1 The result and its context
Foulkes’ conjecture compares two ways of composing symmetric powers. For a finite-dimensional complex vector space V and integers 1\le a\le b, it asks for a \GL(V)-equivariant injection
Over \CC, complete reducibility makes this equivalent to an inequality between the multiplicities of every irreducible polynomial representation. Write \Schur_\lambda V for the Schur module indexed by a partition \lambda, and s_\lambda for its character. If h_r denotes the complete homogeneous symmetric function, then h_a[h_b] is the stable character of \Sym^a(\Sym^b V), where brackets denote plethysm. The conjecture says that h_b[h_a]-h_a[h_b] is Schur-positive: all its Schur coefficients are nonnegative integers. We prove the sixth case.
Theorem 1.1. For every integer b\ge6 and every finite-dimensional complex vector space V, there is a \GL(V)-equivariant injection
Equivalently, h_b[h_6]-h_6[h_b] is Schur-positive for every b\ge6.
Theorem 1.1 resolves this case of Foulkes’ conjecture positively, with no restriction on \dim V. The diagonal case b=6 is immediate; the content is the comparison for every larger b.
History and the canonical map. The conjecture originates in Foulkes’ study of concomitants [11]. For binary forms, classical Hermite reciprocity gives an isomorphism between the two symmetric powers for all a,b; see [22, Introduction]. In higher dimension one seeks a multiplicity inequality instead. The case a=1 is immediate. Thrall’s decomposition formulas imply a=2 [25]; see also [8, p. 237]. Dent and Siemons proved a=3 [8].
A natural approach uses the canonical Foulkes–Howe map
Realize the source as symmetric tensors in V^{\otimes ab}, arrange the tensor positions in a rows of length b, transpose the array, and symmetrize the resulting blocks. McKay proved that injectivity for every V at a seed (a,b_0), with b_0\ge a, propagates to every (a,b) with b\ge b_0 [19, 14]. The computation at (4,4) by Müller and Neunhöffer therefore yields a=4 [20]; Cheung, Ikenmeyer, and Mkrtchyan’s injectivity computation at (5,6) yields a=5, with the diagonal comparison again immediate [7, preprint, Theorem 6(a)]. Canonical injectivity is stronger than Foulkes’ multiplicity comparison. The canonical maps at (5,5) and (6,6) have nonzero kernels in suitable dimensions, although their source and target representations coincide [20, 7]. For the sixth diagonal this already occurs when \dim V\ge6 [7, preprint, Theorem 6(c)]. Our finite comparison does not require injectivity of a prescribed map.
Exact character calculations give another approach. Evseev, Paget, and Wildon verified Foulkes’ multiplicity inequalities when the two parameters sum to at most 19, including the sixth case through b=13 [10, Corollary 5.2]. Their character-deflation recurrence is a predecessor of the exact source calculation used here. The present certificates combine such calculations with bounds on entire families of multiplicities, leaving only a small band for exact comparison.
Brion proved eventual truth for fixed a [2, Corollary 1.3] and later gave a dimension-dependent bound for surjectivity of the canonical map in the opposite direction [3, Theorem 3.3]. That map, \Psi_{b,a,V}, is realized as multiplication in the invariant ring associated with the normalization of the Chow variety [16, Section 7]; [22, Section 2]. The companion paper proves its surjectivity for b\ge a(a-1), uniformly in dimension [21, Theorem 1]. At a=6, an invariant complement to its kernel supplies an embedding for b\ge30. The remaining task is to compare all multiplicities for 6\le b\le29.
For binary forms, recent generalized comparisons concern plethysms with different parameter pairs: Raicu, Sam, Weyman, and Yang prove maximal rank under divisibility conditions [23, Theorem 1.4], and Gangl, Gutiérrez, and Szwej identify the dual map through a substitution construction [13, Theorem 1.2]. These results concern a two-dimensional underlying space; the theorem above treats every finite dimension.
Proof strategy. The source \Sym^6(\Sym^b V) has Schur support of length at most six. Let F(\lambda) and Q(\lambda) denote its multiplicity and the corresponding target multiplicity, respectively, for a partition \lambda of 6b. Section 2 reduces the problem to these six-part partitions and supplies rectangular highest-weight shifts of known comparisons and a chart certificate for partitions with \lambda_2+\cdots+\lambda_6\le b. The canonical multiplication map also proves the main large-degree range in Corollary 2.1.
The finite argument compares upper bounds for F with lower bounds for Q. Section 3 bounds source multiplicities by counting possible intermediate partitions in a signed strip recurrence, and gives exact recurrences for both characters. Section 4 amplifies known target multiplicities: a basis of highest-weight polynomials can be chosen with distinct leading monomials, so a sumset bound supplies many independent products of those polynomials. Multiplication by a single nonzero highest-weight polynomial then transfers these lower bounds to further weights. Section 5 organizes the resulting tests into a finite computation. It checks every partition for 6\le b\le25. In increasing degrees from 26 onward, rectangular shifts reduce the remaining problem to \lambda_6<6; the certificates leave only partitions in the band \lambda_3+\cdots+\lambda_6\le27. Section 6 computes exact differences throughout this band by reusing seven truncated rational-function numerators. Section 7 proves the arithmetic bounds and assembles the range coverage.
An independent route is retained in full. The chart calculation used in the finite comparison, combined with polarization, proves canonical surjectivity for b\ge150 in Proposition 2.7. The polarization argument is related to the successive weight shifts in the proof of McKay’s theorem [14, Section 5]. Both certificate programs run through b=149, so the extended finite verification and this chart bound prove Theorem 1.1 without the quadratic-stabilization input. The main route needs only the finite comparisons through b=29. The complete programs are printed in the appendices and supplied with reference outputs and a reproduction script. The formulas explain what the programs compute; their exact execution supplies the finite nonnegativity assertions.
Throughout, we use characteristic-zero complete reducibility and highest-weight theory for polynomial \GL-representations [12, Theorem 9.19 and Proposition 14.13], Schur functors and the Cauchy decomposition [12, Theorem 6.3 and Exercise 6.11(b)], and the Schur alternant formula [17, Chapter I, (3.1)]. Stable Schur coefficients are unchanged on specializing extra character variables to zero whenever the partition length remains admissible [17, Chapter I, (3.2)–(3.3)]. The character interpretation of plethysm is recalled in [17, Chapter I, Appendix A, Sections 7–8]. We also write e_r for the elementary symmetric function and set h_0=e_0=1.
