# PP296 Davenport–Heilbronn twin: positive rungs through 227

**Claim.** $W_k(x)>0$ for every real $x>0$ and integer $1\le k\le227$.
This uses the independently certified complete T260 zero census. The
finite inequalities for this extension are checked with integers and exact
rationals by verify_227.py; CERTIFICATE_227.json records the strict margins
and upstream SHA-256 identities.

## Analytic setup and multiplicities

Let $a_{n+5}=a_n$, $(a_1,\ldots,a_5)=(1,\kappa,-\kappa,-1,0)$, where

$$
\kappa=(\sqrt{10-2\sqrt5}-2)/(\sqrt5-1),\quad
f(s)=\sum_{n\ge1}a_n n^{-s},\quad
F(s)=(5/\pi)^{s/2}\Gamma((s+1)/2)f(s).
$$

The exact character functional equation gives $F(s)=F(1-s)$; real
coefficients give $F(\bar s)=\overline{F(s)}$. The order-one completion is
entire. For real $s>0$, each grouped block

$$
(5m+1)^{-s}-(5m+4)^{-s}
+\kappa\{(5m+2)^{-s}-(5m+3)^{-s}\}
$$

is strictly positive. Reflection extends $F(s)>0$ across the whole real
axis. Thus there are no real completed zeros.

For an upper-half-plane zero $\rho=\sigma+it$, put
$q=\rho(1-\rho)=t^2+\sigma(1-\sigma)+it(1-2\sigma)$ and
$R_x(q)=4xq/(x+q)^2$. The fold has **one node per upper zero**, with
its zero multiplicity. A simple on-line zero gives one positive real node.
An off-line quartet gives two conjugate upper nodes, each of multiplicity
one, contributing $2\operatorname{Re}R_x(q)^k$. The normalization is

$$
W_k(x)=\frac{(2k-1)!}{(4x)^k}
\sum_{\operatorname{Im}\rho>0}m_\rho R_x(q_\rho)^k.
$$

The prefactor is positive. The order-one zero distribution implies
$\sum_\rho |q_\rho|^{-1}<\infty$, so the folded sum converges absolutely
for each fixed $x>0$ and $k\ge1$.

## Complete low census and analytic high tail

The independent T260 receipts certify 179 upper zeros below height 260:
169 exactly on the line and five simple off-line quartets (ten upper
zeros). A line-centered box with one zero contains a zero fixed on the
line, because functional-equation and conjugation symmetry preserve the
box and its zero is unique. The direct strip/base counts prove
completeness. The separate census receipt auditor verifies those trace
inequalities; verify_227.py pins every source and receipt hash in
REPLAY_SUMMARY.json and rechecks the box geometry it uses.

The positive root of $\kappa^2+2\varphi\kappa-1=0$, with
$\varphi=(1+\sqrt5)/2$, satisfies $0<\kappa<2/7$.
The separate certify_227_strip.py and strip_227_receipt.json certify the
sharper zero-free line $\operatorname{Re}s=2799/2000$; verify_227.py
independently recomputes their majorant by pure-integer 2000th-root
bisection and requires their exact SHA-256 identities. At this line,
the period-five coefficient majorant $c_n=(1,2/7,2/7,1,0)$ satisfies

$$
\sum_{n=2}^{50}c_n n^{-2799/2000}
+\frac{18}{7}\left(50^{-2799/2000}
+\frac{400}{799}50^{-799/2000}\right)
<\frac{999}{1000}<1.
$$

Grouping complete periods and integrating the decreasing power gives
the displayed tail. The checker bounds each 2000th root by adjacent
integer 2000th powers, then compares exact rationals. Thus $f(s)\ne0$ for
$\operatorname{Re}s\ge2799/2000$. Reflection yields
$-799/2000<\operatorname{Re}\rho<2799/2000$, equivalently
$|\sigma-1/2|<1799/2000$. Hence for $t\ge260$,

$$
\tan|\arg q|
\le\frac{(1799/1000)t}{t^2+1/4-(1799/2000)^2}
\le\frac{1870960000}{270397763599}.
$$

