Fixed-degree risk comparisons in every dimension ================================================ This chapter uses the uniform-simplex design, evaluation and noise assumptions of :doc:`skeleton_risk_theory`. Write :math:`M\geq2` for the number of vertices and :math:`D\geq1` for the degree. The optimal risk of dimension-based inductive fitting and the all-at-once risk are .. math:: 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 :math:`M` in :prf:ref:`thm-skeleton-rational` 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 :doc:`skeleton_risk_vertex`. 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 ------------------------------------------ .. prf:theorem:: Fixed-degree comparison for degrees nine through twenty :label: thm-risk-comparisons-nine-twenty For every row of the following table, :math:`R_{\mathrm{ind}}(M,D)h+l`. Therefore, for every integer :math:`M\geq s`, .. math:: 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 :math:`M`; it vanishes when :math:`r>M-s`. Truncation to :math:`0\leq h,l\leq n` is exact because all powers are nonnegative. Define .. math:: 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} Substitution in :eq:`skeleton-scalar-sum` constructs .. math:: 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). Its degree is at most :math:`2(D-s)`, and .. math:: 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 :math:`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 :math:`P_{D,s-1}(M)=\sum_i p_{s,i}M^i`. Fix an integer :math:`L\geq\max(2,D)` and put .. math:: B_s(L)=\frac1{(s!)^2}\sum_{i=0}^{2(D-s)} |p_{s,i}|L^{i-2(D-s)}. For all :math:`M\geq L`, the inequalities :math:`\binom Ms\leq M^s/s!` and :math:`(M)_{2D}\geq M^{2D}` give .. math:: 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 :math:`s>M` has coefficient zero, so also satisfies the bound. By Cauchy--Schwarz, .. math:: R_{\mathrm{ind}}(M,D)\leq D\sum_{s=1}^DB_s(L). Since :math:`\binom{M+D-1}{D}` increases with :math:`M`, the single strict rational inequality .. math:: D\sum_{s=1}^DB_s(L)<\binom{L+D-1}{D} proves that inductive fitting wins for every :math:`M\geq L`. The certificate records each :math:`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 :math:`2\leq MK^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 :math:`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 .. code-block:: console $ .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 :math:`M`, the tail inequalities, and the SHA256 of the source used. Polynomial rows are ordered by :math:`s=1,\ldots,D`, with ascending powers of :math:`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 :math:`D`, or settle the remaining all-degree vertex evaluations. :doc:`skeleton_risk_uniform_dimension_tail` proves the separate all-degree sufficient condition :math:`M\geq9720D`, without constructing these degree-specific polynomials. It does not locate the exact winning boundary.