2 The large-degree range and finite reductions
The proof has a representation-theoretic reduction and a finite multiplicity comparison. We first specify the canonical map supplied by the companion and reduce the remaining comparison to partitions with at most six parts. A chart of the same invariant ring gives one of the finite certificates. The final subsection proves the independent large-degree bound used with the extended computation.
2.1 The multiplication map
For the moment let a\ge2, and set
The symmetric group permutes the tensor factors. We identify \Sym^a W with (W^{\otimes a})^{\SSS_a} by the normalized average
Its inverse is the restriction of the symmetric-product quotient. Let A\subset R be the graded subalgebra generated by R_1=\Sym^a V. For v\in V, put z(v)=v^{\otimes a}\in R_1. Multiplication defines the canonical Foulkes–Howe map
whose image is A_b. Thus \mu_{a,b,V}=\Psi_{b,a,V} in the notation of the introduction. It is equivariant, so A_b=R_b suffices for the desired embedding, by complete reducibility. This is the equal-multidegree invariant construction underlying the Chow normalization; see [22, Section 2]. We will use its explicit polynomial realization.
Corollary 2.1 (The main large-degree range). For every finite-dimensional complex V and every b\ge30, there is a \GL(V)-equivariant injection \Sym^6(\Sym^b V)\hookrightarrow\Sym^b(\Sym^6 V).
Proof. The map in (1), with exactly the averaging convention above, is the canonical multiplication map of [21, Theorem 1]. That theorem gives A_b=R_b for a=6 and b\ge6(6-1)=30, including dimension zero. In positive dimension its kernel has an invariant complement, by complete reducibility of complex polynomial \GL(V)-representations. Restriction to this complement is an isomorphism onto R_b; its inverse gives an equivariant section and hence the stated injection. There is no assertion that this section is a prescribed canonical map.
2.2 Six variables and rectangular shifts
For the finite comparison take a=6. If b\ge0 and \lambda is a partition of 6b, the Cauchy decomposition of P=\Sym(V\otimes\CC^6) [12, Exercise 6.11(b)] gives the source multiplicity
In particular it vanishes for partitions of length greater than six. Stable Schur coefficients allow us to work in V=\CC^6: extra variables are specialized to zero, while longer target constituents already have nonnegative multiplicities. This covers every dimension.
Put S=\Sym(\Sym^6\CC^6) with its ordinary symmetric-algebra grading, and let Q(\lambda) be the multiplicity of \Schur_\lambda\CC^6 in S_b. Pad every partition by zeros to six parts and write
Also put t_s(d)=\sum_{j>s}\lambda(d)_j. We abbreviate F(d)=F(\lambda(d)) and Q(d)=Q(\lambda(d)) when 6\mid w(d); then b=w(d)/6. Gap-vector inequalities are componentwise, and e_i denotes the i-th unit vector in gap coordinates.
Lemma 2.2. Let c,d\in\NN^6 satisfy 6\mid w(c) and 6\mid w(d). If Q(c)>0, then Q(d+c)\ge Q(d). In particular
Proof. Multiplication by one nonzero highest-weight polynomial of weight \lambda(c) injects the highest-weight space at \lambda(d) into that at their sum, because S is a domain. This is the usual highest-weight multiplication argument; compare [5, preprint, Lemma 2.2].
The rectangular occurrence Q(6e_i)>0 is a special case of the even-partition theorem of Bürgisser, Christandl, and Ikenmeyer [4, Section 1.1, Theorem]. We give the determinant-tensor construction of [5, preprint, Corollary 4.8] in the form needed here. Take six copies of the determinant tensor on the first i basis vectors. Regroup the tensor positions into i blocks, each taking the same position from every copy, and symmetrize to \Sym^i(\Sym^6\CC^6). This is a highest vector of weight (6,\ldots,6) with i parts. Pair each block with the same sum of pure sixth powers of the first i dual basis vectors. A surviving term uses the same permutation in all six determinant copies, so its sign is +1; such terms exist. The vector is nonzero.
Adding 6e_6 adds six to every partition part. In (2), this tensors with \det^6, shifts the equal label weight by six, and is trivial on permutation matrices. Thus F(d+6e_6)=F(d).
It follows that, after checking all partitions for 6\le b\le25, induction in 26\le b\le29 need only treat d_6<6. If d_6\ge6, subtract 6e_6; the degree becomes b-6\ge20, and Lemma 2.2 imports the earlier comparison. The same induction works throughout 26\le b\le149, the full range retained in the programs for the independent route.
2.3 A chart certificate for the first row
We return to arbitrary a\ge2 and V for the chart calculation. Its factor bound will show, at a=6, that the multiplication image contains every source highest-weight space with \lambda_1\ge5b. It will also provide the independent large-degree bound in the next subsection. Dimension zero is immediate, so assume V\ne0.
Fix 0\ne v\in V, use it as the first basis vector, and write x_{i0},x_{i1},\ldots for the corresponding polynomial variables in row i of P. Put z=z(v)=\prod_i x_{i0}. After inverting z, every x_{i0} is invertible. Write Y_i=(x_{i1}/x_{i0},x_{i2}/x_{i0},\ldots).
Lemma 2.3. For every integer t\ge0, every row-permutation-invariant polynomial in the lists Y_i of total degree at most t is a linear combination of products of at most t elements of z^{-1}R_1. Consequently, for integers k,L\ge0 and f\in R_k,
Proof. The multisymmetric generators are described in [22, Remark 2.9]; we include the factor-length argument needed here. For the general multisymmetric generator theorems, see [26, Theorem 1] and [24, Corollary 8.4]. For each linear form \ell in one list, the coefficients of
show that e_h(\ell(Y_1),\ldots,\ell(Y_a)), and all their polarizations, lie in z^{-1}R_1, for 1\le h\le a. Here u ranges over the span of the remaining basis vectors.
Give e_h weight h. Newton identities express \sum_i\ell(Y_i)^d as a polynomial in e_1,\ldots,e_a of weighted degree d, including when d>a. Each product has at most d factors. Polarizing in \ell therefore expresses
for every degree-d monomial m using at most d polarized elementary factors.
Invariant monomial orbit sums are scalar multiples of expressions
where the m_\nu have positive degree. Inclusion-exclusion over collisions of the indices expresses this in products of p_m’s; a collision replaces monomials by their product and preserves total degree. Thus total degree at most t requires at most t elementary factors. Constants require no factors.
