Further evaluations of the skeleton risk
This chapter continues Theory of the skeleton risk under exactly its
uniform-simplex design, evaluation and noise assumptions. Here M is
the number of vertices and D is the degree. It separates proved
identities from finite computational checks and remaining evaluations.
The optional reference implementations live in tools/risk_research.
They add no runtime dependencies. An accompanying numerical guard rejects
nonfinite assembled-risk propagation before its condition estimate.
Result |
Scope |
|---|---|
Every fixed |
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The entire two-coordinate kernel is evaluated with harmonic numbers. |
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Every nonvertex source has a trace-correct boundary-transfer inequality. In particular the five-vertex edge has an all-degree rational upper bound; this does not provide an exact codimension-three formula. |
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Every codimension-two coefficient reduces to one angular sum. |
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Every tetrahedron edge coefficient is evaluated using harmonic numbers and two explicitly specified auxiliary recurrences. |
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Every triangular-face coefficient on a five-vertex simplex has an eight-state rational representation and a certified order-eight recurrence. |
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Every tetrahedral-face coefficient on a six-vertex simplex also has an eight-state formula and a certified order-eight recurrence. |
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The tetrahedron vertex has an explicit order-ten degree recurrence certified by four rational telescoping identities. |
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The leading large-degree constant for every fixed codimension-two face with at least two vertices. |
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A vertex norm identity reduces the largest outside-kernel power by two; it is not a completed tetrahedron vertex closed form. |
Theorem 60 additionally proves a sufficient linear vertex threshold at every positive degree: if \(M\geq9720D\), optimal inductive risk is less than four fifths of all-at-once risk. This conservative sufficient region does not determine an exact crossover.
Fixed-dimension degree recurrences exist
Theorem 19 (Holonomicity at fixed dimension)
Fix integers \(M\geq2\) and \(0\leq m<M\). There exist an integer \(r\geq0\) and polynomials \(p_0,\ldots,p_r\in\mathbb Q[D]\), not all zero, such that
for all sufficiently large integer \(D\). A polynomial factor can include any exceptional finite prefix. The order and degrees may depend on \(M,m\).
Proof. Put \(s=m+1\), \(q=M-s\). The explicit logarithmic expression for \(F(x,y)\) in Lemma 5 is multivariate differential D-finite. For fixed nonnegative integer \(q\), the derivatives of \(F^q\) lie in the rational-function span of monomials of total degree at most \(q\) in \(\log(1-x-y),\log(1-x),\log(1-y)\). This is a finite-dimensional span, including the constant function. Thus \(F^q\) is D-finite.
The coefficient dictionary from holonomic power series to holonomic discrete sequences makes \(K_q(h,l)=[x^hy^l]F^q\) holonomic. This uses differential D-finiteness of the generating series, rather than a claim about arbitrary multivariate shift-D-finite sequences. The distinction and coefficient dictionary are discussed in Kauers [Kau23], Section 4.5, printed page 368.
In Theorem 11, all remaining factors are factorial ratios of affine integer expressions, signs and polynomial factors, hence holonomic on the summation domain. Numerator factorials can be extended by zero to negative arguments and their boundary recurrences cleared; denominator factorials use reciprocal-factorial zero extensions. The case \(s=1\) uses the holonomic multiplier \(\mathbf1_{k=0}\). Multiply by the indicators of
All factorial arguments in the original summand are nonnegative on this domain. Holonomic sequences are closed under pointwise products and finite summation over such rational polyhedral ranges (Kauers [Kau23], Theorems 5.24 and 5.33 and printed pages 442–443). Consequently the scalar sum is holonomic in \(D\). Its remaining prefactor is hypergeometric in \(D\). The inactive degrees form only a finite prefix, which preserves P-recursiveness.
Finite products of such fixed-dimension sequences and finite weighted
sums preserve P-recursiveness. Therefore the product-cell formulas and
regular-complex incidence formulas inherit the conclusion when their
combinatorics are fixed and their factor degrees follow fixed affine
integer functions of D that remain positive for all sufficiently large
D. This statement concerns risk coefficients;
it does not assert closure for the square roots in optimal allocation.
Two outside coordinates
Write \(H_n=\sum_{j=1}^n1/j\), with \(H_0=0\).
