RH is Π^0_2 via Stagewise Certificates: A Certificate Calculus for the Riemann Ξ–Function R

1. Main Statement and $\mathrm{\Xi }$-Plane Reformulation of RH

1.1. The Completed Zeta Function and $\mathrm{\Xi }$

Definition 1.1

(Completed zeta function and $\mathrm{\Xi }$). Define

$$\xi \left(s\right):=\frac{1}{2}s(s-1){\pi }^{-s/2}\mathrm{\Gamma }\left({\displaystyle \frac{s}{2}}\right)\zeta \left(s\right)(s\in \mathbb{C}),$$

and

$$\mathrm{\Xi }\left(z\right):=\xi \left(\frac{1}{2}+\mathrm{i}z\right)(z\in \mathbb{C}).$$

Remark 1.1.

It is classical that ξ is entire and satisfies $\xi (1-s)=\xi \left(s\right)$ and $\xi \left(\overline{s}\right)=\overline{\xi \left(s\right)}$. Consequently Ξ is entire and satisfies $\mathrm{\Xi }(-z)=\mathrm{\Xi }\left(z\right)$ and $\mathrm{\Xi }\left(\overline{z}\right)=\overline{\mathrm{\Xi }\left(z\right)}$.

1.2. The RH-Relevant Region in the $\mathrm{\Xi }$-Plane

Definition 1.2

(RH-relevant region). Define

$$\mathcal{U}:=\{z=x+\mathrm{i}y\in \mathbb{C}:x>0,0<y<\frac{1}{2}\}.$$

Lemma 1.1

(RH in the $\mathrm{\Xi }$-plane). RH is equivalent to $Z(\mathrm{\Xi };\mathcal{U})=⌀$.

Proof. 

A nontrivial zero $\rho =\beta +\mathrm{i}\gamma$ of $\zeta$ is a zero of $\xi$ and corresponds to a $\mathrm{\Xi }$-zero

$$z=\gamma +\mathrm{i}\left(\frac{1}{2}-\beta \right),$$

since $\frac{1}{2}+\mathrm{i}z=\beta +\mathrm{i}\gamma$. Thus $\beta =\frac{1}{2}$ holds iff z is real. Conversely, any nonreal zero $z=x+\mathrm{i}y$ of $\mathrm{\Xi }$ yields a zero of $\xi$ off the critical line. By the symmetries $\mathrm{\Xi }(-z)=\mathrm{\Xi }\left(z\right)$ and $\mathrm{\Xi }\left(\overline{z}\right)=\overline{\mathrm{\Xi }\left(z\right)}$, any nonreal zero produces one in $\mathcal{U}$. □

1.3. Main Theorem: RH as a ${\mathrm{\Pi }}_2^0$ Stage Sweep

Theorem 1.1

(RH is equivalent to a ${\mathrm{\Pi }}_2^0$ sweep statement). There exist:

(i)

a countable family of open stage windows ${\left\{{\mathsf{\Omega }}_{j,k}\right\}}_{j\ge 1,k\in \mathbb{Z}}$ covering $\mathcal{U}$ with $\overline{{\mathsf{\Omega }}_{j,k}}\subset \mathcal{U}$ for each $(j,k)$;

(ii)

a decidable predicate $\mathrm{Cert}(j,k,c)$ on $(j,k,c)\in {\mathbb{N}}_{\ge 1}\times \mathbb{Z}\times \mathbb{N}$;

such that:

(a)

(Soundness) if $\mathrm{Cert}(j,k,c)$ holds then $Z(\mathrm{\Xi };{\mathsf{\Omega }}_{j,k})=⌀$;

(b)

(Completeness on zero-free closures) if $Z(\mathrm{\Xi };\overline{{\mathsf{\Omega }}_{j,k}})=⌀$ then there exists c with $\mathrm{Cert}(j,k,c)$;

(c)

defining

$$\mathrm{StageOK}(j,k):⟺\exists c\mathrm{Cert}(j,k,c),\mathsf{CS}:⟺\forall j\ge 1\forall k\in \mathbb{Z}\mathrm{StageOK}(j,k),$$

one has

$$\mathrm{RH}⟺\mathsf{CS}.$$

In particular, $\mathsf{CS}$ has prenex form $\forall j\forall k\exists c\mathrm{Cert}(j,k,c)$ with decidable matrix, hence $\mathsf{CS}$ is ${\mathrm{\Pi }}_2^0$ and RH is ${\mathrm{\Pi }}_2^0$.

Remark 1.2

(What is and is not claimed). 1.1 does not assert RH. It asserts that RH is equivalent to an explicit ${\mathrm{\Pi }}_2^0$ sentence built from a fixed decidable certificate calculus whose soundness and completeness properties are proved here.

2. A Rational Stage Cover of $\mathcal{U}$

(overlapping cover)).Definition 2.1 (Stage parameters and windows Fix a rational base height $T_0\in {\mathbb{Q}}_{>0}$ and define dyadic heights

$$T_k:=2^k{T}_0(k\in \mathbb{Z}).$$

Fix a computable rational sequence ${\left({\sigma }_j\right)}_{j\ge 1}\subset \mathbb{Q}\cap (0,\frac{1}{4})$ with ${\sigma }_j\downarrow 0$.

For $(j,k)\in {\mathbb{N}}_{\ge 1}\times \mathbb{Z}$ define the closed and open stage rectangles

$$R_{j,k}:=\left\{x+\mathrm{i}y:\frac{1}{2}{T}_k\le x\le 2T_k,{\sigma }_j\le y\le \frac{1}{2}-{\sigma }_j\right\},{\mathsf{\Omega }}_{j,k}:=int\left(R_{j,k}\right).$$

Then $\overline{{\mathsf{\Omega }}_{j,k}}=R_{j,k}\subset \mathcal{U}$ and the corners of $R_{j,k}$ lie in $\mathbb{Q}+\mathrm{i}\mathbb{Q}$.

Lemma 2.1

(Stage cover of $\mathcal{U}$). One has

$$\mathcal{U}\subseteq \bigcup _{j\ge 1}\bigcup _{k\in \mathbb{Z}}{\mathsf{\Omega }}_{j,k}.$$

Proof. 

Let $z=x+\mathrm{i}y\in \mathcal{U}$, so $x>0$ and $0<y<\frac{1}{2}$. Choose $k\in \mathbb{Z}$ such that $\frac{1}{2}{T}_k<x<2T_k$ (possible since $T_k\to 0$ as $k\to -\infty$ and $T_k\to \infty$ as $k\to \infty$). Choose j such that ${\sigma }_j<min\{y,\frac{1}{2}-y\}$ (possible since ${\sigma }_j\downarrow 0$). Then ${\sigma }_j<y<\frac{1}{2}-{\sigma }_j$ and $\frac{1}{2}{T}_k<x<2T_k$, hence $z\in {\mathsf{\Omega }}_{j,k}$. □

3. Winding Number, Argument Principle, and Certificate Blueprint

3.1. Winding Number and Argument Principle

Definition 3.1

(Winding number). Let $\mathrm{\Gamma }:[0,1]\to \mathbb{C}\setminus \left\{0\right\}$ be a piecewise $C^1$ loop with $\mathrm{\Gamma }\left(0\right)=\mathrm{\Gamma }\left(1\right)$. Define

$$wind(\mathrm{\Gamma },0):={\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_0^1{\displaystyle \frac{{\mathrm{\Gamma }}^{\prime }\left(t\right)}{\mathrm{\Gamma }\left(t\right)}}dt.$$

Theorem 3.1

(Argument principle (winding form)). Let $R\subset \mathbb{C}$ be a closed rectangle with $\mathsf{\Omega }=int\left(R\right)$. Let F be holomorphic on a neighborhood of $\overline{R}$. If $F\ne 0$ on $\partial R$, then

$$\#Z(F;\mathsf{\Omega })=wind\left(F\right(\partial R),0),$$

where zeros are counted with multiplicity and $\partial R$ is positively oriented.

Proof. 

Standard: $\#Z(F;\mathsf{\Omega })={\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_{\partial R}{F}^{\prime }\left(z\right)/F\left(z\right)dz$, and rewriting via a parametrization yields 3.1. □

3.2. Certificate Idea (Informal)

A stage certificate for $(j,k)$ will certify:

(1)

a finite disk cover of the boundary $\partial R_{j,k}$ by small z-disks,

(2)

validated enclosures of $\mathrm{\Xi }$ on each disk (a $\mathrm{\Xi }$-image disk),

(3)

avoidance of 0 by each image disk (hence $\mathrm{\Xi }\ne 0$ on $\partial R_{j,k}$),

(4)

a certified winding computation concluding $wind(\mathrm{\Xi }(\partial R_{j,k}),0)=0$.

Decidability requires explicit remainder bounds for evaluating $\mathrm{\Xi }$ on disks.

4. Decidability I: Rational Disks with Certified Rational Transcendental Bounds

4.1. Rational Disks

Definition 4.1

(Rational disks). Arational diskis a pair $(c,r)$ with $c\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ and $r\in {\mathbb{Q}}_{\ge 0}$, denoting

$$\overline{B}(c,r):=\{w\in \mathbb{C}:|w-c|\le r\}.$$

4.2. Certified Rational Primitives

Definition 4.2

(Certified rational primitives for $\sqrt{·}$ and $e^{(·)}$). We fix once and for all total computable functions (terminating algorithms)

$$\mathsf{Qsqrtup}:{\mathbb{Q}}_{\ge 0}\times {\mathbb{N}}_{\ge 1}\to {\mathbb{Q}}_{\ge 0},\mathsf{Qexpup}:\mathbb{Q}\times {\mathbb{N}}_{\ge 1}\to {\mathbb{Q}}_{>0},$$

such that for all inputs $(q,n)$ and $(x,n)$:

$$\sqrt{q}\le \mathsf{Qsqrtup}(q,n)\le \sqrt{q}+2^{-n},e^x\le \mathsf{Qexpup}(x,n)\le e^x+2^{-n}.$$

In addition, for verifier-auditable refinement arguments we require the following monotonicity properties:

(i)

(Monotone in the real input) if $x\le y$ then $\mathsf{Qexpup}(x,n)\le \mathsf{Qexpup}(y,n)$;

(ii)

(Monotone in precision) if $n^{\prime }\ge n$ then $\mathsf{Qexpup}(x,n^{\prime })\le \mathsf{Qexpup}(x,n)$ and $\mathsf{Qsqrtup}(q,n^{\prime })\le \mathsf{Qsqrtup}(q,n)$.