For f\in R_k, the polynomial f/z^k is invariant and has total list degree at most ak. A term with m\le ak ratio factors has the form z^{-m}g_1\cdots g_m, with g_\nu\in R_1. Multiplication by z^{k+L} gives an element of A_{k+L} when k+L\ge ak. Equality in the localization descends to P, a domain.
Corollary 2.4 (First-row certificate). For every integer b\ge1 and partition \lambda of 6b with \sum_{i\ge2}\lambda_i\le b, one has
Proof. Partitions of length greater than six have zero source multiplicity, so work in V=\CC^6 with its standard ordered basis and put a=6. Choose v to be the first basis vector in the chart above. For every weight-\lambda polynomial f\in R_b, the total list degree of f/z^b is t=\lambda_2+\cdots+\lambda_6. The first assertion of Lemma 2.3 writes this ratio as a linear combination of terms z^{-m}g_1\cdots g_m, with m\le t\le b and g_\nu\in R_1. Thus every corresponding term of f is
The entire weight-\lambda space of R_b therefore lies in A_b. Since A_b is a \GL(V)-submodule of R_b, their highest-weight spaces at \lambda are equal: intersect the common weight space with the kernels of the positive simple-root operators. The surjection onto A_b induced by (1) then gives Q(\lambda)\ge F(\lambda).
2.4 An independent large-degree bound
We now prove surjectivity of \mu_{a,b,V} for b\ge a(a-1)^2, without using the companion. The chart will clear denominators uniformly once we know that R, as an A-module, is generated in degrees below a. Polarization supplies that finite-generation statement and then turns the chart calculation into surjectivity in large degree. We continue with arbitrary a\ge2; the zero-dimensional case is immediate.
Lemma 2.5 (Polarization). For integers L\ge1 and j\ge aL,
The sums mean linear spans.
Proof. The operator argument is related to the successive weight shifts in [14, Section 5]. Polarize the degree-aL polynomial v\mapsto z(v)^L, and multiply, to obtain a linear map
Its image is the first displayed span. Identify W_j^* with polynomials f(x_1,\ldots,x_a), separately homogeneous of degree j. Introduce one auxiliary vector variable y. The dual map is, up to a nonzero scalar,
Indeed, evaluation of the dual pairing extracts the coefficient of t_1^L\cdots t_a^L in f(x_1+t_1y,\ldots,x_a+t_ay).
On the polynomial space of total degree aj in x_1,\ldots,x_a,y, the operators
form an \mathfrak{sl}_2, with D_i lowering weight by two. The other lists are spectator variables. The standard weight strings are described in [12, Section 11.1, (11.5)]. In an irreducible module of highest weight n, a weight w\ge L is at distance (n+w)/2\ge L from the bottom. Therefore D_i^L is injective on the full weight-w space of any finite-dimensional module. After the first i-1 shifts, the relevant weight is j-(i-1)L\ge L. Every shift is injective on a space containing the preceding image, so their product is injective. Duality proves the first identity. Average its coefficients over \SSS_a; each z(v)^L is invariant, proving the second identity.
Corollary 2.6. The A-module R is generated by R_0,\ldots,R_{a-1}.
Proof. Lemma 2.5 with L=1 gives R_j=R_1R_{j-1} for j\ge a; apply induction on j.
Proposition 2.7 (Independent chart bound). For a\ge2, one has R_b=A_b whenever b\ge a(a-1)^2.
Proof. Take L=(a-1)^2, so that L\ge(a-1)k for every 0\le k<a. Lemma 2.3 therefore applies to each of these degrees. For any fixed nonzero v, multiply an A-linear expression in the generators of Corollary 2.6 by z(v)^L; this gives z(v)^LR\subset A. The same assertion is trivial for v=0. If b\ge aL, Lemma 2.5 now yields
The opposite inclusion is definitional.
For a=6, Proposition 2.7 supplies the independent large-degree range b\ge150. Combined with the retained finite verification through b=149, it proves Theorem 1.1 without the quadratic-stabilization input. The main proof needs only 6\le b\le29, since Corollary 2.1 applies from b=30.
3 An upper bound and two exact recurrences
We compare the source and target multiplicities without computing both at every partition. This section supplies a gap-monotone upper bound U(d)\ge F(d), together with exact recurrences for the target tables and for the source values needed when that bound is inconclusive. All three constructions come from coefficient extraction with Schur alternants.
3.1 The signed strip rule
The coefficient rule below is the complete-symmetric case of the plethystic Murnaghan–Nakayama rule; see [9, pp. 28–29] and [27, arXiv:1408.3554v2, equation (2)]. Its residue-class determinant formulation also appears in [6, Theorem 3.4 and Corollary 3.11]. We include the proof and the containment inequalities in the form needed by the exact calculation.
For n variables put \delta_n=(n-1,\ldots,0). The next rule bounds a single removal step in a product of factors h_b(X^i), and also enumerates that step exactly.
Lemma 3.1 (Signed strips). Let \gamma,\mu be partitions of length at most n, and suppose |\gamma|-|\mu|=ib, where i,b\ge1. Then
For a nonzero coefficient one necessarily has
Here X^i=(x_1^i,\ldots,x_n^i).
Proof. Set l=\gamma+\delta_n and a=\mu+\delta_n. Multiplying by the Schur alternant denominator [17, Chapter I, equation (3.1)] gives
Indeed its determinant expansion assigns the entries of a to positions j, with assigned values \beta_j satisfying
The size condition makes the sum of the decrement quotients equal to b, so there is no further constraint from h_b(X^i).
Group the rows and columns by residue modulo i, preserving decreasing order within each group. Unequal row and column counts in a residue give zero, and empty blocks contribute one. In a square block of size m, each row support is a terminal interval of columns, and the supports decrease with the row index. A zero row or repeated support makes the determinant zero. Otherwise their sizes are m,m-1,\ldots,1: the block is upper triangular with diagonal entries one, and its only allowed matching is the diagonal. Translating this matching back to the original positions gives
There is therefore just one surviving assignment. The sign from sorting the assembled \beta’s is the remaining row-and-column ordering sign, which proves the asserted coefficient values.
Sorting the componentwise inequalities \beta_j\le l_j gives \mu_j\le\gamma_j. For the other inequality, consider the first j+i entries of l. At most one per residue lacks a preceding entry of that residue. Each of the remaining at least j entries has a predecessor position whose assigned value, by (3), is at least l_{j+i}+i. The j-th largest assigned value is therefore at least l_{j+i}+i, which after removing the staircases says \mu_j\ge\gamma_{j+i}.
