The obstacle and the proof strategy.

The stability inequality compares the Dirichlet energy of a test function with a curvature term on the free boundary. The use of the harmonic Hessian has a geometric precedent in Simons’s second-fundamental-form argument for minimal cones [16], a connection developed explicitly by Jerison and Savin. A natural Bernoulli test is a power of the Hessian norm. Jerison and Savin explain why this unweighted choice does not give a uniform instability criterion in dimensions four through six: the interior and boundary estimates impose different restrictions on the power, and their sharp configurations need not agree [13]. Their four-dimensional argument uses a more refined function of the Hessian eigenvalues [13].

Our proof uses the normalized shape of the Hessian to construct both a scalar test and a separate vector field whose divergence controls the test’s energy. The two constructions are chosen together, so the interior and boundary estimates retain the same tensor data. The proof has three stages.

First, suppose the first nonflat minimizing cone occurs in dimension five or six. Blowups away from its vertex reduce the dimension and are therefore flat. Regularity theory makes each component of its spherical positive set smooth, and at least one component has nonzero Hessian. A five-dimensional cone can be made independent of one extra coordinate without losing minimality. Its spherical link then has precisely two exceptional tips. Thus both possible dimensions lead to a domain \(\Omega\subset S^5\) with a smooth regular boundary and, possibly, these two tips (Section [sec:reductions]).

Second, let \(g\) be the restriction of the resulting cone to the sphere, and set \(T=\nabla^2g+g\mathop{\mathrm{Id}}\) and \(w=|T|\). Where \(w>0\), we construct a positive scalar \(h\) from \(w\), the normalized tensor \(T/w\), \(g\), and \(\nabla g\), and a vector field \(Z\) from the same data and \(\nabla T/w\). The target inequalities are \[\mathop{\mathrm{div}}Z\ge |\nabla h|^2+4h^2+\epsilon_*w^{2a}, \qquad N\cdot Z+Hh^2\ge0,\] where \(a\in(1/2,1)\) is fixed, \(\epsilon_*>0\), \(N\) points into \(\Omega\), and \(H=-T(N,N)\). Spherical stability supplies the reverse integrated comparison between the test’s energy and its boundary curvature term. The extra positive mass \(\epsilon_*w^{2a}\) will give the contradiction. The geometry and stability inequality are developed in Section [sec:geometry]. We then state the universal tensor inequality and derive the two geometric comparisons; the full exact verification follows in Section 6.

Two cancellations make this construction possible. A Laplacian contraction and an antisymmetric curvature contraction eliminate fourth derivatives from \(\mathop{\mathrm{div}}Z\). At the boundary, reflection in the normal direction removes the derivative components not determined by the boundary conditions. The remaining inequalities involve only finitely many normalized tensors. We prove them by weighted sums of tensor squares and exact rational remainder bounds, with all coefficients and contraction conventions supplied in Appendix 9.

Finally, neither the normalized Hessian nor its derivatives are defined at \(w=0\), and suspension introduces singular tips. These sets cannot be discarded without an estimate. A regularized Bochner identity gives an integrable weight controlling the derivatives near every Hessian zero. The explicit suspension formulas give an integrable power of the distance parameter at both tips. Smooth product cutoffs therefore remove both obstructions and make all error terms vanish. Section [sec:cutoffs] completes the stability contradiction.

Organization and conventions.

Section [sec:reductions] reduces a hypothetical first nonflat minimizer to a smooth spherical component in dimension six, allowing two suspension tips. Section [sec:geometry] derives the tensor identities and the stability inequality. Section 4 states the universal algebraic estimate, and Section [sec:flux] derives the pointwise geometric comparison from it. Section 6 gives the complete exact verification; Section [sec:cutoffs] then removes the cutoffs to obtain the contradiction. Section 8 gives the regularity and sharpness consequences. Appendix 9 gives all coefficient data. All tensor norms and contractions use orthonormal frames. Repeated indices are summed, \(\Delta=\mathop{\mathrm{tr}}\nabla^2\), and a boundary normal denoted by \(N\) points into the positive phase. The curvature \(H=-T(N,N)\) used below is not divided by the dimension of the boundary.

The regularity inputs

The ballwise convention also gives comparison on every bounded open set \(O\Subset\mathbb R^d\): extend an \(H^1_0(O)\) competitor difference by zero into a ball containing \(\overline O\), apply minimality there, and cancel the energy outside \(O\). Conversely, comparison on bounded open sets includes comparison on balls. It is even enough to have comparison against arbitrary admissible competitors in every origin-centered ball, by the same extension argument. There is no restriction to homogeneous competitors.

The convention that competitors are nonnegative makes no difference to the standard one-phase results. If \(v-u\in H^1_0(O)\) and \(u\geq0\), then \(v_+-u\in H^1_0(O)\) and \[\int_O\bigl(|\nabla v_+|^2+\mathbf 1_{\{v_+>0\}}\bigr) \leq \int_O\bigl(|\nabla v|^2+\mathbf 1_{\{v>0\}}\bigr).\] For completeness, the assertion about \(H^1_0(O)\) follows by approximating \(v-u\) in \(H^1\) by compactly supported smooth functions and applying the continuity of positive-part truncation in \(H^1\); the truncated differences in the approximation have compact support in \(O\).

As in Section 1, a free-boundary point is regular if it has a half-space blowup and singular otherwise. Proposition 3 explains the geometric meaning of regularity. Conversely, a smooth one-sided free boundary with (3) has a half-space blowup by its first-order Taylor expansion. We henceforth use the continuous representative of a minimizer. If it is one-homogeneous almost everywhere, it is one-homogeneous everywhere: for each fixed \(r>0\), both sides of \(u(rx)=ru(x)\) are continuous and agree almost everywhere.

Dimension reduction at a nonzero point

The next two lemmas give the dimension-reduction steps explicitly; compare [17].

In dimension one, every nonnegative one-homogeneous Sobolev function has the form \(u(x)=b_+x_++b_-(-x)_+\) with \(b_+,b_-\ge0\). If both slopes are positive, a nonnegative smooth bump \(\phi\) supported in \((-R,R)\) with \(\phi(0)>0\) preserves positivity volume and gives \[J(u+\tau\phi;(-R,R))-J(u;(-R,R)) =-2\tau(b_++b_-)\phi(0)+\tau^2\int_{-R}^R|\phi'|^2<0\] for small \(\tau>0\). Thus a nonzero minimizing cone has just one positive slope. After reflection write \(u(x)=b x_+\), \(b>0\). The competitors \(v_t(x)=bR(x-t)_+/(R-t)\) on \((-R,R)\), \(|t|<R\), have the same trace and energy \(b^2R^2/(R-t)+(R-t)\). Its derivative at the minimizing parameter \(t=0\) is \(b^2-1=0\), so \(b=1\). This proves flatness in dimension one directly.

The lower-dimensional classification of Jerison–Savin [13] and the dimension-seven minimizing example of De Silva–Jerison [7] give \(5\leq d_*\leq7\). De Silva–Jerison’s Theorem 1.1 uses the same functional and arbitrary ballwise Dirichlet competitors. For the exclusion argument, suppose from now on that \[ d=d_*\in\{5,6\}, \tag{5}\] and fix a nonzero nonflat one-homogeneous global minimizer \(u\) in \(\mathbb R^d\).