Definition 4.3

(Certified square-root lower bound). Define

$$\mathsf{Qsqrtlo}(q,n):=max\{0,\mathsf{Qsqrtup}(q,n)-2^{-n}\}\in {\mathbb{Q}}_{\ge 0}(q\in {\mathbb{Q}}_{\ge 0},n\ge 1).$$

Then for all $q,n$,

$$\mathsf{Qsqrtlo}(q,n)\le \sqrt{q}\le \mathsf{Qsqrtup}(q,n).$$

Definition 4.4

(Certified modulus bounds for rational complex numbers). For $c=a+\mathrm{i}b\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ and $n\in {\mathbb{N}}_{\ge 1}$ define

$${\left|c\right|}_{\mathrm{up},n}:=\mathsf{Qsqrtup}(a^2+b^2,n),{\left|c\right|}_{\mathrm{lo},n}:=\mathsf{Qsqrtlo}(a^2+b^2,n).$$

Then

$${\left|c\right|}_{\mathrm{lo},n}\le \left|c\right|\le {\left|c\right|}_{\mathrm{up},n}{,\left|c\right|}_{\mathrm{up},n}\le \left|c\right|+2^{-n}.$$

Lemma 4.1

(Computability/Termination of the Primitive Bounds). The maps $\mathsf{Qsqrtup}$, $\mathsf{Qsqrtlo}$, $\mathsf{Qexpup}$, and $(c,n)\mapsto {\left|c\right|}_{\mathrm{up},n},{\left|c\right|}_{\mathrm{lo},n}$ are total computable functions. In particular, a verifier may call them and is guaranteed to halt.

4.3. Primitive Disk Operations (Rational Output via Certified Bounds)

Definition 4.5

(Primitive rational-disk operations (precision-parameterized)). Let $D_1=\overline{B}(c_1,r_1)$ and $D_2=\overline{B}(c_2,r_2)$ be rational disks. Fix an integer parameter $n\in {\mathbb{N}}_{\ge 1}$ that controls the tightness of certified rational upper/lower bounds (e.g. errors $\le 2^{-n}$).

Define:

(i)

$D_1\pm D_2:=\overline{B}(c_1\pm c_2,r_1+r_2)$ and $-D_1:=\overline{B}(-c_1,r_1)$;

(ii)

$\lambda D_1:=\overline{B}(\lambda c_1,\left|\lambda \right|r_1)$ for $\lambda \in \mathbb{Q}$;

(iii)

(Product) define $u_1:={\left|c_1\right|}_{\mathrm{up},n}$ and $u_2:={\left|c_2\right|}_{\mathrm{up},n}$ and set

$$D_1·D_2:=\overline{B}(c_1{c}_2,u_1{r}_2+u_2{r}_1+r_1{r}_2).$$

(iv)

(Reciprocal) if $0\notin D_2$ (i.e. $|c_2|>r_2$), define ${\ell }_2:={\left|c_2\right|}_{\mathrm{lo},n}$. If ${\ell }_2\le r_2$ the verifier must reject (insufficient certified separation from 0 at this n). If ${\ell }_2>r_2$, define

$${\displaystyle \frac{1}{D_2}}:=\overline{B}\left({\displaystyle \frac{1}{c_2}},{\displaystyle \frac{r_2}{{\ell }_2({\ell }_2-r_2)}}\right),D_1/D_2:=D_1·(1/D_2).$$

(vi)

(Exponential) let $c_1=a+\mathrm{i}b$ with $a,b\in \mathbb{Q}$, and define $U:=\mathsf{Qexpup}(a,n)$. Let $\mathsf{ExpRat}(c_1,n)\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ be any terminating rational approximation routine satisfying

$$\left|e^{c_1}-\mathsf{ExpRat}(c_1,n)\right|\le 2^{-n}.$$

Define

$$exp\left(D_1\right):=\overline{B}\left(\mathsf{ExpRat}(c_1,n),2^{-n}+U\left(\mathsf{Qexpup}(r_1,n)-1\right)\right).$$

Lemma 4.2

(Certified rational approximation of $e^c$ for $c\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$). There exists a terminating algorithm $\mathsf{ExpRat}:(\mathbb{Q}+\mathrm{i}\mathbb{Q})\times {\mathbb{N}}_{\ge 1}\to (\mathbb{Q}+\mathrm{i}\mathbb{Q})$ such that

$$|e^c-\mathsf{ExpRat}(c,n)|\le 2^{-n}.$$

Proof. 

Compute the Taylor polynomial $S_N\left(c\right)={\sum }_{k=0}^N{c}^k/k!$ in $\mathbb{Q}+\mathrm{i}\mathbb{Q}$. Use the remainder bound

$$\left|e^c-S_N\left(c\right)\right|\le {\displaystyle \frac{{\left|c\right|}^{N+1}}{(N+1)!}}{e}^{\left|c\right|}.$$

Bound $\left|c\right|$ above by ${\left|c\right|}_{\mathrm{up},n}$ and $e^{\left|c\right|}$ above by $\mathsf{Qexpup}\left(\right|c{|}_{\mathrm{up},n},n)$, and choose N by finite search until the bound is $\le 2^{-n}$. Output $S_N\left(c\right)$. □

Lemma 4.3

(Soundness of disk operations). Fix $n\in {\mathbb{N}}_{\ge 1}$. Each operation in Definition 4.5 is sound:

(a)

If $w_1\in D_1$ and $w_2\in D_2$, then $w_1\pm w_2\in D_1\pm D_2$ and $w_1{w}_2\in D_1·D_2$.

(b)

If $0\notin D_2$ and the reciprocal is defined (i.e. the verifier has ${\ell }_2:={\left|c_2\right|}_{\mathrm{lo},n}>r_2$), and if $w_1\in D_1$, $w_2\in D_2$, then $w_1/w_2\in D_1/D_2$.

(c)

If $w\in D_1$, then $e^w\in exp\left(D_1\right)$.

Proof. (a) Addition/subtraction are immediate from the triangle inequality. For multiplication, write $w_j=c_j+{\delta }_j$ with $|{\delta }_j|\le r_j$. Then

$$w_1{w}_2-c_1{c}_2=c_1{\delta }_2+c_2{\delta }_1+{\delta }_1{\delta }_2,$$

so

$$|w_1{w}_2-c_1{c}_2|\le |c_1|r_2+\left|c_2\right|r_1+r_1{r}_2\le u_1{r}_2+u_2{r}_1+r_1{r}_2,$$

since $u_j=|c_j{|}_{\mathrm{up},n}\ge \left|c_j\right|$.

(b) For $w_2\in D_2$ one has

$$|w_2|\ge |c_2|-r_2\ge {\ell }_2-r_2>0,\left|c_2\right|\ge {\ell }_2.$$

Hence

$$\left|{\displaystyle \frac{1}{w_2}}-{\displaystyle \frac{1}{c_2}}\right|={\displaystyle \frac{|w_2-c_2|}{|w_2\left|\right|c_2|}}\le {\displaystyle \frac{r_2}{({\ell }_2-r_2){\ell }_2}},$$

so $1/w_2\in 1/D_2$. Multiply by the product bound in (a) to obtain $w_1/w_2\in D_1/D_2$.

(c) Let $w=c_1+\delta$ with $\left|\delta \right|\le r_1$. Then

$$e^w=e^{c_1}{e}^{\delta },|e^{\delta }-1|\le e^{\left|\delta \right|}-1\le e^{r_1}-1.$$

Hence

$$|e^w-e^{c_1}|\le |e^{c_1}|(e^{r_1}-1)\le e^{Re\left(c_1\right)}(e^{r_1}-1)\le U(\mathsf{Qexpup}(r_1,n)-1),$$

where $U=\mathsf{Qexpup}(Re\left(c_1\right),n)\ge e^{Re\left(c_1\right)}$ and $\mathsf{Qexpup}(r_1,n)\ge e^{r_1}$. Finally,

$$|e^w-\mathsf{ExpRat}(c_1,n)|\le |e^w-e^{c_1}|+|e^{c_1}-\mathsf{ExpRat}(c_1,n)|\le U(\mathsf{Qexpup}(r_1,n)-1)+2^{-n},$$

so $e^w\in exp\left(D_1\right)$. □

5. Decidability II: Euler–Maclaurin Enclosures for $\zeta$ on Rational Disks

This section builds explicit rational-disk enclosures for $\zeta \left(s\right)$ on rational disks $D_s=\overline{B}(c,r_s)$, using: (i) the rational disk operations of Section 4 (Part 1), and (ii) explicit Euler–Maclaurin remainder bounds depending only on a lower bound for $Re\left(s\right)$ on the disk.

A key verifier constraint is that expressions involving transcendental constants are represented only through certified rational interval bounds produced by terminating routines.

5.1. Certified Rational Bounds for $log(·)$ on Integers and Rationals, and for $\pi$

Definition 5.1

(Certified log-interval routine on integers).

Remark 5.1

(Constructibility of $\mathsf{LogInt},\mathsf{LogQInt},\mathsf{PiLow}$) Each of $\mathsf{LogInt}$, $\mathsf{LogQInt}$, and $\mathsf{PiLow}$ may be implemented by an explicit terminating rational-interval algorithm. For example, $logq$ for $q\in {\mathbb{Q}}_{>0}$ can be reduced (by extracting powers of 2 and scaling into $[1,2]$) to evaluating $log(1+x)$ for $x\in [0,1]$ using the alternating Taylor series with a rational tail bound. Similarly, π admits classical rapidly convergent series (e.g. Machin-type formulas or AGM-based algorithms) with explicit rational tail bounds. We treat these as fixed primitives to keep the verifier interface clean.

We fix a total computable function

$$\mathsf{LogInt}:{\mathbb{N}}_{\ge 1}\times {\mathbb{N}}_{\ge 1}\to \mathbb{Q}\times \mathbb{Q},(n,p)\mapsto (L_{n,p},U_{n,p}),$$

such that for all $n\ge 1$ and $p\ge 1$,

$$L_{n,p}\le logn\le U_{n,p},0\le U_{n,p}-L_{n,p}\le 2^{-p}.$$

Define the associatedlog disk

$$D_{log}(n,p):=\overline{B}\left({\displaystyle \frac{L_{n,p}+U_{n,p}}{2}},{\displaystyle \frac{U_{n,p}-L_{n,p}}{2}}\right)\subset \mathbb{R}\subset \mathbb{C}.$$

Then $logn\in D_{log}(n,p)$.

Definition 5.2

(Certified log-interval routine on positive rationals). We fix a total computable function

$$\mathsf{LogQInt}:{\mathbb{Q}}_{>0}\times {\mathbb{N}}_{\ge 1}\to \mathbb{Q}\times \mathbb{Q},(q,p)\mapsto (L_{q,p},U_{q,p}),$$

such that for all $q>0$ and $p\ge 1$,

$$L_{q,p}\le logq\le U_{q,p},0\le U_{q,p}-L_{q,p}\le 2^{-p}.$$

Define the associated real log disk

$$D_{log\mathbb{Q}}(q,p):=\overline{B}\left({\displaystyle \frac{L_{q,p}+U_{q,p}}{2}},{\displaystyle \frac{U_{q,p}-L_{q,p}}{2}}\right)\subset \mathbb{R}\subset \mathbb{C}.$$

Then $logq\in D_{log\mathbb{Q}}(q,p)$.

