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1 \chapter{The symmetric Laplacian in coordinates}\label{app:operator}
2 
3 The universal null construction depends on an exact differential operator. This appendix derives that operator from the Euclidean metric, rather than treating its power-sum formulas as a black box. It also explains the Newton reductions used by the implementation and gives two low-degree harmonic checks. All polynomial identities are on the hyperplane $H=\{v\in\R^5:\sum_i v_i=0\}$, with its induced Euclidean metric. The modularity and valence contracts remain the ones stated in Chapters~\ref{ch:theta} and~\ref{ch:nulls}.
4 
5 Write $s_m=\sum_{i=0}^4v_i^m$, with $s_0=5$ and $s_1=0$, and let $D=\prod_{i<j}(v_i-v_j)$. We distinguish the ordinary Laplacian $\Delta_H$ from the conjugated operator $\mathcal L$ defined by $\Delta_H(DQ)=D\mathcal LQ$. The factor $D$ changes radial coefficients substantially. Confusing the two operators would invalidate the harmonicity test.
6 
7 \section{Two descriptions of the same metric}
8 The orthogonal projection from $\R^5$ to $H$ is
9 \begin{equation}\label{appop:projection}
10 P=I_5-\frac15\mathbf1\mathbf1^T,
11 \end{equation}
12 where $\mathbf1=(1,1,1,1,1)^T$. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies $P^2=P$, and has image $H$. For an ambient polynomial $F$, its gradient along $H$ is $P\nabla F$. Its Laplacian along $H$ is
13 \begin{equation}\label{appop:ambientlap}
14 \Delta_H(F|_H)
15 =\left(\sum_{i=0}^4\partial_i^2F
16 -\frac15\left(\sum_{i=0}^4\partial_i\right)^2F\right)\Big|_H.
17 \end{equation}
18 Only tangential derivatives enter. Thus this expression is independent of the polynomial extension away from $H$.
19 
20 For direct symbolic computation, use four coordinates $x_0,\ldots,x_3$ and substitute
21 \[
22 v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3).
23 \]
24 The coordinate tangent vectors are $b_a=e_a-e_4$. Their Gram matrix and its inverse are
25 \begin{equation}\label{appop:chartmetric}
26 G=I_4+\mathbf1\mathbf1^T,
27 \qquad G^{-1}=I_4-\frac15\mathbf1\mathbf1^T,
28 \end{equation}
29 now with a four-entry vector $\mathbf1$. Matrix multiplication verifies the inverse. The factor $1/5$ remains because the missing fifth coordinate is determined by the four chosen coordinates.
30 
31 The chart Laplacian is consequently
32 \begin{equation}\label{appop:chartlap}
33 \Delta_H=\sum_{a,b=0}^3
34 \left(\delta_{ab}-\frac15\right)\frac{\partial^2}{\partial x_a\partial x_b}.
35 \end{equation}
36 The coordinates $x_a$ are not orthonormal. Using only the four diagonal second derivatives would use a different metric. As a normalization check, $s_2=x^TGx$, so its Hessian is $2G$ and
37 \[
38 \Delta_Hs_2=\operatorname{tr}(G^{-1}2G)=8.
39 \]
40 The rank of the Euclidean space is four, giving twice that rank. Later we obtain $\mathcal Ls_2=48$, an intentionally different number.
41 
42 \section{Conjugating by the Vandermonde}
43 The Vandermonde is alternating under coordinate permutations. Since $\Delta_H$ commutes with those permutations, $\Delta_HD$ is alternating of degree eight. Every alternating polynomial vanishes on each hyperplane $v_i=v_j$. In the polynomial ring of $H$, the ten corresponding linear factors are distinct and divide it. Their product has degree ten. Therefore $\Delta_HD=0$.
44 
45 On the open set where $D\ne0$, the product rule gives
46 \begin{equation}\label{appop:conjugation}
47 \frac{\Delta_H(DQ)}{D}
48 =\Delta_HQ+2\nabla_H\log D\cdot\nabla_HQ.
49 \end{equation}
50 Here $\nabla\log D$ denotes the rational logarithmic derivative $\nabla D/D$. It does not require choosing a complex logarithm or assuming that $D$ is positive. For symmetric $Q$, the apparent poles cancel and the right side extends to a polynomial on all of $H$.
