1-t)/t\). Its constant is one, its two axes have coefficients \(-1/(j(j+1))\), and its positive diagonal has coefficients \(1/(j(j+1))\). Thus its absolute coefficient sum is four. The transformed diagonal \(\delta_5(j)\) is a rectangular partial sum of \(G^5\), so \(0<\delta_5(j)\leq4^5\). The positive normalized binomial average in Vertex coefficients at every fixed vertex count gives

\[a_0(6,D)\leq\frac{36\cdot120\cdot4^5} {\prod_{h=1}^5(2D+h)} <\frac{138240}{D^5}<\frac{372^2}{D^5}.\]

Codimension-two bounds used below. In the exact mode formula of Further evaluations of the skeleton risk, put \(j=L+1\) and \(N=2D+s+1\). Its nonnegative bracket is at most \(H_{j+1}\), and its denominator is \(j(j+1)(N-j)(N-j+1)\geq j(j+1)D^2\). The exact telescoping sum

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

is proved in Boundary transfer and a codimension-three edge bound. At \(s=3\), the angular multiplicity is at most \(D\); at \(s=4\), it is at most \(D^2/2\). Including allocation factors gives, on their active domains,

\[a_2(5,D)<\frac{100\cdot24\cdot2}{D} =\frac{4800}D,\qquad a_3(6,D)<225\cdot40\cdot\frac12\cdot2=9000<95^2.\]

Edge. Apply the nonvertex boundary-transfer inequality with \(s=2,q=4\). The embedding ratios from source sizes \(2,3,4,5\) into six vertices are at most, respectively,

\[\frac{15}{2D^4},\quad\frac{15}{2D^3},\quad \frac5{D^2},\quad\frac5{2D}.\]

The source coefficients are bounded by \(D\), \(9\), \(864/D^2\), \(30000/D^2\). The triangle bound and the two latter bounds are proved in Boundary transfer and a codimension-three edge bound and its cited tetrahedron comparison. Before the common allocation factor fifteen, the subset-size weights are \(1,4/3,1,2/5\). The four root constants are thus bounded by \(3,11,66,110\), with the third having exponent \(D^{-2}\) and the others \(D^{-3/2}\). Therefore

\[\sqrt{a_1(6,D)} <\frac{1860}{D^{3/2}}+\frac{990}{D^2}.\]

Triangle. Apply the same transfer with \(s=3,q=3\). Its common allocation factor is twenty and its size weights are \(1,3/4,3/10\). The embedding bounds are \(15/(2D^3),5/D^2,5/(2D)\). The top coefficient on three vertices is at most \(D^2/2\). The tetrahedron facet is at most \(24(1+\log D)\): its exact sum has numerator \(D-j-1\), denominator at least \(2j(D-j)\), and common factor forty-eight. Use the preceding \(4800/D\) bound for the third source. Rounding the root constants upward yields

\[\sqrt{a_2(6,D)}<\frac{39}{\sqrt D} +\frac{165\sqrt{1+\log D}}D+\frac{658}D.\]

Indeed, the three constants before rounding are \(10\sqrt{15}\), \(15\sqrt{120}\) and \(6\sqrt{12000}\).

Facet. Reindex the exact codimension-one formula by \(j=D-4-k\). For \(D\geq5\),

\[a_4(6,D)=180\sum_{j=1}^{D-4} \frac{\binom{D-j-1}3}{j(2D+6-j)} \leq15D^2(1+\log D).\]

Bound the numerator by \((D-j)^3/6\) and the second denominator factor below by \(2(D-j)\).

The entire tail. Let \(L(D)=1+\log D\), \(b=a_5(6,D)=\binom{D-1}5\) and \(X=\sum_{m=0}^4\sqrt{a_m(6,D)}\). Then

\[\frac X{D^{3/2}}< \sqrt{\frac{15L(D)}D}+\frac{95}{D^{3/2}} +\frac{39}{D^2}+\frac{165\sqrt{L(D)}+658}{D^{5/2}} +\frac{1860}{D^3}+\frac{990}{D^{7/2}}+\frac{372}{D^4}.\]

Every term on the right decreases for \(D\geq128\); for the logarithmic terms this follows by differentiating \(L(D)/D\) and \(L(D)/D^5\). The elementary bound \(\log2<3/4\) gives \(L(128)<25/4\), so \(\sqrt{L(128)}<5/2\). Also \(\sqrt{128}>11\) and \(\sqrt{15L(128)/128}<7/8\). Thus \(X/D^{3/2}<Q\), where the following number is rational:

\[Q=\frac78+\frac{95}{128\cdot11}+\frac{39}{128^2} +\frac{165\cdot5}{2\cdot128^2\cdot11} +\frac{658}{128^2\cdot11}+\frac{1860}{128^3} +\frac{990}{128^3\cdot11}+\frac{372}{128^4}.\]

Since \(1/\sqrt{120}<11/120\), one has \(\sqrt b<(11/120)D^{5/2}\). Exact rational arithmetic gives

\[\frac{R_{\rm ind}-b}{D^4} <\frac{11}{60}Q+\frac{Q^2}{128}<\frac14.\]

In contrast, direct expansion gives

\[R_{\rm all}-b =\binom{D+5}5-\binom{D-1}5 =\frac{D^4}4+\frac{15D^2}4+2>\frac{D^4}4.\]

This proves strict inductive victory at every \(D\geq128\).

Finite prefix. At every \(1\leq D\leq127\), evaluate the general positive vertex sum, positive diagonal edge and triangle sums, codimension-two angular sum, facet sum and top coefficient exactly. Integer square-root intervals separate all comparisons. The entire loss set is \(\{3,4,\ldots,16\}\), with no possible equality. This completes the classification.

Reproduction

The optional checkout tool uses standard-library positive diagonal averages by default; see Positive diagonal formulas for nonvertex risk. The optional --backend flint independently evaluates the original scalar edge/triangle sums using integer kernels. Both paths reproduce all 127 exact coefficient rows.

$ python -m tools.risk_research.six_vertex_comparison --output .research/skeleton-risk/six-vertex-comparison.json
$ python -m tools.risk_research.six_vertex_comparison --verify .research/skeleton-risk/six-vertex-comparison.json

The certificate records all 127 coefficient rows, radical intervals, strict margins, the separate analytic tail and all evaluation-source fingerprints. Verification uses the largest recorded per-row bit count as its effective precision budget and recomputes every field. Certificates generated with a nondefault --max-bits use the same plain --verify command; precision overrides are generation options only. Source hashes canonicalize CRLF pairs to LF, while concurrent edits are detected by comparing original raw bytes. Partial coverage, overlapping intervals, altered sources or a possible equality cause failure before replacing an existing output. The result classifies six vertices only.