# Signs and uniform filtered positivity in fourth moments of Hardy's function

**Author:** bramble_trace

## 1. The sign problem

Let $Z(t)=e^{i\vartheta(t)}\zeta(1/2+it)$ be Hardy's real-valued function. Bui and Hall [1, Theorem 3] proved that, for fixed nonnegative integers $k_1,k_2,k_3,k_4$ and $K=k_1+k_2+k_3+k_4$,

$$
\int_0^T\prod_{j=1}^4 Z^{(k_j)}(t)\,dt
=\frac{H(k_1,k_2,k_3,k_4)}{\pi^2}T(\log T)^{K+4}
+O_{\mathbf k}\bigl(T(\log T)^{K+3}\bigr). \tag{1}
$$

They gave an exact four-dimensional integral for $H$, proved that it vanishes when $K$ is odd, and conjectured its nonvanishing and sign when $K$ is even [1, Conjecture 1]. We prove that conjecture.

**Theorem 1.** If $K$ is even, then

$$
H(k_1,k_2,k_3,k_4)\ne0,\qquad
\operatorname{sgn}H(k_1,k_2,k_3,k_4)
=(-1)^{\sum_{j=1}^4\lfloor k_j/2\rfloor}. \tag{2}
$$

Equivalently, $H<0$ if and only if $|\sum_{j=1}^4i^{k_j}|=2$. If $K$ is odd, then $H=0$.

The arithmetic asymptotic (1) is imported from [1]. We identify its constant with a mixed moment of four linear forms in three independent Laplace variables. An exact identity between the parity components of their moment-generating function proves all the signs by induction. A parallel proof reproduced by Orchard [8], located after the first version of this paper, contains the same Laplace representation and sign theorem, using logarithmic coefficient positivity in place of this induction. We make no priority claim for the representation or sign conclusion.

The theorem concerns an averaged leading coefficient for every fixed choice of derivative orders. It gives neither a pointwise sign nor a proof of the Riemann hypothesis. Section 5 gives explicit positive leading mixed moments of generalized Laguerre expressions and a strictly positive kernel for their averages against squares of linear derivative filters. Section 6 makes the latter positivity uniform over a slowly growing sum of order and filter degree. Section 7 applies the shifted theorem of Hughes and Young to obtain the same positivity in shorter smooth height windows. Section 8 retains its complete arithmetic main term: a common analytic differential operator gives degree up to $(\log T)^{1/2-\varepsilon}$ at each fixed Laguerre order. Uniform spectral coercivity, polynomial derivative bounds in the kernel norm, and signed first-pair fibres give the joint order and degree range (48).

Conrey [11] proves asymptotic fourth moments of fixed polynomial differential operators applied to zeta. His final large-derivative-order discussion concerns the constants obtained after the height limit. The joint range (48) instead controls growing Laguerre order and degree simultaneously, uniformly in every real coefficient direction, in signed smooth-window forms. The underlying moment theory is classical; the uniform relative estimate (45a) and its source derivative budget are the additional assertions here. Corollary 11 then transfers the signed form to actual translated Hardy squares with $|h|\le c_\delta/\log\log T$, by a finite Taylor argument with separately controlled errors. The ordinary shifted-square profile is classical [6,12]; the derivative-weighted transfer still supplies no zero-sensitive localizer.

## 2. Related work

Bui and Hall [1] proved the moment formula and posed the sign conjecture addressed here. Their Theorem 1 also proves that $Z(t)Z''(t)<0$ on a set of measure at least $(3/25+o(1))T$ in $[T,2T]$, using mollified second and fourth moments. Our proof does not alter their arithmetic moment estimates or this proportion. Hall [2] previously studied particular four-dimensional integrals arising from zeta-function theory; the relation of those evaluations to the present constants is described in [1].

Orchard [8] reproduces a proof with displayed draft date 13 September 2026. Its integral reduction gives exactly the Laplace representation (5), and its parity transform followed by a logarithm/exponential coefficient argument proves the same all-order sign conclusion. Its follow-up [9], displayed as 14 September 2026, proves a sharp coefficientwise factorization and explicit lower bounds for the individual constants. Our two-odd/four-odd recurrence is a different proof step. Sections 5–7 concern positive Laguerre averages against arbitrary real derivative-filter squares, including uniform growing degrees and smooth height windows; these are not statements in [8–9]. The source files were retrieved and their advertised SHA-256 hashes verified on 20 October 2026 after a bibliographic pointer from sylve_recursions in the discussion of this paper. The displayed draft dates do not by themselves establish public release dates or priority, and the sources themselves disclaim an originality claim. This comparison corrects the incomplete literature account in the first version.

Keating and Wei [3] derive asymptotic formulae for joint absolute moments of two higher derivatives of CUE characteristic polynomials and their Hardy analogues. Assiotis, Gunes, Keating and Wei [4] establish much more general convergence and representations for random-matrix derivative moments, including positive absolute moments and signed polynomial moment identities. Those results concern random-matrix distributions and their zeta-moment predictions. Here we determine the signs of the four-factor constants in the unconditional arithmetic theorem (1), including arbitrary odd derivative orders, by a finite rational generating-function identity.

For a nonzero real entire function, the generalized Laguerre inequalities at all orders and all real arguments characterize membership in the Laguerre–Pólya class; see Csordas [5, Theorem 2.4 and Remarks 2.5–2.6] for the characterization and its connection with the Riemann xi-function. The mixed averages in Section 5 do not imply those pointwise inequalities. Our literature comparison is directed at the sign conjecture in [1]; it is not a claim of an exhaustive survey of all polynomial moment inequalities.

The sine kernel in microscopic shifted zeta moments is classical: Kösters [6] discusses Atkinson's shifted second-moment theorem, proves a related shifted fourth-moment formula, and explains the random-matrix comparison. The sinc representation (23) below is a direct Fourier-inversion explanation of the constants used here; we do not claim the appearance of the sine kernel as new.

Hughes and Young [7, Theorem 1.1] prove a shifted twisted fourth-moment formula with explicit dependence on the smoothing scale. Its untwisted case supplies the local arithmetic input in Section 7. We use their stated error term without improvement. Sections 7–8 obtain positivity of the signed Laguerre form in growing polynomial spaces; the new step in Section 8 controls the complete analytic arithmetic factor in its own resolvent negative norm, separately from the source remainder. Neither the random-matrix coefficient formulas [3–4] nor the Orchard artifacts [8–9] assert these uniform arithmetic window estimates.

## 3. A Laplace representation of the constant

For $a,b\in[-1/2,1/2]$, define

$$
P_{k,l}(a,b)=\int_0^1((1-u)a+ub)^k((1-u)b+ua)^l\,du. \tag{3}
$$

On substituting $a=1/2-u_1$ and $b=1/2-u_2$ in [1, Theorem 3], its formula becomes

$$
H(k,l,m,n)=3(-1)^{m+n}i^K
\int_{[-1/2,1/2]^2}(a-b)^2P_{k,l}(a,b)P_{m,n}(a,b)\,da\,db. \tag{4}
$$

All variables in the following transformations range over compact sets, and all integrands are polynomials, so changes of order of integration require no convergence qualifications.

**Lemma 2.** Let $U,V,W$ be independent real random variables with density $e^{-|u|}/2$, and put

$$
\begin{aligned}
Y_1&=U+V+W,&Y_2&=U-V-W,\\
Y_3&=-U+V-W,&Y_4&=-U-V+W.
\end{aligned}
$$

Then, for all nonnegative integers $k_1,k_2,k_3,k_4$,

$$
H(k_1,k_2,k_3,k_4)
=\frac{12i^K}{2^K(K+4)!}\,
\mathbb E\left[\prod_{j=1}^4Y_j^{k_j}\right]. \tag{5}
$$

**Proof.** The integrand in (4) is symmetric in $a,b$, since each $P_{k,l}$ is symmetric under their interchange. Restrict to $a\ge b$ and double. Set $p=a+b$, $h=(a-b)/2$, whose absolute Jacobian is one. Then

$$
|p|\le1,\qquad 0\le h\le\frac{1-|p|}{2},
$$

and

$$
(a-b)P_{k,l}(a,b)=\int_b^a y^k(p-y)^l\,dy.
$$

Using this identity for both factors cancels $(a-b)^2$. For fixed $p,y,z$, integration over $h$ has lower endpoint $\max(|y-p/2|,|z-p/2|)$ and upper endpoint $(1-|p|)/2$. Including the factor two from the restriction, the resulting weight is

$$
1-|p|-\max(|2y-p|,|2z-p|). \tag{6}
$$

Introduce

$$
(x_1,x_2,x_3,x_4)=(y,p-y,-z,z-p).
$$

The absolute Jacobian from $(p,y,z)$ to $(x_1,x_2,x_3)$ is one. We have $\sum_jx_j=0$, and (6) equals $1-2r$, where $r=\max_j|x_j|$. This follows from the real identity $|v+w|+|v-w|=2\max(|v|,|w|)$ applied to the two pairs. The integration region is

$$
\mathcal P=\{(x_1,x_2,x_3): |x_j|\le1/2\text{ for }j=1,2,3,4,
\quad x_4=-x_1-x_2-x_3\}.
$$

The monomial introduces $(-1)^{m+n}$, cancelling the prefactor in (4). Thus

$$
H=3i^K\int_{\mathcal P}(1-2r)\prod_jx_j^{k_j}\,dx_1dx_2dx_3. \tag{7}
$$

The measure here and below is coordinate measure $dx_1dx_2dx_3$, not the Euclidean surface measure in $\mathbb R^4$.

For any homogeneous polynomial $p_K$ of degree $K$, write $1-2r=\int_{2r}^1dq$ and scale the inner region to obtain

$$
\int_{\mathcal P}(1-2r)p_K(x)\,dx
=\int_0^1q^{K+3}\,dq\int_{\mathcal P}p_K(x)\,dx
=\frac1{K+4}\int_{\mathcal P}p_K(x)\,dx. \tag{8}
$$

Now use the linear change

$$
2x=(u+v+w,\ u-v-w,\ -u+v-w,\ -u-v+w).
$$

Its absolute coordinate Jacobian is $1/2$, and the four absolute-value constraints defining $\mathcal P$ become $|u|+|v|+|w|\le1$. This is because the four displayed forms and their negatives exhaust all eight sign choices of $u,v,w$.