Definition 5.3

(Certified $\pi$ lower/upper bounds). We fix a total computable function

$$\mathsf{PiLow}:{\mathbb{N}}_{\ge 1}\to {\mathbb{Q}}_{>0}$$

such that for all $p\ge 1$,

$$0<\mathsf{PiLow}\left(p\right)\le \pi ,\pi -\mathsf{PiLow}\left(p\right)\le 2^{-p}.$$

Define also the rational upper bound

$$\mathsf{PiUp}\left(p\right):=\mathsf{PiLow}\left(p\right)+2^{-p},\mathrm{so}\mathrm{that}\mathsf{PiLow}\left(p\right)\le \pi \le \mathsf{PiUp}\left(p\right).$$

Definition 5.4

(A rational upper bound for ${\left(2\pi \right)}^{-r}$). For $r\in {\mathbb{N}}_{\ge 1}$ and $p\in {\mathbb{N}}_{\ge 1}$ define the rational number

$$C_r\left(p\right):={\displaystyle \frac{4}{{\left(2\mathsf{PiLow}\left(p\right)\right)}^r}}\in {\mathbb{Q}}_{>0}.$$

Then

$${\displaystyle \frac{4}{{\left(2\pi \right)}^r}}\le C_r\left(p\right).$$

5.2. Certified Power Upper Bounds $N^q$ for Rational Exponents

Lemma 5.1

(Certified rational upper bound for $N^q$ with $q\in \mathbb{Q}$). Fix $p\ge 1$. Let $N\in {\mathbb{N}}_{\ge 2}$ and $q\in \mathbb{Q}$. Let $(L_{N,p},U_{N,p})=\mathsf{LogInt}(N,p)$ from Theorem 5.1, so

$$L_{N,p}\le logN\le U_{N,p}.$$

Define the rational number

$$T_{N,q,p}:=\left\{\begin{array}{cc}q·U_{N,p},\hfill & q\ge 0,\hfill \\ q·L_{N,p},\hfill & q<0,\hfill \end{array}\right.\in \mathbb{Q},$$

and set

$$\mathsf{PowUp}(N,q,p):=\mathsf{Qexpup}(T_{N,q,p},p)\in {\mathbb{Q}}_{>0}.$$

Then

$$N^q=e^{qlogN}\le \mathsf{PowUp}(N,q,p).$$

Proof. 

If $q\ge 0$ then $qlogN\le q{U}_{N,p}$, hence $e^{qlogN}\le e^{q{U}_{N,p}}\le \mathsf{Qexpup}(q{U}_{N,p},p)$. If $q<0$ then $qlogN\le q{L}_{N,p}$ (inequality reverses when multiplying by q), hence $e^{qlogN}\le e^{q{L}_{N,p}}\le \mathsf{Qexpup}(q{L}_{N,p},p)$. □

5.3. Bernoulli Bounds

Lemma 5.2

(Explicit Bernoulli bound). For $n\ge 1$,

$$|B_{2n}|\le {\displaystyle \frac{4\left(2n\right)!}{{\left(2\pi \right)}^{2n}}}.$$

Proof. 

Classical: use the Fourier series for periodic Bernoulli functions and $\zeta \left(2n\right)\le 2$ for $n\ge 1$. □

Remark 5.2

(Verifier-ready rational Bernoulli bound). In the verifier, 5.2 is used only through the rational inequality

$$|B_{2n}|\le {\displaystyle \frac{4\left(2n\right)!}{{\left(2\pi \right)}^{2n}}}\le {\displaystyle \frac{4\left(2n\right)!}{{\left(2\mathsf{PiLow}\left(p\right)\right)}^{2n}}},$$

so all constants are rationally checkable once p is fixed.

5.4. Pochhammer Disks

Definition 5.5

(Pochhammer symbol). For $s\in \mathbb{C}$ and integer $r\ge 0$ define

$$\mathsf{Poch}sr:=\prod _{j=0}^{r-1}(s+j),\mathsf{Poch}s0:=1.$$

Lemma 5.3

(Disk enclosure for (s)_r). Let $D_s=\overline{B}(c,r_s)$ be a rational disk and let $r\ge 0$. Define disks $D_s^{\left(j\right)}:=D_s+j=\overline{B}(c+j,r_s)$ and define the product disk

$$D_{\mathsf{Poch}}(D_s;r):=\prod _{j=0}^{r-1}{D}_s^{\left(j\right)}$$

using disk multiplication (with the fixed internal precision parameter from 4.5). Then for every $s\in D_s$,

$$\mathsf{Poch}sr\in D_{\mathsf{Poch}}(D_s;r).$$

Proof. 

If $s\in D_s$ then $s+j\in D_s+j=D_s^{\left(j\right)}$. Multiply enclosures using 4.3. □

5.5. Euler–Maclaurin Remainder Bound (Scalar)

Lemma 5.4

(Euler–Maclaurin remainder bound (scalar form)). Fix integers $N\ge 2$ and $m\ge 1$. Let $s\in \mathbb{C}$ satisfy $s\ne 1$ and $Re\left(s\right)>-2m$. Set

$$r_0:=2m+2,\alpha :=Re\left(s\right)+2m+2>1.$$

Then one has the Euler–Maclaurin representation

$$\zeta \left(s\right)=\sum _{n=1}^{N-1}{n}^{-s}+{\displaystyle \frac{N^{1-s}}{s-1}}+{\displaystyle \frac{1}{2}}{N}^{-s}+\sum _{k=1}^m{\displaystyle \frac{B_{2k}}{\left(2k\right)!}}\mathsf{Poch}s2k-1N^{-s-(2k-1)}+R_{m,N}\left(s\right),$$

(1)

with remainder satisfying

$$|R_{m,N}\left(s\right)|\le {\displaystyle \frac{4}{{\left(2\pi \right)}^{r_0}}}·{\displaystyle \frac{|\mathsf{Poch}s{r}_0|}{\alpha -1}}{N}^{1-\alpha }.$$

(2)

Proof. 

Apply Euler–Maclaurin to the tail ${\sum }_{n=N}^{\infty }f\left(n\right)$ with $f\left(x\right)=x^{-s}$, using the truncation that includes the Bernoulli correction terms through $B_{2m}$ and whose remainder is expressed using ${\tilde{B}}_{2m+2}$ and the $(2m+2)$nd derivative of f:

$$R_{m,N}\left(s\right)=-{\displaystyle \frac{1}{(2m+2)!}}{\int }_N^{\infty }{\tilde{B}}_{2m+2}\left(t\right)f^{(2m+2)}\left(t\right)dt.$$

With $r_0:=2m+2$ one has

$$f^{\left(r_0\right)}\left(t\right)={(-1)}^{r_0}\mathsf{Poch}s{r}_0{t}^{-s-r_0}.$$

Bounding $|{\tilde{B}}_{r_0}\left(t\right)|\le |B_{r_0}|$, using 5.2 to bound $|B_{r_0}|$, and integrating $t^{-Re\left(s\right)-r_0}$ from N to ∞ yields (2). □

5.6. Disk Enclosure for $\zeta$ (Verifier-Ready Form)

Definition 5.6

(Lower real part of a disk). For $D_s=\overline{B}(c,r_s)$ define

$${\sigma }_{-}\left(D_s\right):=\underset{w\in D_s}{inf}Re\left(w\right)=Re\left(c\right)-r_s.$$

Definition 5.7

(A computable Pochhammer majorant on a disk). Let $D_s=\overline{B}(c,r_s)$ and $r\ge 0$. Form the Pochhammer disk $D_{\mathsf{Poch}}(D_s;r)=\overline{B}(c_P,r_P)$ as in 5.3. Define the rational majorant

$$P_{max}(D_s;r):={\left|c_P\right|}_{\mathrm{up},n}+r_P,$$

where ${|·|}_{\mathrm{up},n}$ is the certified modulus upper bound from 4.4 and n is the fixed internal precision parameter used by disk operations. Then for all $s\in D_s$, $\left|\mathsf{Poch}sr\right|\le P_{max}(D_s;r)$.

Definition 5.8

(Power disks via certified log-intervals). Fix $p\in {\mathbb{N}}_{\ge 1}$ and let $a\in {\mathbb{N}}_{\ge 2}$. Define the real log disk $D_{log}(a,p)$ by 5.1. For a rational disk $D_s$, define

$$a^{-D_s}:=exp\left(-(D_s·D_{log}(a,p))\right),$$

using disk multiplication and disk exponential (all with internal precision parameter n in 4.5). Then for every $s\in D_s$, one has $a^{-s}\in a^{-D_s}$.

We treat the $a=1$ term exactly as $\overline{B}(1,0)$, and apply 5.8 only for $a\ge 2$.

Lemma 5.5

(Soundness of power disks). With the notation of 5.8, for every $s\in D_s$ one has

$$a^{-s}=e^{-sloga}\in exp\left(-(D_s·D_{log}(a,p))\right)=a^{-D_s}.$$

Proof. 

Since $loga\in D_{log}(a,p)$, we have $sloga\in D_s·D_{log}(a,p)$ by disk multiplication soundness, hence $-sloga\in -(D_s·D_{log}(a,p))$, and exponentiating preserves containment by 4.3. □

Theorem 5.1

(Disk enclosure for $\zeta$ on a rational disk (terminating verifier form)). Fix integers $N\ge 2$ and $m\ge 1$. Fix internal precision parameters $n\in {\mathbb{N}}_{\ge 1}$ (for disk ops) and $p\in {\mathbb{N}}_{\ge 1}$ (for log-intervals and π bounds).

Let $D_s=\overline{B}(c,r_s)$ be a rational disk such that

$$1\notin D_s,{\sigma }_{-}\left(D_s\right)>-2m,\mathrm{and}{|c-1|}_{\mathrm{lo},n}>r_s.$$

Set

$$r_0:=2m+2,{\alpha }_{-}:={\sigma }_{-}\left(D_s\right)+2m+2>1.$$

Let $P_{max}:=P_{max}(D_s;r_0)$.

Define the finite-part disk

$$\begin{array}{cc}\hfill D_{\mathrm{fin}}(D_s;N,m;n,p):=& \overline{B}(1,0)+\sum _{a=2}^{N-1}{a}^{-D_s}+{\displaystyle \frac{N^{1-\left(D_s\right)}}{D_s-1}}+\frac{1}{2}{N}^{-D_s}\hfill \\ \hfill & +\sum _{k=1}^m{\displaystyle \frac{B_{2k}}{\left(2k\right)!}}{D}_{\mathsf{Poch}}(D_s;2k-1)·N^{-(D_s+(2k-1))},\hfill \end{array}$$

(3)

where:

$$a^{-D_s}:=exp\left(-(D_s·D_{log}(a,p))\right),N^{1-\left(D_s\right)}:=exp\left(-((D_s-1)·D_{log}(N,p))\right),$$

$$N^{-D_s}:=exp\left(-(D_s·D_{log}(N,p))\right),N^{-(D_s+(2k-1))}:=exp\left(-((D_s+(2k-1))·D_{log}(N,p))\right),$$

and all disk operations use internal precision parameter n.