Suspension and the working domain

The spherical Hessian and its boundary values

For a smooth solution of \(\Delta g=-mg\) on a domain in \(S^m\), set \[ p=\nabla g,\qquad T=\nabla^2g+g\mathop{\mathrm{Id}},\qquad B_{ijk}=\nabla_iT_{jk},\qquad w=|T|,\qquad s^2=g^2+|p|^2. \tag{11}\] The notation \(s^2\) denotes the displayed smooth function, including at points where it vanishes. If \(u(ry)=rg(y)\) is the homogeneous extension, then in the radial and tangential orthonormal frame \[ D^2u(ry)=r^{-1} \begin{pmatrix}0&0\\0&T(y)\end{pmatrix}. \tag{12}\] Indeed, Euler’s identity gives \(D^2u\,y=0\), and differentiation of \(\nabla_{\mathbb R^{m+1}}u=g y+p\) along a tangent vector gives the tangential block \(\nabla^2g+g\mathop{\mathrm{Id}}\) at radius one. Homogeneity then gives the factor \(r^{-1}\). The same gradient formula identifies \(s^2\) with \(|\nabla_{\mathbb R^{m+1}}u|^2\) on the unit sphere.

The remaining boundary formulas are local. Thus they apply on the original boundary and, once the strict sign has been transported, on the regular side of the suspension.

Transport to a suspension

We now use the unadorned notation \(\Omega,g,p,T,B,w,H\) for the working component in \(S^5\) defined at the end of Section [sec:reductions]. It is either \(\Omega_o\) when \(m_o=5\), or the above suspension when \(m_o=4\). All boundary integrals mean integrals over the regular boundary, with the two tips omitted in the suspended case. The original data retain the subscript \(o\) when needed.

Second variation under compact perturbations

We derive stability directly from the minimizing property, including its sign and test class. The argument works in either original ambient dimension and in the six-dimensional suspension.

The angular constant comes from the sharp radial Hardy threshold, which also enters the cone instability criteria in [13]. We include the radial calculation to specify the compact supports and the order of limits.

Normalized differentiation and the fourth derivatives

On \(\Omega_+\) define \[ A=\frac{T}{w},\qquad X=\frac{B}{w},\qquad q_l=X_{ljk}A_{jk}. \tag{43}\] The identities of Proposition 9 show that \(A\in\mathop{\mathrm{STF}}^2(\mathbb R^5)\), \(|A|=1\), and \(X\in\mathop{\mathrm{STF}}^3(\mathbb R^5)\). Moreover, \(g^2+|p|^2\leq1\), so these are admissible data for the interior certificate. The homogenizing variable in that certificate is set to \(z=1\) in every geometric application.

The square root and the interior comparison

Proposition 15 gives \[ f(A,p,g)\geq\epsilon_* =\frac{438907}{12000000}>0 \tag{46}\] on the entire normalized coefficient domain. The verification in Section 6.5 obtains this bound using the free algebraic data \(X=w=z=0\). Since \(f\) is independent of \(w\), this permitted algebraic substitution does not impose \(w=0\) on the geometric data: the fields below are defined on \(\Omega_+\), where \(w>0\).

Define on \(\Omega_+\) \[ h=w^a\sqrt f,\qquad Z_l=w^{2a}\bigl(G_{lijk}X_{ijk}+wM_l+P_l\bigr). \tag{47}\] These fields are smooth on \(\Omega_+\) and extend smoothly to each regular side boundary point, since \(w>0\) there by Proposition 11.

Boundary reflection and the signed flux

At a regular side boundary point, Proposition 11 gives \[g=0,\quad p=N,\quad |p|=1,\quad Ap=\lambda p, \qquad \lambda=-\frac H w<0.\] As in the boundary certificate, define \[ S_0=\lambda A-A^2,\qquad V_{ijk}=p_i(S_0)_{jk}+p_j(S_0)_{ki}+p_k(S_0)_{ij} +p_ip_jp_k. \tag{52}\]

Uniform estimates for the cutoff argument

Tensor monomials and the fixed polynomials

Here and in Appendix 9, a rank-\(r\) tensor monomial is encoded by a word of decimal digits, partitioned into triples. Its first triple \(hbj\) specifies \[g^h(\mathop{\mathrm{tr}}A^3)^b w^{\mathbf1_{\{j=1\}}} z^{\mathbf1_{\{j=2\}}}.\] Each later triple \(uvk\) specifies an edge carrying the matrix \(A^k\), with \(A^0=\delta\). An endpoint labeled \(i\in\{0,\ldots,r-1\}\) is the \(i\)th free slot, starting with slot zero. An endpoint labeled \(5\) carries one copy of \(p\), and one labeled \(6\) carries one copy of \(\ell\). Three incidences of label \(7\) are contracted against the three slots of one \(X\); three incidences of label \(8\), if present, use a second copy of \(X\). Each occurrence of a vector endpoint is a separate vector factor. Matrix entries, vector entries, and tensor entries are multiplied, and all nonfree slots are summed. Edges are unoriented because \(A\) is symmetric. Permuting the three incidences at an \(X\) does not change its value because \(X\) is symmetric. There are no implicit multiplicities or factorials in this convention.

Write \(\mathcal M_r(s)\) for the tensor encoded by the word \(s\). For example, \[\mathcal M_0(\texttt{200550})=g^2|p|^2,\qquad \mathcal M_4(\texttt{000500520131})_{lijk}=p_lp_jA_{ik},\qquad \mathcal M_1(\texttt{000771700})_l=q_l.\] The digit labels encode slots and must be distinguished from the coordinate indices that are summed from \(1\) to \(5\).

The nonnegative comparison polynomials

The positive tensor squares below have the form of Gram-matrix certificates for polynomial positivity; compare [15]. Here all rational polynomials, projection identities, and remainder bounds are explicit, and we verify the certificate directly.

A header \(\texttt{@ }r\ \mathrm{kind}\ \mathrm{mode}\) in Table II begins a new block; consecutive equal headers still designate different blocks. For a block of nonnegative rank, write its monomial rows as \(s_1,\ldots,s_N\) and its rectangular integer coefficient matrix as \((c_{tc})\). Define \[J_c=\frac1Q\sum_{t=1}^N c_{tc}\mathcal M_r(s_t).\] Its contribution to \(\Gamma\) is \[ \rho\sum_c\langle J_c,\Pi J_c\rangle,\qquad \rho=\begin{cases}1,&\mathrm{mode}=\texttt{I},\\ \chi,&\mathrm{mode}=\texttt{B}. \end{cases} \tag{57}\] Here \(\Pi\) is specified below by \(r\) and \(\mathrm{kind}\). There are nineteen such blocks with a total of \(529\) monomial rows. Every row has degree one in \((w,z,\ell,X)\), assigning degree one to each component of these variables and degree zero to \(A,p,g\). Hence \(\Gamma\) has degree two in these variables.

The two remaining blocks have rank \(-1\) and fourteen rows each. In these blocks the four digits of a row word are an exponent \(\beta=(\beta_1,\ldots,\beta_4)\), encoding \(x^\beta\) for \(x\in\mathbb R^4\). Both coefficient matrices have five columns. Set \[J_c(x)=\frac1Q\sum_t c_{tc}x^{\beta_t},\qquad \operatorname{Av}F(x)=\frac1{24}\sum_{\pi\in S_4} F(x_{\pi(1)},\ldots,x_{\pi(4)}).\] Their sum defines the scalar polynomial \[ \Lambda(x)=\sum_{\text{rank }-1\text{ blocks}} \operatorname{Av}\left(\rho(x)\sum_c J_c(x)^2\right), \qquad \rho(x)=\begin{cases}1,&\mathrm{mode}=\texttt{I},\\ x_1+x_2+x_3+x_4,&\mathrm{mode}=\texttt{H}. \end{cases} \tag{58}\] Consequently \(\Lambda\ge0\) when \(\sum x_i\ge0\).