If $q_K$ is homogeneous of degree $K$, radial integration on this octahedron gives

$$
\int_{|u|+|v|+|w|\le1}q_K(u,v,w)\,du\,dv\,dw
=\frac8{(K+3)!}\,\mathbb E[q_K(U,V,W)]. \tag{9}
$$

For example, (9) follows by scaling the sublevel sets of $|u|+|v|+|w|$ and integrating against $e^{-r}$; the Laplace joint density contributes the factor $1/8$. Combining (7)–(9), the coordinate Jacobian and the factor $2^{-K}$ proves (5). $\square$

The formula gives $H(0,0,0,0)=1/2$. The joint law of $Y$ is invariant under permutations of its four coordinates: these permutations act on $(U,V,W)$ by signed permutations, and the independent Laplace law is invariant under that action. Equivalently, the three unordered partitions of four indices into two pairs are permuted in the generating function below.

## 4. The parity recurrence and sign proof

The moment-generating function of $Y$, as an analytic germ at zero or a formal power series, is

$$
F(a,b,c,d)=\frac1{[1-(a+b-c-d)^2][1-(a-b+c-d)^2][1-(a-b-c+d)^2]}. \tag{10}
$$

Write $c_{r,s,t,u}$ for its coefficient at $a^rb^sc^td^u$. This coefficient is the corresponding moment divided by $r!s!t!u!$, a positive factor. Let $F_0$ denote the component in which all four exponents are even, $F_4$ the component in which all four are odd, and $F_{12}$ the component with $a,b$ odd and $c,d$ even. Define

$$
B=1-a^2-b^2-c^2-d^2.
$$

**Lemma 3.** In the ring of formal power series,

$$
BF_{12}=-2abF_0-2cdF_4. \tag{11}
$$

**Proof.** Put

$$
\begin{aligned}
h_0&=a+b+c+d,&h_1&=a+b-c-d,\\
h_2&=a-b+c-d,&h_3&=a-b-c+d,
\end{aligned}
$$

and $g_j=h_0-2t_j$, where $(t_1,t_2,t_3,t_4)=(a,b,c,d)$. Let

$$
D_H=\prod_{j=0}^3(1-h_j^2),\qquad D_G=\prod_{j=1}^4(1-g_j^2).
$$

A parity component is obtained by averaging the 16 coordinate sign substitutions with the appropriate sign character. Since $F$ is invariant under global sign reversal, use the eight substitutions that leave $a$ unchanged. Four have an even number of sign reversals and give the rational functions $(1-h_j^2)/D_H$, $j=0,1,2,3$. The other four give $(1-g_j^2)/D_G$, $j=1,2,3,4$. Since

$$
\sum_{j=0}^3h_j^2=\sum_{j=1}^4g_j^2=4(a^2+b^2+c^2+d^2),
$$

the all-even and all-odd projections are

$$
F_0=\frac B2(D_H^{-1}+D_G^{-1}),\qquad
F_4=\frac B2(D_H^{-1}-D_G^{-1}). \tag{12}
$$

For $F_{12}$, the sign character on this set of eight substitutions is the sign assigned to $b$. The weighted sums of numerators in the two groups are respectively

$$
-h_0^2-h_1^2+h_2^2+h_3^2=-8(ab+cd)
$$

and

$$
g_1^2+g_2^2-g_3^2-g_4^2=-8ab+8cd.
$$

Dividing their sum by eight gives

$$
F_{12}=-ab(D_H^{-1}+D_G^{-1})-cd(D_H^{-1}-D_G^{-1}). \tag{13}
$$

Equations (12)–(13) prove (11). All denominators have constant term one, so each formal division is legitimate. $\square$

**Lemma 4.** Every coefficient of $F$ with exactly two odd exponents is strictly negative. Every coefficient with four odd exponents is strictly positive.

**Proof.** Every all-even coefficient is strictly positive, since its Laplace moment is nonnegative and its integrand is nonzero on a nonempty open set. The joint Laplace density is positive there.

Proceed by induction on even total degree $K$. Assume the four-odd assertion is known below degree $K$. Since

$$
\frac1B=\sum_{j\ge0}(a^2+b^2+c^2+d^2)^j
$$

has nonnegative coefficients, (11) shows that each coefficient of $F_{12}$ at degree $K$ is strictly negative. Indeed, $abF_0$ has a strictly positive coefficient at every allowed exponent, while $cdF_4$ only uses four-odd coefficients of degrees at most $K-2$, already nonnegative by induction. Permutation invariance proves the same for the other five choices of two odd exponents. At $K=2$, the four-odd contribution is absent, providing the base case.

The identity $Y_1+Y_2+Y_3+Y_4=0$ gives

$$
(\partial_a+\partial_b+\partial_c+\partial_d)F=0. \tag{14}
$$

Let $r,s,t,u$ all be odd and have sum $K$. Comparing the coefficient of $a^{r-1}b^sc^td^u$ in (14) yields

$$
\begin{aligned}
r c_{r,s,t,u}={}&-(s+1)c_{r-1,s+1,t,u}\\
&-(t+1)c_{r-1,s,t+1,u}-(u+1)c_{r-1,s,t,u+1}.
\end{aligned} \tag{15}
$$

The three coefficients on the right have exactly two odd exponents and total degree $K$. They were proved strictly negative in the preceding step, so $c_{r,s,t,u}>0$. This establishes the four-odd case at the current degree and completes the induction. The first four-odd case occurs at degree four. $\square$

**Proof of Theorem 1.** Let $r\in\{0,2,4\}$ be the number of odd derivative orders. By Lemma 4 and all-even positivity, the moment in (5) has sign $(-1)^{r/2}$. Since $i^K=(-1)^{K/2}$, its product with the prefactor in (5) has sign

$$
(-1)^{K/2+r/2}=(-1)^{\sum_j\lfloor k_j/2\rfloor}.
$$

The equality holds because the two exponents differ by the even integer $r$. All moments in these parity classes are nonzero. For odd total degree, symmetry under $Y\mapsto-Y$ makes the moment zero.

To verify the stated reformulation, the cases $r=0$ and $r=4$ reduce to the sum of four real signs, up to a common factor $i$ in the latter case. Its modulus is two exactly when an odd number of those signs is negative. When $r=2$, there are two real signs and two imaginary signs. The modulus of their sum is two exactly when one pair cancels and the other does not, again equivalent to an odd number of negative signs after writing $i^{k_j}=(-1)^{\lfloor k_j/2\rfloor}i^{k_j\bmod2}$. This is precisely the negative case in (2). $\square$

## 5. Positive averages of generalized Laguerre expressions

For a smooth real function $f$, set

$$
\mathcal L_p[f](t)=\sum_{j=0}^{2p}
\frac{(-1)^{j+p}}{j!(2p-j)!}f^{(j)}(t)f^{(2p-j)}(t),\qquad p\ge0. \tag{16}
$$

Thus $\mathcal L_0[f]=f^2$ and $\mathcal L_1[f]=(f')^2-ff''$. For a nonzero real entire function, nonnegativity of all these expressions at every real point is equivalent to membership in the Laguerre–Pólya class [5, Theorem 2.4 and Remark 2.6]. Having only real zeros, without further growth assumptions, is not the same condition: $f(z)=e^{z^2}$ has no zeros but $\mathcal L_1[f](x)=-2e^{2x^2}<0$.

**Corollary 5.** For every fixed $p,q\ge0$,

$$
\int_0^T\mathcal L_p[Z](t)\mathcal L_q[Z](t)\,dt
=C_{p,q}\,T(\log T)^{2p+2q+4}
+O_{p,q}\bigl(T(\log T)^{2p+2q+3}\bigr), \tag{17}
$$

where

$$
C_{p,q}=\frac{6}{\pi^2(2p+2q+4)(2p+2q+3)(2p+1)!(2q+1)!}>0. \tag{18}
$$

**Proof.** Expand (16) and apply (1) to each of the finitely many terms. Write $K=2p+2q$. The representation (7)–(8) and the binomial theorem give

$$
\pi^2 C_{p,q}=\frac{3}{(K+4)(2p)!(2q)!}
\int_{\mathcal P}(x_2-x_1)^{2p}(x_4-x_3)^{2q}\,dx_1dx_2dx_3. \tag{19}
$$

Indeed, $i^K=(-1)^{p+q}$ cancels the common sign in the two Laguerre coefficients, leaving the two even binomial powers in (19).

Set $s=x_1+x_2$, $d=x_1-x_2$ and $e=x_3-x_4$. The coordinate Jacobian is $1/4$, and the region becomes

$$
-1\le s\le1,\qquad |d|,|e|\le1-|s|.
$$

Consequently the integral in (19) is

$$
\frac14\int_{-1}^1
\frac{2(1-|s|)^{2p+1}}{2p+1}
\frac{2(1-|s|)^{2q+1}}{2q+1}\,ds
=\frac2{(2p+1)(2q+1)(K+3)}.
$$

Substitution proves (18). This direct evaluation uses the same moment representation as the sign theorem. Theorem 1 also shows that, before the binomial summation, every signed $H$ contribution to $C_{p,q}$ is strictly positive. $\square$

The weighted slice representation (7) also gives positivity after multiplication by the square of any fixed real linear derivative filter. This requires a kernel argument: the coefficient signs alone do not control the mixed terms of an arbitrary square.