Lemma 6 (Harmonic evaluation of the two-coordinate kernel)
For all nonnegative integers \(h,l\),
Proof. Put \(U=(1-x-y)\log(1-x-y)\), \(A=(1-x)\log(1-x)\), \(B=(1-y)\log(1-y)\). Then \(x^2y^2F^2=U^2-2UA-2UB+2AB+A^2+B^2\). Set \(a=h+2\), \(b=l+2\), \(n=a+b\), and \(C=(n-3)!/(a!b!)\). The last two terms have zero mixed coefficient, and \([x^ay^b]AB=1/(a(a-1)b(b-1))\).
Differentiating the finite binomial products in
once in each parameter at \(\alpha=\beta=1\) gives
For \(b=2\) this follows directly from the derivative at zero; for \(b\geq3\), use the logarithmic derivative of the binomial product at \(2-b\). The displayed expression includes both cases without a negative-index harmonic number. Interchange \(a,b\) for \(UB\). Twice differentiating \((-1)^n\binom\alpha n\) at \(\alpha=2\) gives \([x^ay^b]U^2=4C(H_{n-3}-3/2)\). Combining the terms proves the formula.
A codimension-two face
Theorem 20 (Harmonic norm of every codimension-two mode)
Let \(M=s+2\), \(s\geq1\), \(D\geq s\), \(0\leq k\leq D-s\). Put
The full-simplex squared norm of a propagated angular mode, divided by its source-face squared norm, equals
With \(d_s(k)\) as in Theorem 11,
If \(D<s\), the coefficient is zero.
Proof. Let \(p\) be a propagated mode, with outside coordinates \(y,z\). Let \(g\) be its codimension-one extension with the outside coordinate replaced by \(y+z\). On each facet the traces agree; thus \(p-g\) is a full-simplex bubble. Propagation orthogonality gives \(\langle p,p\rangle=\langle p,g\rangle\).
The common angular factor cancels. The radial factor of \(g\) is \(P_L^{(1,\beta)}(2r-1)/(L+1)\), while the conditional average of \(p\) has coefficients \(A_h=(-1)^h\binom Lh\binom{L+\beta}h\) multiplied by \(2H_{h+1}/((h+1)(h+2))\). The latter is the uniform outside-edge moment of \(T_h\) in the scalar-sum proof.
For
Jacobi orthogonality gives \(I_h=(-1)^hI_0\). Rodrigues’ formula and \(L\) integrations by parts give \(I_0=L!(L+\beta)!/N!\); all boundary terms vanish. The beta density of \(r\) supplies \(s(s+1)\). Hence
Set \(w_h=\binom{L+2}{h+2}\binom{L+\beta}h\). Vandermonde gives \(\sum w_h=W\); differentiating its first upper parameter gives \(\sum w_hH_{h+2}=W(H_{L+2}+H_{L+\beta+2}-H_N)\). The rational certificate
includes both endpoints: \(w_{L+1}=0\) and \(w_0R_0=-1/((L+\beta+1)(L+\beta+2))\). Subtract the telescoped sum from the harmonic Vandermonde identity. Finally use \(I_0W=1/((L+\beta+1)(L+\beta+2))\) and the factor \(1/((L+1)(L+2))\) from rewriting the binomial coefficient.
Tetrahedron edges at every degree
The remaining angular sum for \(s=2\) can itself be evaluated with harmonic numbers and two explicitly specified auxiliary recurrences. This is a fixed-order recurrence representation; it is not a claim that only ordinary harmonic numbers are needed.
Define \(F_0=G_0=0\) and, for \(n\geq1\),
Equivalently, \(F_n=\sum_{j=1}^n1/(j^2\binom nj)\) and \(G_n=\sum_{j=1}^n1/(j\binom nj)\). To see the recurrences directly, put \(S_n=\sum_{j=0}^n1/\binom nj\). Symmetry and Pascal’s identity give \(S_n=(n+1)S_{n-1}/(2n)+1\) and \(G_n=S_{n-1}/n\). The same identity applied to the sum defining \(F_n\) gives its displayed recurrence. In particular, \(G_n=2^{-n}\sum_{j=1}^n2^j/j\).
Theorem 21 (Tetrahedron edge representation)
For \(D\geq1\), set \(N=2D+3\) and define
For \(1\leq j<N\), put
Then
All denominators are nonzero on this range. The same formula gives \(a_1(4,1)=0\); at \(D=2,3,4,5\) it gives \(36/7,33/7,169/44,34/11\).
