Uniform construction for codimension two

The previously evaluated edges, triangular faces and tetrahedral faces have source-face sizes two, three and four. This chapter gives one construction for every fixed source-face size. It uses the uniform-simplex assumptions of Theory of the skeleton risk and the mode formula in Further evaluations of the skeleton risk.

Theorem 25 (A uniform fifteen-state codimension-two construction)

For every integer \(s\geq2\), the coefficient \(a_{s-1}(s+2,D)\) has the following constructive evaluation for every integer \(D\geq1\). It is zero if \(D<s\). On the active domain, set

\[h=D+\lfloor s/2\rfloor,\qquad \epsilon=1+(s\bmod2),\qquad N=2h+\epsilon=2D+s+1.\]

A fifteen-component state, independent of \(s\), has a rational transition from \(h\) to \(h+1\). A finite algorithm depending on \(s\) constructs a rational output row whose product with this state is \(a_{s-1}(s+2,D)\). The algorithm uses polynomial arithmetic, partial fractions with poles of order at most two, and at most \(\lfloor s/2\rfloor+1\) explicit endpoint corrections. Its operation count for constructing the output row at fixed \(s\) is independent of \(D\).

State initialization uses only the fixed values \(h=1,N=3\) or \(h=1,N=4\). State transitions are nonsingular for \(h\geq1\); output construction is nonsingular throughout \(D\geq s\).

This is an upper bound on state dimension, not a claim of minimality. The eight-state formulas at \(s=3,4\) remain useful smaller representations. At \(s=1\) the coefficient is the already evaluated triangle vertex; the polynomial angular multiplicity used below applies to \(s\geq2\). No production sampling API changes are involved.

The common state

Let \(b_j=\binom Nj^{-1}\), and let \(H_r,H_r^{(2)}\) have their usual values, including zero at \(r=0\). Define

\[\begin{split}F_N&=\sum_{j=1}^N\frac{b_j}{j^2},& G_N&=\sum_{j=1}^N\frac{b_j}{j},& P_N&=\sum_{j=0}^N\frac{b_j}{j+1},\\ T_1&=\sum_{j=1}^h\frac{b_j}{j},& T_2&=\sum_{j=1}^h\frac{b_j}{j^2},& U&=\sum_{j=1}^h\frac{b_j}{j+1},\qquad w=b_h.\end{split}\]

The state is

(38)\[\bigl(H_N^2,H_NH_h,H_h^2,H_N,H_h,H_N^{(2)},H_h^{(2)}, F_N,G_N,P_N,T_1,T_2,U,w,1\bigr)^{\mathsf T}.\]

The defining sums establish the identities below; evaluation advances the fixed state and does not recompute them at each degree.

Write \(q=(N+1)(N+2)\) and

\[W=\frac{(h+1)(N-h+1)}q\,w,\qquad S=\frac{N+1}{4}G_N-\frac12 -\mathbf1_{\epsilon=2}\frac{h+1}{2(h+2)}w,\qquad J=\frac N2 S+\frac{(h+1)w-1}{2}.\]

Here \(S=\sum_{j=1}^h b_j\) and \(J=\sum_{j=1}^h j b_j\). The half-range updates are

(39)\[\begin{split}T_1'&=T_1-\frac{2N+3}{q}S+\frac Jq+\frac W{h+1},\\ T_2'&=T_2-\frac{2N+3}{q}T_1+\frac Sq+\frac W{(h+1)^2},\\ U'&=\frac{N+3}{N+1}U-\frac{2S}{N+1}+\frac Jq+\frac W{h+2}, \qquad w'=W.\end{split}\]

Primes denote \(N'=N+2,h'=h+1\). The full inverse-binomial updates can be performed twice using

\[\begin{split}G_{N+1}&=G_N/2+1/(N+1),\\ F_{N+1}&=F_N-G_N/(N+1)+1/(N+1)^2,\\ P_{N+1}&=\frac{N+2}{N+1}P_N-\frac{G_N}{2} -\frac1{N+1}+\frac1{N+2}.\end{split}\]

Finally, \(H_N\) increases by \(a=1/(N+1)+1/(N+2)\), and \(H_h\) by \(c=1/(h+1)\). Their squares and product therefore have affine updates in the listed quadratic basis. The second-order harmonic numbers increase by \(1/(N+1)^2+1/(N+2)^2\) and \(1/(h+1)^2\), respectively. Every displayed denominator is strictly positive on the stated state domain.

