An explicit winning degree tail at every fixed vertex count

Keep the uniform-simplex design, evaluation and noise assumptions of Theory of the skeleton risk. Write

\[R_{\rm ind}(M,D)=\left(\sum_{s=1}^M\sqrt{a_{s-1}(M,D)}\right)^2, \qquad R_{\rm all}(M,D)=\binom{D+M-1}{M-1}.\]

Inactive coefficients are zero. The common positive noise and output factors cancel. The result concerns asymptotic statistical risk.

Theorem 43 (Constructive eventual victory at every fixed vertex count)

For every integer \(M\geq4\), the construction below returns an explicit integer \(H(M)\geq M\) such that, for every integer \(D\geq H(M)\),

\[R_{\rm ind}(M,D)<R_{\rm all}(M,D).\]

The construction uses only rational arithmetic and integer roots. Its constants are deliberately conservative; it does not classify the finite prefix below \(H(M)\) or locate an exact crossover.

The complementary sufficient region \(M\geq9720D\) is proved in An all-degree sufficient region for large vertex counts. The two conditions describe different directions; they do not classify their intervening region.

Coefficient bounds

Vertices. The coefficients of \(g(t)=-(1-t)\log(1-t)/t\) are \(g_0=1\), \(g_j=-1/(j(j+1))\) for \(j\geq1\). In the absolutely summable bivariate series

\[G(x,y)=g(x)+g(y)-g(xy),\]

the constant coefficient is one and the three nonconstant axis/diagonal supports are disjoint. Thus \(\lVert G\rVert_1=4\) and \(\lVert G^q\rVert_1\leq4^q\). The positive transformed diagonal \(\delta_q(j)\) from Vertex coefficients at every fixed vertex count is a rectangular partial sum of these coefficients, so \(0<\delta_q(j)\leq4^q\). The positive binomial weights in that chapter sum to \(Z_D\). Consequently, at \(q=M-1\), for every \(D\geq1\),

(87)\[a_0(M,D)\leq V_M D^{-(M-1)},\qquad V_M=M^2(M-1)!2^{M-1}.\]

This coarse estimate is used only to prove a winning region; the sharper vertex asymptotic and its explicit error remain available separately.

Nonvertex faces with at least two outside coordinates. For \(s\geq2\), define rational constants

(88)\[\begin{split}A_{s,0}&=\frac1{(s-1)!},\\ A_{s,q}&=(2^q-1)\sum_{r=0}^{q-1}\binom qr \frac{(s+q-1)!}{2^{q-r}(s+r-1)!}A_{s,r},\\ C_{s,q}&=\binom{s+q}s^2A_{s,q}.\end{split}\]

For every \(q\geq0,D\geq s\),

(89)\[a_{s-1}(s+q,D)\leq C_{s,q}D^{s-q-1}.\]

At \(q=0\) this is \(\binom{D-1}{s-1}\leq D^{s-1}/(s-1)!\). For the induction step, apply the boundary-transfer inequality in Boundary transfer and a codimension-three edge bound before grouping the proper outside subsets. Cauchy–Schwarz over their \(2^q-1\) terms gives

\[\frac{a_{s-1}(s+q,D)}{\binom{s+q}s^2} \leq(2^q-1)\sum_{r=0}^{q-1}\binom qr \frac{(s+r)_{2D}}{(s+q)_{2D}} \frac{a_{s-1}(s+r,D)}{\binom{s+r}s^2}.\]

The embedding ratio satisfies

\[\frac{(s+r)_{2D}}{(s+q)_{2D}} =\frac{(s+q-1)!}{(s+r-1)!} \prod_{j=s+r}^{s+q-1}\frac1{2D+j} \leq\frac{(s+q-1)!}{2^{q-r}(s+r-1)!}D^{r-q}.\]

Insertion of the induction hypothesis proves the recursion and bound. All quantities are zero on the inactive prefix \(D<s\).