Exact contraction and differentiation rules

We give explicit collection rules both to specify the finite calculation and to justify that it is an identity on all the constrained tensors. A graph key records the rank, powers of \(g,w,z\), a multiset of trace factors \(\mathop{\mathrm{tr}}A^k\), and a multiset of edges with their powers. For purposes of ordering keys, replace endpoint labels \(5,6,7,8\) by \(-1,-2,-3,-4\). Sort the endpoints of each edge, sort the edge list and trace list, and take the lexicographically larger edge list obtained by retaining or exchanging \(-3,-4\). A lone cubic tensor is thereby labeled \(-3\). These choices are only ordering conventions: exchange of two identical copies of \(X\) preserves the value of every graph.

Tensor multiplication takes the disjoint union of graphs, shifts the free slot labels of the second factor, and distinguishes any two cubic tensor copies. A contraction of two free slots joins their incident edges. If these are different edges, their powers add; if they are the same edge, it becomes a trace \(\mathop{\mathrm{tr}}A^k\). The removed free slots are deleted and the others renumbered in order. The identities underlying these operations are precisely \[\sum_i(A^h)_{ai}(A^k)_{ib}=(A^{h+k})_{ab},\qquad \sum_i(A^k)_{ii}=\mathop{\mathrm{tr}}A^k.\] Reduce the trace factors of powers \(0,1,2\) to \(5,0,1\), respectively. An edge of power zero connecting two slots of the same \(X\) makes the monomial zero. Thus every reduction is an equality on the allowed data. Other identities need not be imposed: retaining two different keys for functions that happen to agree cannot invalidate either the expansion or the later triangle inequality.

The derivative of a path and of a trace is \[\begin{align*} D_l(A^k)&=\sum_{h=0}^{k-1}A^hX_lA^{k-1-h}-kq_lA^k, & (X_l)_{ij}&=X_{lij},\tag{62}\\ D_l\mathop{\mathrm{tr}}A^k&=k\,\mathop{\mathrm{tr}}(A^{k-1}X_l)-kq_l\mathop{\mathrm{tr}}A^k. \tag{63}\end{align*}\] For \(k=0\) the sums and derivatives are zero. These formulas follow by differentiating each matrix factor using (39); cyclic invariance of trace makes all \(k\) insertion positions on a trace equal. In graph form, inserting \(X_l\) on an edge of power \(k\) splits it into two paths of lengths \(h,k-1-h\) ending at a new \(X\), whose last slot is the derivative index \(l\). The negative terms from all \(A\) factors combine into minus the total \(A\) degree times \(q_l\) times the original graph. Each \(p\) endpoint is in turn replaced by \(wA_{l\cdot}-gz\delta_{l\cdot}\), and a factor \(g^h\) gives \(hzg^{h-1}p_l\). If an edge has two \(p\) endpoints, these are two product-rule terms, including their multiplicity when they coincide.

These operations are also sound if traces were simplified first, since \[D_l\mathop{\mathrm{tr}}A=0,\qquad D_l\mathop{\mathrm{tr}}A^2=2X_{lij}A_{ij}-2q_l|A|^2=0.\] Only the original Table I monomials are differentiated, exactly once. No already differentiated graph, or graph involving \(X,w,z\), is differentiated again. Thus no unspecified higher derivative enters the calculation.

The linear contribution of each Table I row is obtained by substitution into (40), forming \(C=K-K^{(12)}\) for a \(K\) row. One check on the slot order and coefficients is the single row \(U=A\) with coefficient one. Its interior contribution is \[5z^2+|X|^2+(2a-2)|q|^2,\] and its boundary contribution is \(2\lambda-\mathop{\mathrm{tr}}A^3\). To expand a Gram block, sum over unordered pairs of rows \(s,t\): the coefficient is \(\sum_c c_{sc}c_{tc}/Q^2\), doubled when \(s\ne t\), and the graph polynomial is the contraction of row \(t\) with \(\Pi\) applied to row \(s\), multiplied by \(\rho\). This uses self-adjointness of \(\Pi\) and accounts for every term of (57) exactly once.

Bounds for scalar graphs

Put \(\theta=4/5\). We first establish the tensor estimates that will bound each scalar graph, retaining the dependence between \(p\) and \(g\).

For a nonzero scalar graph key \(t\) of degree two, let \[y=(|w|,|z|,|\ell|,|X|)^T\] and let \(i,j\in\{1,2,3,4\}\) be its two degree-one types, counted with multiplicity and ordered so \(i\le j\). Let \(h\) be its \(g\) exponent and \(b\) its number of \(p\) endpoints. Define \(L_t^2\) as a product of the following rational factors:

1. for each trace \(\mathop{\mathrm{tr}}A^k\), use \(\theta^{k-2}\), replacing this by \(9/20\) when \(k=3\);

2. for an edge of power \(k\) between distinct vertices, use \(\theta^k\);

3. for an edge between two slots of the same \(X\), use \(5/7\) when \(k=1\) and \(\theta^{k-1}\) when \(k\ge2\);

4. if \(h,b>0\), include the right side of (67); otherwise include \(1\).

Here separate occurrences of a vector endpoint count as distinct vertices; thus a \(p\)–\(p\) edge is governed by Item [cert:graph-rule-edge], not Item [cert:graph-rule-self]. Power-zero self-edges at \(X\) have already vanished in the collection rules.

We can now bound every term left after subtracting the nonnegative comparison forms. The graph estimates reduce this remainder to a quadratic form in four magnitudes. The next calculation shows that its error is smaller than the reserved positive diagonal terms.

The interior remainder and its coercive margin

Set \[ d=\left(\frac2{25},\frac2{25},\frac14,1\right),\qquad E=\Gamma+\sum_{i=1}^4d_i y_i^2-\mathcal I. \tag{69}\] The squared norms in Equation (69) are polynomial contractions. Expand and collect \(E=\sum_t e_t\mathcal M(t)\) by Section 6.3, discarding zero coefficients only after their signed contributions have been summed. All coefficients are rational. The denominator \(Q^2\) suffices: Gram terms have that denominator, and the denominators of \(a,d_i\), and all Table I coefficients divide it after multiplication by \(Q^2\).