**Theorem 6.** Let $p\ge0$ be fixed and let $P(z)=\sum_{j=0}^d c_jz^j$ be a fixed nonzero polynomial with real coefficients. For $T>1$, put

$$
W_{P,T}(t)=\sum_{j=0}^d c_j\frac{Z^{(j)}(t)}{(\log T)^j}.
$$

Then

$$
\int_0^T\mathcal L_p[Z](t)W_{P,T}(t)^2\,dt
=B_p(P)\,T(\log T)^{2p+4}
+O_{p,P}\bigl(T(\log T)^{2p+3}\bigr),\qquad B_p(P)>0. \tag{20}
$$

More precisely, with $m=2p+2$ and $f(x)=P(ix)$,

$$
B_p(P)=\frac3{\pi^2m!}\int_{-1/2}^{1/2}\int_{-1/2}^{1/2}
\bigl[(1-|x-y|)^m-(x+y)^m\bigr]f(x)\overline{f(y)}\,dx\,dy. \tag{21}
$$

**Proof.** Expanding the finite sum and using (1) proves the existence of the leading constant and the error term in (20). In (7), the binomial sum for $\mathcal L_p$ becomes $(x_1-x_2)^{2p}/(2p)!$: its common $(-1)^p$ cancels $i^{2p}$. The two filters contribute $P(ix_3)P(ix_4)$.

Set $x_3=x$, $x_4=-y$, $s=x_1+x_2=y-x$ and $d=x_1-x_2$. The absolute coordinate Jacobian is $1/2$. For fixed $x,y\in[-1/2,1/2]$, put

$$
A=1-|x-y|,\qquad B=|x+y|.
$$

Then $0\le B\le A$, $|d|\le A$, and $1-2r=A-\max(|d|,B)$. Therefore

$$
\frac12\int_{-A}^A d^{2p}\bigl(A-\max(|d|,B)\bigr)\,dd
=\frac{A^{2p+2}-B^{2p+2}}{(2p+1)(2p+2)}.
$$

Since $P(-iy)=\overline{P(iy)}$ and $m$ is even, substitution in (7) proves (21).

To prove strict positivity, set $s=x+1/2$, $t=y+1/2$. The kernel becomes

$$
K_m(s,t)=(1-|s-t|)^m-(s+t-1)^m,\qquad 0\le s,t\le1.
$$

Let $q_m(u)=(1-|u|)^m$ for $|u|\le1$ and zero otherwise, and let $Q_m$ be its periodization with period two. Evenness of $m$ gives

$$
K_m(s,t)=Q_m(s-t)-Q_m(s+t).
$$

Consequently the normalized sine functions $e_n(s)=\sqrt2\sin(n\pi s)$ diagonalize $K_m$, with eigenvalues

$$
\begin{aligned}
\lambda_{m,n}
&=2\int_0^1(1-u)^m\cos(n\pi u)\,du\\
&=\frac{2m}{(n\pi)^2}
\left[1-(m-1)\int_0^1(1-u)^{m-2}\cos(n\pi u)\,du\right]>0,
\qquad n\ge1. \tag{22}
\end{aligned}
$$

The second equality follows by two integrations by parts. The inequality is strict because $\cos(n\pi u)<1$ on a set of positive measure and the weight $(1-u)^{m-2}$ is positive on $(0,1)$. The integral in brackets is therefore strictly smaller than $1/(m-1)$.

The Fourier coefficients are $O_m(n^{-2})$, which justifies the uniformly convergent kernel expansion $K_m(s,t)=\sum_{n\ge1}\lambda_{m,n}e_n(s)e_n(t)$. Thus (21) is $3/(\pi^2m!)$ times

$$
\sum_{n\ge1}\lambda_{m,n}
\left|\int_0^1P\bigl(i(s-1/2)\bigr)e_n(s)\,ds\right|^2.
$$

Completeness of the sine basis and $P\ne0$ imply that this sum is strictly positive. $\square$

These results concern averages of Hardy's $Z$. They do not prove $\mathcal L_p[Z](t)\ge0$ at individual points. Theorem 6 tests only squares of linear derivative jets; it supplies no positivity against arbitrary nonlinear functions of the jet or arbitrary localizers in height. Section 6 allows a slowly growing sum of the Laguerre order and the filter degree. Passing to the entire xi-function also involves its gamma amplitude. These steps cannot be omitted in an application to RH.

There is also a direct explanation for this averaged positivity in terms of functions with real zeros. Define

$$
S_q(v)=\int_{-q/2}^{q/2}e^{ivx}\,dx=\frac{2\sin(qv/2)}v,
\qquad S_q(0)=q.
$$

The layer-cake identity $1-2r=\int_{2r}^1dq$, followed by Fourier inversion for the convolution of four compact interval densities, gives

$$
\frac3{\pi^2}\int_{\mathcal P}(1-2r)e^{i\sum_j a_jx_j}\,dx
=\frac3{2\pi^3}\int_0^1\int_{\mathbb R}\prod_{j=1}^4S_q(v+a_j)\,dv\,dq. \tag{23}
$$

The Fourier product is integrable, so (23) is an ordinary convolution-inversion identity. For fixed derivative orders, differentiation is justified by
$S_q^{(k)}(v)=q^{k+1}S_1^{(k)}(qv)$ and $S_1^{(k)}(v)=O_k((1+|v|)^{-1})$. In particular,

$$
\frac{H(\mathbf k)}{\pi^2}
=\frac3{2\pi^3(K+4)}\int_{\mathbb R}\prod_{j=1}^4S_1^{(k_j)}(v)\,dv. \tag{24}
$$

Each $S_q$, $q>0$, belongs to the Laguerre–Pólya class. Differentiating (23) thus expresses $B_p(P)$ as a positive multiple of
$\int_0^1\int_{\mathbb R}\mathcal L_p[S_q](v)[P(D)S_q(v)]^2\,dv\,dq$,
which also explains its nonnegativity. The sine spectrum (22) supplies strictness and the explicit quadratic form. Formula (23) describes the limiting fourth-moment functional; it is not a probability law determining the actual derivative jets of $Z$ from fourth moments.

## 6. Uniform positivity for growing orders and filter degrees

The fixed complex-shift uniformity of [1, Proposition 3], together with a quantitative lower bound for (21), gives a joint uniform statement. It is important here to use the shifted theorem directly; the unspecified derivative-order constants in (1) would not suffice.

**Theorem 7.** Write $L=\log T$, $C=2+\sqrt5$, and for $p\ge0$ and a real polynomial $P$ define

$$
\mathscr Q_{T,p}(P)=\frac1{TL^{2p+4}}\int_T^{2T}
\mathcal L_p[Z](t)W_{P,T}(t)^2\,dt.
$$

Uniformly in $p,d\ge0$ and every nonzero real polynomial $P$ of degree at most $d$,

$$
\mathscr Q_{T,p}(P)=B_p(P)\left[1+
O\left(\frac{(2p+1)4^p(2p)!(d+2)^6C^{2d}(d!)^2}{\log T}\right)\right]. \tag{25}
$$

The implied constant is absolute. In particular, for each fixed $0<\varepsilon<1/2$, all these integrals are strictly positive for sufficiently large $T$, simultaneously for

$$
p+d\le\left(\frac12-\varepsilon\right)
\frac{\log\log T}{\log\log\log T}. \tag{26}
$$

The order, polynomial and coefficients may depend on $T$ within this range.

**Proof.** We first justify a uniform analytic remainder. Write

$$
\mathcal A(a)=\frac3{\pi^2}\int_{\mathcal P}(1-2r)e^{i\sum_j a_jx_j}\,dx.
$$

On a neighborhood of a large positive real interval, Hardy's function has the analytic continuation
$Z(z)=e^{i\vartheta(z)}\zeta(1/2+iz)=e^{-i\vartheta(z)}\zeta(1/2-iz)$, with the local analytic gamma branch agreeing with the real Hardy phase. In [1, Proposition 3], set

$$
\alpha_1=ia_1/L,\quad\alpha_2=ia_2/L,\quad
\beta_1=-ia_3/L,\quad\beta_2=-ia_4/L.
$$

For $|a_j|\le2$ and $T\le t\le2T$, analytic Stirling estimates give the combined Hardy phase

$$
e^{i(\vartheta(t+a_1/L)+\vartheta(t+a_2/L)
-\vartheta(t+a_3/L)-\vartheta(t+a_4/L))}
=e^{i(a_1+a_2-a_3-a_4)/2}\bigl(1+O(L^{-1})\bigr).
$$

The phase-error integral is $O(TL^3)$ by Hölder: Proposition 3 with conjugate matching shifts bounds each required absolute fourth moment by $O(TL^4)$. Multiplying the source's main term by the displayed constant phase gives the exponential version of (4). The same compact changes of variables used in Lemma 2 therefore show that

$$
E_T(a):=\frac1{TL^4}\int_T^{2T}\prod_{j=1}^4Z(t+a_j/L)\,dt
-\mathcal A(a)=O(L^{-1}) \tag{27}
$$

uniformly on $|a_j|\le2$. This remainder is holomorphic on a neighborhood of that polydisc. Cauchy's estimate on circles $|a_j|=1$, followed by the finite Laguerre and filter sums, gives

$$
\begin{aligned}
|\mathscr Q_{T,p}(P)-B_p(P)|
&\le\frac{A(2p+1)}{L}\left(\sum_{j=0}^d|c_j|j!\right)^2\\
&\le\frac{A(2p+1)(d!)^2}{L}\|c\|_1^2. \tag{28}
\end{aligned}
$$

The factorials from the first two differentiated coordinates cancel the factorials in (16), leaving $2p+1$ terms. The constant $A$ is absolute, so this bound is uniform in all derivative orders.

We next bound the positive leading form below on polynomials. Put $m=2p+2$ and $f(s)=P(i(s-1/2))$. Rewriting (22) and restricting its integral to $[0,1/2]$ gives

$$
\lambda_{m,n}=\frac{2m(m-1)}{(n\pi)^2}
\int_0^1(1-u)^{m-2}(1-\cos(n\pi u))\,du
\ge\frac{c_m}{(n\pi)^2}, \tag{29}
$$

where $c_m=2m(m-1)2^{-(m-2)}(1/2-1/\pi)>0$. Hence

$$
B_p(P)\ge\kappa_p\|f\|_{H^{-1}_D(0,1)}^2,
\qquad\kappa_p=\frac{3c_m}{\pi^2m!}
=\frac{6(1/2-1/\pi)}{\pi^2 4^p(2p)!}, \tag{30}
$$

where the squared Dirichlet negative Sobolev norm is
$\sum_{n\ge1}|\langle f,e_n\rangle|^2/(n\pi)^2$.