Proof. The first two full inverse-binomial updates are already proved in Further evaluations of the skeleton risk. The ratios

\[\frac{b_{N+1,j}}{b_{N,j}}=\frac{N+1-j}{N+1},\qquad \frac{b_{N+2,j}}{b_{N,j}} =\frac{(N+1-j)(N+2-j)}q\]

give the \(P_N\) update and the three half-range updates by polynomial division by \(j,j^2,j+1\); the new endpoint is exactly \(W\). To express the zero-th and first half moments, observe

\[\frac{N+1}{j}b_j=b_j+b_{j-1},\qquad (2j-N)b_j=(N-j)b_{j+1}-(N-j+1)b_j.\]

Summing the first identity over \(1\leq j\leq N\) gives \(\sum_{j=0}^N b_j=(N+1)G_N/2+1\). Symmetry and, when \(\epsilon=2\), subtraction of the middle term give \(S\). The second identity telescopes to the displayed expression for \(J\). This proves all the transitions and their initialization by the fixed defining sums at \(h=1\).

From arbitrary face size to polynomial moments

Set

\[\begin{split}x&=j(N-j),\qquad \Delta=x(x+N+1),\\ B_j&=H_{j+1}+H_{N-j+1}-H_N,\\ g_N(j)&=\frac{B_j}{\Delta} -\frac{(N+1)(x-1)}{\Delta^2} -\frac{b_j}{x\Delta}.\end{split}\]

Reindexing (27) by \(j=L+1\) and using \(d_s(k)=\binom{k+s-2}{s-2}\) gives

(40)\[a_{s-1}(s+2,D) =K_s\sum_{j=1}^{D-s+1}p(j)g_N(j),\quad K_s=\binom{s+2}s^2\,2s(s+1),\quad p(j)=\binom{D-j-1}{s-2}.\]

In the reduction, \(p\) is the polynomial binomial product, including its values at negative upper arguments. Extending the sum to \(h\) adds zero-weight terms at \(D-s+2,\ldots,D-1\) and the explicit terms at \(j=D+t\), \(0\leq t\leq\lfloor s/2\rfloor\), with weights

\[p(D+t)=(-1)^{s-2}\binom{t+s-2}{s-2}.\]

For such an endpoint, write \(j=h-r\). Its two harmonics differ from \(H_h\) by a fixed number of increments, while

\[b_{h-r}=w\prod_{i=1}^r\frac{N-h+i}{h-i+1}.\]

Consequently every correction is a rational row in the same state. On the active domain \(h\geq s+\lfloor s/2\rfloor\), none of these denominators can vanish.

Reduction of the harmonic part

The rational function \(p(j)/\Delta\) has a polynomial part and only simple poles at \(0,-1,N,N+1\). The pole part is a linear combination of

\[\left(\frac1j\mathbin\pm\frac1{N-j}\right)B_j, \qquad \left(\frac1{j+1}\mathbin\pm\frac1{N-j+1}\right)B_j.\]

For the symmetric parts, full-range summation and symmetry give the half-range sums. The necessary full sums are

\[\begin{split}\sum_{j=1}^{N-1}\frac{B_j}{j} &=\frac{H_N^2-H_N^{(2)}}2+1-\frac1N +\frac{2H_N-1-1/N}{N+1},\\ \sum_{j=1}^{N-1}\frac{B_j}{j+1} &=\frac{H_N^2-H_N^{(2)}}2 +2\left(\frac1{N+1}+\frac1{N+2}\right)H_N +\frac2{(N+1)(N+2)}-\frac2{N+1}-1.\end{split}\]

These follow from \(\sum_{k=1}^r H_k/k=(H_r^2+H_r^{(2)})/2\) and \(\sum_{k=1}^{r-1}H_{r-k}/k=H_r^2-H_r^{(2)}\). When \(N\) is even, subtract the central summand before halving the full symmetric sum. The central harmonic is \(2H_{h+2}-H_N\), so this uses the listed state.

For an arbitrary constant \(H\), define

\[\begin{split}A(C,L;H)={}&\frac12\left(H_L^2+H_L^{(2)} -H_{C-1}^2-H_{C-1}^{(2)} +H_{C-L-1}^2+H_{C-L-1}^{(2)}\right)\\ &+H_LH_{C-L-1}-H(H_L-H_{C-1}+H_{C-L-1}).\end{split}\]

The following prefix identity handles both antisymmetric parts:

(41)\[A(C,L;H)=\sum_{j=1}^L \left(\frac1j-\frac1{C-j}\right)(H_j+H_{C-j}-H).\]

Indeed both sides vanish at \(L=0\), and their differences at successive \(L\) agree by \(H_L-H_{L-1}=1/L\). In particular, the cross terms telescope using

\[\sum_{j=1}^L\left(\frac{H_{C-j}}j-\frac{H_j}{C-j}\right) =H_LH_{C-L-1}.\]

For poles at \(j+1,N-j+1\), take \(C=N+2\), \(L=h+1\), \(H=H_N\), and remove the \(j=1\) term. For poles at \(j,N-j\), take \(C=N,L=h,H=H_N\), and add the rational correction

\[\sum_{j=1}^h\frac{(N+2)(N-2j)}\Delta.\]

This correction changes \(H_j+H_{N-j}\) into \(H_{j+1}+H_{N-j+1}\). At the half-range endpoints, every harmonic index in these formulas is \(N\) or \(h\) plus a fixed integer. Hence the formulas use only the quadratic harmonic components of the state.

