Positive diagonal formulas for nonvertex risk

Under the uniform-simplex assumptions of Theory of the skeleton risk, every nonvertex coefficient admits a positive construction using the same univariate transformed diagonals as vertex risk. At codimension three these diagonals are explicit harmonic expressions. This gives an exact positive double sum for every source-face size, including the five-vertex edge and six-vertex triangle, without a bivariate kernel convolution. The number of terms still grows with the degree; no bounded classical special-function expression, fixed-state bound or minimal order is claimed.

A Laguerre identity with arbitrary weight

For integers \(L,\beta\geq0\), put

\[A_h=(-1)^h\binom Lh\binom{L+\beta}h,\qquad W_j=\binom Lj\binom{L+\beta}j,\qquad C=L!(L+\beta)!.\]

Lemma 8 (Arbitrary-weight binomial Gram diagonalization)

For arrays \(u,v\) indexed by \(0,\ldots,L\), let \(\widehat u_j=\sum_{h=0}^j(-1)^h\binom jh u_h\), and similarly for \(v\). Then

(49)\[\sum_{h,l=0}^L A_hA_lh!l!(2L+\beta-h-l)!u_hv_l =C\sum_{j=0}^LW_j\widehat u_j\widehat v_j.\]

Proof. Define \(P_u(t)=\sum_{h=0}^LA_hh!u_ht^{L-h}\). Binomial inversion and the finite Laguerre expansion give

\[P_u(t)=(-1)^LL!\sum_{j=0}^L(-1)^j \binom{L+\beta}j\widehat u_jL_{L-j}^{(\beta)}(t).\]

To check this identity, compare the coefficient of \(t^{L-h}\) and use ordinary binomial inversion on \(u_h\); thus it also holds at \(L=0\). Integrate \(P_uP_ve^{-t}t^\beta\) on \([0,\infty)\). Its monomial moments are the factorials on the left of (49). Laguerre orthogonality removes all unequal indices and gives

\[L!^2\binom{L+\beta}j^2 \frac{(L+\beta-j)!}{(L-j)!} =CW_j.\]

This proves the identity, including all endpoints.

For a bivariate formal series \(G\), let

\[\mathscr T(G)(x,y) =\frac{G(-x/(1-x),-y/(1-y))}{(1-x)(1-y)}.\]

By linearity, the lemma evaluates the corresponding bivariate functional as \(C\sum_jW_j[x^jy^j]\mathscr T(G)\). Only coefficients through \(L\) in each variable are used, so this statement concerns finite sums and requires no analytic interchange.

One universal diagonal for every source face

Use \(g(t)=-(1-t)\log(1-t)/t\) and the positive sequences \(\delta_q(j)\) of Vertex coefficients at every fixed vertex count:

\[\delta_q(j)=[x^jy^j] \frac{(g(x)+g(y)-g(xy))^q}{(1-x)(1-y)}.\]

For convenience put \(\delta_0(j)=1\). At every \(q\geq1\) the existing univariate convolution construction evaluates these diagonals exactly, and they are strictly positive.

Theorem 28 (Positive nonvertex diagonal averages)

Let \(M\geq s\geq2\), \(q=M-s\), \(D\geq s\), and \(0\leq k\leq D-s\). Define

\[\begin{split}L&=D-s-k,&\beta&=3s+2k-1,\\ Z_{L,\beta}&=\binom{2L+\beta}L,& \mathbb P(J=j)&=\frac{\binom Lj\binom{L+\beta}j}{Z_{L,\beta}} \quad(0\leq j\leq L).\end{split}\]

Every source-face angular mode of degree \(k\) has propagated squared norm divided by its source norm equal to

(50)\[\rho_{s,q,k,D}=\frac{(s)_q}{(2D+s)_q}\, \mathbb E[\delta_q(J)].\]

Consequently

(51)\[a_{s-1}(M,D)=\binom Ms^2\frac{(s)_q}{(2D+s)_q} \sum_{k=0}^{D-s}\binom{k+s-2}{s-2} \mathbb E[\delta_q(J)].\]

The coefficient is zero when \(1\leq D<s\).

Proof. The radial coefficients in Theorem 11 are exactly \(A_h\) with the displayed \(L,\beta\). Moreover \(2L+\beta=2D+s-1\), independent of \(k\). Apply Lemma 8 to its outside kernel \(F^q\). The formal substitution already proved in Tetrahedron vertex risk gives \(F(-x/(1-x),-y/(1-y))=g(x)+g(y)-g(xy)\). Thus its transformed diagonal is precisely \(\delta_q(j)\); it does not depend on \(s,k\) or \(D\).