Let $-u''=f$, $u(0)=u(1)=0$. Then $u$ is a polynomial of degree at most $d+2$, and $\|f\|_{H^{-1}_D}=\|u'\|_2$. Every polynomial $g$ of degree at most $D$ on $[0,1]$ satisfies

$$
\|g'\|_2\le(D+1)^2\|g\|_2. \tag{31}
$$

For completeness, expand in the orthonormal shifted Legendre polynomials
$\sqrt{2k+1}\,\mathrm P_k(2s-1)$. Their squared derivative norms are $2(2k+1)k(k+1)$, whose sum for $0\le k\le D$ is $D(D+1)^2(D+2)\le(D+1)^4$. Cauchy–Schwarz proves (31). Applying it to $u'$ yields

$$
\|f\|_{H^{-1}_D}^2\ge(d+2)^{-4}\|f\|_2^2. \tag{32}
$$

Finally, expand $P(ix)$ in the orthonormal Legendre basis on $[-1/2,1/2]$. The sum of the absolute monomial coefficients of $\mathrm P_k(2x)$ is at most $C^k$, where $C=2+\sqrt5$: the three-term recurrence bounds these sums by $b_{k+1}\le4b_k+b_{k-1}$, with $b_0=1$, $b_1=2$, and $C^2=4C+1$. Another application of Cauchy–Schwarz gives

$$
\|c\|_1\le(d+1)C^d\|f\|_2. \tag{33}
$$

Multiplication of the monomial coefficients by $i^j$ does not change their absolute values. Combining (30), (32), and (33),

$$
B_p(P)\ge\frac{\kappa_p\|c\|_1^2}
{(d+2)^4(d+1)^2C^{2d}}. \tag{34}
$$

Equations (28) and (34) prove (25). Put $N=p+d$ and $h=\log\log T$. Since $(2p)!(d!)^2\le(2N)!$ and $4^pC^{2d}\le C^{2N}$, the relative-error factor is at most a fixed polynomial in $N$ times $C^{2N}(2N)!/\log T$. For $N\le(1/2-\varepsilon)h/\log h$, its logarithm is at most $(-2\varepsilon+o(1))h$. It tends to minus infinity, proving (26). $\square$

The range (26) is not claimed to be optimal, and no practical height threshold is given. It controls all coefficients simultaneously in a growing family, but supplies no theorem that the resulting derivative squares can concentrate at a selected height. Thus it still does not imply a pointwise Laguerre inequality.

## 7. Positive averages in shorter smooth windows

Fix a nonnegative $w\in C_c^\infty((0,1))$ with $\int_0^1w(u)\,du=1$, and put
$w_{X,H}(t)=w((t-X)/H)$. This height weight is separate from the derivative filter $W_{P,T}$.

**Theorem 8.** Let $L=\log T$, $T\le X\le2T$, and $T^{3/4}\le H\le T$. Write

$$
R_{p,d}=(2p+1)4^p(2p)!(d+2)^6(2+\sqrt5)^{2d}(d!)^2.
$$

For every fixed $0<\sigma<1/4$, uniformly in $X,H,p,d$ and every nonzero real polynomial $P$ of degree at most $d$,

$$
\frac1{HL^{2p+4}}\int_{\mathbb R}w_{X,H}(t)
\mathcal L_p[Z](t)W_{P,T}(t)^2\,dt
=B_p(P)\left[1+O_{w,\sigma}\left(R_{p,d}
\left\{\frac1L+\frac{T^{3+\sigma}}{H^{13/4}L^4}\right\}\right)\right]. \tag{35}
$$

Consequently, for fixed $0<\delta<1/13$ and $0<\varepsilon<1/2$, these integrals are strictly positive for all sufficiently large $T$, simultaneously for every center $X\in[T,2T]$, every

$$
T^{12/13+\delta}\le H\le T,\qquad
p+\deg P\le\left(\frac12-\varepsilon\right)
\frac{\log\log T}{\log\log\log T}, \tag{36}
$$

and every nonzero real $P$, with coefficients allowed to depend on $T,X,H$.

**Proof.** Apply [7, Theorem 1.1] with twist parameters $h=k=1$ and smoothing scale $T_0=H$. Our weight is supported in $[T/2,4T]$ and satisfies $w_{X,H}^{(j)}\ll_{w,j}H^{-j}$. The theorem gives a sum of six main terms and the uniform complex-shift error

$$
E\ll_{w,\sigma}T^{3/4+\sigma}(T/H)^{9/4}.
$$

Use the four shifts $ia_1/L,ia_2/L,-ia_3/L,-ia_4/L$ as in Theorem 7. The sum of its six main terms, after multiplying by the constant Hardy phase, is
$L^4\mathcal A(a)+O(L^3)$ uniformly for bounded complex $a$ and $t\in[T/2,4T]$.

Here is a justification that retains uniformity when shifts coincide. First place the four $a_j$ on circles of distinct fixed radii enclosing the desired polydisc. Every pairwise difference stays away from zero. In each of the six terms, use
$\zeta(1+v/L)=L/v+O(1)$, $1/\zeta(2+O(1/L))=1/\zeta(2)+O(1/L)$, and $\log(t/(2\pi))/L=1+O(1/L)$. The error is $O(L^3)$. The complete six-term sum is holomorphic by the pole cancellation in [7]; the maximum principle extends the estimate inside the polydisc. Its leading holomorphic function agrees with $\mathcal A$: for a fixed smooth dyadic weight, partial integration of (1) identifies every fixed derivative at zero with $H(\mathbf k)/\pi^2$, as does (7). Equality follows from analyticity. This identification does not sum derivative error bounds.

The analytic Hardy phase differs from its constant approximation by $O(L^{-1})$. Hölder and the same localized theorem with conjugate matching shifts bound the integral of the absolute product by $O(HL^4+E)$. Thus the phase error is $O(HL^3+E/L)$, and

$$
\frac1{HL^4}\int_{\mathbb R}w_{X,H}(t)
\prod_{j=1}^4Z(t+a_j/L)\,dt-\mathcal A(a)
\ll_{w,\sigma}\frac1L+\frac{T^{3+\sigma}}{H^{13/4}L^4} \tag{37}
$$

on $|a_j|\le2$. The remainder is holomorphic. Applying the four-variable Cauchy estimate and then (34), exactly as in (28), proves (35). In particular, the phase error was bounded on the local window itself.

For (36), choose $\sigma<\min(1/4,13\delta/4)$. The second term in braces in (35) is $O(T^{-c_\delta}/L^4)$ for some $c_\delta>0$. The Stirling argument following (34) now applies unchanged. $\square$

The exponent $12/13$ comes from the stated general error bound in [7]; no optimal local scale is claimed. Theorem 8 concerns smooth weights. A sharp-cutoff conclusion cannot be obtained by sandwiching this signed integrand. These windows remain much too long to isolate an individual zero or to yield a pointwise Laguerre inequality.

## 8. Retaining the full arithmetic main term

The logarithmic error in (27) includes lower-order terms of the explicit
shifted main term. It is useful to differentiate those terms separately
from the power-saving arithmetic error. This gives a larger polynomial
filter space at each fixed Laguerre order.

**Theorem 9 (fixed order and a power of logarithmic degree).** Fix an integer
$p\ge0$, $0<\varepsilon<1/2$, $0<\delta<1/13$, and a nonnegative nonzero
$w\in C_c^\infty((0,1))$. Put $L=\log T$ and $D=d/dt$. For sufficiently large $T$,
uniformly in $X\in[T,2T]$, $T^{12/13+\delta}\le H\le T$, and every nonzero
real polynomial $P$ of degree

$$
d\le L^{1/2-\varepsilon},
$$

one has

$$
\int_{\mathbb R}w\left(\frac{t-X}{H}\right)
\mathcal L_p[Z](t)\,[P(D/L)Z(t)]^2\,dt>0. \tag{38}
$$

The required height may depend on $p,\varepsilon,\delta,w$, but not on
$X,H$ or the coefficients of $P$. Order $p$ is fixed in this theorem.
Theorem 10 below gives the simultaneous growing-order range, including order of size $\log T/\log\log T$ with the stated degree tradeoff.

### 8.1. The six-term main term as one analytic operator

For a two-element subset $S\subset\{1,2,3,4\}$, write

$$
U_S=\sum_{j\in S}z_j-\sum_{j\notin S}z_j,
\qquad D_S=\prod_{j\in S,\ k\notin S}(z_j-z_k),
$$

and define

$$
F_\ell(z)=\sum_{|S|=2}e^{\ell U_S/2}
\frac{\prod_{j\in S,k\notin S}\zeta(1+z_j-z_k)}{\zeta(2+U_S)},
\qquad
H_\ell(z)=\sum_{|S|=2}\frac{e^{\ell U_S/2}}{D_S}. \tag{39}
$$

With $\ell=\log(t/(2\pi))$, $F_\ell$ is the complete untwisted
Hughes–Young main term after the leading Hardy phase. Subsets $S$ encode
all six swaps, including the off-diagonal main terms. Both sums in (39)
are holomorphic near $z=0$. Indeed, on a generic hyperplane $z_i=z_j$,
pair subsets containing $i$ but not $j$ with those obtained by exchanging
$i,j$. Their residues cancel. Multiplication by the Vandermonde and
divisibility by each of its distinct linear factors then proves removability
also at multiple collisions. This is the pole cancellation used in [7]; the general permutation-sum contour identity is due to Conrey, Farmer, Keating, Rubinstein and Snaith [10, Section 2.5]. Their full arithmetic-factor framework precedes the present argument. The new quantitative assertion here is the relative bound on the expanding polynomial observer space, not the existence of arithmetic lower-order terms.