Proof. Reindex (27) by \(j=L+1\). Its summand is \(432g_N(j)\). Both \(f_N\) and \(v_N\) are symmetric under \(j\mapsto N-j\). Since \(N=2D+3\), half the full range \(1\leq j<N\) is \(1\leq j\leq D+1\). Remove the central terms \(j=D,D+1\).
To evaluate the harmonic range, use
The usual harmonic convolution identities give
For the rational correction, use
Summation of these rational terms and the two harmonic identities yields \(\frac12\sum f_N(j)=Q_N\).
For the reciprocal range, write \(w_j=1/\binom Nj\),
Direct multiplication verifies the rational identity
The telescoping boundary is \(R(N)-R(1)/N=-2(N-1)(N^2+2N-4)/(N^4(N+1)(N+2))\). Finally \(\sum_{j=1}^{N-1}w_j=(N+1)G_N/2-1\) and \(\sum_{j=1}^{N-1}w_j/j^2=F_N-1/N^2\). Substitution gives \(\sum v_N(j)=V_N\) and proves the result.
Explicit scalar degree recurrences
Corollary 10 (Certified tetrahedron edge recurrences)
The sequence \(a_1(4,D)\) satisfies both of the following polynomial-coefficient recurrences for every integer \(D\geq1\):
One operator has \(r=7\) and maximum polynomial degree 67; another
has \(r=8\) and maximum polynomial degree 40. Their complete
integer coefficients, in ascending powers of \(D\), are stored in
tools/risk_research/examples/tetrahedron_edge_recurrence_order7.json
and tetrahedron_edge_recurrence_order8.json.
The leading polynomial of each operator has no positive integer root. Their initial values are the corresponding prefix of
Proof. This is a computer-assisted proof by exact rational identities, independently of the finite data that suggested the order-eight operator. Put \(N=2D+3\) and
The finite representation gives a rational row \(r(D)\) with \(a_1(4,D)=432r(D)v(D)\). The state satisfies \(v(D+1)=T(D)v(D)\) over \(\mathbb Q(D)\). In detail, writing \(a=1/(N+1)+1/(N+2)\) and \(b=1/(N+1)^2+1/(N+2)^2\), its nonconstant updates are
For each supplied operator, exact multiplication verifies the eight component identities
where the empty product is the identity matrix. All state and output denominators are nonzero for \(D\geq1\); the verifier checks that these nonzero polynomials have nonnegative coefficients. Thus these are identities for all stated degrees, rather than a finite list of zero residuals. The order-seven operator is obtained directly from a left kernel of these state rows, without numerical fitting.
Every coefficient of the order-seven leading polynomial is strictly
positive. Exact rational-root determination also excludes positive
integer roots for the supplied operators. The finite representation
supplies the initial values above, completing forward determination.
tools/risk_research/edge_recurrence.py rebuilds the rational state
and verifies the operators and initial values from their coefficient
files; its reports include file hashes and the eight zero identities.
Triangular faces on a five-vertex simplex
The angular multiplicity for \(s=3\) is \(d_3(k)=k+1\). One additional auxiliary sequence evaluates this weighted angular sum. Use the same \(F_n,G_n,Q_N,V_N,g_N(j)\) as in the tetrahedron-edge representation; their displayed definitions make sense for even \(N\) as well as odd \(N\).
Define \(J_1=3/4\) and, for \(n\geq1\),
Theorem 22 (Eight-state triangular-face representation)
For every integer \(D\geq1\), put \(n=D+2\), \(N=2n\), \(w_n=\binom{2n}{n}^{-1}\) and
Then
Every denominator is nonzero on this domain. In particular the same formula gives the inactive values \(a_2(5,1)=a_2(5,2)=0\) and
The expression is a rational row on the eight-state vector
It therefore evaluates each new degree with a fixed number of rational operations on its state. This concerns arithmetic operation counts; integer bit lengths still grow with the degree.
Proof. Reindex (27) by \(j=L+1\). Its normalized summand is \((n-3-j)g_N(j)\), so
This also gives zero for \(D=1,2\) by cancellation. Symmetry gives \(\sum_{j=1}^{n-1}g_N(j)=Q_N-V_N/2-g_N(n)/2\). It remains to evaluate the weighted half range as \(E_n\).
