A quadratic sufficient degree region, uniform in vertex count

Use the uniform-simplex design, evaluation and noise assumptions of Theory of the skeleton risk. As in An explicit winning degree tail at every fixed vertex count, write \(R_{\rm ind}=(\sum_{s=1}^M\sqrt{a_{s-1}})^2\) and \(R_{\rm all}=\binom{D+M-1}{M-1}\). Common positive output and noise factors are omitted.

Theorem 62 (A uniform quadratic winning degree region)

For every pair of integers \(M\geq4\) and \(D\geq8M^2\),

(121)\[R_{\rm all}(M,D)-R_{\rm ind}(M,D) >\frac{239}{3200}\frac{M}{(M-2)!}D^{M-2}>0.\]

This is a simultaneous sufficient region. It does not identify the crossover or classify the remaining finite degrees at a given \(M\).

The result improves the constructive but very large \(H(M)\) of An explicit winning degree tail at every fixed vertex count. It is compatible with the complementary region \(M\geq9720D\) in An all-degree sufficient region for large vertex counts. Neither condition asserts victory throughout the intervening region. For \(M=4,5,6\), the new thresholds are respectively \(128,200,288\); the existing special-case classifications can give smaller sufficient thresholds.

The new threshold is smaller than the older construction for every \(M\geq4\): its \(B_M\geq M\) implies \(t_M\geq12(M-2)!\geq M\), hence \(H(M)\geq M^4>8M^2\). Individual coefficient majorants need not all be smaller; the improvement comes from their joint binomial structure and from retaining the facet’s sharper estimate.

A direct diagonal majorant

For \(s\geq2\), \(q=M-s\), the positive-average formula of Theorem 28 and the absolutely summable generating series of Vertex coefficients at every fixed vertex count give

\[0<\delta_q(j)\leq4^q,\qquad \sum_{k=0}^{D-s}\binom{k+s-2}{s-2}=\binom{D-1}{s-1}.\]

Indeed the diagonal is a rectangular partial sum of coefficients of \(G(x,y)^q\), where \(\|G\|_1=4\); its absolute value is at most \(4^q\). Therefore, for \(D\geq s\),

(122)\[a_{s-1}(M,D) \leq\binom Ms^2\frac{(M-1)!\,2^{M-s}}{((s-1)!)^2} D^{2s-M-1}.\]

Here \((2D+s)_q\geq(2D)^q\), \((s)_q=(M-1)!/(s-1)!\), and \(\binom{D-1}{s-1}\leq D^{s-1}/(s-1)!\). At \(s=1\), exactly the same displayed bound follows from the vertex majorant (118). On an inactive face \(D<s\), its actual coefficient and its implementation majorant are both zero. This avoids the repeated Cauchy–Schwarz factors in the older recursive constants. The facet will use its sharper logarithmic estimate instead of this direct bound.

Uniformly sum the lower skeletons

Set

\[b_0=\frac{D^{M-1}}{(M-1)!},\qquad p=\frac{M(M-1)}D\leq\frac18,\qquad X=\sum_{s=1}^{M-1}\sqrt{a_{s-1}(M,D)}.\]

For every nonfacet, put \(q=M-s\geq2\). Dividing (122) by \(b_0\) gives

\[\sqrt{a_{s-1}/b_0} \leq\binom Mq\frac{(M-1)!}{(M-q-1)!} \left(\frac{\sqrt2}{D}\right)^q \leq\frac{(\sqrt2p)^q}{q!}.\]

For \(q\geq2\), \(q!\geq2\,3^{q-2}\). With \(x=\sqrt2p\leq3p/2<3\), it follows that

\[\sum_{s=1}^{M-2}\sqrt{a_{s-1}/b_0} \leq\sum_{q=2}^{\infty}\frac{x^q}{q!} \leq\frac{x^2}{2(1-x/3)} \leq\frac{p^2}{1-p/2}\leq\frac{2p}{15}.\]

For the facet, (120) is replaced by the sharper facet estimate in An explicit winning degree tail at every fixed vertex count:

\[\sqrt{a_{M-2}/b_0} \leq p\sqrt{\frac{(M-2)(1+\log D)}{2D}}.\]

The function \((1+\log D)/D\) decreases for \(D\geq1\), and \((1+\log(8M^2))/M\) decreases for \(M\geq4\). Consequently, throughout the asserted domain,

\[\frac{(M-2)(1+\log D)}{2D} <\frac{1+\log(8M^2)}{16M} \leq\frac{1+\log128}{64}<\frac{3}{32} <\left(\frac5{16}\right)^2.\]

The elementary \(\log128<5\) can be checked without a numerical logarithm: \(e^5>\sum_{k=0}^7 5^k/k!>128\). Combining the facet and all nonfacets yields

\[T:=\frac{X}{\sqrt{b_0}} <\left(\frac5{16}+\frac2{15}\right)p =\frac{107}{240}p<\frac9{20}p.\]

Compare with the all-at-once gap

Let \(b=a_{M-1}(M,D)=\binom{D-1}{M-1}\leq b_0\). The positive elementary-symmetric expansion in An explicit winning degree tail at every fixed vertex count gives, for \(D\geq M\),

\[R_{\rm all}-b\geq\frac{M}{(M-2)!}D^{M-2}=p b_0.\]

On the other hand,

\[\frac{R_{\rm ind}-b}{b_0} =\frac{2\sqrt b X+X^2}{b_0} \leq2T+T^2 <\left(\frac9{10}+\frac{81}{400}p\right)p \leq\frac{2961}{3200}p.\]

Subtracting proves (121). All estimates are simultaneous in \(M,D\); no fixed-dimension asymptotic remainder or finite scan has been substituted into a growing-dimension limit.

Exact specialization and limits

The standard-library checkout tool records the region, rational margin, direct majorants and proof/source fingerprints:

$ python -m tools.risk_research.quadratic_degree_tail --vertices 6 --output .research/skeleton-risk/quadratic6.json
$ python -m tools.risk_research.quadratic_degree_tail --verify .research/skeleton-risk/quadratic6.json

It never calculates an actual risk or infers a theorem from a finite scan. The normalized quantitative margin is relative to \(b_0\) and applies only for \(M\geq4,D\geq8M^2\). Below that domain the tool’s coefficient majorants remain valid, but they give no new classification. Verification recomputes the complete record with exact JSON types and rejects duplicate keys, extra fields, modified domains or changed sources. Fingerprints canonicalize CRLF to LF; execution guards compare raw source bytes before and after. Failed generation preserves an existing output. Output paths resolving to any fingerprinted evaluator or proof source, including symlink aliases, are rejected before evaluation or writing. The destination is checked again after calculation and immediately before writing. The writer receives the resolved parent and original filename, so retargeting the supplied parent alias at writer entry cannot redirect the output. Independent Gram/scalar coefficient checks, integer-root comparisons and frozen holdouts validate implementation; the proof establishes the infinite region.