3.2 A monotone upper certificate
Averaging the permutations of six tensor factors projects onto \Sym^6(\Sym^b V). Taking traces gives the cycle formula
We bound a coefficient of each product by removing its factors one at a time. The signed strip rule restricts the intermediate partitions; the bound below counts their possible coordinates while discarding their signs and some of their constraints.
For a nonnegative gap vector d, define
For an ordered list I=(i_1,\ldots,i_v) with positive entries summing to six put
where I\sim\pi means any ordering of the cycle lengths of \pi. This definition makes sense even when 6\nmid w(d).
Proposition 3.2. The function U is nondecreasing in every gap. For every nonnegative gap vector d with 6\mid w(d), one has F(d)\le U(d).
Proof. Each N_s is the minimum of the products obtained by omitting one of the widths h_{sj}. Every width is nondecreasing in the gaps, proving monotonicity through the products, minima, sum, and floor.
The zero vector has F(0)=U(0)=1, so assume b=w(d)/6\ge1. Remove the factors of one product in (4) in an order I. After lengths totaling 6-s have been removed, the remaining product is the trace of a permutation of the s factors on (\Sym^b V)^{\otimes s}, together with the diagonal \GL(V)-action. The permutation commutes with that action, so its Schur support lies among the constituents of the tensor power. Those have length at most s by Pieri’s rule [17, Chapter I, equation (5.16)]. Iterating Lemma 3.1, every possible intermediate partition \mu satisfies
Fix all but one coordinate; the sum fixes the remaining one. Thus there are at most N_s(d) choices for 2\le s\le5, and at most one for s=0,1. Each chain has absolute coefficient at most one. The coefficient of the product consequently has absolute value at most B_I(d), for every order I. Average these bounds in (4) and use the integrality of F(d).
In Appendix A, init_masks lists the eleven cycle types with their class sizes and the remaining lengths s\in\{2,3,4,5\} for each removal order. The routine ub_cycle forms the corresponding products of N_s, takes their minima, and performs the weighted sum and division by 720.
3.3 Exact coefficients
The target tables and the remaining source comparisons use Newton’s identity in place of a bound. For an integer vector v of length n, a signed lookup is zero if v has a negative entry or a repeated entry. Otherwise sort v decreasingly, subtract \delta_n, and multiply the coefficient at that partition by the sorting sign. A strictly decreasing nonnegative integer list is at least \delta_n componentwise, so this subtraction always gives a partition.
Write Q_b(\lambda)=[s_\lambda]h_b[h_6] and \widetilde Q_b(v) for its signed lookup in n variables. The initial value is Q_0(0)=1. Newton’s identity [17, Chapter I, equation (2.11)], multiplied by the alternating denominator, gives
Indeed h_6(X^i)=\sum_{|\alpha|=6}X^{i\alpha}. Every nonzero previous partition \mu in this formula satisfies \mu\le\lambda componentwise: subtracting a nonnegative vector cannot increase any decreasing order statistic, and the same staircase is then subtracted. In particular, a bound on a partition’s tail is preserved in all previous lookups.
For the source, fix b and set
Newton’s identity now gives the source recurrence; compare the character formulation in [10, Proposition 5.1]:
At stage j, n=j variables suffice. Lemma 3.1 gives an economical enumeration. Put l=\gamma+\delta_j. In each residue modulo i, its last assigned value must be that residue’s least nonnegative representative: the partition \mu has length at most j-i, so its shifted list ends in i-1,\ldots,0. If any residue is absent, there is no term. At every other position h, whose next position in its residue is h', choose
The sum of all decrement quotients, including the fixed last ones, is b. These choices have distinct values: different residues cannot coincide, and in the same residue \beta_h>l_{h'}\ge\beta_{h'}. The fixed representatives 0,\ldots,i-1 are the smallest i values. Thus sorting and subtracting \delta_j gives a partition \mu of length at most j-i, with the sorting sign. Every such choice contributes once. This is exactly the routine strip in Appendix A. Equation (5) is implemented by gen, Make, and QTail; (6) by Source.
4 A power bound from leading monomials
Rectangular shifts give a first lower bound for Q. The following bound amplifies a known multiplicity.
The sumset estimate is the repeated-set case of the Matolcsi–Ruzsa inequality [18, preprint, Theorem 1.5]; take A=B=D and r=k-1 there when k\ge2. See also [1, author version, Corollary 2]. We include a self-contained triangulation proof. Its generic-point assignment is closely related to the half-open-simplex count used in [1, author version, Theorem 20], where the regions are instead obtained from a shelling.
Lemma 4.1 (Sumsets). Let D be a nonempty finite set of C points in a real vector space, of affine dimension h. Write kD=\{v_1+\cdots+v_k:v_1,\ldots,v_k\in D\}. For this k-fold sumset, k\ge1, one has
Proof. The case h=0 is immediate. Triangulate \conv D using every point of D as a vertex and no other vertices. Such a triangulation can be built by starting with a full-dimensional simplex from D, then inserting points. For a point in the current hull, subdivide the simplices through its minimal containing face in the current triangulation by coning from it, retaining the induced common face subdivisions. For a point outside the hull, cone from it to the boundary simplices on strictly visible facets. Rays from the new point give the intersection and coverage assertions in the latter construction. Both operations preserve all previous vertices.
Choose p in the relative interior of the hull, off all hyperplanes of facets of full-dimensional simplices. Assign x\in\conv D to the unique full simplex whose interior contains (1-\epsilon)x+\epsilon p for all sufficiently small positive \epsilon. There are finitely many facet hyperplanes, so this rule is well defined.
In barycentric coordinates of one full simplex, this assigned region requires nonnegative coordinates, with strict positivity exactly at the indices where the corresponding coordinate of p is negative. Assign a sum of k vertices according to its average, which lies in \conv D. If there are e strict indices, the sums of k vertices of this simplex whose averages are assigned to it number \binom{k-e+h}{h}, interpreted as zero if k<e. Affine independence makes these points distinct; the assignment makes the counted sets for different simplices disjoint subsets of kD.
There is exactly one simplex with e=0, namely the one containing p. At k=1 the counting rule counts precisely the C vertices. To see this, a vertex of the complex lying in a full simplex must be a vertex of that simplex, by the common-face property. Its assigned simplex therefore counts it. A simplex with e=0 contributes h+1, one with e=1 contributes 1, and all others contribute zero. Thus exactly C-h-1 simplices have e=1. Keeping just the e=0,1 contributions for arbitrary k proves the bound.