For each retained key let \(n_t\) be the least nonnegative integer satisfying \[ n_t^2\ge10^{10}L_t^2, \qquad \widehat L_t=\frac{n_t}{10^5}. \tag{70}\] There is no floating-point square root in this prescription. For a rational \(s=10^{10}L_t^2\), compute \(n=\lfloor\sqrt{\lfloor s\rfloor}\rfloor\) by integer arithmetic and increase it by one exactly when \(n^2<s\). Form a symmetric matrix \(D_0\) by adding \[ \frac{|e_t|\widehat L_t}{1+\mathbf1_{\{i\ne j\}}} \tag{71}\] to its \(ij\) entry and also its \(ji\) entry if \(i\ne j\), starting with zero. The factor \(1/2\) in an off-diagonal entry is required because the associated term in \(y^TD_0y\) is \(2(D_0)_{ij}y_iy_j\). Lemma 24 and the triangle inequality therefore give \[ |E|\le y^TD_0y. \tag{72}\]

For clarity, the exact finite calculation needed here is the following. Decode all rows of Tables I and II; expand the nineteen Gram blocks by unordered row pairs and the finite projections (59)–(61); subtract each Table I contribution to (40); and add the four contractions with coefficients \(d_i\). Use the contraction, derivative, and key ordering rules above after each operation. Combine all signed coefficients at equal keys. For each nonzero key compute its rational \(L_t^2\), perform (70), and add (71). This gives all ten upper-triangular entries as follows: \[ \begin{array}{c|r|r} (i,j)&\text{nonzero keys}&2\cdot10^5Q^2(D_0)_{ij}\\\hline (1,1)&89&321268243322644\\ (1,2)&53&96616799870252\\ (1,3)&123&294707669526536\\ (1,4)&170&385792585960684\\ (2,2)&64&269583307398596\\ (2,3)&85&115717471031956\\ (2,4)&106&125155603107368\\ (3,3)&238&605509501361296\\ (3,4)&414&618424270356936\\ (4,4)&372&1004504025266456 \end{array} \tag{73}\] In particular there are \(1714\) nonzero keys. These finite sums use only integer and rational operations and comparisons; the complete literal input is in Appendix 9. The accompanying program verification/checks/reproduce.py implements the specified calculation and records the individual coefficients as well as these sums.

Entrywise, (73) implies \(D_0\le\overline D\), where \[ \overline D=\frac1{10^5} \begin{pmatrix} 2571&773&2358&3087\\ 773&2157&926&1002\\ 2358&926&4845&4948\\ 3087&1002&4948&8037 \end{pmatrix}. \tag{74}\] For example, this entrywise check consists of comparing each last-column integer in (73) with the corresponding integer in (74) multiplied by \(2Q^2\). Let \(\nu=(2484,1440,996,304)^T\). The exact diagonal slacks are \[ \left(d_i-\frac{(\overline D\nu)_i}{\nu_i}\right)_{i=1}^4 =\left(\frac{673}{18400},\frac{438907}{12000000}, \frac{2844869}{24900000},\frac{3479389}{7600000}\right). \tag{75}\] For \(y_i\ge0\), weighted Young inequalities give \[2y_iy_j\le\frac{\nu_j}{\nu_i}y_i^2+ \frac{\nu_i}{\nu_j}y_j^2, \qquad y^TD_0y\le y^T\overline D y \le\sum_i\frac{(\overline D\nu)_i}{\nu_i}y_i^2.\] Use \(\Gamma\ge0\) in (69) and then (72). It follows that \(\mathcal I\) is at least the sum of \(y_i^2\) times the four positive slacks in (75). In particular the coefficients of \(z^2\) and \(|\ell|^2\) are at least \(\epsilon_*\), proving (40). To obtain the stated positivity of \(f\), keep \(A,p,g\) fixed, set \(X=w=z=0\), and choose \(|\ell|=1\). Then \(q=0\), every \(D\) term is zero, and \(\mathcal I=f\).

Boundary reduction and the eleven coefficients

It remains to verify (41). Orthogonal equivariance allows us to take \(p=e_5\) and to diagonalize \(A\) on \(p^\perp\). Write \[ A=\operatorname{diag}(x_1,x_2,x_3,x_4,\lambda),\qquad \lambda=-\sum_{i=1}^4x_i, \qquad R(x):=\sum_{i=1}^4x_i^2+\left(\sum_{i=1}^4x_i\right)^2=1. \tag{76}\] No sign or distinctness condition is imposed on the \(x_i\); the normal eigenvalue condition is exactly \(\sum_i x_i>0\). In these coordinates \((S_0)_{55}=0\) and \((S_0)_{\alpha\alpha}=b_\alpha:=\lambda x_\alpha-x_\alpha^2\). The only nonzero entries of \(V\) are \(V_{555}=1\) and the three permutations of \(V_{5\alpha\alpha}=b_\alpha\). Since \(C_{55jk}=0\), a direct component formula for the boundary polynomial is therefore \[ \mathcal S=-\lambda f+U_{55}+M_5+ \sum_{\alpha=1}^4 b_\alpha (U_{\alpha\alpha}+C_{5\alpha5\alpha} +C_{5\alpha\alpha5}), \tag{77}\] where the fixed polynomials are evaluated at (76), \(p=e_5,g=0\). This formula does not assume additional slot symmetries of \(U\) or \(K\).

Equivalently, one may contract the boundary expression as a scalar graph and then substitute \[ p\cdot A^kp=\left(-\sum_i x_i\right)^k,\qquad \mathop{\mathrm{tr}}A^k=\sum_i x_i^k+\left(-\sum_i x_i\right)^k. \tag{78}\] For \(k=0\), the latter is \(5\) and the former is \(1\). The trace reductions \(\mathop{\mathrm{tr}}A=0\), \(\mathop{\mathrm{tr}}A^2=1\) remain valid on \(R=1\). Thus either (77) or (78) evaluates the same scalar on all the allowed data. Orthogonal invariance under permutations of \(p^\perp\) shows that \(\mathcal S\) is symmetric in \(x_1,\ldots,x_4\).

Define the boundary remainder on this constraint set by \[E_b=\Lambda+\frac14-\mathcal S.\] Expanding the two boundary Gram blocks and the Table I expression gives a polynomial representative with monomials of degree at most five. A finite way to check the degree assertion is to use (77): discard positive \(g\) powers, substitute the diagonal matrix and normal vector in each of the \(231\) rows, and add the displayed terms. All remaining monomials have degree at most five. Each boundary \(J_c\) has degrees one and two, so its squared terms have degree at most four, or five after the \(\texttt{H}\) weight.

The following convention gives a small exact list of its coefficients. For an ordinary monomial \(x^\alpha\) of degree \(j\le5\), replace it by \[ x^\alpha R^{\lfloor(5-j)/2\rfloor}. \tag{79}\] This preserves its value on \(R=1\) and gives degree four if \(j\) is even and degree five if \(j\) is odd. Perform the replacement before collecting exponents. Because \(R\) is symmetric, it commutes with permutation averaging. For a nondecreasing exponent vector \(\beta\) put \[m_\beta(x)=\operatorname{Av}(x^\beta).\] Expand (79) and sum the coefficients of all exponent vectors whose nondecreasing rearrangement is \(\beta\). Their sum is the coefficient of \(m_\beta\) after averaging: indeed \(\operatorname{Av}(x^\alpha)=m_\beta\) for every member \(\alpha\) of that orbit, including when exponents repeat. Thus sorting exponents and summing coefficients introduces no extra factor of \(24\). Since \(\mathcal S\) and the constant are symmetric and \(\Lambda\) was defined by averaging, the resulting averaged polynomial still equals \(E_b\) on \(R=1\).

