Fixed-degree risk comparisons in every dimension

This chapter uses the uniform-simplex design, evaluation and noise assumptions of Theory of the skeleton risk. Write \(M\geq2\) for the number of vertices and \(D\geq1\) for the degree. The optimal risk of dimension-based inductive fitting and the all-at-once risk are

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

The polynomial dependence on \(M\) in Theorem 18 permits a complete comparison at each listed degree without requiring a bounded-length tetrahedron vertex formula or an explicit vertex degree recurrence. The positive single sum is proved separately in Tetrahedron vertex risk. The proof below uses exact integers and rational numbers, including the square-root enclosures; it needs no floating-point error estimate or paid computer-algebra system.

Complete comparisons through degree twenty

Theorem 58 (Fixed-degree comparison for degrees nine through twenty)

For every row of the following table, \(R_{\mathrm{ind}}(M,D)<R_{\mathrm{all}}(M,D)\) holds exactly in the two displayed ranges. In the remaining integer range the reverse inequality is strict. There are no ties.

Table 15 All vertex counts, not a finite dimension cutoff

Degree \(D\)	First winning range	Second winning range	Certified infinite tail starts at
9	\(2\leq M\leq4\)	\(M\geq71\)	115
10	\(2\leq M\leq4\)	\(M\geq81\)	131
11	\(2\leq M\leq4\)	\(M\geq92\)	146
12	\(2\leq M\leq5\)	\(M\geq103\)	162
13	\(2\leq M\leq5\)	\(M\geq114\)	178
14	\(2\leq M\leq5\)	\(M\geq126\)	193
15	\(2\leq M\leq5\)	\(M\geq137\)	209
16	\(2\leq M\leq5\)	\(M\geq148\)	225
17	\(2\leq M\leq6\)	\(M\geq160\)	240
18	\(2\leq M\leq6\)	\(M\geq171\)	256
19	\(2\leq M\leq6\)	\(M\geq182\)	272
20	\(2\leq M\leq6\)	\(M\geq194\)	288

The result is computer-assisted. The final column is a sufficient bound used in the proof; it is not the actual crossover. The classifications for \(D\leq8\) are in Proposition 5.

Proof. The next three steps give the exact polynomial construction, a finite radical comparison, and a bound for all remaining dimensions. Their implementation is tools/risk_research/comparison.py. Applied to the twelve rows, they certify 2,391 finite comparisons and twelve infinite tails, with no unresolved cases. Each degree has \(D\) polynomial rows; the twelve degrees contain 174 rows and 2,666 rational coefficients. No shape for the losing set is assumed.

Complete comparisons at degrees twenty-one through thirty

Theorem 59 (Fixed-degree comparison for degrees twenty-one through thirty)

For every row below, inductive fitting wins exactly in the two displayed ranges. All-at-once fitting wins strictly at every remaining integer vertex count. There are no ties.

Table 16 Complete classifications for every integer vertex count

Degree \(D\)	First winning range	Second winning range	Certified infinite tail starts at
21	\(2\leq M\leq6\)	\(M\geq205\)	304
22	\(2\leq M\leq6\)	\(M\geq217\)	319
23	\(2\leq M\leq7\)	\(M\geq228\)	335
24	\(2\leq M\leq7\)	\(M\geq240\)	351
25	\(2\leq M\leq7\)	\(M\geq252\)	367
26	\(2\leq M\leq7\)	\(M\geq263\)	382
27	\(2\leq M\leq7\)	\(M\geq275\)	398
28	\(2\leq M\leq7\)	\(M\geq286\)	414
29	\(2\leq M\leq8\)	\(M\geq298\)	430
30	\(2\leq M\leq8\)	\(M\geq310\)	446

Proof. Apply the exact construction, finite radical enclosures and infinite-tail argument below. These ten degrees give 255 polynomial rows with 6,585 rational coefficients. All 3,726 finite comparisons separate at a 32-bit fractional grid using exact integers; the ten rational tail inequalities cover every remaining vertex count. No shape for the losing set is assumed.

A separate Newton interpolation of the original scalar angular-mode reference, using the proved degree bounds, reproduced every polynomial. Additional vertex counts beyond those used in the interpolation agree exactly. Independent rational radical enclosures and tail bounds reproduce all ten classifications. Finite checks verify these computations; the infinite tail is covered by the displayed rational inequality.

The last column is a sufficient proof threshold, rather than the exact crossover. Reproduce this extension with --degrees 21:30 in the command below. The earlier degree-nine-through-twenty certificates remain unchanged.

Exact construction of the dimension polynomials

Put \(s=m+1\), \(n=D-s\), and define

\[F(x,y)=\sum_{h,l\geq0} \frac{\binom{h+l}{h}}{(h+1)(l+1)}x^hy^l, \qquad T_r(h,l)=[x^hy^l](F-1)^r.\]

Since \(F-1\) has no constant term, \(T_r(h,l)=0\) when \(r>h+l\). Therefore, for every integer \(M\geq s\),

\[K_{M-s}(h,l)=[x^hy^l]F^{M-s} =\sum_{r=0}^{h+l}\binom{M-s}{r}T_r(h,l).\]

The binomial on the right is its falling-factorial polynomial in \(M\); it vanishes when \(r>M-s\). Truncation to \(0\leq h,l\leq n\) is exact because all powers are nonnegative. Define

\[\begin{split}A_{k,h}=(-1)^h\binom{n-k}{h}\binom{n+k+3s-1}{h}, \qquad d_s(k)=\begin{cases} \binom{k+s-2}{s-2},&s\geq2,\\ \mathbf1_{k=0},&s=1. \end{cases}\end{split}\]