The leading-monomial argument in the next proposition is an elementary instance of the value-semigroup method: dimensions are counted by distinct values, and multiplication adds values; see [15, preprint, Propositions 2.3, 2.6, and 2.10]. We give the finite-degree argument and the dimension estimate explicitly, since their quantitative form is used in the certificate.
Proposition 4.2 (Power certificate). Suppose Q(x)\ge C>0, where C is an integer and w(x)=6d_0 with d_0>0. Put
Then for all k\ge1,
The cap at 18 only weakens the lower bound; it limits the integer arithmetic in Section 7.
Proof. Choose C independent highest-weight polynomials at x in the polynomial algebra S=\Sym(\Sym^6\CC^6). Row reduction with respect to a fixed monomial order gives a basis with distinct leading monomials. Their exponent vectors form a set D of size C, all with nonnegative integer entries summing to d_0. For each point of kD, choose a product of k basis elements with that leading exponent. The products have distinct leading monomials and hence are independent highest-weight polynomials at kx. Thus Q(kx)\ge|kD|.
Let h=\dim\aff D. Some projection onto h coordinate positions is injective on \aff D, by a nonzero maximal minor of its direction space. The projected integer points are nonnegative with sum at most d_0, so
Lemma 4.1 applies. Finally, for 0\le j<h one computes
This proves f(C,h,k)\ge f(C,z,k). The formula also covers C=1, when h=z=0, and k=1.
5 Certificates in degrees below 150
We now combine the upper bound F\le U, exact target multiplicities, and highest-weight multiplication. The base interval 6\le b\le25 is checked directly. Beyond it, exact target values supply lower-bound seeds: they are computed from the target recurrence without assuming the desired comparison. The resulting certificates leave a set of partitions whose exact differences will be checked in Section 6. We retain the full interval through b=149 for the independent chart proof; only b\le29 is needed with quadratic stabilization.
5.1 Initial tables and the base interval
Compute Q using (5) on the following four sets. All partitions have length at most six.
Previous-degree lookups remain in the applicable set by Section 3. The first table supplies the direct base comparisons; all four supply lower-bound seeds for larger degrees. Overlaps are immaterial.
For 6\le b\le25, every partition is first tested against Q(d)\ge U(d). When this fails, compute F(d)=H_6(\lambda(d)) exactly by (6). Only these requested values at the last stage j=6 are needed. Table 1 records all the fallback comparisons and their equalities.
| b | exact comparisons | equalities | b | exact comparisons | equalities |
| 6 | 2428 | 2428 | 16 | 7370 | 354 |
| 7 | 4476 | 315 | 17 | 7599 | 369 |
| 8 | 6820 | 255 | 18 | 7801 | 384 |
| 9 | 6899 | 250 | 19 | 8057 | 399 |
| 10 | 6886 | 264 | 20 | 8233 | 414 |
| 11 | 6814 | 279 | 21 | 8427 | 429 |
| 12 | 6850 | 294 | 22 | 8650 | 444 |
| 13 | 6872 | 309 | 23 | 8850 | 459 |
| 14 | 6973 | 324 | 24 | 9031 | 474 |
| 15 | 7131 | 339 | 25 | 9268 | 489 |
Table 1: The base interval: 145435 exact fallback comparisons, 9272 equalities, and no negative difference. All other partitions pass Q\ge U.
5.2 Residues and transport of lower bounds
For 26\le b\le149 we need only consider d_6<6, by Lemma 2.2. There is a unique expression
Write b_0(r)=w(r)/6, so that b=b_0(r)+w(m). There are 7776 residues: choose r_2,\ldots,r_6, and r_1 is uniquely determined modulo six. Moreover b_0(r)\le17. The grid \mathcal G has 6611697 points. We call its indices m modes: fixing m groups the 7776 possible residues for simultaneous testing. Pairs whose degree lies outside 26\le b\le149 will not require a test.
For an array on a downward-closed set, propagation replaces its value at m by the maximum of all values at indices u\le m. Extending an array means filling new positions by zero and propagating. An increasing-w sweep taking maxima with the five immediate predecessors performs this operation. A reverse sweep through the same predecessors forms the downward closure of a set. Lemma 2.2 shows that propagation preserves every lower bound for Q(r+6m), with r fixed.
The same multiplication lemma also transports a multiplicity from residue zero to residue r. Suppose J(u)\le Q(6u) for u\in\mathcal G, M(m)\le Q(r+6m), and E\subseteq\mathcal G is a set of indices satisfying Q(r+6c)>0 for c\in E. Multiplication by one nonzero highest-weight polynomial at r+6c gives
The right side certifies the comparison whenever it is at least U(r+6m). No product of two multiplicity dimensions is used: a single nonzero factor gives each injection.
We first construct J and choices of M,E valid for every residue, so that one test can discard a whole mode. For the modes left over we use more precise choices depending on r. The following construction specifies all arrays and tests; the common cap Z=2^{100} is applied only to lower bounds.
5.3 Certificates shared by all residues
A shared power table. Initialize J(m)=1 on \mathcal G, using the rectangle occurrences and the nonzero constant at m=0. For every gap vector x with x_6=0, Q(x)>1, and known value in one of the first three sets of (8), put d_0=w(x)/6 and C=\min(Q(x),Z). For each integer k\ge1 with kd_0\le149 and all entries of kx divisible by six, update
where z is as in Proposition 4.2. Then propagate. It follows that
Common multiplicities and occurrences. For each r, initialize the exact array Q(r+6u) on w(u)\le25-b_0(r), then extend it to w(m)\le42. Take the pointwise minimum of these propagated arrays over all r, with a cap at Z, to obtain M_0 on that grid. Let E_0 be the minimal points where M_0>0, before extending M_0 to all of \mathcal G. Then
The exact entries supplying the minimum may come from different predecessors for different r. Their common consequence is precisely the two universal inequalities in (11).
Selecting modes. Keep a mode m\in\mathcal G unless either w(m)+17\le25, or
The first condition puts every residue in the already settled base range whenever its degree is at least six. For the second, use (9) with M=M_0 and E=E_0. Proposition 3.2 bounds F(r+6m) by U(6m+(5,5,5,5,5,5)), because r\le(5,5,5,5,5,5). The high-residue gap vector need not be admissible; U is defined and monotone there nonetheless.