Define the remainder radius

$${\rho }_{\mathrm{rem}}(D_s;N,m;n,p):=C_{r_0}\left(p\right)·{\displaystyle \frac{P_{max}}{{\alpha }_{-}-1}}\mathsf{PowUp}(N,1-{\alpha }_{-},p),$$

(4)

where $C_{r_0}\left(p\right)=4/{\left(2\mathsf{PiLow}\left(p\right)\right)}^{r_0}$ is as in 5.4 and $\mathsf{PowUp}$ is from 5.1.

Set

$$D_{\zeta }(D_s;N,m;n,p):=D_{\mathrm{fin}}(D_s;N,m;n,p)+\overline{B}\left(0,{\rho }_{\mathrm{rem}}(D_s;N,m;n,p)\right).$$

Then for every $s\in D_s$,

$$\zeta \left(s\right)\in D_{\zeta }(D_s;N,m;n,p).$$

Proof. 

Fix $s\in D_s$.

Finite part. For each $a\ge 1$, by 5.1 we have $loga\in D_{log}(a,p)$, so $-sloga\in -(D_s·D_{log}(a,p))$ and thus

$$a^{-s}=e^{-sloga}\in exp\left(-(D_s·D_{log}(a,p))\right)=a^{-D_s}$$

by 5.5. The same reasoning gives disk containment for each N-power term and for the Bernoulli correction terms using 5.3 and disk arithmetic soundness (4.3). Summing finitely many disks preserves containment. Therefore the finite Euler–Maclaurin expression lies in $D_{\mathrm{fin}}(D_s;N,m;n,p)$.

Remainder. For $s\in D_s$ we have $Re\left(s\right)\ge {\sigma }_{-}\left(D_s\right)$, hence $\alpha =Re\left(s\right)+2m+2\ge {\alpha }_{-}$. Thus ${(\alpha -1)}^{-1}\le {({\alpha }_{-}-1)}^{-1}$ and

$$N^{1-\alpha }\le N^{1-{\alpha }_{-}}\le \mathsf{PowUp}(N,1-{\alpha }_{-},p).$$

Also $|\mathsf{Poch}s{r}_0|\le P_{max}$ by 5.7. Finally,

$${\displaystyle \frac{4}{{\left(2\pi \right)}^{r_0}}}\le C_{r_0}\left(p\right)$$

by 5.4. Insert these bounds into 5.4 to obtain

$$|R_{m,N}\left(s\right)|\le {\rho }_{\mathrm{rem}}(D_s;N,m;n,p).$$

Hence $R_{m,N}\left(s\right)\in \overline{B}(0,{\rho }_{\mathrm{rem}})$.

Combine. Using (1), $\zeta \left(s\right)$ is the sum of the finite part and the remainder, hence lies in $D_{\mathrm{fin}}+\overline{B}(0,{\rho }_{\mathrm{rem}})=D_{\zeta }(D_s;N,m;n,p)$. □

Remark 5.3

(Termination/decidability of the $\zeta$-disk routine). Every object in 5.1 is obtained from finitely many rational operations, finitely many calls to total computable interval routines $\mathsf{LogInt}$ and $\mathsf{PiLow}$, and finitely many primitive disk operations. Hence the computation terminates and yields a rational disk.

Remark 5.4

(Interface for later sections). All later special-function disk routines (for ${\zeta }^{\prime },{\zeta }^{\prime \prime }$, and then $\mathrm{\Gamma },\psi ,{\psi }^{\prime }$ and Ξ) use the same two precision parameters:

• n: internal precision knob for certified modulus/exponential bounds in disk arithmetic;
• p: precision knob for certified rational intervals for $log(·)$ and π-dependent constants.

These will be included in the certificate parameter record in Part 4.

6. Decidability III: Differentiated Euler–Maclaurin Enclosures for ${\zeta }^{\prime }$ and ${\zeta }^{\prime \prime }$

This section gives explicit, computable rational-disk enclosures for ${\zeta }^{\prime }\left(s\right)$ and ${\zeta }^{\prime \prime }\left(s\right)$ on a rational disk $D_s=\overline{B}(c,r_s)$, using differentiated Euler–Maclaurin with explicit tail-integral majorants.

All occurrences of $N^{1-\alpha }$ with $\alpha >1$ are implemented via the certified routine $\mathsf{PowUp}(N,1-\alpha ,p)$ from 5.1. Note that here $1-\alpha <0$; the sign-aware definition of $\mathsf{PowUp}$ is therefore essential for soundness.

6.1. Tail-Integral Majorants with Log Factors (Verifier Form)

Definition 6.1

(Upper bounds for $I_q(\alpha ,N)$). Fix $N\ge 2$, $p\ge 1$, and $\alpha \in \mathbb{Q}$ with $\alpha >1$. Let $(L_{N,p},U_{N,p})=\mathsf{LogInt}(N,p)$ from 5.1, so $logN\le U_{N,p}$. Define the rational numbers

$$A:={\displaystyle \frac{1}{\alpha -1}}\in {\mathbb{Q}}_{>0},P:=\mathsf{PowUp}(N,1-\alpha ,p)\in {\mathbb{Q}}_{>0}.$$

Define

$$I_0^{\mathrm{up}}(\alpha ,N;p):=P·A,$$

$$I_1^{\mathrm{up}}(\alpha ,N;p):=P·A\left(U_{N,p}+A\right),$$

$$I_2^{\mathrm{up}}(\alpha ,N;p):=P·A\left(U_{N,p}^2+2U_{N,p}A+2A^2\right).$$

Lemma 6.1

(Correctness of the tail-integral majorants). For $\alpha >1$, $N\ge 2$ and $q\in \{0,1,2\}$, let

$$I_q(\alpha ,N):={\int }_N^{\infty }{x}^{-\alpha }{(logx)}^{q}dx.$$

Then for each $q\in \{0,1,2\}$,

$$I_q(\alpha ,N)\le I_q^{\mathrm{up}}(\alpha ,N;p).$$

Proof. 

The exact closed forms are

$$I_0(\alpha ,N)={\displaystyle \frac{N^{1-\alpha }}{\alpha -1}},I_1(\alpha ,N)={\displaystyle \frac{N^{1-\alpha }}{\alpha -1}}\left(logN+{\displaystyle \frac{1}{\alpha -1}}\right),$$

$$I_2(\alpha ,N)={\displaystyle \frac{N^{1-\alpha }}{\alpha -1}}\left({(logN)}^2+{\displaystyle \frac{2logN}{\alpha -1}}+{\displaystyle \frac{2}{{(\alpha -1)}^2}}\right).$$

Now $N^{1-\alpha }\le \mathsf{PowUp}(N,1-\alpha ,p)=P$ (by 5.1) and $logN\le U_{N,p}$, so each expression is bounded above by the corresponding definition in 6.1. □

6.2. Derivative Bounds for Pochhammer Factors on Disks

Lemma 6.2

(Scalar identities for $\mathsf{Poch}sr$ and its derivatives). Let $r\ge 1$ and set $P_r\left(s\right):=\mathsf{Poch}sr={\prod }_{j=0}^{r-1}(s+j)$. Then wherever $s\notin \{-0,-1,\dots ,1-r\}$,

$$P_r^{\prime }\left(s\right)=P_r\left(s\right)\sum _{j=0}^{r-1}{\displaystyle \frac{1}{s+j}},P_r^{\prime \prime }\left(s\right)=P_r\left(s\right)\left[{\left(\sum _{j=0}^{r-1}{\displaystyle \frac{1}{s+j}}\right)}^2-\sum _{j=0}^{r-1}{\displaystyle \frac{1}{{(s+j)}^2}}\right].$$

Consequently,

$$|P_r^{\prime }\left(s\right)|\le |P_r\left(s\right)|\sum _{j=0}^{r-1}{\displaystyle \frac{1}{|s+j|}},$$

$$|P_r^{\prime \prime }\left(s\right)|\le |P_r\left(s\right)|\left[{\left(\sum _{j=0}^{r-1}{\displaystyle \frac{1}{|s+j|}}\right)}^2+\sum _{j=0}^{r-1}{\displaystyle \frac{1}{{|s+j|}^2}}\right].$$

Definition 6.2

(Computable uniform Pochhammer derivative majorants on a disk). Let $D_s=\overline{B}(c,r_s)$ be a rational disk and let $r\ge 1$. Fix internal precision parameter $n\ge 1$ (as in 4.5, Part 1).

Form the Pochhammer disk $D_{\mathsf{Poch}}(D_s;r)=\overline{B}(c_P,r_P)$ using disk multiplication. Define

$$P_{max}(D_s;r):={\left|c_P\right|}_{\mathrm{up},n}+r_P.$$

For each $j=0,\dots ,r-1$ define the certified lower bound

$$d_j:={|c+j|}_{\mathrm{lo},n}-r_s\in \mathbb{Q}.$$

Assume $d_j>0$ for all j. Define

$$S_1:=\sum _{j=0}^{r-1}{\displaystyle \frac{1}{d_j}},S_2:=\sum _{j=0}^{r-1}{\displaystyle \frac{1}{d_j^2}},$$

and

$$P_{max}^{\prime }(D_s;r):=P_{max}(D_s;r)S_1,P_{max}^{\prime \prime }(D_s;r):=P_{max}(D_s;r)(S_1^2+S_2).$$

Lemma 6.3

(Correctness of the derivative majorants). With the notation and assumptions of 6.2, for all $s\in D_s$,

$$\left|\mathsf{Poch}sr\right|\le P_{max}(D_s;r),|{\left(\mathsf{Poch}sr\right)}^{\prime }|\le P_{max}^{\prime }(D_s;r),\left|{\left(\mathsf{Poch}sr\right)}^{\prime \prime }\right|\le P_{max}^{\prime \prime }(D_s;r).$$

Proof. 

If $s\in D_s$, then $|s+j|\ge |c+j|-r_s\ge {|c+j|}_{\mathrm{lo},n}-r_s=d_j$. Also $\left|\mathsf{Poch}sr\right|\le P_{max}$ by construction of the Pochhammer disk. Apply 6.2 and substitute ${\sum |s+j|}^{-1}\le \sum d_j^{-1}=S_1$ and ${\sum |s+j|}^{-2}\le S_2$. □

6.3. Uniform Remainder Bounds for ${\zeta }^{\left(q\right)}$ on Disks, $q=0,1,2$

Theorem 6.1

(Verifier-computable remainder radii for ${\zeta }^{\left(q\right)}$ on disks). Fix integers $N\ge 2$ and $m\ge 1$, set $r_0:=2m+2$. Fix internal precision parameters $n\ge 1$ (disk ops) and $p\ge 1$ (log/π intervals).