Vandermonde’s identity gives \(\sum_jW_j=Z_{L,\beta}\) and \(CZ_{L,\beta}=(2D+s-1)!\). The remaining normalization in the scalar mode formula is

\[\frac{(2D+s-1)!}{(s-1)!(M)_{2D}} =\frac{(s)_q}{(2D+s)_q}.\]

This proves (50). Summing the \(\binom{k+s-2}{s-2}\) angular modes and multiplying by \(\binom Ms^2\) proves the coefficient formula. For \(D<s\) there are no source bubbles. For \(q=0\), every average is one and the formula recovers \(a_{s-1}(s,D)=\binom{D-1}{s-1}\).

The exact harmonic codimension-three family

Write \(H_j=\sum_{r=1}^j1/r\), \(H_j^{(2)}=\sum_{r=1}^j1/r^2\), with both zero at \(j=0\). The tetrahedron vertex identity in Tetrahedron vertex risk gives

(52)\[\delta_3(j)=6\left( \frac{H_j(H_j-j/(j+2))}{(j+1)(j+2)(j+3)} +\frac{H_j^{(2)}}{(j+1)(j+3)} +\frac{2}{(j+1)^2(j+2)^2(j+3)}\right)>0.\]

At \(j=0\) this is one. At positive \(j\), \(H_j\geq1>j/(j+2)\) proves positivity term by term.

Corollary 13 (Every nonvertex codimension-three coefficient)

For all integers \(s\geq2,D\geq s\),

(53)\[a_{s-1}(s+3,D)= \binom{s+3}s^2\frac{s(s+1)(s+2)} {(2D+s)(2D+s+1)(2D+s+2)} \sum_{k=0}^{D-s}\binom{k+s-2}{s-2} \frac{\displaystyle\sum_{j=0}^{D-s-k} \binom{D-s-k}j\binom{D+2s+k-1}j\delta_3(j)} {\binom{2D+s-1}{D-s-k}}.\]

It is zero on the inactive prefix \(1\leq D<s\). In particular this evaluates \(a_1(5,D)\) and \(a_2(6,D)\) at every positive degree.

Proof. Substitute \(q=3\) and (52) in (51). Every denominator is a positive integer on the active domain. The formula has exactly \((D-s+1)(D-s+2)/2\) inner terms; harmonic tables and binomial weights have nonsingular updates.

Fixed-codimension large-degree behavior

The same positive formula determines the true leading order at every fixed nonvertex source size and codimension at least two.

Theorem 29 (Nonvertex asymptotic at every fixed codimension)

For each fixed pair of integers \(s\geq2,q\geq2\),

(54)\[a_{s-1}(s+q,D)\sim A_{s,q}D^{s-q-2},\qquad A_{s,q}=\binom{s+q}s^2 \frac{(s)_q}{2^q(s-2)!}\frac{c_{q+1}}{q+1}>0.\]

Here \(c_{q+1}\) is the positive constant of Theorem 39. In particular,

(55)\[a_{s-1}(s+3,D)\sim \frac{s(s+1)^3(s+2)^3(s+3)^2}{24(s-2)!} (\zeta(2)-\zeta(3))D^{s-5}.\]

Thus \(a_1(5,D)\sim3600(\zeta(2)-\zeta(3))D^{-3}\) and \(a_2(6,D)\sim36000(\zeta(2)-\zeta(3))D^{-2}\). No finite-degree error bound or simultaneous growth of \(s,q,D\) is asserted by this theorem.

Proof. First the univariate construction in Vertex coefficients at every fixed vertex count gives, for \(q\geq2\),

\[\sum_{j\geq0}\delta_q(j) =\sum_{j\geq0}e_q(j)=\frac{c_{q+1}}{q+1}.\]

Indeed all sequences in that construction are absolutely summable. The sum of \(\phi_q\) is zero, by its generating function \(g(t)^q/(1-t)\) at \(t=1\). For \(k\geq1\), \(\sum_jb_k(j)=0\), so the sum of every convolution \(b_k*e_{q-k}\) is zero. Only the \(k=0\) term remains. This also proves that the displayed sum is finite and positive.