51 
52 There is a second explanation of the extension. The polynomial $\Delta_H(DQ)$ is alternating, so $D$ divides it. Its quotient is symmetric. Thus the operator $\mathcal L$ is well defined on symmetric polynomials even at coordinate collisions. The packet shell itself has no collisions because its residues are distinct. The polynomial extension is nevertheless needed when constructing and comparing symbolic identities.
53 
54 Translation invariance of $D$ gives
55 \[
56 \sum_i\frac{\partial_iD}{D}=0,
57 \qquad
58 \frac{\partial_iD}{D}=\sum_{j\ne i}\frac1{v_i-v_j}.
59 \]
60 Its ambient logarithmic gradient is already tangential. This removes a potential extra projection term in the derivation below.
61 
62 \section{Deriving the power-sum operator}
63 For $m\ge2$, differentiation and projection give
64 \begin{align}
65 (\nabla_Hs_m)_i
66  &=m\left(v_i^{m-1}-\frac15s_{m-1}\right),\label{appop:gradsm}\\
67 \Delta_Hs_m
68  &=\frac45m(m-1)s_{m-2}.\label{appop:lapsm}
69 \end{align}
70 In the logarithmic term, the coordinate-mean subtraction vanishes because the logarithmic gradient has zero sum. Pair the ordered indices $i,j$ and $j,i$ to obtain
71 \[
72 2\nabla_H\log D\cdot\nabla_Hs_m
73 =2m\sum_{i<j}\frac{v_i^{m-1}-v_j^{m-1}}{v_i-v_j}.
74 \]
75 The divided difference is the finite polynomial
76 \[
77 \frac{v_i^{m-1}-v_j^{m-1}}{v_i-v_j}
78 =\sum_{a=0}^{m-2}v_i^av_j^{m-2-a}.
79 \]
80 The sum over $a$ is symmetric in the two coordinates. Hence twice the sum over $i<j$ equals the sum over $i\ne j$. For each $a$, that ordered sum is $s_as_{m-2-a}-s_{m-2}$. There are $m-1$ values of $a$. We obtain
81 \[
82 2\nabla_H\log D\cdot\nabla_Hs_m
83 =m\sum_{a=0}^{m-2}s_as_{m-2-a}
84 -m(m-1)s_{m-2}.
85 \]
86 Adding \eqref{appop:lapsm} proves
87 \begin{equation}\label{appop:lsm}
88 \boxed{\mathcal Ls_m
89 =m\sum_{a=0}^{m-2}s_as_{m-2-a}
90 -\frac{m(m-1)}5s_{m-2}.}
91 \end{equation}
92 This formula is asserted for $m\ge2$. The boundary values are separately $\mathcal Ls_0=0$ and $\mathcal Ls_1=0$, since $s_0$ is constant and $s_1$ vanishes on $H$. No negative-index power sum is introduced.
93 
94 The first values are
95 \begin{equation}\label{appop:firstoperator}
96 \mathcal Ls_2=48,\qquad
97 \mathcal Ls_3=0,\qquad
98 \mathcal Ls_4=\frac{188}{5}s_2,\qquad
99 \mathcal Ls_5=46s_3.
100 \end{equation}
101 For example, at $m=2$ the first term is $2s_0^2=50$, while the subtraction is $2s_0/5=2$. The additional $40$ compared with $\Delta_Hs_2=8$ comes from differentiating the degree-ten factor $D$.
102 
103 \section{The chain-rule metric and polynomial assembly}
104 The same projected gradients give
105 \begin{equation}\label{appop:chainmetric}
106 \Gamma_{mn}:=\nabla_Hs_m\cdot\nabla_Hs_n
107 =mn\left(s_{m+n-2}-\frac15s_{m-1}s_{n-1}\right).
108 \end{equation}
109 Expanding the product in \eqref{appop:gradsm} gives the first power sum once and the coordinate-mean product with coefficient $-1/5$. This identity includes $m,n\ge1$, with the evident zero tangential gradient for $s_1$.
110 
111 If $Q=F(s_2,s_3,s_4,s_5)$, the ordinary second-order chain rule and \eqref{appop:conjugation} yield
112 \begin{equation}\label{appop:chainrule}
113 \mathcal LQ
114 =\sum_{m=2}^5\frac{\partial F}{\partial s_m}\mathcal Ls_m
115 +\sum_{m,n=2}^5
116 \frac{\partial^2F}{\partial s_m\partial s_n}\Gamma_{mn}.