Substitution in (12) constructs

\[\begin{split}P_{D,s-1}(M) ={}&\frac1{(s-1)!}\sum_{k=0}^n d_s(k) \sum_{h,l=0}^{n-k} A_{k,h}A_{k,l}(2D+s-h-l-1)!h!l!\\ &\hspace{2em}\cdot\sum_{r=0}^{h+l}\binom{M-s}{r}T_r(h,l).\end{split}\]

Its degree is at most \(2(D-s)\), and

\[a_{s-1}(M,D)=\binom Ms^2\frac{P_{D,s-1}(M)}{(M)_{2D}} \qquad(M\geq s).\]

This is a direct symbolic construction, not interpolation. For \(s>M\) the face does not exist and its coefficient is set to zero; one does not evaluate the polynomial as a physical risk outside its stated range.

A rational bound for the infinite tail

Write \(P_{D,s-1}(M)=\sum_i p_{s,i}M^i\). Fix an integer \(L\geq\max(2,D)\) and put

\[B_s(L)=\frac1{(s!)^2}\sum_{i=0}^{2(D-s)} |p_{s,i}|L^{i-2(D-s)}.\]

For all \(M\geq L\), the inequalities \(\binom Ms\leq M^s/s!\) and \((M)_{2D}\geq M^{2D}\) give

\[0\leq a_{s-1}(M,D) \leq\frac1{(s!)^2}\sum_i|p_{s,i}|M^{i-2(D-s)} \leq B_s(L).\]

The last inequality follows because every exponent is nonpositive. More generally an absent face \(s>M\) has coefficient zero, so also satisfies the bound. By Cauchy–Schwarz,

\[R_{\mathrm{ind}}(M,D)\leq D\sum_{s=1}^DB_s(L).\]

Since \(\binom{M+D-1}{D}\) increases with \(M\), the single strict rational inequality

\[D\sum_{s=1}^DB_s(L)<\binom{L+D-1}{D}\]

proves that inductive fitting wins for every \(M\geq L\). The certificate records each \(B_s(L)\), both sides, and their strictly positive difference. The displayed tail thresholds satisfy this inequality exactly.

Strict comparisons for the finite prefix

For every remaining integer \(2\leq M<L\), evaluate the active coefficients as exact fractions. For \(a=u/v\geq0\), choose \(K=2^b\) and compute the integer

\[t=\operatorname{isqrt}\left(\left\lfloor\frac{uK^2}{v}\right\rfloor\right).\]

Then \(t/K\leq\sqrt a\leq(t+1)/K\). If \(t^2v=uK^2\), the two endpoints coincide at \(t/K\). Let \(A\) and \(B\) be the sums of the lower and upper integer numerators. Positivity gives

\[A^2\leq K^2R_{\mathrm{ind}}(M,D)\leq B^2.\]

Thus \(B^2<K^2R_{\mathrm{all}}\) certifies an inductive win, and \(A^2>K^2R_{\mathrm{all}}\) certifies an all-at-once win. Overlapping bounds require higher precision. If the configured precision limit is reached, certification fails and no all-dimension result is reported. An exact equality is therefore never silently included among strict wins or losses.

All 2,391 finite comparisons for degrees nine through twenty and all 3,726 for degrees twenty-one through thirty separate at \(b=32\), giving 6,117 comparisons in total. This is a 32-bit fractional grid for the square roots, with arbitrarily large integer arithmetic throughout; it is not a 32-bit floating-point calculation.

Reproduction and verification

From a source checkout with the development environment available, run

$ .venv/bin/python -m tools.risk_research.comparison --degrees 9:20 --output .research/skeleton-risk/comparisons.json

The command needs only the Python standard library. --degrees 9 restricts it to the first new degree. --max-tail bounds the search for a sufficient infinite-tail inequality, and --max-bits bounds finite comparison precision. A failure preserves an existing output file.

The JSON certificate contains exact numerator/denominator decimal strings for the polynomials, the dyadic endpoints and strict sign margin for every finite \(M\), the tail inequalities, and the SHA256 of the source used. Polynomial rows are ordered by \(s=1,\ldots,D\), with ascending powers of \(M\) within each row. The output is deterministic for fixed source and arguments.

Independent validation includes equality with the orbit-counted polynomials through degree eight, the original scalar-sum reference, the rational Gram-propagation solver at degree nine on a triangle, all triangle closed forms at degree twenty, and the independently derived large-dimension limits. A separate exact interpolation of the scalar reference, using the proved degree bounds and additional check points, also reproduced all 174 polynomial rows and all comparison certificates for degrees nine through twenty.

The extension to degrees twenty-one through thirty independently reproduces another 255 polynomial rows, 6,585 rational coefficients, 3,726 finite comparisons and ten infinite tails by Newton interpolation of the original scalar reference. The degree-thirty regression also checks the previously proved triangle, tetrahedron-vertex, five-vertex-vertex and five-vertex-face formulas, together with every independently derived large-dimension limit.

These results give complete fixed-degree comparisons through thirty. They do not assert a uniform formula for the two winning ranges as functions of \(D\), or settle the remaining all-degree vertex evaluations.

An all-degree sufficient region for large vertex counts proves the separate all-degree sufficient condition \(M\geq9720D\), without constructing these degree-specific polynomials. It does not locate the exact winning boundary.