Five-vertex vertex degree recurrence

Under the uniform-simplex assumptions of Theory of the skeleton risk, the harmonic single sum in Five-vertex vertex risk admits an explicit scalar recurrence. The construction uses eight rational telescoping identities. It does not fit finitely many coefficient values.

Theorem 65 (Certified five-vertex vertex degree recurrence)

Put \(a_D=a_0(5,D)\). The artifact tools/risk_research/examples/five_vertex_vertex_recurrence_order17.json gives integer polynomials \(p_0,\ldots,p_{17}\), each of degree 188, such that

\[\sum_{i=0}^{17}p_i(D)a_{D+i}=0\qquad(D=1,2,\ldots).\]

Every coefficient of \(p_{17}\) is strictly positive. There are no exceptional positive integer indices. The artifact supplies the seventeen initial values \(a_1,\ldots,a_{17}\), starting with \(5/3,5/8,5/24\), so the recurrence determines every degree. Minimality of its order or polynomial degree is not asserted.

The coefficient arrays are in ascending powers of \(D\). They are stored in full rather than reproduced as thousands of large integers in this chapter. The following certificate construction independently checks the full arrays over \(\mathbb Q(D)\).

The triangular harmonic state

Use the notation from Five-vertex vertex risk and order the row of harmonic monomials as

\[T(j)=\left(H_j^3,H_j^2,H_jH_j^{(2)},H_j^{(3)},A_j, H_j,H_j^{(2)},1\right)^{\mathsf T}.\]

With \(t=(j+1)^{-1}\), its shift matrix \(S(j)\) is defined by the following increments of its eight components:

\[T(j+1)-T(j)=\left( 3tH_j^2+3t^2H_j+t^3,\; 2tH_j+t^2,\; tH_j^{(2)}+t^2H_j+t^3,\; t^3,\;t^2H_j,\;t,\;t^2,\;0 \right)^{\mathsf T}.\]

Thus \(T(j+1)=S(j)T(j)\). Write the known diagonal weight as \(\Psi_j=b(j)T(j)\); the eight rational entries of \(b\) are exactly (123) in this ordering.

Normalization to the largest shift

For a candidate order \(r\), set \(N=D+r-1\) and

\[\begin{split}w_{N,j}&=\binom Nj\binom{N+2}j,\\ U_m(j)&=(N-j)^{\underline m}(N+2-j)^{\underline m},\\ B_m&=N^{\underline m}(N+2)^{\underline m},\\ P(j)&=\sum_{m=0}^{r}\lambda_m(N)U_m(j).\end{split}\]

Falling factorials use a nonnegative integer number of factors. The weight for degree \(D+i\), divided by \(w_{N,j}\), is \(U_{r-i}(j)/B_{r-i}\). This identity also respects the different supports: \(U_m\) vanishes at the extra upper indices.

Put \(A(j)=(N-j)(N+2-j)\). A row certificate \(c(j)\) must satisfy

(127)\[\frac{A(j)}{(j+1)^2}c(j+1)S(j)-c(j)=P(j)b(j).\]

Multiplication by \(w_{N,j}T(j)\) then gives a telescoping difference.

Eight polynomial equations

Choose \(c_i(j)=j^2Q_i(j)/E_i(j)\), with denominator types

\[\begin{split}E_{\mathrm L}(j)&=(j+1)(j+2)(j+3),\\ E_{\mathrm M}(j)&=(j+1)^2(j+2)^2(j+3),\\ E_{\mathrm H}(j)&=(j+1)^3(j+2)^2(j+3).\end{split}\]

The types in state order are \((\mathrm L,\mathrm M,\mathrm L,\mathrm L,\mathrm L, \mathrm M,\mathrm M,\mathrm H)\). Let \(\Omega_i=\operatorname{lcm}(E_i(j),E_i(j+1))\). After the previous \(Q_l\) are known, equation \(i\) of (127) becomes

\[\mathcal L_iQ_i=\Omega_i\left(Pb_i -A\sum_{l<i}\frac{Q_l(j+1)}{E_l(j+1)}S_{li}(j)\right).\]

The three polynomial difference operators are

\[\begin{split}\mathcal L_{\mathrm L}Q &=A(j)(j+1)Q(j+1)-j^2(j+4)Q(j),\\ \mathcal L_{\mathrm M}Q &=A(j)(j+1)^2Q(j+1)-j^2(j+3)(j+4)Q(j),\\ \mathcal L_{\mathrm H}Q &=A(j)(j+1)^3Q(j+1)-j^2(j+2)(j+3)(j+4)Q(j).\end{split}\]

On \(j^m\), their leading degrees are \(m+2,m+3,m+4\) and their leading coefficients are \(m-2N-5,m-2N-7,m-2N-8\). Triangular polynomial reduction therefore leaves at most \((2,3,2,2,2,3,3,4)\) remainder coefficients, twenty-one in total.

Reduce \(U_0,\ldots,U_{17}\) separately over \(\mathbb Q(N)\). Their remainder columns have a one-dimensional kernel. Its vector supplies \(\lambda_0,\ldots,\lambda_{17}\). The verifier reconstructs the resulting eight \(Q_i\), checks all polynomial remainders are zero, and separately substitutes the rational row into (127). The certificate degrees are \((32,34,33,33,34,35,36,35)\).

Domain, boundaries and scalar coefficients

All \(j\)-poles of the row lie at \(-1,-2,-3\), outside the integer summation range. The quotient degrees are at most \(2r+2\). For \(N=D+r-1\), the largest possible leading coefficient of any triangular reduction is at most \(-2D-1<0\); the other two types are bounded still farther below zero. The verifier additionally checks every one of the 280 parameter denominators directly: after the shift \(N=D+16\), each has nonnegative coefficients and is nonzero. Thus none vanishes at any \(D\geq1\).

The lower boundary vanishes through the common factor \(j^2\) at zero. The upper boundary is \(w_{N,N+1}c(N+1)T(N+1)=0\): the weight vanishes and the remaining factors are finite. There is no missing endpoint term.

Finally put \(d_D=(2D+4)!/((D-1)!(D+1)!)\). The proved single sum is \(a_D=600\sum_jw_{D-1,j}\Psi_j/d_D\). Therefore the operator entries before common-denominator clearing are

\[p_i(D)=\lambda_{r-i}(D+r-1)B_{r-i}(D+r-1) \frac{d_{D+i}}{d_D},\]

up to a shared nonzero scalar. Primitive normalization produces the recorded integer arrays. The seventeen initial fractions are checked against the existing harmonic formula. The strict positivity of all 189 coefficients of \(p_{17}\) proves the absence of exceptional positive integers independently of finite root searches.

Reproduction and independent checks

Use the prepared Sage environment to derive or verify the full operator:

$ python -m tools.risk_research.vertex_five_recurrence --verify tools/risk_research/examples/five_vertex_vertex_recurrence_order17.json --report .research/skeleton-risk/five-vertex-proof.json
$ python -m tools.risk_research.vertex_five_recurrence --output .research/skeleton-risk/five-vertex-rederived.json

recurrence_coefficients(maximum_degree) evaluates the certified artifact with ordinary integer and fraction arithmetic and does not import Sage. It is an evaluator, not a replacement for the symbolic verification of an untrusted or edited operator.

Tests compare the recurrence with the independently proved harmonic sum through degree 48 and at new degrees 263, 277, 293 and 307, as well as the original scalar and rational Gram references at small degrees. Optional native Sage tests rederive every coefficient, reject altered operator/initial/domain data, and round-trip both CLI actions and their source-bound proof reports. Finite comparisons check the implementation; the eight exact identities and domain argument prove all degrees.