Write \(B_j=H_{j+1}+H_{N-j+1}-H_N\) and \(\Delta_j=j(j+1)(N-j)(N-j+1)\). For \(r_j=1/(j(N-j+1))\), \(t_j=1/(j(N-j+2))\), direct rational identities give
Summation by parts, including both endpoints, yields \(\sum_{j=1}^{n-1}(n-j)B_j/\Delta_j=A_n\). The rational partial fractions used in the edge proof similarly give
For the reciprocal term put \(b_j=\binom Nj^{-1}\) and \(R_j=1/(2j^2(N-j+1))\). Exact multiplication verifies
Define \(T_n=\sum_{j=1}^{n-1}b_j/j^2\). The boundary is \(b_nR_n-b_1R_1=w_n/(2n^2(n+1))-1/(2N^2)\) and symmetry gives
The partial first and zeroth moments are
The first follows by telescoping \(b_j(1/j-1/(N-j))\); the second follows from symmetry and the full reciprocal sum in the edge proof. The ratio \(\binom{N+2}{j}^{-1}/b_j=(N+2-j)(N+1-j)/((N+1)(N+2))\) and the added endpoint \(j=n\) now give
Consequently \(J_n=H_n^{(2)}-T_n+(F_N-H_N^{(2)})/2\) has initial value \(3/4\) and satisfies (30). Combining the three weighted ranges gives exactly \(E_n\).
Finally \(w_{n+1}=(n+1)w_n/(2(2n+1))\). The other state transitions follow from harmonic increments and the displayed \(F,G\) recurrences. In \(Q_N-V_N/2\), the coefficients of \(H_N^2,H_N^{(2)},F_N\) sum to zero, so the two differences in \(v_n\) suffice. Each central term uses \(H_{n+r}=H_n+\sum_{i=1}^r1/(n+i)\) and \(\binom{N}{n-r}^{-1}=w_n\prod_{i=1}^r(n+i)/(n-i+1)\) for \(r=0,1,2\); only a fixed number of shifts occurs. Thus all output coefficients and all state transitions are rational in \(n\), proving the asserted fixed-state representation.
Corollary 11 (Certified triangular-face degree recurrence)
The sequence \(a_2(5,D)\) satisfies
where the complete integer coefficient lists are stored in
tools/risk_research/examples/five_vertex_triangle_recurrence_order8.json.
The order is eight and the maximum polynomial degree is 76. The leading
coefficient is strictly positive for every real \(D\geq1\), so
there are no positive-integer exceptional indices. Its initial values are
Proof. This is an exact rational-state certificate, with the same construction
as Corollary 10.
tools/risk_research/triangle_recurrence.py reconstructs the eight
state transitions and rational output row of
Theorem 22 over \(\mathbb Q(D)\).
Exact elimination supplies the displayed operator; multiplication of
its coefficient row by the shifted state-output rows gives eight zero
rational functions. After \(n=D+2\), every output and transition
denominator is a nonzero polynomial with nonnegative coefficients,
proving validity for \(D\geq1\). Every coefficient of
\(p_8(1+z)\) is strictly positive, proving the leading-coefficient
claim. The fixed-state representation supplies the eight initial values.
Large-degree constants
Corollary 12 (Codimension-two large-degree risk)
For every fixed integer \(s\geq2\),
In particular,
Proof. In (27), put \(j=L+1\). For fixed \(j\), the multiplicity divided by \(D^{s-2}\) tends to \(1/(s-2)!\), and
The bracket in (26) is nonnegative because it is a squared norm. It is at most \(H_{j+1}\): the other harmonic difference is nonpositive, and both subtracted terms are nonnegative. Also \((N-j)(N-j+1)\geq D^2\), and the scaled multiplicities are bounded by \(1/(s-2)!\). A constant multiple of the summable sequence \(H_{j+1}/(j(j+1))\) dominates the scaled summands. Dominated convergence applies, with
Multiplication by \(\binom{s+2}s^2\) proves the first result. For the sharper tetrahedron error, \(Q_N=\pi^2/(6N^2) +O(\log^2N/N^3)\), each central \(f_N\) is \(O(\log N/N^4)\), and \(0\leq\sum_{j=1}^{D-1}v_N(j)\leq16/N^4\). Substitute \(N=2D+3\) in the finite formula.