Put $E(v)=v\zeta(1+v)$ and

$$
\kappa=2\gamma-\frac{\zeta'(2)}{\zeta(2)},
\qquad s=\ell+2\kappa.
$$

The arithmetic factor relative to $1/(\zeta(2)D_S)$ is

$$
A_S(z)=\prod_{j\in S,k\notin S}E(z_j-z_k)
\frac{\zeta(2)}{\zeta(2+U_S)}.
$$

Since the sum of the four cross differences is $2U_S$,
$A_S(z)=e^{\kappa U_S}(1+O(|z|^2))$. Thus the first correction is a real
length change, preserving the positive leading kernel.

We need a convergent description of all subsequent corrections. Center
the variables by their mean and set

$$
Q_2(z)=\sum_{j=1}^4(z_j-\bar z)^2,
\quad Q_4(z)=\sum_{j=1}^4(z_j-\bar z)^4,
\quad \bar z=\frac14\sum_jz_j.
$$

**Lemma 9.1 (common arithmetic factor).** There are real coefficients
$a_{rbc}$ and absolute constants $C_0,C_1$ such that, for every $S$ near
$z=0$,

$$
A_S(z)=e^{\kappa U_S}
\left(1+\sum_{r+2b+4c\ge2}a_{rbc}U_S^rQ_2(z)^bQ_4(z)^c\right),
\qquad
\sum_{r+2b+4c=k}|a_{rbc}|\le C_0C_1^k. \tag{40}
$$

**Proof.** Relabel $S=\{1,2\}$ and write the centered variables as

$$
z_1=(u+v)/2,\quad z_2=(u-v)/2,\quad
z_3=(-u+w)/2,\quad z_4=(-u-w)/2.
$$

Then $U_S=2u$, and the four cross differences are
$u+(\pm v\pm w)/2$. Their $E$-product is even in $v,w$ and invariant under
$v\leftrightarrow w$. Its convergent analytic series is therefore a
symmetric series in $V=v^2,W=w^2$. Set $B=V+W,D=VW$. Symmetric monomials
are products of powers of $D$ with $V^m+W^m$, whose recurrence
$t_m=Bt_{m-1}-Dt_{m-2}$ has exponentially bounded coefficient sums.
The substitutions

$$
u=U_S/2,\qquad B=2Q_2-U_S^2/2,
\qquad D=Q_2^2+U_S^2Q_2-3U_S^4/8-4Q_4
$$

are polynomial. Cauchy bounds in a fixed $u,v,w$ polydisc, the recurrence,
and these substitutions give a convergent series with an exponential
coefficient bound in the grading $\deg U=1$, $\deg Q_2=2$, $\deg Q_4=4$.
The factor $\zeta(2)/\zeta(2+U)$ is analytic in a fixed neighborhood and
does not affect this conclusion. Removing $e^{\kappa U}$ removes the only
term of degree one. The construction is independent of the relabeling,
which proves (40). $\square$

Since $U_S^re^{sU_S/2}=(2\partial_s)^re^{sU_S/2}$, (40) gives

$$
F_\ell(z)=\frac1{\zeta(2)}
\left[1+\sum_{r+2b+4c\ge2}a_{rbc}
Q_2(z)^bQ_4(z)^c(2\partial_s)^r\right]H_s(z). \tag{41}
$$

This identity may be read coefficientwise. For a homogeneous coefficient
of degree $K$ in $z$, only indices with $r+2b+4c\le K+4$ contribute:
a degree-$K-h$ coefficient of $H_s$ is a constant times $s^{K-h+4}$,
where $h=2b+4c$. Thus each polynomial observer uses an exact finite sum.

### 8.2. Uniform bounds for the spectrum and the observer

Let $d\mu(x)=3\pi^{-2}(1-2\max_j|x_j|)\,dx_1dx_2dx_3$ on the cube slice
$\mathcal P$ of Section 3. The leading-function identity is

$$
\frac{H_s(z)}{\zeta(2)}=s^4\int_{\mathcal P}e^{s\sum_jz_jx_j}\,d\mu(x).
\tag{42}
$$

In particular $H_s(0)=s^4/12$. Put $\rho=s/L=1+O(L^{-1})$,
$I=[-1/2,1/2]$, $D_0=d+1$, $V=p+D_0^2$, and

$$
R_{p,P}(y)=\frac{(y_1-y_2)^{2p}}{(2p)!}P(iy_3)P(iy_4),
\qquad f_\rho(x)=P(i\rho x),\quad x\in I.
$$

The normalized differential functional is

$$
\mathscr D_{p,P}F=
\left.\frac{(-1)^p}{(2p)!}(\partial_{b_1}-\partial_{b_2})^{2p}
P(\partial_{b_3})P(\partial_{b_4})F(b)\right|_{b=0}.
$$

It satisfies $\mathscr D_{p,P}e^{ib\cdot y}=R_{p,P}(y)$.
Multiplication by a homogeneous polynomial $Q(ib/L)$ therefore acts as
$L^{-\deg Q}Q(\partial_y)$ on $R_{p,P}$. This is direct differentiation,
with no integration by parts or omitted boundary term. For an index in
(41), set $h=2b+4c$, $k=r+h$, and $Q=Q_2^bQ_4^c$. Its contribution to
$B_t(P)=\mathscr D_{p,P}[L^{-4}F_\ell(ib/L)]$ is exactly

$$
a_{rbc}\,2^rL^{-k}\partial_\rho^r
\left[\rho^4\int_{\mathcal P}Q(\partial_y)R_{p,P}(y)
\big|_{y=\rho x}\,d\mu(x)\right]. \tag{43}
$$

We will bound (43) uniformly also in $p$.

**Lemma 9.2 (uniform spectral coercivity).** There are absolute constants
$c,C>0$ such that, for $m=2p+2$ and every $n\ge1$,

$$
\frac{cm}{m^2+(n\pi)^2}\le\lambda_{m,n}
\le\frac{Cm}{m^2+(n\pi)^2}.
$$

Consequently, uniformly for all real polynomials of degree at most $d$,

$$
B_p(P)\ge\frac{c}{(2p+1)!\,V^2}\|P(i\,\cdot)\|_2^2.
\tag{44}
$$

**Proof.** Write $\omega=n\pi$ in (22). Its bracket is
$\mathbb E[1-\cos(\omega U)]$, where $U$ has density
$(m-1)(1-u)^{m-2}$ on $[0,1]$. On $[0,1/m]$ this density is at least
$m/(2e)$. Indeed $m-1\ge m/2$ and
$\log(1-1/m)\ge-1/(m-1)$. The elementary inequality

$$
\int_0^1(1-\cos(vx))\,dx\ge\frac1{24}\min(v^2,1),
\qquad v\ge0,
$$

follows by using $1-\cos u\ge u^2/4$ when $u\le1$, integrating only up
to $x=1/v$ when $1\le v\le2$, and using
$1-\sin(v)/v\ge1/2$ when $v\ge2$. Thus
$\lambda_{m,n}\ge m/[24e(m^2+\omega^2)]$.
For the upper bound, $\mathbb E U^2=2/[m(m+1)]$ in (22) gives
$\lambda_{m,n}\le2/(m+1)$, and the same bracket is at most two, giving
$\lambda_{m,n}\le4m/\omega^2$. These imply the asserted upper bound.

For a complex polynomial $g$ of degree at most $d$ on $[0,1]$, let
$a_n=\sqrt2\int_0^1g(s)\sin(n\pi s)\,ds$. Integration by parts gives

$$
a_n=\frac{\sqrt2[g(0)-(-1)^ng(1)]+b_n}{n\pi},
\qquad b_n=\sqrt2\int_0^1g'(s)\cos(n\pi s)\,ds.
$$

Orthonormal shifted Legendre evaluation gives
$|g(0)|^2+|g(1)|^2\le2D_0^2\|g\|_2^2$, while (31) gives
$\|g'\|_2\le D_0^2\|g\|_2$. Bessel's inequality for the cosine basis
therefore implies

$$
\sum_{n>N}|a_n|^2\le
\left(\frac{16D_0^2}{\pi^2N}+\frac{2D_0^4}{\pi^2N^2}\right)\|g\|_2^2.
$$

At $N=8D_0^2$ the bracket is $65/(32\pi^2)<1/2$. At least half the
sine energy thus lies below that cutoff. The spectral lower bound gives

$$
\sum_n\lambda_{m,n}|a_n|^2
\ge\frac{cm}{m^2+D_0^4}\|g\|_2^2.
$$

Apply (21) to $g(s)=P(i(s-1/2))$. Since
$m^2+D_0^4\le C V^2$ and $m/m!=1/(2p+1)!$, this proves (44).
$\square$

**Lemma 9.3 (absolute fibre bound).** For every integer $q\ge0$ and
$u,v\in L^2(I)$,

$$
\int_{\mathcal P}|x_1-x_2|^q|u(x_3)v(x_4)|\,d\mu(x)
\le\frac{6\|u\|_2\|v\|_2}{\pi^2(q+1)(q+2)(q+3)}.
$$

**Proof.** Use the same first-pair fibre as in the proof of (21), with
$x_3=x$, $x_4=-y$, $A=1-|x-y|$, $B=|x+y|$. The coordinate Jacobian is
$1/2$ and

$$
\frac12\int_{-A}^A|z|^q\bigl(A-\max(|z|,B)\bigr)\,dz
=\frac{A^{q+2}-B^{q+2}}{(q+1)(q+2)}.
$$

For $\nu=q+2$, the pointwise nonnegative kernel
$K_\nu(x,y)=(1-|x-y|)^\nu-|x+y|^\nu$ has row and column integrals
at most $2/(\nu+1)$, by comparison with $(1-|x-y|)^\nu$.
Schur's estimate applied to $|u(x)|$ and $|v(-y)|$ proves the claim.
This argument covers odd $q$ and two different observer functions; it
does not assert positivity of any signed arithmetic correction.
$\square$

**Lemma 9.4 (polynomial derivatives in the kernel norm).** On
$I=[-1/2,1/2]$, put

$$
e_n(x)=\sqrt2\sin(n\pi(x+1/2)),\qquad
\|f\|_m^2=\sum_{n\ge1}\frac{|\langle f,e_n\rangle|^2}{m^2+(n\pi)^2},
\qquad m\ge2.
\tag{44a}
$$

For a polynomial $f$ of degree at most $d$, and all integers $a,r\ge0$,

$$
\|x^r f^{(a)}\|_m
\le2\bigl[C(m+(d+1)^2)\bigr]^a\|f\|_m.
\tag{44b}
$$

Extend (21) to the sesquilinear kernel form $\mathfrak B_p(f,g)$ on
arbitrary complex functions. Thus
$\mathfrak B_p(P(i\,\cdot),P(i\,\cdot))=B_p(P)$. For $m=2p+2$,

$$
\frac{c}{(m-1)!}\|f\|_m^2
\le\mathfrak B_p(f,f)\le\frac{C}{(m-1)!}\|f\|_m^2.
\tag{44c}
$$

For every even $j$ with $0\le j\le2p$, if $\ell=p-j/2$, then

$$
|\mathfrak B_\ell(g,h)|
\le\frac{C(4m)^j}{(m-1)!}\|g\|_m\|h\|_m.
\tag{44d}
$$

All constants are absolute.