Facets. Put \(s=M-1\geq3\), \(L(D)=1+\log D\) and \(F_M=sM^2/(2(s-2)!)\). The exact codimension-one formula, with \(j=D-s-k+1\), gives

\[a_{s-1}(M,D) =sM^2\sum_{j=1}^{D-s+1} \frac{\binom{D-j-1}{s-2}}{j(2D+s+1-j)} \leq F_M D^{M-4}L(D).\]

Indeed, the numerator is at most \((D-j)^{s-2}/(s-2)!\), and \(2D+s+1-j\geq2(D-j)>0\) on the summation range. Since \(s-3\geq0\), bound the remaining power by \(D^{s-3}\) and use \(H_{D-s+1}\leq L(D)\). This is zero when \(D<s\).

Threshold and proof of the theorem

Let \(\lceil\sqrt{x}\rceil\) mean the least integer whose square is at least the positive rational \(x\). Set

\[B_M=\lceil\sqrt{V_M}\rceil+\lceil\sqrt{F_M}\rceil +\sum_{s=2}^{M-2}\lceil\sqrt{C_{s,M-s}}\rceil, \qquad c_M=\frac{M}{(M-2)!}.\]

The exponents of all the nonfacet bounds are at most \(M-5\); the vertex exponent is also at most \(M-4\) for \(M\geq4\). Therefore, for \(D\geq M\),

\[X:=\sum_{s=1}^{M-1}\sqrt{a_{s-1}(M,D)} \leq B_M D^{(M-4)/2}\sqrt{L(D)}.\]

The elementary inequality \(L(D)\leq2\sqrt D\) follows from \(\log u\leq u-1\) at \(u=\sqrt D\). For \(b=\binom{D-1}{M-1}\) one has \(\sqrt b\leq D^{(M-1)/2}/\sqrt{(M-1)!}\). Hence

\[\frac{R_{\rm ind}-b}{D^{M-2}} =\frac{2\sqrt bX+X^2}{D^{M-2}} <3B_M D^{-1/4}+2B_M^2 D^{-3/2}.\]

Here \(2\sqrt2/\sqrt{(M-1)!}<3\). Choose the integer

\[t_M=\max\left(1, \left\lceil\frac{12B_M}{c_M}\right\rceil, \left\lceil(8B_M^2/c_M)^{1/6}\right\rceil\right), \qquad H(M)=\max(M,t_M^4).\]

At every \(D\geq H(M)\), the upper bound is at most

\[\frac{3B_M}{t_M}+\frac{2B_M^2}{t_M^6} \leq\frac{c_M}4+\frac{c_M}4=\frac{c_M}2<c_M.\]

Finally, put \(n=M-1\) and expand

\[R_{\rm all}-b =\frac{\prod_{j=1}^n(D+j)-\prod_{j=1}^n(D-j)}{n!}.\]

All the surviving elementary-symmetric terms have positive coefficients. The first is \(2(1+\cdots+n)D^{n-1}/n!=c_MD^{M-2}\). Thus \(R_{\rm all}-b\geq c_MD^{M-2}\), which proves strict victory throughout the claimed tail.

Reproduction and limits

The optional standard-library checkout tool specializes this general proof without calculating any risk coefficients:

$ python -m tools.risk_research.fixed_vertex_tail --vertices 6 --output .research/skeleton-risk/fixed6-tail.json
$ python -m tools.risk_research.fixed_vertex_tail --verify .research/skeleton-risk/fixed6-tail.json

The certificate records every majorant, integer-root bound, threshold, strict rational margin, and source/proof fingerprint. Verification recomputes the full specialization and rejects altered fields. Hashes canonicalize CRLF pairs to LF; the concurrent-edit guard compares the original raw bytes. A failed computation preserves an existing output.

The proof establishes eventual victory for each fixed \(M\), with an explicit but large threshold. Sharper special-case comparisons can give substantially smaller thresholds. It gives neither the exact finite loss set at a general vertex count nor a joint asymptotic formula when \(M,D\) both grow.