It remains to sum \(Q(j)B_j\) for a polynomial \(Q\). Construct a polynomial \(R\) with \(R(0)=0\) and \(R(j)-R(j-1)=Q(j)\). Its coefficients follow by triangular elimination, dividing only by positive integers. Summation by parts gives

\[\sum_{j=1}^h Q(j)B_j=R(h)B_h -\sum_{j=1}^{h-1}R(j) \left(\frac1{j+2}-\frac1{N-j+1}\right).\]

The remaining sums are rational, with fixed simple poles. Their partial fractions evaluate using harmonic numbers and polynomial antidifferences. This completes the harmonic reduction for every polynomial weight.

Rational and inverse-binomial parts

The rational part \(p(j)(N+1)(x-1)/\Delta^2\) has only the same four poles, of order at most two. Its sum therefore uses only polynomials, first-order harmonics and second-order harmonics.

The inverse-binomial part has denominator \(j^2(j+1)(N-j)^2(N-j+1)\). After partial fractions, its proper part uses \(T_1,T_2,U\) and the reflected half sums. Reflection expresses these through \(G_N,F_N,P_N\), the same half sums and, for even \(N\), the central term. For example,

\[\sum_{j=1}^h\frac{b_j}{N-j} =G_N-\frac1N-T_1 -\mathbf1_{\epsilon=2}\frac{w}{h+2}.\]

For a polynomial part \(Q\), introduce the operator

\[(\mathcal LR)(j)=(j+1)R(j+1)-(N-j+1)R(j).\]

If \(R(j)=j^k\), its leading term is \(2j^{k+1}\). Triangular elimination thus gives, for every polynomial \(Q\), a polynomial \(R\) and a constant \(c\) with \(Q=\mathcal LR+c\). The ratio \(b_{j+1}/b_j=(j+1)/(N-j)\) yields

\[\sum_{j=1}^h Q(j)b_j =cS+(h+1)wR(h+1)-R(1).\]

Thus even arbitrarily high polynomial moments introduce no new state. Polynomial degrees grow with \(s\); the number of residual sequences does not.

Completion of the construction and its scope

Proof. Apply the three reductions above to the polynomial in (40), subtract its finite endpoint corrections, and multiply by \(K_s\). Every step is finite polynomial arithmetic or an explicit partial-fraction identity. Harmonic products are at most quadratic in \(H_N,H_h\); all inverse-binomial residuals are in the listed state. This constructs the rational output row.

Partial fractions divide only by differences of the four pole locations, hence by nonzero powers of \(N,N+1,N+2\) and constants. Polynomial antidifferences divide by positive integers; inverse-binomial polynomial reduction divides by two. The endpoint increments divide by positive shifts of \(h\) on the active domain. All formulas are therefore nonsingular. The inactive prefix is explicitly zero. Together with the established rational transitions this proves the theorem for every \(s\geq2\), rather than only the finitely many tested face sizes.

The construction is implemented by tools.risk_research.codimension_two_family using only standard-library exact fractions. family_coefficients(s, maximum) returns a complete degree prefix, while family_coefficient(s, D) advances the state and constructs only the requested output row. The number of rational operations per degree at fixed \(s\) is bounded; numerator and denominator bit lengths still increase with \(D\).

The fifteen rational output rows obtained by advancing the state, together with one further row, are linearly dependent over \(\mathbb Q(D)\). They therefore also give a constructive scalar recurrence of order at most fifteen. This chapter supplies the common state construction, not a claim that the scalar order is minimal or that an eliminated leading coefficient has no exceptional indices. An explicit scalar operator must check those indices separately.

Reproduction and independent verification

$ .venv/bin/python -m tools.risk_research.codimension_two_family --face-size 5 --max-degree 50 --output .research/skeleton-risk/family-m7.json

The optional native-Sage construction verifier checks the emitted rational identities over \(\mathbb Q(h)(j)\), polynomial antidifferences, boundary weights and positivity of all output and transition denominators. It also uses a separate dense linear solve for polynomial inverse-binomial reduction.

$ python -m tools.risk_research.codimension_two_family_certificate --face-size 6 --output .research/skeleton-risk/family-m8-proof.json

Run that command in the prepared Sage environment. The report records the output row, both source hashes and the exact residuals. Native regressions are selected by BSF_SAGE_PYTHON as for the earlier recurrence verifiers.

Independent tests compare all fifteen transitions to the defining sums in both parities, polynomial kernel moments through degree eight, original scalar coefficients and low-degree Gram propagation, and the original angular formula at larger degrees and new face sizes. The committed fresh holdout artifact includes faces larger than the first construction examples. These finite tests check the implementation; the reductions above establish the all-size, all-degree theorem.