History and the canonical map.

The conjecture originates in Foulkes’ study of concomitants (Foulkes 1950). For binary forms, classical Hermite reciprocity gives an isomorphism between the two symmetric powers for all \(a,b\); see (Raicu et al. 2022, Introduction). In higher dimension one seeks a multiplicity inequality instead. The case \(a=1\) is immediate. Thrall’s decomposition formulas imply \(a=2\) (Thrall 1942); see also (Dent and Siemons 2000, 237). Dent and Siemons proved \(a=3\) (Dent and Siemons 2000).

A natural approach uses the canonical Foulkes–Howe map \[\Psi_{a,b,V}:\mathop{\mathrm{Sym}}^a(\mathop{\mathrm{Sym}}^b V)\longrightarrow\mathop{\mathrm{Sym}}^b(\mathop{\mathrm{Sym}}^a V).\] 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 \(b\ge b_0\) (McKay 2008; Ikenmeyer 2015). The computation at \((4,4)\) by Müller and Neunhöffer therefore yields \(a=4\) (Müller and Neunhöffer 2005); Cheung, Ikenmeyer, and Mkrtchyan’s injectivity computation at \((5,6)\) yields \(a=5\), with the diagonal comparison again immediate (Cheung et al. 2017, 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 (Müller and Neunhöffer 2005; Cheung et al. 2017). For the sixth diagonal this already occurs when \(\dim V\ge6\) (Cheung et al. 2017, 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 \(b=13\) (Evseev et al. 2014, 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 \(a\) (Brion 1993, Corollary 1.3) and later gave a dimension-dependent bound for surjectivity of the canonical map in the opposite direction (Brion 1997, 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 (Landsberg 2015, sec. 7); (Raicu et al. 2022, sec. 2). The companion paper proves its surjectivity for \(b\ge a(a-1)\), uniformly in dimension (OpenAI 2026, 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 (Raicu et al. 2026, Theorem 1.4), and Gangl, Gutiérrez, and Szwej identify the dual map through a substitution construction (Gangl et al. 2026, Theorem 1.2). These results concern a two-dimensional underlying space; the theorem above treats every finite dimension.

Proof strategy.

The source \(\mathop{\mathrm{Sym}}^6(\mathop{\mathrm{Sym}}^b V)\) 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.

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 8. The polarization argument is related to the successive weight shifts in the proof of McKay’s theorem (Ikenmeyer 2015, sec. 5). Both certificate programs run through \(b=149\), so the extended finite verification and this chart bound prove Theorem 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 \(\mathop{\mathrm{GL}}\)-representations (Fulton and Harris 1991, Theorem 9.19 and Proposition 14.13), Schur functors and the Cauchy decomposition (Fulton and Harris 1991, Theorem 6.3 and Exercise 6.11(b)), and the Schur alternant formula (Macdonald 1995, I, (3.1)). Stable Schur coefficients are unchanged on specializing extra character variables to zero whenever the partition length remains admissible (Macdonald 1995, I, (3.2)–(3.3)). The character interpretation of plethysm is recalled in (Macdonald 1995, I, Appendix A, Sections 7–8). We also write \(e_r\) for the elementary symmetric function and set \(h_0=e_0=1\).

The multiplication map

For the moment let \(a\ge2\), and set \[P=(\mathop{\mathrm{Sym}}V)^{\otimes a},\qquad W_j=P_{(j,\ldots,j)},\qquad R_j=W_j^{\mathfrak S_a},\qquad R=\bigoplus_{j\ge0}R_j.\] The symmetric group permutes the tensor factors. We identify \(\mathop{\mathrm{Sym}}^a W\) with \((W^{\otimes a})^{\mathfrak S_a}\) by the normalized average \[w_1\cdots w_a\mathrel{% \BeginAccSupp{method=hex,unicode,ActualText=27FC}% \OriginalLongmapsto\EndAccSupp{}} \frac1{a!}\sum_{\sigma\in\mathfrak S_a} w_{\sigma(1)}\otimes\cdots\otimes w_{\sigma(a)}.\] Its inverse is the restriction of the symmetric-product quotient. Let \(A\subset R\) be the graded subalgebra generated by \(R_1=\mathop{\mathrm{Sym}}^a V\). For \(v\in V\), put \(z(v)=v^{\otimes a}\in R_1\). Multiplication defines the canonical Foulkes–Howe map \[ \mu_{a,b,V}:\mathop{\mathrm{Sym}}^b(\mathop{\mathrm{Sym}}^a V)\longrightarrow R_b=\mathop{\mathrm{Sym}}^a(\mathop{\mathrm{Sym}}^b V), \tag{1}\] 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 (Raicu et al. 2022, sec. 2). We will use its explicit polynomial realization.