Let $D_s=\overline{B}(c,r_s)$ be a rational disk such that:

$$1\notin D_s,{\sigma }_{-}\left(D_s\right){>-2m,|c-1|}_{\mathrm{lo},n}>r_s,\mathrm{and}{d}_j:={|c+j|}_{\mathrm{lo},n}-r_s>0(0\le j\le r_0-1).$$

Set the rational lower bound

$${\alpha }_{-}:={\sigma }_{-}\left(D_s\right)+2m+2>1.$$

Let $P_{max},P_{max}^{\prime },P_{max}^{\prime \prime }$ be the majorants from 6.2 applied to $(D_s,r_0)$.

Define

$$C_{r_0}\left(p\right):={\displaystyle \frac{4}{{\left(2\mathsf{PiLow}\left(p\right)\right)}^{r_0}}}\in {\mathbb{Q}}_{>0},$$

so that $4/{\left(2\pi \right)}^{r_0}\le C_{r_0}\left(p\right)$. Let $I_q^{\mathrm{up}}:=I_q^{\mathrm{up}}({\alpha }_{-},N;p)$ be as in 6.1.

Then for each $q\in \{0,1,2\}$ the Euler–Maclaurin remainder $R_{m,N}^{\left(q\right)}\left(s\right)$ in

$${\zeta }^{\left(q\right)}\left(s\right)={\zeta }_{\mathrm{fin}}^{\left(q\right)}(s;N,m)+R_{m,N}^{\left(q\right)}\left(s\right)$$

admits the uniform bound

$$|R_{m,N}^{\left(q\right)}\left(s\right)|\le {\rho }_{\mathrm{rem}}^{\left(q\right)}(D_s;N,m;n,p)(s\in D_s),$$

where therationalremainder radii are

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{rem}}^{\left(0\right)}& =C_{r_0}\left(p\right)P_{max}{I}_0^{\mathrm{up}},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{rem}}^{\left(1\right)}& =C_{r_0}\left(p\right)\left(P_{max}^{\prime }{I}_0^{\mathrm{up}}+P_{max}{I}_1^{\mathrm{up}}\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{rem}}^{\left(2\right)}& =C_{r_0}\left(p\right)\left(P_{max}^{\prime \prime }{I}_0^{\mathrm{up}}+2P_{max}^{\prime }{I}_1^{\mathrm{up}}+P_{max}{I}_2^{\mathrm{up}}\right).\hfill \end{array}$$

(7)

Proof. 

Differentiate the standard Euler–Maclaurin integral remainder representation under the integral sign. Absolute values yield bounds in terms of $|\mathsf{Poch}s{r}_0|$, $|{\left(\mathsf{Poch}s{r}_0\right)}^{\prime }|$, $|{\left(\mathsf{Poch}s{r}_0\right)}^{\prime \prime }|$ times the tail integrals $I_q(\alpha ,N)$.

Uniformize on $D_s$ by:

$$|\mathsf{Poch}s{r}_0|\le P_{max},|{\left(\mathsf{Poch}s{r}_0\right)}^{\prime }|\le P_{max}^{\prime },\left|{\left(\mathsf{Poch}s{r}_0\right)}^{\prime \prime }\right|\le P_{max}^{\prime \prime }$$

from 6.3, and

$$I_q(\alpha ,N)\le I_q^{\mathrm{up}}({\alpha }_{-},N;p)$$

from 6.1, and

$${\displaystyle \frac{4}{{\left(2\pi \right)}^{r_0}}}\le C_{r_0}\left(p\right)$$

from 5.4. Substituting yields (5)–(7). □

6.4. Disk Enclosures for ${\zeta }^{\prime }$ and ${\zeta }^{\prime \prime }$

6.4.1. Verifier-Ready Differentiated Finite-Part Disks

Definition 6.3

(Harmonic-sum disks for Pochhammer derivatives). Let $D_s=\overline{B}(c,r_s)$ be a rational disk and let $r\ge 1$. Fix internal precision parameter $n\ge 1$.

For each $j=0,\dots ,r-1$ form the shifted disk $D_{s+j}:=D_s+j=\overline{B}(c+j,r_s)$. The verifier attempts to form reciprocals

$$D_{1/(s+j)}:=1/D_{s+j}$$

using the reciprocal rule of 4.5 (Part 1). If any required reciprocal is rejected, then the construction is rejected.

Define the sum disks

$$D_{H_1}(D_s;r):=\sum _{j=0}^{r-1}{D}_{1/(s+j)},D_{H_2}(D_s;r):=\sum _{j=0}^{r-1}\left(D_{1/(s+j)}·D_{1/(s+j)}\right).$$

Lemma 6.4

(Soundness of $H_1,H_2$ disks). If the construction in 6.3 does not reject, then for all $s\in D_s$,

$$\sum _{j=0}^{r-1}{\displaystyle \frac{1}{s+j}}\in D_{H_1}(D_s;r),\sum _{j=0}^{r-1}{\displaystyle \frac{1}{{(s+j)}^2}}\in D_{H_2}(D_s;r).$$

Proof. 

By soundness of reciprocal and multiplication disks (4.3, Part 1) and finite sums. □

Definition 6.4

(Pochhammer derivative disks). Let $D_s=\overline{B}(c,r_s)$ be a rational disk and let $r\ge 1$. Fix internal precision parameter $n\ge 1$.

Form the Pochhammer disk $D_{\mathsf{Poch}}(D_s;r)$ from 5.3. Form $D_{H_1}(D_s;r)$ and $D_{H_2}(D_s;r)$ by 6.3 (reject if needed). Define

$$D_{{\left(\mathsf{Poch}\right)}^{\prime }}(D_s;r):=D_{\mathsf{Poch}}(D_s;r)·D_{H_1}(D_s;r),$$

$$D_{{\left(\mathsf{Poch}\right)}^{\prime \prime }}(D_s;r):=D_{\mathsf{Poch}}(D_s;r)·\left(D_{H_1}(D_s;r)·D_{H_1}(D_s;r)-D_{H_2}(D_s;r)\right).$$

Lemma 6.5

(Soundness of Pochhammer derivative disks). If the construction in 6.4 does not reject, then for all $s\in D_s$,

$${\left(\mathsf{Poch}sr\right)}^{\prime }\in D_{{\left(\mathsf{Poch}\right)}^{\prime }}(D_s;r),{\left(\mathsf{Poch}sr\right)}^{\prime \prime }\in D_{{\left(\mathsf{Poch}\right)}^{\prime \prime }}(D_s;r).$$

Proof. 

Use the scalar identities of 6.2 together with 6.4 and disk-operation soundness. □

Definition 6.5

(Differentiated Euler–Maclaurin finite-part disks). Fix $N\ge 2$, $m\ge 1$, precision parameters $n,p\ge 1$, and let $D_s$ be a rational disk. Let $D_{log}(a,p)$ be as in 5.1 (Part 2) and $a^{-D_s}$ as in 5.8.

Define the basic building blocks:

$$D_{a^{-s}}:=a^{-D_s}(2\le a\le N-1),D_{loga}:=D_{log}(a,p),D_{logN}:=D_{log}(N,p).$$

Let $D_{1/(s-1)}:=1/(D_s-1)$ be formed using the reciprocal rule (reject if it rejects), and define

$$D_{1/{(s-1)}^2}:=D_{1/(s-1)}·D_{1/(s-1)},D_{1/{(s-1)}^3}:=D_{1/{(s-1)}^2}·D_{1/(s-1)}.$$

Define the N-power disk

$$D_{N^{1-s}}:=exp\left(-((D_s-1)·D_{logN})\right),$$

and the auxiliary term

$$D_T:=D_{N^{1-s}}·D_{1/(s-1)}.$$

For Bernoulli correction terms, for each $k=1,\dots ,m$ set $r_k:=2k-1$ and define

$$D_{P_k}:=D_{\mathsf{Poch}}(D_s;r_k),D_{P_k^{\prime }}:=D_{{\left(\mathsf{Poch}\right)}^{\prime }}(D_s;r_k),D_{P_k^{\prime \prime }}:=D_{{\left(\mathsf{Poch}\right)}^{\prime \prime }}(D_s;r_k),$$

(reject if any required reciprocal in these definitions is rejected), and define

$$D_{N^{-(s+r_k)}}:=exp\left(-((D_s+r_k)·D_{logN})\right).$$

Now define the finite-part disks:

$$D_{\zeta ,\mathrm{fin}}:=\overline{B}(1,0)+\sum _{a=2}^{N-1}{D}_{a^{-s}}+D_T+\frac{1}{2}{N}^{-D_s}+\sum _{k=1}^m{\displaystyle \frac{B_{2k}}{\left(2k\right)!}}{D}_{P_k}·D_{N^{-(s+r_k)}}.$$

$$D_{{\zeta }^{\prime },\mathrm{fin}}:=\sum _{a=2}^{N-1}\left(-D_{loga}·D_{a^{-s}}\right)+\left(-D_{logN}·D_T-D_{N^{1-s}}·D_{1/{(s-1)}^2}\right)+\frac{1}{2}\left(-D_{logN}·N^{-D_s}\right)$$

$$+\sum _{k=1}^m{\displaystyle \frac{B_{2k}}{\left(2k\right)!}}\left(D_{P_k^{\prime }}·D_{N^{-(s+r_k)}}+D_{P_k}·\left(-D_{logN}·D_{N^{-(s+r_k)}}\right)\right).$$

$$D_{{\zeta }^{\prime \prime },\mathrm{fin}}:=\sum _{a=2}^{N-1}\left((D_{loga}·D_{loga})·D_{a^{-s}}\right)+\left((D_{logN}·D_{logN})·D_T+2D_{logN}·D_{N^{1-s}}·D_{1/{(s-1)}^2}+2D_{N^{1-s}}·D_{1/{(s-1)}^3}\right)$$

$$\begin{array}{cc}& +\frac{1}{2}\left((D_{logN}·D_{logN})·N^{-D_s}\right)\hfill \\ & +\sum _{k=1}^m{\displaystyle \frac{B_{2k}}{\left(2k\right)!}}(D_{P_k^{\prime \prime }}·D_{N^{-(s+r_k)}}\hfill \\ & +2D_{P_k^{\prime }}·\left(-D_{logN}·D_{N^{-(s+r_k)}}\right)\hfill \\ & +D_{P_k}·\left((D_{logN}·D_{logN})·D_{N^{-(s+r_k)}}\right)).\hfill \end{array}$$

Lemma 6.6

(Soundness of the differentiated finite-part disks). Assume the construction in 6.5 does not reject. Then for all $s\in D_s$,

$${\zeta }_{\mathrm{fin}}(s;N,m)\in D_{\zeta ,\mathrm{fin}},{\zeta }_{\mathrm{fin}}^{\prime }(s;N,m)\in D_{{\zeta }^{\prime },\mathrm{fin}},{\zeta }_{\mathrm{fin}}^{\prime \prime }(s;N,m)\in D_{{\zeta }^{\prime \prime },\mathrm{fin}},$$

where ${\zeta }_{\mathrm{fin}}^{\left(q\right)}$ denotes the differentiated finite Euler–Maclaurin expression (i.e. the finite part of (1) and its first two derivatives).