Precisely, using the row-pair expansion of Section 6.3 for the two rank-\(-1\) blocks, subtracting the boundary contributions from Table I with denominator \(Q\), and adding the homogenized constant \(1/4\), one obtains \[ E_b(x)=\sum_\beta e^{(b)}_\beta m_\beta(x)\quad\text{on }R=1, \tag{80}\] with exactly the following eleven nonzero coefficients: \[ \begin{array}{c|r@{\qquad}c|r} \beta&Q^2 e^{(b)}_\beta&\beta&Q^2 e^{(b)}_\beta\\\hline (0,0,0,4)&-13479110 &(0,0,0,5)&17712712\\ (0,0,1,3)&-93201124 &(0,0,1,4)&151038620\\ (0,0,2,2)&-78543848 &(0,0,2,3)&244636976\\ (0,1,1,2)&-139986162&(0,1,1,3)&310091962\\ (0,1,2,2)&445177656 &(1,1,1,1)&-4861942\\ (1,1,1,2)&157995170&& \end{array} \tag{81}\] This coefficient computation consists solely of the explicitly stated substitutions, multiplication of ordinary four-variable polynomials, and coefficient addition. It is also included in the accompanying exact reproducer.

On \(R=1\) every \(|x_i|\le1\), hence \(|m_\beta(x)|\le1\). Adding the eleven absolute integer coefficients gives \[Q^2\sum_\beta|e^{(b)}_\beta|=1656725282.\] Since \(\sum_i x_i=-\lambda>0\), (58) is nonnegative. It follows from (80) that \[\mathcal S=\Lambda+\frac14-E_b \ge\frac14-\frac{1656725282}{62500000000} =\frac{6984137359}{31250000000}=\delta_*.\] This completes the proof of Proposition 15. The estimate also persists at \(\lambda=0\) by continuity; repeated tangent eigenvalues cause no singularity in any of the formulas.

The pointwise comparisons and field bounds of Section [sec:flux] are therefore established. Section [sec:cutoffs] now justifies their integration across Hessian zeros and near the suspension tips.

Weighted integrability on the original component

Smooth angular cutoffs and the integrated comparison

Let \(\vartheta\in C^\infty([0,\infty))\) satisfy \(0\leq\vartheta\leq1\), \(\vartheta=0\) on \([0,1]\), and \(\vartheta=1\) on \([2,\infty)\). Fix this profile once and for all, and set \[L_\vartheta=\|\vartheta'\|_\infty, \qquad \kappa=2L_\vartheta.\] For integers \(j\geq3\) define \[ \alpha_j(y)=\vartheta(jw_o(y)),\qquad \xi_j= \begin{cases} \alpha_j,&m_o=5,\\ \alpha_j(y)\beta_j(t),\quad \beta_j(t)=\vartheta(j\sin t),&m_o=4. \end{cases} \tag{85}\] Here \(\alpha_j\) is defined to be zero at \(\mathcal Z_o\), consistently with the displayed formula. It is smooth on \(\overline\Omega_o\): near any zero of \(w_o\) it is identically zero, and all its transitions occur where \(w_o\) is smooth and positive. In the suspension, \(\beta_j\) vanishes on whole neighborhoods of both tips.

Figure 1 displays the two exclusions made by the product cutoff in the suspended case. Its transition bands shrink toward the two tips and the original Hessian zero set. The weighted estimate above will control the errors produced in all these bands at once.

On \(\Omega_+=\{x\in\Omega:w(x)>0\}\), use the fields \[h=w^a\sqrt{f(A,p,g)},\qquad Z_l=w^{2a}(G_{lijk}X_{ijk}+wM_l+P_l)\] constructed in Section [sec:flux]. By Lemma 20, there is a fixed \(C_F\geq1\) such that \[ |h|\leq C_Fw^a,\qquad |\nabla h|\leq C_Fw^a(1+w+|X|),\qquad |Z|\leq C_Fw^{2a}(1+w+|X|). \tag{86}\] The constant depends only on \(a\), the dimension \(5\), the finitely many coefficients defining \(f,G,M,P\), and the positive lower bound \(f\geq\epsilon_*\). In particular \(C_F\) is independent of the cone, \(w\), \(X\), and \(j\).

For each fixed \(j\), define \(v_j=\xi_jh\) and \(F_j=\xi_j^2Z\) on \(\Omega_+\), and extend them by zero where \(w=0\). These are smooth on the regular closure of \(\Omega\) and are supported away from the tips when present. The same neighborhood argument as for \(\alpha_j\) proves smoothness of these extensions; it does not require \(h\) or \(Z\) themselves to extend smoothly across \(w=0\). All products below containing one of these cutoff factors or its derivative are likewise understood to be zero on \(w=0\). The mass \(w^{2a}\), whose exponent is positive, has its ordinary value zero there and involves no normalized tensor.

Every cancellation above is made before removing the angular cutoff. For fixed \(j\) and radial index \(k\), the tests \(\zeta_k(r)v_j\) from Corollary 14 have compact support on the regular Euclidean cone. First take the variation parameter \(\varepsilon\) to zero on this fixed support; then let \(k\to\infty\) in Equation (37) to obtain the constant \(4\). The smooth extension collar and the allowed variation size may depend on \(j\) and \(k\). We now take \(j\to\infty\) only in Equation (87), which contains the positive mass and three cutoff errors. The energy and boundary terms have already canceled for each fixed \(j\).

Vanishing of all three errors

On \(\Omega_{o,+}\), differentiation of \(\alpha_j\) and \(|\nabla w_o|\leq|B_o|\) give \[ |\nabla_o\alpha_j| \leq\kappa\frac{|B_o|}{w_o}. \tag{88}\] To see the constant, wherever \(\vartheta'(jw_o)\ne0\) one has \(1<jw_o<2\), so \(j\leq2/w_o\); outside this transition region the derivative is zero. The same argument gives \[ |\beta_j'(t)|\leq\frac{\kappa}{\sin t}. \tag{89}\] At every fixed point of \(\Omega_{o,+}\), \(\alpha_j=1\) and \(\nabla_o\alpha_j=0\) for all sufficiently large \(j\). At every fixed \(t\in(0,\pi)\) the corresponding statements hold for \(\beta_j\).

Write \(b_o=|B_o|/w_o\) on \(\Omega_{o,+}\). Let \(E_{1,j}\), \(E_{2,j}\), and \(E_{3,j}\) denote, in order, the three signed integrands on the right of (87).

First suppose \(m_o=5\), so \(\Omega=\Omega_o\), and set \(K_n=1+M_o\). Since \(X=B_o/w_o\), \[1+w+|X|\leq K_n(1+b_o),\qquad |\nabla\xi_j|\leq\kappa(1+b_o).\] Equations (86) and the preceding bounds give \[ |E_{1,j}|+|E_{2,j}|+|E_{3,j}| \leq C_F^2(\kappa^2+4K_n\kappa)D_o. \tag{90}\] Indeed the three coefficient bounds are respectively \(C_F^2\kappa^2\), \(2C_F^2K_n\kappa\), and \(2C_FK_n\kappa\), and \(C_F\geq1\). The dominator is integrable by Lemma 25. Each error is eventually zero at every point of \(\Omega_{o,+}\), and is defined to be zero on \(\mathcal Z_o\). Dominated convergence therefore makes the right side of (87) tend to zero.