For summable domination we use the following exact inverse-factorial moment of the binomial weights. Write \(B=L+\beta\geq L\). Vandermonde’s identity gives

\[\begin{split}\mathbb E\!\left[\frac1{(J+1)(J+2)}\right] &=\frac{(L+B+1)(L+B+2)} {(L+1)(L+2)(B+1)(B+2)},\\ \mathbb E[(J+1)^{-2}] &\leq\frac8{(L+1)(L+2)}.\end{split}\]

For the first identity, replace \(\binom Bj/((j+1)(j+2))\) by \(\binom{B+2}{j+2}/((B+1)(B+2))\) and sum. For the inequality use \((j+1)^{-2}\leq2/((j+1)(j+2))\) and \((L+B+i)/(B+i)\leq2\) for \(i=1,2\).

For each fixed \(q\geq2\), constants \(K,p\) exist with

\[0<\delta_q(j)\leq K\frac{(1+\log(j+1))^p}{(j+1)^2}\quad(j\geq0).\]

At \(q=2\), this follows directly from \(\delta_2=-\phi_2+2\phi_1^2\), with \(p=1\). At \(q\geq3\), it follows from the explicit diagonal error (102), with \(p=q-1\). Since \(0\leq J\leq L\), the inverse-square bound therefore gives

\[\mathbb E[\delta_q(J)]\leq \frac{8K(1+\log(L+1))^p}{(L+1)(L+2)},\]

a summable sequence in \(L\), independent of \(D\).

Reindex the angular sum by \(L=D-s-k\). For each fixed \(L\), \(B=2D+s-L-1\) tends to infinity, and \(\mathbb P(J=L)=\binom BL/\binom{B+L}L\to1\). Thus its average tends to \(\delta_q(L)\). Also

\[\frac{\binom{k+s-2}{s-2}}{D^{s-2}} \longrightarrow\frac1{(s-2)!},\qquad 0\leq\frac{\binom{k+s-2}{s-2}}{D^{s-2}} \leq\frac1{(s-2)!}.\]

Set terms outside \(0\leq L\leq D-s\) to zero. Dominated convergence and \(D^q(s)_q/(2D+s)_q\to(s)_q/2^q\) prove (54). At \(q=3\), use \(c_4=48(\zeta(2)-\zeta(3))\) and simplify. At \(q=2\) the theorem recovers the existing codimension-two asymptotic.

Evaluation and independent verification

tools.risk_research.codimension_three_exact evaluates the harmonic double sum using exact fractions. Its generic nonvertex evaluator uses the same theorem with the existing univariate \(\delta_q\) construction. Neither path constructs \(F^q\). The q3 path uses quadratically many rational operations in \(D-s+1\), with growing integer bit lengths. The generic diagonal construction adds \(O(q^2(D-s+1)+q(D-s+1)^2)\) rational operations.

$ .venv/bin/python -m tools.risk_research.codimension_three_exact --face-size 2 --degree 40 --output .research/skeleton-risk/q3-edge-exact.json
$ .venv/bin/python -m tools.risk_research.codimension_three_exact --face-size 3 --degree 40 --output .research/skeleton-risk/q3-triangle-exact.json
$ .venv/bin/python -m tools.risk_research.codimension_three_exact --face-size 2 --degree 40 --verify .research/skeleton-risk/q3-edge-exact.json --output .research/skeleton-risk/q3-edge-rechecked.json

Records bind the evaluator and its direct source dependencies. SHA256 hashes normalize CRLF pairs to LF so that the same source record can be checked on Windows. The concurrent-edit guard compares the original raw bytes and therefore still rejects line-ending edits during calculation. Invalid inputs, changed records and calculation failures preserve the previous destination through atomic output. Recomputing a record checks its exact value and provenance; the all-degree proof is the theorem above.

nonvertex_leading_constant_enclosure(s,q,terms) scales the existing vertex-constant interval by the exact positive rational factor

\[\frac{A_{s,q}}{C_{q+2}}= \frac{\binom{s+q}s^2(s)_q} {2(s-2)!(q+1)(q+2)^2(q+1)!}.\]

It encloses the leading constant for fixed \(s,q\geq2\); it does not enclose the finite-degree coefficient by itself.

Tests compare every mode with independent polynomial moments at genuine full-bubble degrees, recover existing q0/q1/q2 formulas, and compare small values with the original scalar formula and separate Gram propagation. Frozen larger cases are generated by the original integer outside-kernel path, independently of this diagonal formula. No finite comparison is used in place of the all-parameter proof.