Proof. 

Differentiate the scalar finite Euler–Maclaurin expression term-by-term. Each scalar operation is mirrored by a sound disk operation, and each scalar identity involving Pochhammer derivatives is enclosed by 6.5. Summing finitely many sound enclosures preserves containment. □

Theorem 6.2

(Rational disk enclosures for $\zeta ,{\zeta }^{\prime },{\zeta }^{\prime \prime }$ on $D_s$). Fix $N\ge 2$, $m\ge 1$, and internal precision parameters $n,p\ge 1$. Let $D_s=\overline{B}(c,r_s)$ satisfy the hypotheses of 6.1 (in particular, ${|c-1|}_{\mathrm{lo},n}>r_s$).

Then one can compute rational disks

$$D_{\zeta }(D_s;N,m;n,p),D_{{\zeta }^{\prime }}(D_s;N,m;n,p),D_{{\zeta }^{\prime \prime }}(D_s;N,m;n,p)$$

such that for all $s\in D_s$,

$$\zeta \left(s\right)\in D_{\zeta }(D_s;N,m;n,p),{\zeta }^{\prime }\left(s\right)\in D_{{\zeta }^{\prime }}(D_s;N,m;n,p),{\zeta }^{\prime \prime }\left(s\right)\in D_{{\zeta }^{\prime \prime }}(D_s;N,m;n,p).$$

Moreover, the verifier computes the finite-part disks explicitly as in 6.5 (and rejects if any required certified reciprocal guard fails), and then adds the remainder disks of radii (5)–(7) from 6.1.

Proof. 

For $q=0$ this is 5.1 (Part 2), and the certified pole-separation hypothesis ${|c-1|}_{\mathrm{lo},n}>r_s$ is included among our assumptions via 6.1. For $q=1,2$, differentiate each finite Euler–Maclaurin term:

$${\displaystyle \frac{d}{ds}}{a}^{-s}=-(loga)a^{-s},{\displaystyle \frac{d^2}{d{s}^2}}{a}^{-s}={(loga)}^2{a}^{-s},$$

and similarly for the $N^{1-s}/(s-1)$ and correction terms (finite combinations of $a^{-s}$, ${(s-1)}^{-1}$ and ${(s-1)}^{-2}$, and Pochhammer factors). Implement each scalar factor $loga$ as the real disk $D_{log}(a,p)$. Finally add the remainder disks with radii from 6.1. □

7. Decidability IV: Stirling-Type Enclosures for $\mathrm{\Gamma }$, $\psi$, and ${\psi }^{\prime }$ on Disks

We give a terminating enclosure mechanism for $\mathrm{\Gamma }$ and its logarithmic derivatives on rational disks. The verifier uses only:

• rational arithmetic and rational disks;
• certified $log(·)$ intervals ($\mathsf{LogInt},\mathsf{LogQInt}$), certified $\pi$ bounds ($\mathsf{PiLow},\mathsf{PiUp}$), and certified exp bounds ($\mathsf{Qexpup},\mathsf{ExpRat}$);
• rational Stirling remainder bounds depending only on ${\sigma }_{-}\left(D\right)$ (in a right half-plane regime), tightened by an explicit shift parameter h.

7.1. Verifier-Ready Disks for $log\pi$ and $log\left(2\pi \right)$

Definition 7.1

(A sound $log\pi$ disk from $\mathsf{PiLow}/\mathsf{PiUp}$). Fix $p\ge 1$. Define

$$\mathsf{PiLow}\left(p\right)\le \pi \le \mathsf{PiUp}\left(p\right):=\mathsf{PiLow}\left(p\right)+2^{-p}.$$

Let

$$(L_{-},U_{-}):=\mathsf{LogQInt}(\mathsf{PiLow}\left(p\right),p),(L_{+},U_{+}):=\mathsf{LogQInt}(\mathsf{PiUp}\left(p\right),p).$$

Since log is increasing on ${\mathbb{R}}_{>0}$,

$$log\pi \in [log\left(\mathsf{PiLow}\left(p\right)\right),log\left(\mathsf{PiUp}\left(p\right)\right)]\subseteq [L_{-},U_{+}].$$

Define the real disk

$$D_{log\pi }\left(p\right):=\overline{B}\left({\displaystyle \frac{L_{-}+U_{+}}{2}},{\displaystyle \frac{U_{+}-L_{-}}{2}}\right)\subset \mathbb{R}\subset \mathbb{C}.$$

Then $log\pi \in D_{log\pi }\left(p\right)$.

Definition 7.2

(A sound $log\left(2\pi \right)$ disk). Fix $p\ge 1$. Define

$$D_{log2}\left(p\right):=D_{log\mathbb{Q}}(2,p)\mathrm{from},D_{log\left(2\pi \right)}\left(p\right):=D_{log2}\left(p\right)+D_{log\pi }\left(p\right).$$

Then $log\left(2\pi \right)\in D_{log\left(2\pi \right)}\left(p\right)$.

7.2. Elementary-Log Enclosures: $Log\left(c\right)$ for Rational Complex c and $Log\left(D\right)$ on RHP Disks

Definition 7.3

(Certified principal-log approximation on rational inputs away from the branch cut). We fix a total computable function

$$\mathsf{LogRat}:\left((\mathbb{Q}+\mathrm{i}\mathbb{Q})\setminus (-\infty ,0]\right)\times {\mathbb{N}}_{\ge 1}\to (\mathbb{Q}+\mathrm{i}\mathbb{Q}),$$

such that for all $c\in (\mathbb{Q}+\mathrm{i}\mathbb{Q})\setminus (-\infty ,0]$ and all $p\ge 1$,

$$\left|Log\left(c\right)-\mathsf{LogRat}(c,p)\right|\le 2^{-p},$$

where Log denotes the principal branch on $\mathbb{C}\setminus (-\infty ,0]$.

Definition 7.4

(A disk enclosure for Log on a right-half-plane disk). Fix $p\ge 1$. Let $D=\overline{B}(c,r)$ be a rational disk with

$${\sigma }_{-}\left(D\right):=Re\left(c\right)-r>0$$

(so D lies in the open right half-plane). Since ${\sigma }_{-}\left(D\right)>0$, the center c lies in the open right half-plane, hence $c\notin (-\infty ,0]$ and $\mathsf{LogRat}(c,p)$ is defined. Define

$$M_{log}^{\mathrm{RHP}}\left(D\right):={\displaystyle \frac{r}{{\sigma }_{-}\left(D\right)}}\in {\mathbb{Q}}_{\ge 0},Log\left(D\right):=\overline{B}\left(\mathsf{LogRat}(c,p),2^{-p}+M_{log}^{\mathrm{RHP}}\left(D\right)\right).$$

Lemma 7.1

(Soundness of $Log\left(D\right)$ on right-half-plane disks). If ${\sigma }_{-}\left(D\right)>0$, then for all $z\in D$ one has $Log\left(z\right)\in Log\left(D\right)$.

Proof. 

On the right half-plane, ${Log}^{\prime }\left(w\right)=1/w$ and $|1/w|\le 1/Re\left(w\right)$. Thus for $z\in D$,

$$|Log\left(z\right)-Log\left(c\right)|\le \underset{w\in D}{sup}{\displaystyle \frac{1}{Re\left(w\right)}}·|z-c|\le {\displaystyle \frac{r}{{\sigma }_{-}\left(D\right)}}.$$

Combine with $|Log\left(c\right)-\mathsf{LogRat}(c,p)|\le 2^{-p}$. □

7.3. Right-Half-Plane Stirling Remainder Bounds (Rationalized for the Verifier)

Lemma 7.2

(A right-half-plane lower bound for $|t+z|$). Let $z\in \mathbb{C}$ with $Re\left(z\right)\ge a>0$. Then for all $t\ge 0$,

$$|t+z|\ge t+Re\left(z\right)\ge t+a.$$

Theorem 7.1

(Stirling truncation for $log\mathrm{\Gamma }$ with a right-half-plane remainder bound). Fix $M\ge 1$ and let $z\in \mathbb{C}$ satisfy $Re\left(z\right)\ge a>0$. Define the Stirling truncation

$$L_M\left(z\right):=\left(z-\frac{1}{2}\right)Logz-z+\frac{1}{2}Log\left(2\pi \right)+\sum _{k=1}^{M-1}{\displaystyle \frac{B_{2k}}{2k(2k-1)z^{2k-1}}}.$$

Then

$$log\mathrm{\Gamma }\left(z\right)=L_M\left(z\right)+E_M\left(z\right),$$

where the remainder admits an integral representation and satisfies

$$|E_M\left(z\right)|\le {\displaystyle \frac{|B_{2M}|}{2M(2M-1)}}{a}^{1-2M}.$$

(8)

Proof. 

A standard Euler–Maclaurin/Stirling integral remainder form gives

$$E_M\left(z\right)={\int }_0^{\infty }{\displaystyle \frac{{\tilde{B}}_{2M}\left(t\right)}{2M{(t+z)}^{2M}}}dt.$$

Bound $|{\tilde{B}}_{2M}\left(t\right)|\le |B_{2M}|$ and use 7.2: ${|t+z|}^{-2M}\le {(t+a)}^{-2M}$. Then

$$|E_M\left(z\right)|\le {\displaystyle \frac{|B_{2M}|}{2M}}{\int }_0^{\infty }{(t+a)}^{-2M}dt={\displaystyle \frac{|B_{2M}|}{2M}}·{\displaystyle \frac{a^{1-2M}}{2M-1}},$$

which is (8). □

Theorem 7.2

(Stirling expansions for $\psi$ and ${\psi }^{\prime }$ with right-half-plane remainder bounds). Fix $M\ge 1$ and let $z\in \mathbb{C}$ satisfy $Re\left(z\right)\ge a>0$. Then

$$\psi \left(z\right)=Logz-{\displaystyle \frac{1}{2z}}-\sum _{k=1}^{M-1}{\displaystyle \frac{B_{2k}}{2k{z}^{2k}}}+R_M^{\psi }\left(z\right),$$

$${\psi }^{\prime }\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{1}{2z^2}}+\sum _{k=1}^{M-1}{\displaystyle \frac{B_{2k}}{z^{2k+1}}}+R_M^{{\psi }^{\prime }}\left(z\right),$$

with bounds

$$|R_M^{\psi }\left(z\right)|\le {\displaystyle \frac{|B_{2M}|}{2M}}{a}^{-2M},|R_M^{{\psi }^{\prime }}\left(z\right)|\le |B_{2M}|a^{-2M-1}.$$

(9)

Proof. 