Now suppose \(m_o=4\). Write \(s_t=\sin t\) and use the suspension coordinates \((t,y)\in(0,\pi)\times\Omega_o\). The exact formulas in Proposition 12 give \[ w(t,y)=\frac{w_o(y)}{s_t},\qquad |B(t,y)|^2 =\frac{|B_o(y)|^2+3\cos^2(t)w_o(y)^2}{s_t^4},\qquad dV=s_t^4\,dt\,dV_o. \tag{91}\] In particular \(\Omega_+=(0,\pi)\times\Omega_{o,+}\), and there \[|X|=\frac{\sqrt{b_o^2+3\cos^2t}}{s_t} \leq\frac{\sqrt3}{s_t}(1+b_o).\] The warped metric gives the exact cutoff identity \[|\nabla\xi_j|^2 =\alpha_j^2|\beta_j'|^2 +s_t^{-2}\beta_j^2|\nabla_o\alpha_j|^2,\] so (88)–(89) imply \[ |\nabla\xi_j|\leq\frac{\kappa}{s_t}(1+b_o),\qquad 1+w+|X|\leq\frac{K_s}{s_t}(1+b_o),\qquad K_s=1+M_o+\sqrt3. \tag{92}\] Multiplying the bounds for the three errors by the volume factor in (91) now gives \[ s_t^4\bigl(|E_{1,j}|+|E_{2,j}|+|E_{3,j}|\bigr) \leq C_F^2(\kappa^2+4K_s\kappa) s_t^{2-2a}D_o(y). \tag{93}\] This inequality is initially read on \(\Omega_{o,+}\), and both sides are assigned zero on \(\mathcal Z_o\). Its constant depends only on the fixed certificate, \(a\), \(\vartheta\), and \(M_o\), and is independent of \(j\), \(t\), and \(y\). It also shows explicitly why the original weighted \(|B_o|^2\) estimate suffices: by (83), \[s_t^{2-2a}D_o(y) \leq 2s_t^{21/25} \bigl(w_o^{29/25}+w_o^{-21/25}|B_o|^2\bigr) \quad\hbox{on }\Omega_{o,+}.\] The right side is integrable with respect to \(dt\,dV_o\), because \(21/25>-1\) and (82) holds. Every error is eventually zero at each fixed \((t,y)\in(0,\pi)\times\Omega_{o,+}\). Dominated convergence applied to (93) therefore makes the right side of (87) tend to zero also in the suspension.

The known lower-dimensional classification gives \(d_*\geq5\), and the dimension-seven globally minimizing example gives \(d_*\leq7\), as stated in Section 1. Theorem 27 excludes the only two remaining possibilities below \(7\). Hence \(d_*=7\), completing the proof of Theorem 1.

Decoding and block conventions

For nonnegative rank, read a monomial code in consecutive triples. Its first triple \(h\,b\,j\) contributes \(g^h(\mathop{\mathrm{tr}}A^3)^b w^{[j=1]}z^{[j=2]}\). Each subsequent triple \(u\,v\,k\) is an unoriented edge carrying \(A^k\), with \(A^0=\mathop{\mathrm{Id}}\). Endpoints \(0,\ldots,r-1\) denote the free slots in order, \(5\) denotes a copy of \(p\), and \(6\) a copy of \(\ell\); three occurrences of \(7\) are contracted with one \(X\), and three occurrences of \(8\) with a second copy. Multiply all factors and contract all nonfree slots. There are no additional multiplicity factors; see Section 6.1.

In Table I the first field specifies the polynomial, the second its monomial, and the third its coefficient numerator. The free-slot orders are \(U_{jk}\), \(K_{lijk}\), \(M_l\), and \(P_l\), while \(f\) has rank zero. The numbers of rows for \((f,U,M,P,K)\) are \((34,57,74,32,34)\), for a total of \(231\).

Each line of Table II beginning with @ starts a separate block and gives its rank, projection label, and multiplier mode. For each coefficient column, sum the monomials with the coefficients in that column divided by \(Q\) to obtain a tensor \(J_c\). For nonnegative rank the block contributes \(\rho\sum_c\langle J_c,\Pi J_c\rangle\) to \(\Gamma\), where the modes I and B mean \(\rho=1\) and \(\rho=\chi=1-|p|^2-g^2\), respectively. The labels none, anti, sym, and hook specify exactly the operators of Section 6.2.1; they do not instruct any further normalization of the coefficients. Identical headers in different positions still define distinct blocks.

For rank \(-1\), the four code digits instead give the four individual exponents of \(x_1,x_2,x_3,x_4\). The block contributes to \(\Lambda\) the average over all \(24\) permutations of \(\rho\sum_c J_c^2\). Here I means \(\rho=1\), while H means \(\rho=\sum_i x_i=-\lambda\). The averaging and boundary normalization \(R=\sum_i x_i^2+(\sum_i x_i)^2=1\) are those of Section 6.2. Table II contains \(21\) blocks: \(19\) interior blocks with \(529\) rows and two boundary blocks with \(14\) rows each, hence \(557\) coefficient rows in total. Each block retains its own number of coefficient columns.

Table I: the multiplier coefficients

The columns below read from top to bottom, first on the left and then on the right.