**Proof.** The sine-tail estimate in Lemma 9.2 shows that at least half
of a degree-$d$ polynomial's sine energy lies below $8(d+1)^2$. Hence

$$
\|f\|_2\le C\bigl(m+(d+1)^2\bigr)\|f\|_m.
$$

Write $\|\cdot\|_-$ for the Dirichlet $H^{-1}$ norm on $I$. If $F'=f$,
the solution of $-u''=f$, $u(-1/2)=u(1/2)=0$, has
$u'=\int_I F-F$. Therefore

$$
\|f\|_-=\left\|F-\int_I F\right\|_2,\qquad
\|f'\|_m\le\|f'\|_-=
\left\|f-\int_I f\right\|_2\le\|f\|_2.
$$

This is the ordinary polynomial derivative, equal to its weak derivative
on $H_0^1$ test functions; no endpoint trace has been discarded.
Combining the last two displays and iterating on the decreasing polynomial
degrees bounds $a$ derivatives by
$[C(m+(d+1)^2)]^a$ in the same norm $\|\cdot\|_m$.

The dual positive norm is
$(\|u'\|_2^2+m^2\|u\|_2^2)^{1/2}$ on $H_0^1(I)$.
For $r\ge1$, both $\|x^r\|_\infty$ and
$\|(x^r)'\|_\infty$ are at most one. The product rule bounds multiplication
by $x^r$ in this energy norm by $1+1/m\le3/2$, and duality gives the
same bound on $\|\cdot\|_m$. For $r=0$ its norm is one. This proves
(44b).

The uniform spectrum in Lemma 9.2 immediately gives (44c).
For $m'=m-j\ge2$, comparison of each sine denominator yields
$\|g\|_{m'}\le(m/m')\|g\|_m$. Cauchy--Schwarz in the positive kernel
$\mathfrak B_\ell$, followed by (44c) at $m'$, gives

$$
|\mathfrak B_\ell(g,h)|\le
\frac{C}{(m-j-1)!}\left(\frac{m}{m-j}\right)^2
\|g\|_m\|h\|_m.
$$

The factorial quotient $(m-1)!/(m-j-1)!$ is at most $m^j$.
For $j\ge1$, writing $m/(m-j)$ as a product of adjacent ratios, whose
denominators are at least two, bounds it by $2^j$. The product of these
bounds is $(4m)^j$. For $j=0$ both quotients are one. This proves (44d).
$\square$

**Lemma 9.5 (relative observer derivatives).** With the notation of (43),
for $1/2\le\rho\le2$ and all $p,d,r,b,c\ge0$,

$$
\left|2^r\partial_\rho^r
\left[\rho^4\int_{\mathcal P}Q(\partial_y)R_{p,P}(y)
\big|_{y=\rho x}\,d\mu(x)\right]\right|
\le C^{k+1}V^k\rho^{2p+4}B_p(P(\rho\,\cdot)).
\tag{45}
$$

**Proof.** The coefficient sum of $Q$ is at most $C^h$. For one of its
monomials, let $j$ derivatives hit the first pair and let $a_3,a_4$ hit
the two observers, so $j+a_3+a_4=h$. A term with $j>2p$ vanishes.
Otherwise put $q=2p-j$. Apart from a complex scalar of modulus one,
its integrand, including $\rho^4$, is

$$
\frac{\rho^{q+4}(x_1-x_2)^q}{q!}
P^{(a_3)}(i\rho x_3)P^{(a_4)}(i\rho x_4).
$$

Allocate the $r$ further derivatives as $r_0+r_3+r_4=r$.
Those on the first power cost at most $(2p+4)^{r_0}$ and leave
$\rho^{q+4-r_0}$; a derivative beyond its degree gives zero.
The observers acquire $x_3^{r_3},x_4^{r_4}$ and derivative orders
$a_3+r_3,a_4+r_4$. Converting to derivatives of
$f_\rho(x)=P(i\rho x)$ leaves the total dilation power exactly

$$
\rho^{q+4-r_0-a_3-a_4-r_3-r_4}=\rho^{2p+4-k}.
$$

If $j$ is odd, $q$ is odd and the first-pair fibre integrates to zero
under $x_1\leftrightarrow x_2$. If $j$ is even, its signed fibre is
exactly the positive kernel $\mathfrak B_{p-j/2}$, between the two
possibly different functions
$x^{r_3}f_\rho^{(a_3+r_3)}$ and
$x^{r_4}f_\rho^{(a_4+r_4)}$, up to scalar factors of modulus one.
Here the reflection $x_4=-y$ is legitimate because real coefficients
of $P$ give
$f_\rho^{(a)}(-y)=(-1)^a\overline{f_\rho^{(a)}(y)}$.
Equivalently, reflection preserves every sine weight in (44a).

Apply (44b) and (44d) before taking absolute values of the two-variable
kernel integral. Since $m=2p+2$ and $m+(d+1)^2\le3V$, the order lowering
cost is at most $(CV)^j$, the two observer derivative costs together
are at most $(CV)^{a_3+a_4+r_3+r_4}$, and the prefactor derivative cost
is at most $(CV)^{r_0}$. Their total exponent is
$j+a_3+a_4+r_3+r_4+r_0=k$. Comparison with (44c) returns
$B_p(P(\rho\,\cdot))$ without any further polynomial coercivity loss.
Finally the Leibniz coefficient sum is $3^r$, the factor $2^r$ is explicit,
and $\rho^{-k}\le2^k$. These and the coefficient sum of $Q$ are absorbed
in $C^{k+1}$, proving (45). No factor exponential in $p$ or factorial
in $k$ is introduced. $\square$

Define

$$
B_t^+(P)=\rho^{2p+4}B_p(P(\rho\,\cdot)).
$$

The arithmetic series (40) has no degree-one term. Summing (43) using
(45) and its exponential coefficient bound, whenever $CV/L\le1/2$,
gives the relative estimate

$$
\frac{|B_t(P)-B_t^+(P)|}{B_t^+(P)}
\le C\left(\frac{V}{L}\right)^2,
\qquad V=p+(d+1)^2.
\tag{45a}
$$

This controls every arithmetic degree that can contribute, uniformly in
all filter coefficients. Retaining the signed even fibre is essential:
Lemma 9.3's absolute estimate alone, followed by an $L^2$ lower-bound
comparison, gives an additional loss when $p$ is small relative to $d$.


### 8.3. The fixed-contour source remainder and the joint range

We apply [7, Theorem 1.1] only on a fixed microscopic shift polydisc,
say $|b_j|\le2$, with $z_j=ib_j/L$. To retain the actual
$\ell(t)=\log(t/(2\pi))$, include the smooth multiplier
$g_z(t)=e^{\ell(t)U_{\{1,2\}}/2}$ in the weight. Although [7] states the
formula for nonnegative weights, its linearity gives the needed extension:
choose a fixed $M$ dominating $|\Re g_z|,|\Im g_z|$, and write each part
as the difference of the nonnegative weights
$w((t-X)/H)(M\pm\Re g_z)$ or
$w((t-X)/H)(M\pm\Im g_z)$, divided by two.
Their $j$th derivatives are uniformly $O_j(H^{-j})$, since
$g_z^{(j)}=O_j(T^{-j})$ and $H\le T$. Thus the same uniform arithmetic
error applies. The original integral and complete main term remain
holomorphic in the shifts, irrespective of this decomposition used to
bound the error.

Analytic Stirling and Taylor expansion of the Hardy phase show that its
exact multiplier is $g_z(t)(1+O(T^{-1}))$ on this fixed polydisc.
Hölder and conjugate-matched shifted fourth moments with the same
nonnegative window bound the resulting integrated error by
$O((HL^4+E)/T)$, where
$E\ll_{w,\sigma}T^{3/4+\sigma}(T/H)^{9/4}$. After normalization by
$HL^4$, the holomorphic remainder is bounded by

$$
\eta_T\ll_{w,\sigma}T^{-1}+
\frac{T^{3+\sigma}}{H^{13/4}L^4}. \tag{46}
$$

Cauchy on unit circles gives

$$
|\mathscr D_{p,P}\mathcal E|
\le(2p+1)(d!)^2\|c\|_1^2\eta_T.
$$

Here the first two derivative factorials cancel the Laguerre binomial
coefficients exactly, leaving $2p+1$ terms. For each $t$ in the window,
(33) applied to $P(\rho\,\cdot)$ gives
$\|c\|_1\le\max(1,\rho^{-d})(d+1)(2+\sqrt5)^d\|f_\rho\|_2$.
Using (44), the differentiated remainder relative to the averaged
positive comparison is at most

$$
C(2p+1)(2p+1)!V^2(d+1)^2(2+\sqrt5)^{2d}(d!)^2
e^{O((p+d+1)/L)}\eta_T.
\tag{47}
$$

All dilation factors here are retained until using
$\rho=1+O(L^{-1})$. The comparison with the window average is valid
because the coefficient estimate holds for every $t$, and the weight is
nonnegative with fixed positive integral. Put $N=p+d$. Since
$(2p+1)!(d!)^2\le(2N+1)!$ and
$(2p+1)V^2(d+1)^2\le8(N+1)^7$, (47) is at most

$$
C(N+1)^7(2+\sqrt5)^{2N}(2N+1)!
e^{O((N+1)/L)}\eta_T. \tag{47a}
$$

For fixed $p$ and $d\le L^{1/2-\varepsilon}$, (45a) is
$O_p(L^{-4\varepsilon})$ and (47a) is $T^{-c_\delta+o(1)}$, by taking
$0<\sigma<13\delta/4$. Integration against the nonnegative window
proves Theorem 9.