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 \(P=\mathop{\mathrm{Sym}}(V\otimes\mathbb C^6)\) (Fulton and Harris 1991, Exercise 6.11(b)) gives the source multiplicity \[ F(\lambda)=\dim\left((\mathbf S_\lambda\mathbb C^6)_{(b,\ldots,b)}^{\mathfrak S_6}\right). \tag{2}\] In particular it vanishes for partitions of length greater than six. Stable Schur coefficients allow us to work in \(V=\mathbb C^6\): extra variables are specialized to zero, while longer target constituents already have nonnegative multiplicities. This covers every dimension.

Put \(S=\mathop{\mathrm{Sym}}(\mathop{\mathrm{Sym}}^6\mathbb C^6)\) with its ordinary symmetric-algebra grading, and let \(Q(\lambda)\) be the multiplicity of \(\mathbf S_\lambda\mathbb C^6\) in \(S_b\). Pad every partition by zeros to six parts and write \[d_i=\lambda_i-\lambda_{i+1},\quad \lambda_7=0,\qquad \lambda(d)_i=\sum_{j=i}^6d_j,\qquad w(d)=\sum_{i=1}^6 i\,d_i.\] 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.

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 3 imports the earlier comparison. The same induction works throughout \(26\le b\le149\), the full range retained in the programs for the independent route.

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 \(Y_i=(x_{i1}/x_{i0},x_{i2}/x_{i0},\ldots)\).

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.

For \(a=6\), Proposition 8 supplies the independent large-degree range \(b\ge150\). Combined with the retained finite verification through \(b=149\), it proves Theorem 1 without the quadratic-stabilization input. The main proof needs only \(6\le b\le29\), since Corollary 2 applies from \(b=30\).

The signed strip rule

The coefficient rule below is the complete-symmetric case of the plethystic Murnaghan–Nakayama rule; see (Désarménien et al. 1994, 28–29) and (Wildon 2016, arXiv:1408.3554v2, equation (2)). Its residue-class determinant formulation also appears in (Cao et al. 2025, 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.

A monotone upper certificate

Averaging the permutations of six tensor factors projects onto \(\mathop{\mathrm{Sym}}^6(\mathop{\mathrm{Sym}}^b V)\). Taking traces gives the cycle formula \[ h_6[h_b](X)=\frac1{720}\sum_{\pi\in\mathfrak S_6} \prod_{\text{cycles }c\text{ of }\pi}h_b(X^{|c|}). \tag{4}\] 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 \[h_{sj}=1+\lambda(d)_j-\lambda(d)_{j+6-s},\qquad N_s(d)=\frac{\prod_{j=1}^s h_{sj}}{\max_{1\le j\le s}h_{sj}} \quad(2\le s\le5),\qquad N_0=N_1=1.\] For an ordered list \(I=(i_1,\ldots,i_v)\) with positive entries summing to six put \[B_I(d)=\prod_{u=1}^v N_{6-i_1-\cdots-i_u}(d),\qquad U(d)=\left\lfloor\frac1{720} \sum_{\pi\in\mathfrak S_6}\min_{I\sim\pi}B_I(d)\right\rfloor ,\] where \(I\sim\pi\) means any ordering of the cycle lengths of \(\pi\). This definition makes sense even when \(6\nmid w(d)\).

In Appendix 8, 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 \(720\).