The middle expression decreases for $t\ge260$. For $x>0$ and
$\operatorname{Re}q>0$, the angle of $x+q$ lies between zero and that of
$q$, giving $|\arg R_x(q)|\le|\arg q|$. Also
$|\arg q|=\arctan(\tan|\arg q|)<\tan|\arg q|$ for a non-real node.
Exact rational checks establish

$$
2(227)\frac{1870960000}{270397763599}
<\frac{1231847548}{392109375}<\pi.
$$

The rational lower bound on $\pi$ follows from
$\pi=16\arctan(1/5)-4\arctan(1/239)$ and
$z-z^3/3+z^5/5-z^7/7<\arctan z<z$ for $0<z<1$.
Every high-tail conjugate pair therefore
has positive real contribution through $k=227$. The four off-line
quartets other than the lowest have $\tan|\arg q|<31/10000$, satisfying
the same sector test. All on-line terms are positive.

## The sole exceptional quartet

The remaining east/west boxes have indices 170/171 and height near
85.699. The neighboring line boxes have indices 43 and 44. Exact
$10^{-7}$ box arithmetic, including both coordinates, establishes

$$
6907<l<6908,\quad 7682<u<7683,\quad
7344<a=\operatorname{Re}q<|q|<7345,\quad
\beta=|\operatorname{Im}q|>0.
$$

Here $l,u$ are positive real folded nodes. For any line node $v>0$,

$$
\frac{R_x(v)}{|R_x(q)|}
=\frac{v|x+q|^2}{|q|(x+v)^2}.
$$

The integer quadratics below are conservative lower bounds for the
cross-multiplied assertion that this ratio exceeds $101/100$:

$$
\begin{aligned}
D_l(x)&=-51145x^2-104328920x+1851258259120,\\
D_u(x)&=26355x^2-115868670x-2357632507005.
\end{aligned}
$$

$D_l$ is concave and positive at both endpoints of $[0,3600]$:
the values are 1851258259120 and 812834947120. Also
$D_u(15000)=1834212442995>0$, its derivative there is
674781330, and it is convex; hence it is positive for every
$x\ge15000$. In these outer ranges, one line term exceeds
$(101/100)^k|R_x(q)|^k$. For $k\ge219$, Bernoulli's inequality gives
$(101/100)^k\ge1+k/100>2$. One positive line term therefore strictly
dominates the sum of magnitudes of **both** exceptional conjugate terms.

For $3600\le x\le15000$, choose $q=a+i\beta$. Its phase is

$$
\phi(x)=\arg R_x(q)
=\arctan(\beta/a)-2\arctan(\beta/(x+a)),\qquad
\phi'(x)=\frac{2\beta}{(x+a)^2+\beta^2}>0.
$$

Exact box endpoints and the rational inequalities
$z-z^3/3<\arctan z<z$ for $0<z<1$ give
$-2464/10^6<\phi(x)<2467/10^6$. Therefore
$|\phi(x)|<3/1000$ and $2(227)(3/1000)<\pi$; the exceptional
pair itself contributes positively throughout the middle range.

All other nodes contribute positively in each $x$ range. For earlier
rungs, the checker takes the maximum exact-rational angle bound over
**all 179** below-260 boxes, including the exceptional pair, and proves
it is less than $9/1250$. The high-tail bound is smaller too, and
$2(218)(9/1250)<\pi$. This completes every $x>0$ and every integer
$1\le k\le227$.

## Reproduction and limit

Run: python verify_227.py. The default T260 input is a sibling
independent_mpfr (vendored package) or independent_zero_eval (working
checkout); --evidence-dir PATH overrides it. The sharper-strip source
and receipt may sit beside the checker, in sibling strip_227 (vendored
package), or in sibling next_census (working checkout); --strip-dir PATH
overrides that location.
Python's standard library is sufficient. The output is
CERTIFICATE_227.json. This extension uses
the already audited complete census and exact functional equation as
inputs; it is not a second zero-evaluation implementation.

The present coarse high-tail sector estimate stops at 227:
$2(228)(1870960000/270397763599)>22/7>\pi$. This is a limitation of
this sufficient bound and says nothing about the sign of $W_{228}$ or the
first failed rung.