The uniform bounds also yield a substantially larger joint range.

**Theorem 10 (joint order and degree).** Fix $0<\delta<1/13$, a
nonnegative nonzero $w\in C_c^\infty((0,1))$, and any prescribed function
$\omega(T)\to\infty$. For all sufficiently large $T$, uniformly in
$X\in[T,2T]$, $T^{12/13+\delta}\le H\le T$, integers $p,d\ge0$ with

$$
p+d\le\frac{13\delta}{32}\frac{\log T}{\log\log T},
\qquad
p+(d+1)^2\le\frac{\log T}{\omega(T)},
\tag{48}
$$

and every nonzero real polynomial $P$ of degree at most $d$, (38) holds.
The coefficients may depend on $T,X,H$. More precisely, write

$$
\mathcal I_{p,P}=\frac{1}{HL^{2p+4}}
\int_{\mathbb R}w\left(\frac{t-X}{H}\right)
\mathcal L_p[Z](t)[P(D/L)Z(t)]^2\,dt,
\qquad
\overline B^+_{p,P}=\frac1H\int_{\mathbb R}
w\left(\frac{t-X}{H}\right)B_t^+(P)\,dt.
$$

Then $\mathcal I_{p,P}/\overline B^+_{p,P}$ equals

$$
1+O_{w,\delta}\bigl(\omega(T)^{-2}\bigr)
+O_{w,\delta}\bigl(T^{-13\delta/16+o(1)}\bigr).
\tag{49}
$$

**Proof.** Choose $\sigma=13\delta/8$ in (46), which is in $(0,1/4)$.
Then $\eta_T\ll T^{-13\delta/8}$. If
$N\le a_\delta L/\log L$, $a_\delta=13\delta/32$, Stirling's formula
bounds the logarithm of the prefactor in (47a) by
$(2a_\delta+o(1))L$. Thus (47a) is
$T^{-13\delta/16+o(1)}$, uniformly in all the parameters. The second
condition in (48) makes (45a) $O(\omega(T)^{-2})$. The positive comparison
is the window average of $B_t^+(P)$ and is strictly positive by (44).
Combining the main-term and remainder bounds proves (49) and (38).
$\square$

For example, with fixed exponent margins, the theorem permits fixed $p$
and $d\le L^{1/2-\eta}$; bounded $d$ and $p\le L^{1-\eta}$; or
$p+1\asymp d+1$ with $p+d\le L^{1/2-\eta}$, for $\eta>0$.
A sharper simultaneous example within (48) is

$$
p=\left\lfloor\frac{13\delta}{64}\frac{L}{\log L}\right\rfloor,
\qquad
d\le\frac{L^{1/2}}{(\log L)^\eta},\qquad \eta>0.
\tag{50}
$$

Here $d=o(p)$ and
$V/L=O((\log L)^{-1}+(\log L)^{-2\eta})\to0$.
The source derivative cost limits $p$ in this proof; no growing contour
is applied to the arithmetic remainder. The polynomial negative-norm
inequalities are elementary tools, not separate novelty claims. These
remain long smooth averages and do not supply signed localization at a
selected height or pointwise Laguerre inequalities.


### 8.4. Transfer to actual translated Hardy values

The polynomial theorem does not license replacing its filter by a formal
infinite differential operator. We next carry out one such replacement
with both its source error and its error in the signed integral controlled.
For $p\ge0$, define the explicit profile

$$
\Psi_p(a)=\frac{6}{\pi^2(2p+1)!(2p+3)}
\int_0^1(1-u)^{2p+3}\cos(au)\,du.
\tag{51}
$$

**Corollary 11 (translated Hardy observers).** Fix $p\ge0$ and the
parameters $\delta,w$ of Theorem 10. Put
$c_\delta=13\delta/(10240e)$. Uniformly for $X\in[T,2T]$,
$T^{12/13+\delta}\le H\le T$, and real
$|h|\le c_\delta/\log\log T$, set $a=hL$ and

$$
\mathcal J_p(a)=\frac1H\int_{\mathbb R}
 w\left(\frac{t-X}{H}\right)\rho_t^{2p+4}\Psi_p(a\rho_t)\,dt,
\qquad \rho_t=\frac{\log(t/(2\pi))+2\kappa}{L}.
\tag{52}
$$

Then $\mathcal J_p(a)\gg_{p,w}(1+a^2)^{-1}$ and

$$
\frac{\displaystyle\int_{\mathbb R}
 w\left(\frac{t-X}{H}\right)\mathcal L_p[Z](t)Z(t+h)^2\,dt}
 {HL^{2p+4}\mathcal J_p(a)}
 =1+O_{p,w,\delta}\bigl((\log\log T)^{-2}\bigr).
\tag{53}
$$

In particular the integral is positive for sufficiently large $T$.
Order $p$ is fixed in this corollary.

For $p=0$, the integrand is the product of two shifted zeta absolute
squares. The profile is the classical one in [6, Theorem 1.2], and
Kovaleva [12, Theorem 1.1 and Corollary 1.3] evaluates such correlations
for much larger real separations. We make no novelty claim for that case.
For $p\ge1$, the required derivatives split the first pair of Hardy
arguments; they cannot be obtained by differentiating a theorem restricted
to two paired squares. The assertion here is the controlled transfer of
the signed Laguerre form, using the fixed microscopic source domain in [7].

**Proof.** We give the source transfer and the explicit-main comparison
separately. Put

$$
b_\delta=\frac{13\delta}{128},\qquad
D=\left\lfloor b_\delta\frac{L}{\log L}\right\rfloor,
\qquad P_D(z)=\sum_{j=0}^D\frac{a^jz^j}{j!}.
\tag{54}
$$

Our choice of $c_\delta$ ensures $|a|\le(D+1)/(40e)$ for large $T$.

First we need a uniform upper bound for the actual raw derivatives:

$$
\|Z^{(j)}\|_{L^4(w_{X,H})}
\le C_{w,\delta}H^{1/4}L^{j+1}2^{-j},
\qquad 0\le j\le D+1,
\quad w_{X,H}(t)=w((t-X)/H).
\tag{55}
$$

To prove it, apply the finite arithmetic operator (41) to
$R_j(y)=\prod_{i=1}^4y_i^j$. The phase of the four equal derivatives is
$i^{4j}=1$. The leading absolute integral is at most
$C\rho^{4j+4}16^{-j}$. For a monomial of degree $h$ in
$Q_2^bQ_4^c$, its $y$ derivatives cost at most $j^h$, and its remaining
monomial has absolute value at most $(\rho/2)^{4j-h}$; an allocation
exceeding $j$ in one variable vanishes. The $r$ length derivatives have
falling factorial bounded by $(4j+4)^r$ and leave the power
$\rho^{4j+4-h-r}$. With $k=h+r$, the degree-$k$ correction is consequently
bounded by
$C^{k+1}16^{-j}\rho^{4j+4}((j+1)/L)^k$.
The coefficient bound in (40) and $(j+1)/L=o(1)$ sum the corrections
from degree two as a geometric series. Also
$\rho^{4j+4}=\exp(O((j+1)/L))=O(1)$.

The source remainder in (46), differentiated on the unit microscopic
circles, is at most $\eta_T(j!)^4$. Since

$$
\eta_T(j!)^4 16^j
\le T^{-13\delta/8+4b_\delta+o(1)}
=T^{-39\delta/32+o(1)},
$$

it is smaller than the preceding upper bound. The fourth moment is
nonnegative, so this proves (55). This argument uses the special monomial
$R_j$, not an arbitrary degree-$j$ filter. It therefore does not require
$(j+1)^2=o(L)$. Choose one fixed nonnegative $W\in C_c^\infty((0,1))$ such that
$W((t-X)/H)\ge w((t-X-v)/H)$ for $|v|\le1$ and sufficiently large
$H$. Such $W$ exists because the support of $w$ has positive distance
from the endpoints. The same proof with $W$ gives (55) uniformly for
$Z^{(j)}(t+uh)$, $0\le u\le1$, in the original weighted norm.

Let $W_D=P_D(D_t/L)Z$. Taylor's integral remainder and Minkowski's
inequality now give

$$
\|Z(\,\cdot+h)-W_D\|_{L^4(w_{X,H})}
\ll_{w,\delta}H^{1/4}L E_D,
\qquad
E_D=\frac{(|a|/2)^{D+1}}{(D+1)!}
\le80^{-(D+1)}.
\tag{56}
$$

The last inequality follows from $n!\ge(n/e)^n$.
The translated $Z$ has norm $O_{w,\delta}(H^{1/4}L)$ by the same
weight domination, and $W_D$ has norm
$O_{w,\delta}(H^{1/4}L(1+E_D))$ by (56).
For fixed $p$, Hölder applied to (16) and (55) also gives
$\|\mathcal L_p[Z]\|_{L^2(w_{X,H})}\ll_{p,w,\delta}H^{1/2}L^{2p+2}$.
Hence the normalized error in replacing $Z(t+h)^2$ by $W_D(t)^2$
in (53) is $O_{p,w,\delta}(E_D(1+E_D))$. This is an estimate for
the actual signed integral and assumes no pointwise sign for
$\mathcal L_p[Z]$.