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 (Macdonald 1995, I, equation (2.11)), multiplied by the alternating denominator, gives \[ bQ_b(\lambda)= \sum_{i=1}^b\ \sum_{\substack{\alpha\in\mathbb Z_{\ge 0}^n\\|\alpha|=6}} \widetilde Q_{b-i}(\lambda+\delta_n-i\alpha). \tag{5}\] 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 \[H_j(\gamma)=[s_\gamma]h_j[h_b],\qquad H_0(0)=1.\] Newton’s identity now gives the source recurrence; compare the character formulation in (Evseev et al. 2014, Proposition 5.1): \[ jH_j(\gamma)=\sum_{i=1}^j\ \sum_{\mu} [s_\gamma]\bigl(h_b(X^i)s_\mu(X)\bigr)H_{j-i}(\mu), \qquad |\mu|=(j-i)b. \tag{6}\] At stage \(j\), \(n=j\) variables suffice. Lemma 9 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 \[\beta_h=l_h-ix_h,\qquad 0\le x_h\le(l_h-l_{h'})/i-1.\] 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 8. Equation (5) is implemented by gen, Make, and QTail; (6) by Source.

Initial tables and the base interval

Compute \(Q\) using (5) on the following four sets. All partitions have length at most six. \[ \begin{array}{c|l|c} \text{degrees}&\text{additional restriction}&\text{variables used}\\ \hline 0\le b\le25&\text{none}&6\\ 0\le b\le149&\operatorname{length}(\lambda)\le3&3\\ 0\le b\le55&\operatorname{length}(\lambda)\le4&4\\ 0\le b\le45&t_3(d)\le18&6 \end{array} \tag{8}\] 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.

Residues and transport of lower bounds

For \(26\le b\le149\) we need only consider \(d_6<6\), by Lemma 3. There is a unique expression \[d=r+6m,\qquad r\in\mathcal R=\{0,\ldots,5\}^6\cap\{6\mid w(r)\},\qquad m\in\mathcal G=\{m\in\mathbb Z_{\ge 0}^6:m_6=0,\ w(m)\le149\}.\] 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 3 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 \[ Q(r+6m)\ge \max\left(\{M(m)\}\cup \{J(m-c):c\in E,\ c\le m\}\right). \tag{9}\] 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.

Certificates shared by all residues

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 \[t_1(d)\le b \quad\text{or}\quad \max\left(\{M_r(m)\}\cup \{J(m-c):c\in E_r,\ c\le m\}\right)\ge U(d).\] Flag it otherwise. The first test is Corollary 5. 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 8, 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.

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 13 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 \(26\le b\le149\).

Coefficient extraction and truncation

The normalized alternating Schur formula gives \[ Q(\lambda)-F(\lambda) =[q^By^\eta]\mathcal C(q,y)\mathcal A(y) (\mathcal Q_b-\mathcal F_b)(1,q,y). \tag{13}\] 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 \(\mathbb Q(q)\).

Seven numerators independent of the degree

For \(0\le j\le6\) define \[L_j(q,y)= \prod_{\substack{\tau\in\mathbb Z_{\ge 0}^4,\ 1\le|\tau|\le6\\0\le u\le6-|\tau|}} (1-y^\tau q^{u-j})^{-1}.\] Then \[ \mathcal Q_b(1,q,y)\equiv_{27} \frac1{D(q)}\sum_{j=0}^6q^{jb}(-1)^j e_j(q,q^2,\ldots,q^6)L_j(q,y). \tag{14}\] 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 \[h_{b-k}(1,q,\ldots,q^6)= \prod_{h=1}^6\frac{1-q^{b-k+h}}{1-q^h} \quad(b-k\ge0).\] 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 \[ h_b(1,q,y)\equiv_{27} \frac{\prod_\ell(1-y_\ell)^{-1} -q^{b+1}\prod_\ell(1-y_\ell/q)^{-1}}{1-q}. \tag{15}\] 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 \[n_I=\frac{720}{\prod_{r\ge1}r^{m_r}m_r!},\qquad E_I(q)=\frac{D(q)}{\prod_{h=1}^v(1-q^{i_h})},\] 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 \[ 720D(q)(\mathcal Q_b-\mathcal F_b)(1,q,y) \equiv_{27}\sum_{j=0}^6q^{jb}G_j(q,y), \tag{16}\] where the seven numerators are explicitly \[\begin{align*} G_j(q,y)={}&720(-1)^je_j(q,\ldots,q^6)L_j(q,y)\\ &-\sum_{I\vdash6}n_I \sum_{\substack{J\subseteq\{1,\ldots,v\}\\\sum_{h\in J}i_h=j}} (-1)^{|J|}q^j E_I(q) \prod_{\ell=1}^4\prod_{h=1}^v (1-y_\ell^{i_h}q^{-i_h\boldsymbol 1_{h\in J}})^{-1}. \tag{17}\end{align*}\] Here \(\boldsymbol 1_{h\in J}\) 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 \(b\).