Differentiate the Stirling remainder integral for $log\mathrm{\Gamma }$ under the integral sign:

$${(log\mathrm{\Gamma })}^{\prime }=\psi ,{(log\mathrm{\Gamma })}^{\prime \prime }={\psi }^{\prime }.$$

Each differentiation introduces an extra factor ${(t+z)}^{-1}$, and no trigonometric sector constant is needed because $|t+z|\ge t+a$. Integrate ${(t+a)}^{-2M-1}$ and ${(t+a)}^{-2M-2}$ to obtain (9). □

Definition 7.5

(Verifier-ready Bernoulli upper bounds for Stirling remainders). Fix $p\ge 1$ and $M\ge 1$. Define the rational bound

$$|B_{2M}|\le {\displaystyle \frac{4\left(2M\right)!}{{\left(2\pi \right)}^{2M}}}\le {\displaystyle \frac{4\left(2M\right)!}{{\left(2\mathsf{PiLow}\left(p\right)\right)}^{2M}}}=:B_{2M}^{\mathrm{up}}\left(p\right)\in {\mathbb{Q}}_{>0},$$

using 5.2 and $\mathsf{PiLow}\left(p\right)\le \pi$.

Remark 7.1.

The verifier uses the exact $B_{2k}$ only in thefiniteStirling truncation sums (for $k=1,\dots ,M-1$); the remainder radii use only the rational bound $B_{2M}^{\mathrm{up}}\left(p\right)$.

7.4. Shifting Into a Uniform Right-Half-Plane Regime (Explicit Shift Knob)

Definition 7.6

(Right-half-plane shifting by a chosen integer). Let $D=\overline{B}(c,r)$ be a rational disk and let $h\in \mathbb{N}$. Define

$$D+h:=\overline{B}(c+h,r).$$

Then ${\sigma }_{-}(D+h)={\sigma }_{-}\left(D\right)+h$.

7.5. Disk Enclosures for $log\mathrm{\Gamma }$, $\mathrm{\Gamma }$, $\psi$, ${\psi }^{\prime }$

Theorem 7.3

(Disk enclosure for $log\mathrm{\Gamma }$ on a right-half-plane disk (verifier form)). Fix $M\ge 1$ and precision parameters $n,p\ge 1$. Let $D=\overline{B}(c,r)$ be a rational disk such that ${\sigma }_{-}\left(D\right)>0$ and ${\left|c\right|}_{\mathrm{lo},n}>r$. Set $a:={\sigma }_{-}\left(D\right)\in {\mathbb{Q}}_{>0}$.

Define the truncation disk $L_M\left(D\right)$ by evaluating the formula of 7.1 using disk arithmetic (with internal precision parameter n), using $Log\left(D\right)$ from 7.4, and using the constant disk $\frac{1}{2}{D}_{log\left(2\pi \right)}\left(p\right)$ from 7.2.

Define the rational remainder radius

$$R_{log\mathrm{\Gamma }}^{\mathrm{up}}(D;M;p):={\displaystyle \frac{B_{2M}^{\mathrm{up}}\left(p\right)}{2M(2M-1)}}{a}^{1-2M}\in {\mathbb{Q}}_{>0},$$

where $B_{2M}^{\mathrm{up}}\left(p\right)$ is from 7.5. Then

$$log\mathrm{\Gamma }\left(D\right)\subseteq L_M\left(D\right)+\overline{B}(0,R_{log\mathrm{\Gamma }}^{\mathrm{up}}(D;M;p)).$$

Proof. 

For each $z\in D$, 7.1 gives $log\mathrm{\Gamma }\left(z\right)=L_M\left(z\right)+E_M\left(z\right)$ with $|E_M\left(z\right)|\le {\displaystyle \frac{|B_{2M}|}{2M(2M-1)}}{a}^{1-2M}\le R_{log\mathrm{\Gamma }}^{\mathrm{up}}(D;M;p)$. The disk computation $L_M\left(D\right)$ encloses $L_M\left(z\right)$ for all $z\in D$ by soundness of disk arithmetic and 7.1. □

Theorem 7.4

(Disk enclosures for $\mathrm{\Gamma },\psi ,{\psi }^{\prime }$ on a right-half-plane disk). Fix $M\ge 1$ and precision parameters $n,p\ge 1$. Let $D=\overline{B}(c,r)$ be a rational disk with ${\sigma }_{-}\left(D\right)>0$ and ${\left|c\right|}_{\mathrm{lo},n}>r$.

Then there is a terminating algorithm that outputs rational disks

$$D_{\mathrm{\Gamma }}(D;M;n,p),D_{\psi }(D;M;n,p),D_{{\psi }^{\prime }}(D;M;n,p)$$

satisfying

$$\mathrm{\Gamma }\left(D\right)\subseteq D_{\mathrm{\Gamma }}(D;M;n,p),\psi \left(D\right)\subseteq D_{\psi }(D;M;n,p),{\psi }^{\prime }\left(D\right)\subseteq D_{{\psi }^{\prime }}(D;M;n,p).$$

Proof. 

Compute $L_M\left(D\right)$ and $R_{log\mathrm{\Gamma }}^{\mathrm{up}}$ as in 7.3, so $log\mathrm{\Gamma }\left(D\right)\subseteq \overline{B}(c_L,r_L)$ for some rational disk $\overline{B}(c_L,r_L)$. Then $\mathrm{\Gamma }\left(D\right)\subseteq exp\left(\overline{B}(c_L,r_L)\right)$ by disk exponential soundness (4.3).

For $\psi$ and ${\psi }^{\prime }$, compute the truncated Stirling disks using $Log\left(D\right)$, $1/D$, and powers $D^{-k}$ via disk arithmetic. Use the remainder bounds (9) with $|B_{2M}|$ replaced by $B_{2M}^{\mathrm{up}}\left(p\right)$ (from 7.5) and $a={\sigma }_{-}\left(D\right)$, and add the corresponding remainder disks. All operations terminate. □

Theorem 7.5

(Disk enclosures for $\mathrm{\Gamma },\psi ,{\psi }^{\prime }$ on a general disk by shifting). Fix $M\ge 1$, an integer shift $h\in \mathbb{N}$, and precision parameters $n,p\ge 1$. Let $D=\overline{B}(c,r)$ be a rational disk such that

$${\sigma }_{-}(D+h)=Re\left(c\right)+h-r>0,$$

and such that for each $j=0,1,\dots ,h$ the verifier has the certified separation

$${|c+j|}_{\mathrm{lo},n}>r(\mathrm{equivalently}0\notin D+j\mathrm{with}\mathrm{certified}\mathrm{margin}).$$

Compute enclosures on the shifted disk $D+h$ using 7.4. Then pull back to D using the identities

$$\mathrm{\Gamma }\left(z\right)={\displaystyle \frac{\mathrm{\Gamma }(z+h)}{{\prod }_{j=0}^{h-1}(z+j)}},\psi \left(z\right)=\psi (z+h)-\sum _{j=0}^{h-1}{\displaystyle \frac{1}{z+j}},{\psi }^{\prime }\left(z\right)={\psi }^{\prime }(z+h)+\sum _{j=0}^{h-1}{\displaystyle \frac{1}{{(z+j)}^2}},$$

implemented by disk arithmetic with reciprocal guards as in 4.5 (Part 1). This yields terminating rational-disk enclosures for $\mathrm{\Gamma }\left(D\right)$, $\psi \left(D\right)$, and ${\psi }^{\prime }\left(D\right)$.

Proof. 

Shifting by h gives $D+h$ with ${\sigma }_{-}(D+h)>0$, so 7.4 applies. For pullback, the hypotheses ${|c+j|}_{\mathrm{lo},n}>r$ ensure each reciprocal $1/(D+j)$ is defined with certified separation from 0 (Part 1, reciprocal rule). Each pullback identity is a finite composition of sound disk operations, so soundness follows from 4.3. Termination is immediate from finiteness. □

8. Decidability V: Enclosures for $\xi$ and $\mathrm{\Xi }$ on Disks

In this section we assemble the special-function enclosures into a terminating rational-disk routine

$$D_z⟼D_{\mathrm{\Xi }}(P;D_z)$$

enclosing $\mathrm{\Xi }\left(D_z\right)$ for rational z-disks $D_z$ and integer parameter packages

$$P=(N,m,M,h,n,p)\in {\mathbb{N}}^6.$$

Here $(N,m)$ are Euler–Maclaurin parameters, M is the Stirling truncation order, h is an explicit right-half-plane shift for $\mathrm{\Gamma },\psi ,{\psi }^{\prime }$, and

$$n\in {\mathbb{N}}_{\ge 1}\mathrm{controls}\mathrm{disk}-\mathrm{ops}\mathrm{certified}\mathrm{bounds},p\in {\mathbb{N}}_{\ge 1}\mathrm{controls}\log /\pi \mathrm{interval}\mathrm{tightness}.$$

Throughout this section we use the verifier-ready real disk $D_{log\pi }\left(p\right)$ enclosing $log\pi$ from 7.1 (Part 3).

8.1. The Archimedean Factor and Its Logarithmic Derivatives

Definition 8.1

(Archimedean factor). Define

$$A\left(s\right):=\frac{1}{2}s(s-1){\pi }^{-s/2}\mathrm{\Gamma }\left({\displaystyle \frac{s}{2}}\right),\xi \left(s\right)=A\left(s\right)\zeta \left(s\right),\mathrm{\Xi }\left(z\right)=\xi \left(\frac{1}{2}+\mathrm{i}z\right).$$

Lemma 8.1

(Logarithmic derivatives of A). Let $L\left(s\right):=A^{\prime }\left(s\right)/A\left(s\right)$ and $L^{\prime }\left(s\right):={(A^{\prime }/A)}^{\prime }\left(s\right)$. Then

$$L\left(s\right)={\displaystyle \frac{1}{s}}+{\displaystyle \frac{1}{s-1}}-{\displaystyle \frac{1}{2}}log\pi +{\displaystyle \frac{1}{2}}\psi \left({\displaystyle \frac{s}{2}}\right),$$

$$L^{\prime }\left(s\right)=-{\displaystyle \frac{1}{s^2}}-{\displaystyle \frac{1}{{(s-1)}^2}}+{\displaystyle \frac{1}{4}}{\psi }^{\prime }\left({\displaystyle \frac{s}{2}}\right).$$

Proof. 