f 000 172575762
f 200 -265322011
f 400 301491831
f 600 -44526835
f 800 14565793
f 000550 -54772531
f 200550 219932290
f 400550 52506144
f 600550 9630205
f 000550550 137906751
f 200550550 106469818
f 000550550550 -131450161
f 200550550550 46298458
f 000550550550550 19137894
f 000551 119093719
f 200551 -79672895
f 400551 -9715716
f 000550551 -60378744
f 200550551 14546676
f 400550551 -57731025
f 000550550551 156426055
f 200550550551 -335208469
f 000550550550551 -114479609
f 000552 36551855
f 200552 -61391750
f 400552 -54261369
f 000550552 47470176
f 000551551 38322040
f 200550552 -123645111
f 200551551 21745017
f 000550550552 -27358875
f 000550551551 -10633583
f 000551552 -63061863
f 200551552 -7335033
U 000500510 209038273
U 200500510 -16767570
U 400500510 -283063959
U 600500510 -30248846
U 000550500510 -136083135
U 200550500510 -427782476
U 400550500510 -31623665
U 000550550500510 10893857
U 200550550500510 546331605
U 000550550550500510 427088076
U 000011 170760340
U 200011 -1206356
U 400011 121148296
U 600011 41094485
U 000550011 159740708
U 000500511 -12972332
U 200550011 -371328968
U 200500511 158604412
U 400550011 323446445
U 400500511 -150167309
U 600550011 -71599174
U 000550550011 -98427304
U 000550500511 -66406904
U 000551500510 161528440
U 200550550011 360663252
U 200550500511 -17938869
U 200551500510 84273648
U 400550550011 -202347674
U 400550500511 -24865978
U 400551500510 20187308
U 000550550550011 -95478557
U 000550550500511 243609372
U 000550551500510 -260094552
U 200550550550011 -175753712
U 200550550500511 -115703729
U 200550551500510 -18269297
U 000550550550550011 52505668
U 000550550550500511 -136286395
U 000550550551500510 124924734
U 000551011 -34002037
U 200551011 -87426923
U 400551011 -16110396
U 000550551011 -51017991
U 000551500511 182661478
U 000552500510 -212756173
U 200550551011 98857514
U 200551500511 -82275068
U 200552500510 -68411390
U 000550550551011 -365742
U 000550551500511 121156685
U 000550552500510 65143542
U 000551551500510 -118149741
U 000551551011 20474336
U 000552500511 13163650
U 200551551011 19565853
U 200552500511 -13163652
U 000550552500511 -13163652
M 000500 119808440
M 200500 -78615281
M 400500 58172228
M 600500 117063991
M 000550500 -8257673
M 200550500 -246670620
M 400550500 546437417
M 000550550500 -304458502
M 200550550500 147099465
M 000550550550500 -259741510
M 000501 13135177
M 200501 230900882
M 400501 503642015
M 600501 113276750
M 000550501 284895091
M 000551500 -41618173
M 200550501 895262372
M 200551500 -64307015
M 400550501 -501973998
M 400551500 203881659
M 000550550501 -251554675
M 000550551500 -68782868
M 200550550501 -641088126
M 200550551500 179691016
M 000550550550501 139784474
M 000550550551500 -40969083
M 000502 -24304787
M 200502 -622936552
M 400502 268923235
M 600502 168388434
M 000550502 -108635150
M 000551501 318683040
M 000552500 33990338
M 200550502 361921940
M 200551501 153090065
M 200552500 -747048669
M 400550502 558178192
M 400551501 -1665598002
M 400552500 1296406233
M 000550550502 411493367
M 000550551501 -121808407
M 000550552500 11773023
M 000551551500 67252403
M 200550550502 -446325581
M 200550551501 79505560
M 200550552500 125885807
M 200551551500 -19251826
M 000550550550502 -408356086
M 000550550551501 2958960216
M 000550550552500 -2765855048
M 000550551551500 -39050858
M 000503 31451284
M 010500 -111968845
M 200503 461057708
M 210500 221824980
M 400503 -412407021
M 410500 -101679590
M 000550503 121978129
M 000551502 -465236572
M 000552501 -85409256
M 000553500 501586157
M 010550500 189809541
M 200550503 -457984830
M 200551502 -131436746
M 200552501 1156439419
M 200553500 -438673045
M 210550500 -199478216
M 000550550503 -37100467
M 000550551502 376993983
M 000550552501 216559416
M 000550553500 -236527028
M 000551551501 118549720
M 000551552500 -289837183
M 010550550500 -92636597
P 100500 311718636
P 300500 -206555629
P 500500 -10345809
P 700500 69636002
P 100550500 -335187565
P 300550500 141480631
P 100550550500 -48726916
P 300550550500 -59898908
P 100550550550500 -130179927
P 700501 664922213
P 100551500 37613817
P 500550501 2068537473
P 100550551500 -80492409
P 300550550501 -3956993330
P 300550551500 2941040012
P 100550550550501 -4535791726
P 100550550551500 2719912082
P 100552500 59422497
P 300552500 -112216670
P 100550552500 160359392
P 100551551500 -210782200
P 100551502 -955698012
P 100552501 289629865
P 100553500 289477038
P 300551502 290419431
P 300552501 -244526547
P 300553500 -89919120
P 100550551502 372091130
P 100550552501 -196017256
P 100550553500 -130830861
P 100551551501 223993677
P 100551552500 -287868455
K 000500520131 -395492688
K 200500520131 557130979
K 400500520131 -208630461
K 000550500520131 265035499
K 000500511520530 177271543
K 200550500520131 -288587200
K 200500511520530 -18030270
K 000550550500520131 -29990138
K 000550500511520530 -153812921
K 000500511231 142329663
K 200500511231 -265174093
K 400500511231 209730769
K 000550500511231 -47161546
K 000551500520131 126782841
K 000500512520530 -310431451
K 200550500511231 134860957
K 200551500520131 -72013133
K 200500512520530 90786861
K 000550550500511231 -155714998
K 000550551500520131 4534988
K 000550500512520530 171186722
K 000551500511520530 -64357492
K 000500512231 380733573
K 200500512231 -425302994
K 400500512231 164988206
K 000550500512231 -325733710
K 000551500511231 -84676007
K 000552500520131 6581825
K 200550500512231 449829666
K 200551500511231 7194909
K 200552500520131 -6581826
K 000550550500512231 153898314
K 000550551500511231 46595618
K 000550552500520131 -6581826

Table II: the comparison coefficients

The block headers are part of the data.

@ 0 none I
001 12

@ 0 none I
002 184

@ 1 none I
000771700 -638806
000750750701 -983284
000750751700 1320922
000750750750501 805293
000750750701550 1754296
000600 503057
000600551 -1946630
000600550551 1923242
000651501 -2068563
000601551 -1021342
001502 -3593596
001550502 4994642

@ 1 none B
000602 1754266
000601551 -1522955
000602550 -2522248

@ 2 sym I
000750700710 1391786 1177679
200750700710 -635584 -1188129
000750750700510 -669474 -759226
200771700510 145856 439798
000771750500510 -365548 1663410
000771700550510 333085 -1276634
000750750750011 898927 557074
000750750700511 -311872 -891024
000750750701510 -41287 -1733462
000750751700510 -93961 1393977
000750700710551 113136 227458
000771750011 57792 -1227345
000771700511 -2146827 33344
000771701510 -214285 799004
200771750011 1112073 309389
200771700511 1556631 843960
200771701510 -294789 -594312
000771750550011 -13597 795429
000771750500511 -329731 1039673
000771751500510 -836011 1028480
000771700550511 216212 -603835
000771700551510 1434254 -827393
000771701550510 656139 -279125
000600510 -388425 -131346
000600550510 -477571 -239908
000650011 1059131 288654
000601510 252356 -522796
200650011 -1055033 202800
200601510 422375 425949
000650550011 -1586221 -893296
000650500511 393362 557037
000651500510 573436 -1067580
000600550511 -290759 380780
000600551510 -876849 -172942
000651011 16748 -630547
000601511 18986 878826
000602510 -32503 767092
200651011 -572999 -197116
200602510 -62643 182099
000651500511 -99714 99762
000652500510 1165974 -802821
000601551510 -1273792 -46716
000602550510 151518 380083
001500510 -818099 -26218
201500510 551865 2425759
201011 807336 716540
401011 -1809265 -594
001550011 1952714 -432372
001500511 -970041 4080275
201550011 -3494733 -1591876
201500511 -2942057 1178858
001550550011 -2197984 152320
001550500511 -1126597 -3610821
001551500510 1302795 2046516
201012 453912 -635196
001550012 802719 479394
001500512 1149385 1541894
001501511 -570358 863506
102550011 2753121 -35533
102551011 -7449352 0
102500512 712433 -1155568
102501511 -648523 1532248

@ 2 anti I
000750750700510 -971778 -593070 595799
000750750700511 1851218 390321 -1032828
000750751700510 -2221492 -384831 773149
000771700511 1671121 1696523 -468940
000771701510 1285532 -64095 -352045
200771700511 362947 -2582045 -2929678
200771701510 -181856 1690707 -1313034
000771750500511 977807 -1459534 1360532
000771700550511 -1653489 -1171267 -3601252
000771700551510 1692989 13676 305417
000600511 912880 512840 1024400
000600550511 1022451 2751390 2442625
000601511 2261978 -1322826 -210765
000602510 2192414 -515944 -180511
000651500511 2409290 -1068889 -1803516
000601550511 -1177522 2817530 -34700
000601551510 1957710 -3140417 -720539
000602550510 -261398 1630593 -956020
001500512 -130602 5301163 0
102500512 -5486646 0 0