5.4 Tests for each remaining residue
Form the downward closure of the retained modes. On this closure, initialize Q(r+6m) wherever it is available in (8), use zero elsewhere, and propagate to obtain M_r. Let E_r be the minimal positive locations of the exact array Q(r+6u) on w(u)\le25-b_0(r). For each originally retained mode with 26\le b_0(r)+w(m)\le149, put d=r+6m and clear the pair if either
Flag it otherwise. The first test is Corollary 2.4. The second is (9) with M=M_r and E=E_r.
Minimal positive locations can be found by checking the immediate predecessors. Indeed positive exact multiplicities form an upper set within the small grid, by the rectangular shifts; the propagated common array has the same property. The set used for M_r retains every predecessor needed for propagation. Each difference m-c in a mixed test is nonnegative and lies in \mathcal G.
In Appendix A, powerAll stores J, uniBase and uOcc store M_0 and E_0. The routine setup retains the modes and their downward closure, init_occ constructs E_r, and TestSpec propagates M_r and tests the remaining pairs. The routine run performs the base-interval comparisons.
Finite verification 5.1. The complete program in Appendix A verifies the base comparisons in Table 1. Its mode list has 87354 points, or 114314 after taking the downward closure. It flags 4969698 pairs in the middle interval, all with t_2(d)\le27. Their counts by degree range appear in Table 2.
Every pair in 26\le b\le149 cleared by these tests satisfies Q\ge F, by the proved upper and lower bounds or the chart. Verification 5.1 supplies the separate finite assertion that every pair left over has t_2\le27. Thus it remains only to check the entire band t_2\le27 in 26\le b\le149.
6 Exact differences in the small-tail band
We compute the entire band
In particular it contains every flagged partition of Verification 5.1. For each fixed tail \eta, we will construct seven Laurent series independent of b. Their coefficients at B,B-b,\ldots,B-6b will sum to 720(Q(\lambda)-F(\lambda)). The same seven rows therefore serve every degree and every second part in the displayed range. Write \mathcal Q_b=h_b[h_6], \mathcal F_b=h_6[h_b], and set
We first expand the characters and geometric products in the tail variables y, and then expand their rational coefficients in q as Laurent series at q=0. In particular, D(q)^{-1} has only nonnegative powers.
6.1 Coefficient extraction and truncation
The normalized alternating Schur formula gives
Indeed in six variables it extracts the coefficient of x^\lambda after multiplication by \prod_{i<j}(1-x_j/x_i). Distinct decreasing shifted partitions cannot be permutations of each other, so only the Schur term at \lambda contributes. All Laurent monomials have total degree 6b; setting the first variable to one loses no information about its exponent once the remaining five are fixed.
Give each y_\ell degree one and q degree zero. The monomials of \mathcal A have total y-degree zero; those of \mathcal C have nonnegative total y-degree. Character terms above degree 27 therefore cannot contribute to (13). This uses total degree even when individual exponents in the alternant are negative. Write \equiv_{27} for equality of coefficients of nonnegative y-monomials through total degree 27, with coefficients in \mathbb Q(q).
6.2 Seven numerators independent of the degree
For 0\le j\le6 define
Then
To prove this, fix a multiset of k inner monomials with nonzero tail. The remaining b-k entries come from 1,q,\ldots,q^6, with generating polynomial
This Gaussian polynomial counts partitions in a 6-by-(b-k) rectangle. In general their generating polynomial P_{r,s}, with at most r rows of length at most s, satisfies P_{r,s}=P_{r,s-1}+q^sP_{r-1,s}, by removing a row of length s when one is present, and P_{0,s}=P_{r,0}=1. The displayed product with r=b-k,s=6 satisfies the same recurrence and boundary values, proving the formula. Expanding the numerator gives the e_j in (14); the factor q^{-jk} contributes q^{-j} per selected tail monomial, and hence produces L_j. Within the cutoff k\le27 and b\ge26, so the only possible negative value of b-k is -1. Its numerator has the factor 1-q^0 and vanishes, exactly as required. Thus the formula includes the boundary b=26,k=27; its cancellation is across the seven j-terms.
Similarly, the binary complement of one tail monomial gives
At tail degree t, its coefficient is (1-q^{b+1-t})/(1-q). It counts the binary monomials when t\le b and is zero at t=b+1, the only extra possible degree.
For a cycle type I=(i_1,\ldots,i_v)\vdash6, let
where m_r counts cycles of length r. The quotient E_I is an integer polynomial of degree 15: the multiplicity of the cyclotomic factor \Phi_d in its denominator is the number of cycle lengths divisible by d, which is at most \lfloor6/d\rfloor, its multiplicity in D. Apply (15) with variables raised to each cycle length in (4). Together with (14) this yields
where the seven numerators are explicitly
Here \one_{h\in J} is the indicator of membership. Every selected second term of (15) contributes -q^{i_h(b+1)}, explaining the sign and the residual q^j after q^{jb} is factored out. Equal-length cycles still have distinct positions in the subset sum. These formulas involve eleven cycle types and seven values of j, regardless of b.
6.3 Computing the numerators
Write L_{j,a}=[y^a]L_j. Euler differentiation of the product gives
For example, this follows by differentiating the logarithmic series \sum_{i\ge1}(y^\tau q^{u-j})^i/i. Its denominator i cancels the extra i from differentiation. Every lookup has smaller total tail degree, and the division by |a| is exact.
For the source product in (17), put
At a single tail-coordinate exponent m, its Laurent coefficient is
Multiply these four one-coordinate polynomials and then (-1)^{|J|}n_Iq^jE_I(q). These are the ways arrays and convolutions of source in Appendix B. The routines target and gen_source implement (18) and the cycle-type enumeration.
All raw arrays are symmetric in the four tail variables, so they need only be stored at decreasing nonnegative exponent vectors; arbitrary orders are looked up by sorting, without a sign. To multiply by the cross factors in \mathcal C, the coefficient at a shift v\in\{0,1,2\}^4 is
An in-place sweep in decreasing tail degree reads unmodified lower-degree rows. A subsequent decreasing-exponent sweep multiplies by 1-q. This is the routine cross.