117 \end{equation}
118 The second sum runs over ordered pairs. Thus a mixed derivative contributes twice when $m\ne n$. If one instead sums over $m\le n$, the off-diagonal terms require a factor two. These are two equivalent conventions, but their factors must not be mixed.
119 
120 Useful instances are
121 \begin{align}
122 \mathcal L(s_2^2)&=104s_2,\label{appop:lradialtwo}\\
123 \mathcal L(s_2^3)&=168s_2^2,\label{appop:lradialthree}\\
124 \mathcal L(s_2s_4)&=64s_4+\frac{188}{5}s_2^2,\label{appop:ls2s4}\\
125 \mathcal L(s_3^2)&=18s_4-\frac{18}{5}s_2^2.\label{appop:ls3square}
126 \end{align}
127 For the first identity, $2s_2\mathcal Ls_2+2\Gamma_{22}=96s_2+8s_2$. For the mixed product, $s_4\mathcal Ls_2+s_2\mathcal Ls_4+2\Gamma_{24}$ gives the displayed coefficients. Each calculation is a check of the ordered-pair convention.
128 
129 The metric formula may produce $s_6,s_7,s_8$. They are not new independent variables. The invariant ring on $H$ is generated by $s_2,s_3,s_4,s_5$, and Newton's identities reduce higher indices exactly.
130 
131 \section{Newton reductions with zero first power sum}
132 Let $e_j$ be the elementary symmetric polynomials in the five coordinates. Their characteristic polynomial is
133 \[
134 \prod_i(T-v_i)=T^5+e_2T^3-e_3T^2+e_4T-e_5,
135 \]
136 because $e_1=s_1=0$. The first Newton identities give
137 \begin{align}
138 e_2&=-\frac12s_2,& e_3&=\frac13s_3,\label{appop:e23}\\
139 e_4&=\frac18s_2^2-\frac14s_4,&
140 e_5&=\frac15s_5-\frac16s_2s_3.\label{appop:e45}
141 \end{align}
142 For instance, the degree-five identity is $s_5+e_2s_3-e_3s_2-5e_5=0$. The two middle terms combine to $-5s_2s_3/6$, fixing the sign in $e_5$.
143 
144 Every coordinate satisfies its characteristic polynomial. Multiplying that equality by $v_i^{k-5}$ and summing, for $k\ge6$, gives
145 \begin{equation}\label{appop:newtonrec}
146 s_k=-e_2s_{k-2}+e_3s_{k-3}-e_4s_{k-4}+e_5s_{k-5}.
147 \end{equation}
148 This recurrence uses only lower indices. Substitution at $k=6,7$ yields
149 \begin{align}
150 s_6&=\frac34s_2s_4+\frac13s_3^2-\frac18s_2^3,\label{appop:s6}\\
151 s_7&=\frac7{10}s_2s_5+\frac7{12}s_3s_4
152 -\frac7{24}s_2^2s_3.\label{appop:s7}
153 \end{align}
154 The same rule gives
155 \begin{equation}\label{appop:s8}
156 s_8=\frac14s_4^2+\frac14s_2^2s_4
157 -\frac1{16}s_2^4+\frac8{15}s_3s_5.
158 \end{equation}
159 Each term has the required invariant degree. This supplies a useful local check before substitution into a larger operator expression.
160 
161 At the first packet $v=(0,1,2,-2,-1)$, the first power sums are $s_2=10$, $s_3=0$, $s_4=34$, $s_5=0$. Formula \eqref{appop:s6} gives $255-125=130$, matching the direct sum of sixth powers. Formula \eqref{appop:s7} gives zero, matching cancellation of the opposite coordinates. Formula \eqref{appop:s8} gives $514$. These examples fix normalizations. The polynomial recurrences, rather than these examples, establish the identities on every vector in $H$.
162 
163 \section{Radial factors and harmonic projection}
164 Let $h_\ell$ be homogeneous harmonic of degree $\ell$ on the rank-four space $H$. Euler's identity is $v\cdot\nabla_Hh_\ell=\ell h_\ell$. The product rule for the radial factor $s_2^j$ gives
165 \begin{equation}\label{appop:radial}
166 \Delta_H(s_2^jh_\ell)
167 =4j(\ell+j+1)s_2^{j-1}h_\ell,\qquad j\ge1.
168 \end{equation}
169 At $j=0$ the left side is zero. At $j=1,\ell=0$ the coefficient is $8$, agreeing with $\Delta_Hs_2$. At $j=1,\ell=10$ it is $48$, agreeing with $\Delta_H(s_2D)=48D$.