@ 2 sym I
100750700710 386912 590109 2476735 909989
300750700710 -324620 -1836020 1070522 -709211
100750750700510 207036 867875 -1315712 -1562386
100750700710550 -2292260 -2320517 -877238 -137767
100771700510 679665 129424 -1701690 711883
300771700510 -691965 -2367998 -600979 1614358
100771700550510 71429 -500416 812821 -488846
100750750750011 425332 -329290 685878 -37805
100750750700511 1670160 798393 1099878 1675685
100750750701510 618399 661946 -1024107 -525490
100750751700510 82393 -436965 -872804 890466
100750700710551 -1425953 113042 -858760 -800516
100771750011 625914 918398 -1118557 -279579
100771700511 631990 1190528 -803355 693822
100600510 747288 856865 765295 19368
300600510 -196289 641398 -318973 -1435522
100600550510 -2242557 1145358 1750835 25513
300600511 -350390 -359430 -160354 -354520
300601510 -998743 -242982 -243886 303678
100650500511 1317416 -1625911 1342770 527410
100651500510 620553 -221636 -278822 -584012
100600550511 -563526 294802 -1407035 -675807
100601550510 -1165326 -290961 158430 23084
100651011 -1073379 74678 -692146 131418
100601511 610649 -693839 -174161 54662
100602510 1468661 -1066444 257457 -1866082
101011 891879 -816951 -355999 -242185
301011 -128043 -1656108 -3338499 0
101550011 262135 -1773969 -1081346 1502914
101012 247618 992865 441378 -984847
301012 273293 -835560 84484 79079
101550012 -968245 -1440718 -66260 1176036
101551011 -540581 4211626 0 0
101500512 571373 -2763887 -288603 -351329
101501511 -633764 1126754 22804 -757142
002500510 2095639 323489 -1897011 484510
002550500510 -2013271 -345687 1721490 -160235
002011 -22315 614400 440061 106088
202011 -789708 -1586444 435018 1192156
402011 176801 1776607 -935958 110464
002550011 -752649 2387292 158922 -11589
002500511 965395 122624 -1704548 -1591006
202550011 713344 -2161421 1573837 -1092257
202500511 4263736 0 0 0
002550550011 651775 -2930070 -692641 103706
002550500511 -241365 -1063259 672383 277457
002551500510 -1497924 882585 1136032 805365
002551011 14258 -733558 -562699 359850
002500512 -1418407 941987 1726713 -1244268
002501511 598151 -558439 -599397 139693

@ 2 anti I
300771700510 -4701036 0
100750750701510 -1924807 -3971273

@ 2 sym B
000750700710 1123954 88221 -1701307 30701 738750
200750700710 270992 1585448 -437804 -115089 -193670
000750750700510 1172241 -871881 1057083 321028 -649097
000750700710550 -524706 1366647 827758 -1278231 500667
000771700510 864484 287922 -68675 62361 -539022
200771700510 775795 2191982 950393 914569 -166597
000771700550510 277543 -125661 661263 -1877406 74958
000750750750011 -334486 169170 -871041 -791399 -94389
000750750700511 -975320 58917 -494669 -25254 -1254928
000750750701510 -1503410 -485996 1281582 -196803 63872
000750751700510 -184521 435306 51628 1182804 106124
000750700710551 852497 -216529 665736 60101 1484575
000771750011 169826 196263 532547 536554 -474267
000771700511 -1010968 -842722 1104099 -283384 -997589
000771701510 652964 -776198 -18629 -4645 767925
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302011 -893937 -2987523 1695406 937674 871113
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102500511 -5673921 0 0 0 0

@ 2 anti B
000771700510 -470106 -1867823
200771700510 -1539213 -3753948
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000600550511 -1157549 1248692
001500511 7658597 984973
001500512 9501737 0

@ 2 sym B
100750700710 -430860 3401943
100771700510 2216725 -501573
100771750011 -324461 488844
100771700511 -460201 -3052123
100602510 -29622 190280
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101011 1458366 -2486198
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002500510 2853790 -389130
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202011 2486645 -2061003
002500512 226083 1738130

@ 3 sym I
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@ 3 hook I
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102501121 764808 -802563 -1980310 422533 -196982

@ 3 sym I
100700710720 763442 201492 207522
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202501121 -793881 1173902 348366
002550501121 -1007658 503344 -581407

@ 3 hook I
100750700710520 825602 -647783 -380956 627925
100750750700121 -609729 889144 939275 -551336
100771700121 553682 -757497 47709 76332
300771700121 52101 1440744 -29071 462803
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002551500121 -2047579 -591242 193597 116743
002500510522 -323862 385594 269999 -545727
002500511521 1649181 -126174 -960119 202680

@ 3 sym B
000700710720 846861 -24321
200700710720 -587033 -427093
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001500510521 -1212241 -484563
001500122 1303787 -240027
102500121 470182 -1041354
102501121 -2194212 0

@ 3 hook B
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102500121 -3095117 0 0 0
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@ 3 sym B
100700710720 489692 598090
300700710720 -395941 -186631
100750700710520 156230 -1702877
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002550500121 -1490277 -1501412
002500510521 859145 -683392
002501121 -259132 1352466

@ 3 hook B
100750700710520 -130387 1549320
100771700510520 -325730 2482101
100750750700121 509197 -841014
100771700121 -1556447 1491379
100600121 90411 -1200728
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202500121 2603006 -45060
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002500510521 -221322 -2086794
002501121 -1023409 -1933433

@ -1 none I
1000 -7510150 892635 2182767 23202 -4016
0100 -5873297 -966404 1065127 891160 -4153
0010 -7594462 3164488 32982 23257 -4265
0001 -6664095 -676125 -2074840 -90371 -141368
2000 10527925 -1288669 -3350633 0 0
1100 13562510 -782596 -3072444 -1185008 -525859
1010 14954300 -3513649 -2045232 175643 711822
1001 16001742 0 0 0 0
0200 7976754 1609001 -1608780 -1353027 0
0110 15395603 -1696042 -847566 -840285 -117247
0101 12758609 1366427 714605 -471973 827299
0020 10659228 -4829347 0 0 0
0011 14311972 -2973496 1455043 -343953 -411843
0002 9209493 1156789 3285152 176907 213479

@ -1 none H
1000 2926888 -771909 -672812 -28439 -32476
0100 2472139 -1224863 -681989 -18744 51971
0010 2504046 -1123530 -477116 133871 -30159
0001 2879832 -887584 -486032 142684 52873
2000 -3960308 1436663 856832 -73562 -8956
1100 -5950295 2321626 989491 -2064346 0
1010 -6842585 3562264 2077026 553336 -610276
1001 -4035267 4405894 0 0 0
0200 -3909355 1486027 856034 -75355 -19599
0110 -9334809 0 0 0 0
0101 -7349996 2169698 1941569 669597 456042
0020 -3913415 1476575 835966 -92637 -12106
0011 -6055392 2242925 3499526 0 0
0002 -3956252 1450146 834863 -93150 -20809

Reproduction of the finite arithmetic

In the complete source directory accompanying this article, run make -C build check; dependencies and the preserved calculation records are in verification/checks/README.md. The entry point enables assertions, fixes \(a=29/50\), and checks source pins, literal table transcription, and fresh exact outputs from two interior expansions and the boundary, projection, and graph calculations. The interior implementations agree on every key, signed coefficient, magnitude assignment, rounded bound, and aggregate. The retained provenance records describe the second implementation as separately authored from the mathematical specification and frozen before comparison with the first engine’s output. That specification already contained the claimed aggregate totals. These checks verify finite arithmetic; correspondence between formulas and code, and the analytic argument, require mathematical review.