The cross factors preserve symmetry in the four tail variables. To extract a Schur coefficient, we now apply the alternating factor \mathcal A; the permutation signs enter at this step, rather than in the sorted lookups. Put \rho=(3,2,1,0), and let M_{j,a}=[y^a]\mathcal C G_j, with value zero at a vector with a negative coordinate. The exact row produced by schur_row is
This follows from \mathcal A(y)=y^{-\rho}\sum_\pi\sgn(\pi)y^{\pi\rho}. Every accepted index in (19) has nonnegative entries and total degree |\eta|\le27; in particular every coordinate is at most 27. There are 1908 decreasing four-coordinate tails in this range. Equations (13) and (16) show that the final integer tested is precisely
The notation on the right specifies partitions rather than gap vectors.
6.4 The Laurent window
At total tail degree t, the coefficients and individual Euler summands of L_j have q-support in [-6t,5t]. The elementary factor adds at most 21. A cross term consuming s tail units shifts down by at most s, while its raw input has degree t-s; the factor 1-q adds at most one. Thus all target numerator summands lie in
The source has raw support in [-t,21] and final support in [-t,22], using \deg E_I=15 and 0\le j\le6. Alternation does not change total tail degree or q-support. At t\le27 all these supports are therefore contained in
the interval stored by the program. Its clipping operation discards only zero terms. In particular all terms of the boundary cancellation at b=26,k=27 are retained separately.
Since D(0)=1, its inverse is a power series with no negative exponents. Starting from exponent -168, six forward passes a_n\gets a_n+a_{n-h}, for h=1,\ldots,6, divide by D exactly. The numerator is zero above 161. Expansion through 447 suffices because
Entries below -168 are zero; every accessed nonnegative array index is at most 168+447=615, within the 616 allocated entries. Neither division nor the shift q^{jb} can bring a discarded high exponent down to a requested one. The program converts to arbitrary-precision integers before the denominator passes and uses them for all final sums and divisibility tests.
Finally, run_thin visits each of the 1908 tails once, constructs its seven rows, and loops over the displayed ranges of b and B. For a fixed b,\eta, the number of permitted second parts is
This is positive throughout the band. Summing it over the tails and 26\le b\le149 gives 57065668 distinct partitions; there is no dependence on which partitions the first program flagged.
Finite verification 6.1. The complete program in Appendix B checks 57065668 coefficients in this band. Of these, 177134 are zero and none are negative; every integer in (20) is divisible by 720. The grouped counts appear in Table 2.
7 Finite arithmetic, reproduction, and conclusion
7.1 Indexing and arithmetic in the certificate program
Appendix A lists the complete program used for Verification 5.1; it has no external data input. Its partition order reads parts from last to first. If p_j(u) counts partitions of u into at most j parts, the prefix table counts those whose last padded part is <k, by
The second summand subtracts the last part from all j positions. Negative arguments count as zero and p_0(0)=1. Subtracting two prefix counts ranks the next part after the already fixed smaller neighbor. A shorter partition padded by zeros has the same rank. For the tail-restricted table, first group by its exact last three parts, subtract their largest part from the preceding three, and use the three-part rank. The grid similarly groups by w(m) and ranks the partition corresponding to its first five gaps.
The allocated prefix array has size parameter 1250. Its terminal weight need not supply a complete unrestricted count; all counts used for multiplicities and grid indices have weight at most 894, strictly below that terminal boundary. The prefix entries are bounded by \binom{1254}{5}<2^{63}. The actual largest complete levels used in six, four, and three variables have respectively 1229120, 261072, and 67051 partitions. The largest tail-restricted level has 1263138 entries, and the grid has 6611697. These ranks, offsets, and predecessor indices fit signed 32-bit integers. Independently of these evaluated counts, coarse bounds for the three complete levels, the grid, and any tail level are
These count unrestricted weak compositions. For the grid, w(m)\le149 implies \sum_i m_i\le149; for a tail level there are at most \binom{21}{3} ordered tails of size at most 18, and each remaining three-part prefix has weight at most 270.
Multiplicity arithmetic is signed 128-bit. The target recurrences check every accumulation for overflow, and test integrality and nonnegativity before division. For the source, only b\le25 is needed. Each intermediate multiplicity is at most 151^{10}: it is bounded by its multiplicity in (\Sym^b V)^{\otimes j}, j\le6, and the successive interlacing partitions have at most 151^{0+1+2+3+4} choices. In one source Newton step, there are fewer than 6\cdot151^4+6 strip terms, by fixing one coordinate with the size constraint. Thus every signed accumulation is bounded absolutely by
For U, every width in every actual or high-residue call is at most 920. A chain has at most 1+2+3+4=10 width factors, so
The minimum initialized at Z therefore never clips an upper bound, and summing class weights costs a factor at most 720.
For the power bound, S has \dim\Sym^6\CC^6=\binom{11}{5}=462 polynomial generators. Consequently \binom{d_0+461}{461}\ge Q(x), so the binomial search for z, before its cap at 18, stops by 461. With C\le Z and d_0\le149, its first crossing is at most 150Z; predivision multiplication costs at most another factor 610, still below 2^{127}. After the cap, the other binomials are at most \binom{167}{18}. Products in the final lower bound are screened against Z before multiplication. Capping can only weaken a lower certificate. These observations cover the unchecked arithmetic as well as the program’s explicit assertions.
OpenMP distributes distinct coefficient destinations within one degree. All previous degrees are read-only, and each work-sharing loop has its barrier before the next degree. The mathematical result is therefore independent of the thread count.
7.2 Integer bounds for the band program
It remains to bound the signed 128-bit arithmetic before the arbitrary-precision division described in Section 6. For a Laurent series truncated to total tail degree at most 27, use the sum of absolute coefficients as its norm. The norm of L_j is bounded by
Here (M_1,\ldots,M_6)=(24,50,80,105,112,84). There is a convenient purely elementary estimate
The first inequality follows from nonnegative coefficients. For the second, put r_h=4^h/2. Bernoulli’s inequality gives (1-4^{-h})^{r_h}>1/2. Thus the h-th factor of P(1/4) is at most 2^{\lceil M_h/r_h\rceil}, with strict total inequality. The six ceilings are 12,7,3,1,1,1, whose sum is 25; combine this with 4^{27}=2^{54}.
Each four-coordinate source product has at most 24 geometric factors, each consuming at least one tail unit, so its norm is at most
Using \|D\|_1\le64, the quotient E_I has norm at most
Indeed dominate all reciprocal cycle factors by (1-q)^{-6} and sum through degree 15. The partial quotient calculations through degree 21 have bound 64\binom{27}{6}=18944640, and the one-coordinate arrays A_J have coefficients at most \binom{32}{5}=201376. Their additions fit signed 32-bit integers. Class-size divisions are exact: for each partial cycle multiset of size r\le6, its centralizer order divides r!, hence divides 720.
