Six-vertex risk comparison at every degree ========================================== Under the uniform-simplex design, evaluation and noise assumptions of :doc:`skeleton_risk_theory`, put .. math:: 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. .. prf:theorem:: Complete six-vertex comparison :label: thm-risk-six-vertex-comparison For every integer :math:`D\geq1`, optimal inductive skeleton fitting on six vertices has strictly smaller asymptotic risk exactly when .. math:: D\in\{1,2\}\quad\text{or}\quad D\geq17. All-at-once fitting strictly wins at :math:`3\leq D\leq16`. There are no ties. The finite prefix is computer-assisted. .. prf:proof:: We prove the entire tail :math:`D\geq128` analytically, and certify every smaller degree over exact rational square-root intervals. No monotonicity or single-crossing assumption is made. **Vertex.** Write :math:`G(x,y)=g(x)+g(y)-g(xy)`, where :math:`g(t)=-(1-t)\log(1-t)/t`. Its constant is one, its two axes have coefficients :math:`-1/(j(j+1))`, and its positive diagonal has coefficients :math:`1/(j(j+1))`. Thus its absolute coefficient sum is four. The transformed diagonal :math:`\delta_5(j)` is a rectangular partial sum of :math:`G^5`, so :math:`0<\delta_5(j)\leq4^5`. The positive normalized binomial average in :doc:`skeleton_risk_vertex_general` gives .. math:: 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 :doc:`skeleton_risk_extensions`, put :math:`j=L+1` and :math:`N=2D+s+1`. Its nonnegative bracket is at most :math:`H_{j+1}`, and its denominator is :math:`j(j+1)(N-j)(N-j+1)\geq j(j+1)D^2`. The exact telescoping sum .. math:: \sum_{j\geq1}\frac{H_{j+1}}{j(j+1)}=2 is proved in :doc:`skeleton_risk_codimension_three`. At :math:`s=3`, the angular multiplicity is at most :math:`D`; at :math:`s=4`, it is at most :math:`D^2/2`. Including allocation factors gives, on their active domains, .. math:: 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 :math:`s=2,q=4`. The embedding ratios from source sizes :math:`2,3,4,5` into six vertices are at most, respectively, .. math:: \frac{15}{2D^4},\quad\frac{15}{2D^3},\quad \frac5{D^2},\quad\frac5{2D}. The source coefficients are bounded by :math:`D`, :math:`9`, :math:`864/D^2`, :math:`30000/D^2`. The triangle bound and the two latter bounds are proved in :doc:`skeleton_risk_codimension_three` and its cited tetrahedron comparison. Before the common allocation factor fifteen, the subset-size weights are :math:`1,4/3,1,2/5`. The four root constants are thus bounded by :math:`3,11,66,110`, with the third having exponent :math:`D^{-2}` and the others :math:`D^{-3/2}`. Therefore .. math:: \sqrt{a_1(6,D)} <\frac{1860}{D^{3/2}}+\frac{990}{D^2}. **Triangle.** Apply the same transfer with :math:`s=3,q=3`. Its common allocation factor is twenty and its size weights are :math:`1,3/4,3/10`. The embedding bounds are :math:`15/(2D^3),5/D^2,5/(2D)`. The top coefficient on three vertices is at most :math:`D^2/2`. The tetrahedron facet is at most :math:`24(1+\log D)`: its exact sum has numerator :math:`D-j-1`, denominator at least :math:`2j(D-j)`, and common factor forty-eight. Use the preceding :math:`4800/D` bound for the third source. Rounding the root constants upward yields .. math:: \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 :math:`10\sqrt{15}`, :math:`15\sqrt{120}` and :math:`6\sqrt{12000}`. **Facet.** Reindex the exact codimension-one formula by :math:`j=D-4-k`. For :math:`D\geq5`, .. math:: 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 :math:`(D-j)^3/6` and the second denominator factor below by :math:`2(D-j)`. **The entire tail.** Let :math:`L(D)=1+\log D`, :math:`b=a_5(6,D)=\binom{D-1}5` and :math:`X=\sum_{m=0}^4\sqrt{a_m(6,D)}`. Then .. math:: \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 :math:`D\geq128`; for the logarithmic terms this follows by differentiating :math:`L(D)/D` and :math:`L(D)/D^5`. The elementary bound :math:`\log2<3/4` gives :math:`L(128)<25/4`, so :math:`\sqrt{L(128)}<5/2`. Also :math:`\sqrt{128}>11` and :math:`\sqrt{15L(128)/128}<7/8`. Thus :math:`X/D^{3/2}\frac{D^4}4. This proves strict inductive victory at every :math:`D\geq128`. **Finite prefix.** At every :math:`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 :math:`\{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 :doc:`skeleton_risk_codimension_three_exact`. The optional ``--backend flint`` independently evaluates the original scalar edge/triangle sums using integer kernels. Both paths reproduce all 127 exact coefficient rows. .. code-block:: console $ 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.