Computing the numerators

Write \(L_{j,a}=[y^a]L_j\). Euler differentiation of the product gives \[ L_{j,0}=1,\qquad |a|L_{j,a}= \sum_{\tau,u}\ \sum_{\substack{i\ge1\\i\tau\le a}} |\tau|q^{i(u-j)}L_{j,a-i\tau}\quad(|a|>0). \tag{18}\] 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 \[A_J(m)=[z^m]\prod_{h\in J}(1-z^{i_h})^{-1}.\] At a single tail-coordinate exponent \(m\), its Laurent coefficient is \[\sum_{p=0}^m A_J(p)A_{\bar J}(m-p)q^{-p}.\] 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 9. 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 \[(-1)^{n_1}q^{-n_2}(1+q^{-1})^{n_1}, \qquad n_s=\#\{\ell:v_\ell=s\}.\] 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 \[ R_{j,\eta}(q)=\frac1{D(q)} \sum_{\pi\in\mathfrak S_4}\mathop{\mathrm{sgn}}(\pi) M_{j,\eta+\rho-\pi\rho}(q). \tag{19}\] This follows from \(\mathcal A(y)=y^{-\rho}\sum_\pi\mathop{\mathrm{sgn}}(\pi)y^{\pi\rho}\). 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 \[ \sum_{j=0}^6[q^{B-jb}]R_{j,\eta}(q) =720\bigl(Q(6b-B-|\eta|,B,\eta)-F(6b-B-|\eta|,B,\eta)\bigr). \tag{20}\] The notation on the right specifies partitions rather than gap vectors.

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 \[[-6t,5t+22].\] 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 \[[-162,157]\subset[-168,161],\] 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 \[B-jb\le B\le3b\le447.\] 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 \[3b-\left\lceil\frac{|\eta|}{2}\right\rceil-\eta_1+1.\] 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.

Indexing and arithmetic in the certificate program

Appendix 8 lists the complete program used for Verification 13; 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 \[N_j(u,k)=N_j(u,k-1)+p_{j-1}(u-j(k-1)).\] 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 \[\max\left\{\binom{155}{5},\binom{333}{3},\binom{896}{2}, \binom{154}{5},\binom{21}{3}\binom{272}{2}\right\}<2^{31}.\] 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 \((\mathop{\mathrm{Sym}}^b V)^{\otimes j}\), \(j\le6\), 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 \[151^{10}(6\cdot151^4+6)<2^{127}.\] 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 \[B_I\le920^{10}<Z=2^{100}.\] 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 \(\dim\mathop{\mathrm{Sym}}^6\mathbb C^6=\binom{11}{5}=462\) 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.

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 \[N=\sum_{n=0}^{27}[z^n]P(z),\qquad P(z)=\prod_{h=1}^6(1-z^h)^{-M_h},\qquad M_h=(7-h)\binom{h+3}{3}.\] Here \((M_1,\ldots,M_6)=(24,50,80,105,112,84)\). There is a convenient purely elementary estimate \[ N\le4^{27}P(1/4)<2^{79}. \tag{21}\] 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 \[S_0=\binom{51}{24}<2^{51}.\] Using \(\|D\|_1\le64\), the quotient \(E_I\) has norm at most \[Q_0=64\binom{21}{6}=3472896<2^{22}.\] 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 \[\sum_{I\vdash6}n_I2^{v(I)} =\sum_{\sigma\in\mathfrak S_6}2^{\#\text{cycles}(\sigma)} =2\cdot3\cdot4\cdot5\cdot6\cdot7=5040.\] 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 \[ 512\cdot24\bigl(27\cdot720\cdot64N+5040Q_0S_0\bigr) <2^{115}<2^{127}. \tag{22}\] 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.

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, 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 8. 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.

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 2

The 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.

Completion of the proof