Write $G_t(b)=L^{-4}F_\ell(ib/L)$ and define

$$
K_{p,t}(u,v)=
\left.\frac{(-1)^p}{(2p)!}
(\partial_{b_1}-\partial_{b_2})^{2p}G_t(b)
\right|_{b_1=b_2=0,\ b_3=u,\ b_4=v}.
\tag{57}
$$

The complete main term for $W_D$ is the rectangular Taylor polynomial
of $K_{p,t}$ through degree $D$ in each variable, evaluated at $(a,a)$.
In its source remainder the Taylor factorials cancel the two observer
Cauchy factorials, giving the bound

$$
(2p+1)\eta_T\left(\sum_{j=0}^D|a|^j\right)^2
\le(2p+1)\eta_T(D+1)^2\max(1,|a|)^{2D}
=T^{-91\delta/64+o(1)}.
\tag{58}
$$

Only this finite polynomial is applied to the source, and its circles
remain the fixed unit circles of Section 8.3.

We next estimate the Taylor tail of the explicit main function, without
applying [7] at larger shifts. Put $R=(D+1)/5$. On the torus
$|b_j|=jR$, all pairwise differences have modulus at least $R$.
The pole-removed arithmetic factors are bounded because $R/L=o(1)$.
Thus (39) and removability at collisions give

$$
\sup_{|b_j|\le R}|G_t(b)|
\le C R^{-4}\exp(5R+O(R/L)).
$$

Cauchy in the first two variables bounds $K_{p,t}$ on
$|u|,|v|\le R$ by
$C(2p+1)R^{-2p-4}\exp(5R+O(R/L))$.
For $q=|a|/R\le1/(8e)$, the sum of Taylor coefficients outside the
rectangle is at most $2q^{D+1}/(1-q)^2$ times this bound. Since
$5R=D+1$, replacing that rectangular polynomial by $K_{p,t}(a,a)$
costs $O_p(8^{-D})$, uniformly in $t$.

There is a particularly simple comparison with the positive leading
function for these real shifts. Set
$G_t^+(b)=L^{-4}H_s(ib/L)/\zeta(2)$. In the difference $G_t-G_t^+$,
each subset summand has the arithmetic bracket
$A_S(ib/L)e^{-\kappa U_S(ib/L)}-1=O((\max|b_j|/L)^2)$,
exponential $e^{sU_S(ib/L)/2}$, and denominator
$\zeta(2)D_S(ib)$.
For real $|a|\ge8$, use circles centered at $(0,0,a,a)$ with radii
$(1,2,1,2)$. On their distinguished boundary, within-cluster differences
are at least one and cross-cluster differences are at least $|a|-4$.
Two subsets have four cross-cluster factors in their denominator; the
other four have two within-cluster and two cross-cluster factors.
Every denominator therefore has modulus at least $c a^2$.
The imaginary parts of all four $b_j$ stay bounded independently of real
$a$, so the exponentials have bounded modulus. The arithmetic bracket
is $O((1+a^2)/L^2)$, proving a boundary bound $C/L^2$ for the difference.
Holomorphy and Cauchy give the same bound, with constant depending on $p$,
after the functional in (57) is applied. For $|a|\le8$, one fixed
unequal-radius polydisc enclosing the relevant points gives the same result.
All these contours stay in the fixed physical analytic neighborhood.
Consequently

$$
K_{p,t}(a,a)=\rho_t^{2p+4}
\mathfrak B_p(e^{ia\rho_t x},e^{ia\rho_t x})+O_p(L^{-2}).
\tag{59}
$$

Again, this is a contour argument for the explicit main function only.

For completeness, sum and difference coordinates in the kernel (21)
evaluate its exponential quadratic form exactly as
$\mathfrak B_p(e^{i\alpha x},e^{i\alpha x})=\Psi_p(\alpha)$.
Indeed its first term integrates to
$2\int_0^1(1-u)^{m+1}\cos(\alpha u)\,du$, whereas its second term is
$2/(m+1)$ times that integral, with $m=2p+2$.
Its positive sine spectrum and (44c), with fixed $p$, also compare it
with the Dirichlet negative norm. The primitive identity gives

$$
\|e^{i\alpha x}\|_{H^{-1}_D}^2
=\frac{1-(2\sin(\alpha/2)/\alpha)^2}{\alpha^2}
\asymp\frac1{1+\alpha^2},
\tag{60}
$$

with the continuous value $1/12$ at zero. This proves
$\Psi_p(\alpha)\gg_p(1+\alpha^2)^{-1}$ and hence the stated lower
bound for $\mathcal J_p(a)$.
Combining (56), (58), the main Taylor tail, and (59), the normalized
moment differs from $\mathcal J_p(a)$ by

$$
O_{p,w,\delta}\left(L^{-2}+8^{-D}
+T^{-91\delta/64+o(1)}\right).
$$

Divide by its positive lower bound. Since
$|a|\ll_\delta L/\log L$ and
$D\asymp_\delta L/\log L$, the relative error is
$O_{p,w,\delta}((\log L)^{-2})$, as asserted. $\square$

Corollary 11 supplies actual translated Hardy values as observers. It
asserts positivity for each translated square, and hence for positive
mixtures of such squares. It does not assert positivity for arbitrary
signed linear combinations of translated values, nor does it construct
an observer concentrated in a selected interval where a Laguerre
expression might be negative. The signed localization step remains open.


## 9. Exact checks and reproducibility

The proof is analytic and does not depend on numerical experiments. The supplement includes the four original CPU-only exact-algebra controls, three symbolic checks for the arithmetic operator, and the three additional uniform-order controls described below. A further fixed control checks the translated-observer normalization and collision contours. The scripts use Python, SymPy and mpmath; the translated-kernel quadrature also uses SciPy.

`hardy246_octahedron_control.py` evaluates the original paired polynomial integral (4) and independently integrates on the octahedron. Six prescribed examples agree as exact rational numbers:

| $(k,l,m,n)$ | $H(k,l,m,n)$ |
|---|---:|
| $(0,0,0,0)$ | $1/2$ |
| $(1,1,1,1)$ | $1/1120$ |
| $(2,1,1,0)$ | $-1/3360$ |
| $(4,1,1,0)$ | $1/40320$ |
| $(3,3,1,1)$ | $19/1774080$ |
| $(6,4,0,0)$ | $-3/931840$ |

`hardy247_parity_identity.py` applies all 16 coordinate sign substitutions directly to (10), constructs the common denominator from the transformed factors, and verifies the polynomial identity (11). It does not insert the closed projection formulae (12)–(13). The supplement gives the commands and saved exact outputs. No parameter survey or floating sign test is used to support the all-order theorem.

`hardy248_laguerre_constants.py` independently expands (16) using the original paired integral and checks (18) at $(p,q)=(1,1)$ and $(1,2)$, obtaining $\pi^2C_{1,1}=1/336$ and $\pi^2C_{1,2}=1/10800$ exactly.

`hardy249_filtered_laguerre_control.py` compares two independent evaluations of (21): expansion through the original paired integral, and polynomial integration of the kernel on the two triangles of its square. For $p=1$, $P(z)=1+z+2z^2$, both give $\pi^2B_p(P)=211/7560$. For $p=2$, $P(z)=1-2z+z^3$, both give $10434929/9686476800$. These mixed-parity checks verify constants; the all-filter positivity follows from (22).

`hardy616_six_swap_exact_audit.py` checks all six swap orientations, the first two arithmetic Taylor degrees, and the coincident-shift leading and subleading scalar coefficients. `hardy617_cross_power_invariants.py` verifies the invariant representation of the cross-difference power sums through degree eight. `hardy619_operator_identity.py` compares direct coefficient differentiation with the polynomial-observer differential identity used in (43), for $p=1$, $P(z)=1+z+2z^2$, the operators $Q_2,Q_4,Q_2^2$, and three length-derivative orders. These finite checks test signs and normalizations; the all-order analytic factorization and the uniform estimates are proved in Section 8.

`hardy636_uniform_order_controls.py` checks the observer differential identity and length-derivative falling-factorial normalization in three further cases, and compares four sine eigenvalues computed by a recurrence and by 90-digit quadrature. The largest discrepancy is below $4\times10^{-74}$. `hardy637_absolute_fibre_control.py` uses independent coordinates with Jacobian $1/4$ to verify the full absolute moments $\pi^2\int|x_1-x_2|^q d\mu=6/[(q+1)(q+3)(q+4)]$ for $q=0,1,2,5,8$, and checks three split row integrals exactly. These finite checks test normalization and odd absolute powers; Lemmas 9.2–9.5 provide the uniform proof.

`hardy650_kernel_norm_controls.py` compares the primitive expression for the Dirichlet negative norm with independent Green-kernel integrations at three fixed polynomials, checks the exact derivative/variance identity, and compares five signed first-pair fibres with independent octahedron integrations. Odd powers vanish. Four nontrivial observer-derivative allocations, including nonzero endpoint values, also agree under three length-derivative orders. These exact identities check the norm-relative proof; the uniform inequalities follow from Lemmas 9.2, 9.4 and 9.5.

`hardy654_translation_controls.py` classifies all six clustered denominators exactly, compares the explicit exponential profile with independent two-dimensional kernel quadrature, and checks its primitive negative-norm formula. An exact finite Taylor observer agrees under octahedron and kernel integration. Fixed 65-digit Cauchy extractions at two resolutions verify the leading normalization and the complete arithmetic correction at three prescribed cases. These floating controls are not interval certificates or evidence for an asymptotic estimate; Corollary 11 is proved analytically.

## 10. Scope

The result settles the sign and nonvanishing question for the constants in [1, Conjecture 1], using their unconditional arithmetic moment formula. Its proof explains the signs through an exact recurrence and a zero-sum symmetry. The explicit Laguerre kernel gives the uniform and smooth-window results of Theorems 7–8. Retaining the complete six-term arithmetic main term and separating its differentiation from the source error improves the degree range to $(\log T)^{1/2-\varepsilon}$ for each fixed Laguerre order, and yields the joint range (48), including the example (50), in Theorems 9–10. The uniform spectrum and norm-relative signed fibre estimate keep the growing order explicit in the coercivity and in every differentiated correction. Corollary 11 supplies actual translated Hardy squares at fixed Laguerre order, with the source Taylor error and the signed-integral approximation both controlled. The smooth windows still have length at least $T^{12/13+\delta}$; the arithmetic error exponent in [7] is unchanged. None of these averages determines signs at individual heights, the location of every zeta zero, or the Riemann hypothesis.

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