Vertex boundary-pair reduction
Let \(q=M-1\), \(f(x)=-\log(1-x)/x\), and
Then \(a_0(M,D)=M^2\mathcal L_D(F^q)/(M)_{2D}\) by Theorem 11.
Theorem 23 (Two fewer outside-kernel factors for vertex risk)
For all \(M\geq2,D\geq1\),
For \(q=1\) the sum is empty. In particular, symmetry gives
Proof. Write \(x\) for the source-vertex coordinate. For a subset \(S\) of the \(q\) outside coordinates, let \(p_S\) be the homogeneous degree-\(D\) lift of the vertex error on its face with \(x\). The scalar-mode extension has coefficients independent of the ambient number of coordinates. Dirichlet moments therefore give
The inclusion-exclusion lift \(b=\sum_{S\subsetneq[q]}(-1)^{q-|S|+1}p_S\) agrees with \(p_{[q]}\) on every facet. Their difference is a full bubble, so \(\langle p_{[q]},p_{[q]}\rangle=\langle p_{[q]},b\rangle\). Grouping subsets by size yields
For \(1\leq r<q\), condition on the sum of the coordinates of the corresponding face. Its normalized angular coordinates are uniform and independent of that sum and the remaining coordinates. The terms of \(p_{[q]}\) positive in all \(r\) selected outside coordinates also have positive source-coordinate \(x\) exponent. They are face bubbles of angular degree at most \(D\); they are orthogonal to \(p_S\). Removing the zero first-exponent axis in each coordinate kernel therefore gives
If \(D<r+1\), the same statement holds because the relevant monomials do not exist. Expand this identity in the preceding subset sum and put \(a=q-r\), \(b=r-j\). The resulting signs and binomial factors are exactly (33).
Positive diagonals at every nonvertex source face
Positive diagonal formulas for nonvertex risk extends binomial Gram diagonalization to every angular mode. It gives an exact positive harmonic double sum for every nonvertex codimension-three coefficient, and a leading asymptotic for every fixed nonvertex source size and codimension at least two. The remaining sums grow with the degree.
Verification and remaining questions
tools/risk_research/kernel_identities.py independently evaluates
(25); its tests compare exact polynomial
convolution and the axis boundary through degree 100.
tools/risk_research/edge_closed.py evaluates
(29) and the general codimension-two
mode formula. Tests compare the pre-existing scalar sum, independent Gram
propagation and direct reciprocal-binomial sums, including degree one.
vertex_boundary.py checks the reduced vertex formula with exact
coefficient arithmetic. Finite checks detect errors; the all-degree
claims rely on the proofs above.
triangle_faces.py evaluates (31)
using a fixed state and checks independent scalar and Gram references.
triangle_recurrence.py verifies its complete scalar operator by eight
exact rational identities, including the initial values and denominator
conditions, independently of finite sequence residual checks.
Tetrahedron vertex degree recurrence supplies a certified order-ten tetrahedron vertex recurrence at every positive integer degree. A bounded classical-special-function expression remains open. The boundary-pair identity reduces its mixed part to \(\mathcal L_D(Ff(x)f(y))\); Tetrahedron vertex risk further evaluates the complete vertex coefficient by one positive harmonic sum and proves its leading large-degree constant. Bounded polynomial/harmonic ansatz spaces can be excluded by exact modular rank, but such exclusions say nothing about formulas outside the recorded spaces. A general elementary or classical-special-function evaluation of the remaining codimension-two angular sums is also still open beyond the fixed-state evaluations above and the common fifteen-state construction for every nonvertex codimension-two face in Uniform construction for codimension two. That construction has a state bound independent of face size; it does not assert a bounded classical-special-function expression or a minimal scalar order. Explicit vertex errors at every fixed vertex count supplies an explicit finite-degree remainder for every fixed vertex count at least four, with a computable constant and the common degree domain \(D\geq256\). It distinguishes the analytic error from truncating the leading constant and from arithmetic rounding. Fixed-degree risk comparisons in every dimension settles the complete comparison at every fixed degree through thirty. A general crossover law and the nonlinear self-identification questions of Theory of the skeleton risk remain. The fixed-tetrahedron comparison is completely classified over all degrees in Tetrahedron risk comparison at every degree. The five-vertex comparison is also completely classified over all positive degrees in Five-vertex risk comparison at every degree, using the nonvertex boundary-transfer inequality and an independently certified finite prefix.