170 
171 For a homogeneous polynomial $P$ of degree $M$, a concrete top-degree harmonic projection in dimension four is
172 \begin{equation}\label{appop:hprojection}
173 \mathcal H_M(P)=\sum_{j=0}^{\lfloor M/2\rfloor}
174 \frac{(-1)^j(M-j)!}{4^j j!M!}\,s_2^j\Delta_H^jP.
175 \end{equation}
176 For $M=0$ or $1$ the sum contains only $j=0$. In general, if $a_j$ denotes its coefficient, the product rule gives the cancellation condition
177 \[
178 a_j+4(j+1)(M-j)a_{j+1}=0.
179 \]
180 The factorial coefficients satisfy this condition. The last possible iterated Laplacian has no further nonzero derivative, so the resulting polynomial is harmonic. Its difference from $P$ is divisible by $s_2$.
181 
182 Repeating the construction on that lower-degree quotient produces the harmonic decomposition
183 \begin{equation}\label{appop:decomposition}
184 P=\sum_{j=0}^{\lfloor M/2\rfloor}s_2^jh_{M-2j}.
185 \end{equation}
186 It is unique. If such a sum were zero, apply the Laplacian as many times as the largest occupied radial exponent. All smaller-exponent terms vanish. Formula \eqref{appop:radial} leaves a nonzero scalar times the last harmonic polynomial, so that polynomial must be zero. Descending removes every term.
187 
188 When $P=DQ$ is alternating, each projected harmonic component is alternating because the Laplacian and multiplication by $s_2$ commute with permutations. Each nonzero component is divisible by $D$ and has degree at least ten. Thus for a correction of homogeneous degree $d$, only $j\le d/2$ can occur. This is the degree bound used in the quasimodular argument for nonhomogeneous content conservation.
189 
190 Harmonic projection is not preservation of shell values. The elementary counterexample is $P=s_2D$, of degree $12$. Since $\Delta_HP=48D$ and the next Laplacian is zero,
191 \[
192 \mathcal H_{12}(s_2D)=s_2D-\frac1{48}s_2(48D)=0.
193 \]
194 Yet on $\Vn$ the original polynomial equals $10nD$, which is nonzero at every packet. The operation removes radial information. It cannot silently turn a shell interpolant into a universally null polynomial with the same values.
195 
196 \section{Two explicit harmonic moment checks}
197 The degree-four correction
198 \[
199 K_4=130s_4-47s_2^2
200 \]
201 has
202 \[
203 \mathcal LK_4
204 =130\frac{188}{5}s_2-47(104s_2)=0.
205 \]
206 Thus $DK_4$ is harmonic. At the first packet, $W=1$ and $K_4=130\cdot34-47\cdot100=-280$. Its theta form has weight $16$, whose cusp space is spanned by $\Delta E_4$ with first coefficient one. Consequently the theta form is $-280\Delta E_4$, giving the moment identity already stated in Chapter~\ref{ch:nulls}.
207 
208 Likewise, set
209 \[
210 K_6=90s_2s_4-27s_2^3-320s_3^2.
211 \]
212 Using \eqref{appop:lradialthree}--\eqref{appop:ls3square}, its operator image is
213 \begin{align*}
214 \mathcal LK_6={}&(5760-5760)s_4\\
215 &+(3384-4536+1152)s_2^2=0.
216 \end{align*}
217 At the first packet its value is $90\cdot10\cdot34-27\cdot1000=3600$. The weight-$18$ cusp space is spanned by $\Delta E_6$, again normalized with first coefficient one. Its theta form is therefore $3600\Delta E_6$.
218 
219 These checks have different roles from the null certificates. Both polynomials are harmonic, but neither is universally null: their first coefficients are nonzero. Harmonicity supplies a modular-form space. The initial coefficients then identify the form or, when enough initial coefficients vanish, force it to be zero by valence. Skipping that second step would misidentify a nonzero moment as a permitted null correction.
220 
221 The retained programs organize these same operations using sparse rational polynomials. A direct four-coordinate implementation can instead substitute $v_4=-\sum_{a=0}^3x_a$ into $DQ$ and apply \eqref{appop:chartlap}. Agreement between these organizations checks the symbolic construction through different algebraic representations. The universal justification remains the metric calculation, the exact polynomial identities and the stated harmonic theta and valence contracts. None of those statements alone excludes the unresolved zero prime parameter of $\Delta$.

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