The undivided Euler step costs at most 27N. The absolute weight of all target elementary masks is 720\cdot64, and the class-subset weight is
The last identity follows inductively by inserting the new largest element into an existing cycle or making it a new cycle. Cross multiplication including 1-q has norm at most 512; alternation costs at most 24. A bound for every predivision coefficient, partial sum, and product is therefore
The source convolutions have nonnegative factors before their final signs, so this also bounds their intermediate products. Potentially large products in the code already have a 128-bit operand. The row divisions and final sums then use arbitrary precision. Each output counter covers at most six degrees, 1908 tails, and 448 second parts per degree and tail. Thus even the coarse bound 1908\cdot6\cdot448<2^{31} keeps these counters within signed 32-bit range.
7.3 Recorded computations and reproduction
The reference output streams contain 20 base rows, 21 flag rows, and 21 band rows. Tables 1 and 2 present all 62 numeric rows, including every tested degree. The reproduction script checks these streams against fixed hashes and compares their numeric contents with both manuscript tables.
A full reproduction compiled the two printed sources with GNU C++ 13.3.0 and executed both programs over their entire stated ranges, with assertions enabled. The first used two OpenMP threads and the second was serial. Both exited with status zero, and all four output streams matched the reference streams byte for byte. This execution reproduced all 62 numeric rows in the tables.
The programs retain the complete ranges 6\le b\le149. For the main proof using Corollary 2.1, only the base comparisons 6\le b\le25 and the first row of Table 2, for 26\le b\le29, are required. That row contains 29814 flags and 467068 band coefficients; the band differences are nonnegative, with 2114 equalities. The later rows give the additional finite checks for the independent route through Proposition 2.7. Exact target tables at higher degrees are computed from the recurrences, not assumed from Foulkes’ conjecture, so their use in lower certificates introduces no circularity.
| b range | flagged pairs | largest t_2 | band coefficients | zero differences |
| 26–29 | 29814 | 25 | 467068 | 2114 |
| 30–35 | 46805 | 25 | 872322 | 3621 |
| 36–41 | 50091 | 25 | 1078386 | 4161 |
| 42–47 | 54443 | 25 | 1284450 | 4701 |
| 48–53 | 63027 | 25 | 1490514 | 5241 |
| 54–59 | 74729 | 20 | 1696578 | 5781 |
| 60–65 | 90684 | 18 | 1902642 | 6321 |
| 66–71 | 109505 | 18 | 2108706 | 6861 |
| 72–77 | 130920 | 19 | 2314770 | 7401 |
| 78–83 | 155274 | 20 | 2520834 | 7941 |
| 84–89 | 182871 | 21 | 2726898 | 8481 |
| 90–95 | 214444 | 22 | 2932962 | 9021 |
| 96–101 | 248073 | 23 | 3139026 | 9561 |
| 102–107 | 285382 | 24 | 3345090 | 10101 |
| 108–113 | 324614 | 25 | 3551154 | 10641 |
| 114–119 | 366623 | 26 | 3757218 | 11181 |
| 120–125 | 410723 | 26 | 3963282 | 11721 |
| 126–131 | 457715 | 26 | 4169346 | 12261 |
| 132–137 | 506088 | 26 | 4375410 | 12801 |
| 138–143 | 557588 | 26 | 4581474 | 13341 |
| 144–149 | 610285 | 27 | 4787538 | 13881 |
| total | 4969698 | 27 | 57065668 | 177134 |
Table 2: Full retained verification range. The first row is the residual range required with quadratic stabilization; subsequent rows support the independent chart route. All flags lie in the checked band, and no negative difference is found. The program’s first row is labeled 24, the multiple of six below the first tested degree; the explicit degree ranges here avoid that convention.
The supporting package contains the complete program sources, reference output streams, their hashes and numeric tables, and a reproduction script. A new full run creates its own compiler and execution metadata. The commands, run from the paper directory, are
python3 -B verification/verify_computations.py --check
python3 -B verification/verify_computations.py --run --threads 2The first checks source and reference-stream identities, all 62 numeric rows, agreement with the manuscript tables, and the numerical bounds above; it does not execute the mathematical programs. The second compiles both unchanged sources with -std=c++17 -O3 -UNDEBUG, enables OpenMP only for the first, executes them, and requires both complete output streams to match the reference streams byte for byte. It keeps each run in a new external directory, with the compiler version, build and execution commands, exit statuses, timings, hashes, and captured streams. The option –serial may replace –threads 2 to compile and run the first program without OpenMP. The two C++ files are also printed in full below, so every finite step is specified in this paper. The coefficient formulas, indexing arguments, and arithmetic bounds establish what they check; successful execution supplies the finite nonnegativity assertions.
7.4 Completion of the proof
Proof of Theorem 1.1. Corollary 2.1 settles b\ge30. Verification 5.1 settles 6\le b\le25. Proceed by increasing b from 26 through 29. If d_6\ge6, Lemma 2.2 reduces the comparison to degree b-6\ge20. Otherwise d=r+6m lies in the exhaustive classification of Section 5. Every pair is either cleared there by a proved certificate or the chart, or flagged. The flags have t_2\le27 and are covered by Verification 6.1. Hence Q(\lambda)\ge F(\lambda) for every source partition and every b\ge6. The source has no constituents of length greater than six; specialization at extra variables zero transfers the comparisons to every finite dimension. Complete reducibility gives the required equivariant injection.
Remark 7.1 (The independent full-range route). The same finite argument, with no change of predicate or arithmetic, continues from b=30 through b=149, as recorded in the full tables. Combining it with Proposition 2.7 for b\ge150 proves the theorem without using the quadratic companion. The chart and certificate methods are therefore retained as a complete independent proof, while Corollary 2.1 gives the shorter large-degree route.
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A The bounded certificate program
The following is the complete first program. It uses GNU C++17 signed 128-bit integers and optional OpenMP; assertions must remain enabled. The source file is verification/programs/appendix-a.cpp. See Sections 3–5 for the construction and Section 7 for arithmetic and reproduction.
B The tail-band program
The following is the complete second program. It uses GNU C++17 and Boost.Multiprecision for the final coefficient sums. The source file is verification/programs/appendix-b.cpp. See Section 6 for the construction and Section 7 for arithmetic and reproduction.