Differentiate $logA\left(s\right)=log\frac{1}{2}+logs+log(s-1)-{\displaystyle \frac{s}{2}}log\pi +log\mathrm{\Gamma }(s/2)$ and use ${(log\mathrm{\Gamma })}^{\prime }=\psi$ and ${\displaystyle \frac{d}{ds}}\psi (s/2)={\displaystyle \frac{1}{2}}{\psi }^{\prime }(s/2)$. □

Lemma 8.2

(Derivatives of $\xi$ without dividing by $\zeta$). With $L,L^{\prime }$ as above,

$${\xi }^{\prime }\left(s\right)=A\left(s\right)\left(L\left(s\right)\zeta \left(s\right)+{\zeta }^{\prime }\left(s\right)\right),$$

$${\xi }^{\prime \prime }\left(s\right)=A\left(s\right)\left((L{\left(s\right)}^2+L^{\prime }\left(s\right))\zeta \left(s\right)+2L\left(s\right){\zeta }^{\prime }\left(s\right)+{\zeta }^{\prime \prime }\left(s\right)\right).$$

Proof. 

Differentiate $\xi =A\zeta$ twice and substitute $A^{\prime }=LA$ and $A^{\prime \prime }=(L^2+L^{\prime })A$. □

8.2. Induced s-Disks and the Parameter Package

Definition 8.2

(Induced s-disk from a z-disk). Let $D_z=\overline{B}(c_z,r_z)$ be a rational disk with $c_z\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ and $r_z\in {\mathbb{Q}}_{\ge 0}$. Define the induced s-disk under the affine map $z\mapsto s=\frac{1}{2}+\mathrm{i}z$ by

$$D_s:=\frac{1}{2}+\mathrm{i}{D}_z:=\overline{B}\left(\frac{1}{2}+\mathrm{i}{c}_z,r_z\right).$$

Lemma 8.3

(Soundness of the induced s-disk). With notation as in 8.2, for every $z\in D_z$ one has

$$s=\frac{1}{2}+\mathrm{i}z\in D_s.$$

Moreover, if $D_z=\overline{B}(c_z,r_z)$ then the induced disk has center $c_s=\frac{1}{2}+\mathrm{i}{c}_z$ and radius $r_s=r_z$.

Proof. 

Write $z=c_z+\delta$ with $\left|\delta \right|\le r_z$. Then

$$\frac{1}{2}+\mathrm{i}z=\left(\frac{1}{2}+\mathrm{i}{c}_z\right)+\mathrm{i}\delta ,$$

and $\left|\mathrm{i}\delta \right|=\left|\delta \right|\le r_z$, hence $\frac{1}{2}+\mathrm{i}z\in \overline{B}(\frac{1}{2}+\mathrm{i}{c}_z,r_z)=D_s$. □

Lemma 8.4

(Stage region avoids the guarded singularities used by the verifier). Let $z\in \overline{{\mathsf{\Omega }}_{j,k}}=R_{j,k}\subset \mathcal{U}$ and set $s=\frac{1}{2}+\mathrm{i}z$. Then

$${\sigma }_j\le Re\left(s\right)\le \frac{1}{2}-{\sigma }_j.$$

In particular, $s\ne 0$ and $s\ne 1$, and $Re(s/2)\ge {\sigma }_j/2>0$, so $s/2\notin \{0,-1,-2,\dots \}$.

Proof. 

Write $z=x+\mathrm{i}y$ with $y\in [{\sigma }_j,\frac{1}{2}-{\sigma }_j]$ on $R_{j,k}$. Then $s=\frac{1}{2}+\mathrm{i}(x+\mathrm{i}y)=(\frac{1}{2}-y)+\mathrm{i}x$, so $Re\left(s\right)=\frac{1}{2}-y\in [{\sigma }_j,\frac{1}{2}-{\sigma }_j]$. The stated consequences follow immediately. □

Definition 8.3

(Parameter package). Aparameter packageis

$$P=(N,m,M,h,n,p)\in {\mathbb{N}}^6$$

with $N\ge 2$, $m\ge 1$, $M\ge 1$, $h\ge 0$, $n\ge 1$, $p\ge 1$, interpreted as:

• $(N,m)$: Euler–Maclaurin truncation parameters for $\zeta ,{\zeta }^{\prime },{\zeta }^{\prime \prime }$;
• M: Stirling truncation order for $\mathrm{\Gamma },\psi ,{\psi }^{\prime }$;
• h: explicit right-half-plane shift used in 7.5 (Part 3);
• n: internal precision knob for disk operations (certified modulus/exp bounds);
• p: certified-interval knob for $log(·)$ and π-dependent constants.

8.3. Parameterized $\mathrm{\Xi }$-Disk Routine

Definition 8.4

(Parameterized $\mathrm{\Xi }$-disk routine). Fix a parameter package $P=(N,m,M,h,n,p)$. Given a rational disk $D_z$:

(1)

Form the induced s-disk $D_s$ from $D_z$ as in 8.2. Write $D_s=\overline{B}(c_s,r_s)$.

(2)

Compute disks enclosing $\zeta \left(D_s\right)$, ${\zeta }^{\prime }\left(D_s\right)$, ${\zeta }^{\prime \prime }\left(D_s\right)$ using (Parts 2–3), producing rational disks

$$D_{\zeta },D_{{\zeta }^{\prime }},D_{{\zeta }^{\prime \prime }}.$$

If any hypothesis required by those routines fails (e.g. the certified pole-separation guard $|c_s{-1|}_{\mathrm{lo},n}>r_s$ fails, or a certified reciprocal guard fails at some intermediate step), the routine rejects.

(3)

Form $D_{s/2}:=\frac{1}{2}{D}_s=\overline{B}(\frac{1}{2}{c}_s,\frac{1}{2}{r}_s)$ and compute disks enclosing $\mathrm{\Gamma }\left(D_{s/2}\right)$, $\psi \left(D_{s/2}\right)$, ${\psi }^{\prime }\left(D_{s/2}\right)$ using 7.5 (Part 3) with shift h, producing rational disks

$$D_{\mathrm{\Gamma }},D_{\psi },D_{{\psi }^{\prime }}.$$

Concretely, this call performs (and may reject on failure of) the decidable guard checks:

$${\sigma }_{-}(D_{s/2}+h)=Re\left(c_{s/2}\right)+h-r_{s/2}>0,$$

and for each $j=0,1,\dots ,h$ the certified separation

$$|c_{s/2}{+j|}_{\mathrm{lo},n}>r_{s/2}(\mathrm{ensuring}\mathrm{the}\mathrm{shifted}-\mathrm{disk}\mathrm{reciprocal}\mathrm{and}\mathrm{the}\mathrm{pullback}\mathrm{reciprocals}1/(D_{s/2}+j)\mathrm{are}\mathrm{verifier}-\mathrm{defined}).$$

(4)

Compute a disk enclosure for ${\pi }^{-D_s/2}$ as follows. Let $D_{log\pi }:=D_{log\pi }\left(p\right)$ from 7.1 (Part 3). Define

$$D_{{\pi }^{-s/2}}:=exp\left(-(D_{s/2}·D_{log\pi })\right),$$

using disk multiplication and disk exponential (with internal precision parameter n).

(5)

Compute $A\left(D_s\right)$ by

$$D_A:=\frac{1}{2}·D_s·(D_s-1)·D_{{\pi }^{-s/2}}·D_{\mathrm{\Gamma }}.$$

(6)

Compute $L\left(D_s\right)$ and $L^{\prime }\left(D_s\right)$ using 8.1:

$$D_L:={\displaystyle \frac{1}{D_s}}+{\displaystyle \frac{1}{D_s-1}}-\frac{1}{2}{D}_{log\pi }+\frac{1}{2}{D}_{\psi },$$

Let $D_{1/s}:=1/D_s$ and $D_{1/(s-1)}:=1/(D_s-1)$ be computed via the certified reciprocal rule (4.5, Part 1); if either reciprocal is rejected, reject. Define

$$D_{L^{\prime }}:=-(D_{1/s}·D_{1/s})-(D_{1/(s-1)}·D_{1/(s-1)})+\frac{1}{4}{D}_{{\psi }^{\prime }}.$$

Any reciprocal invoked here is computed using the certified lower-modulus guard from 4.5 (Part 1), and the routine rejects if the guard test fails.

(7)

Assemble disks for $\xi \left(D_s\right),{\xi }^{\prime }\left(D_s\right),{\xi }^{\prime \prime }\left(D_s\right)$ using 8.2:

$$D_{\xi }:=D_A·D_{\zeta },$$

$$D_{{\xi }^{\prime }}:=D_A·(D_L·D_{\zeta }+D_{{\zeta }^{\prime }}),$$

$$D_{{\xi }^{\prime \prime }}:=D_A·\left((D_L^2+D_{L^{\prime }})·D_{\zeta }+2D_L·D_{{\zeta }^{\prime }}+D_{{\zeta }^{\prime \prime }}\right).$$

(8)

Output

$$D_{\mathrm{\Xi }}(P;D_z):=D_{\xi },D_{{\mathrm{\Xi }}^{\prime }}(P;D_z):=\mathrm{i}{D}_{{\xi }^{\prime }},D_{{\mathrm{\Xi }}^{\prime \prime }}(P;D_z):=-D_{{\xi }^{\prime \prime }}.$$

Theorem 8.1

(Termination and soundness of the $\mathrm{\Xi }$-disk routine). For each fixed parameter package $P=(N,m,M,h,n,p)$, the routine in 8.4 terminates on every rational disk input $D_z$ and either rejects by a decidable hypothesis failure, or outputs rational disks satisfying:

$$\mathrm{\Xi }\left(D_z\right)\subseteq D_{\mathrm{\Xi }}(P;D_z),{\mathrm{\Xi }}^{\prime }\left(D_z\right)\subseteq D_{{\mathrm{\Xi }}^{\prime }}(P;D_z),{\mathrm{\Xi }}^{\prime \prime }\left(D_z\right)\subseteq D_{{\mathrm{\Xi }}^{\prime \prime }}(P;D_z).$$

Proof. 

Termination: the routine calls only terminating subroutines and then performs finitely many primitive disk operations. The certified primitives $\mathsf{LogInt}$, $\mathsf{LogQInt}$, $\mathsf{PiLow}$, $\mathsf{PiUp}$, $\mathsf{LogRat}$, and $\mathsf{ExpRat}$ are total computable by definition.

Soundness: each successful subroutine call provides a sound enclosure for its special function over the input disk; disk arithmetic preserves containment (4.3); and the algebraic identities for $A,L,L^{\prime },{\xi }^{\left(q\right)}$ and ${\mathrm{\Xi }}^{\left(q\right)}$ preserve inclusion under sound operations. □

Remark 8.1

(Why the verifier may reject some disks). The ${\zeta }^{\left(q\right)}$ routines require certified pole-separation hypotheses (not merely set-theoretic avoidance), in particular the verifier needs $|c_s{-1|}_{\mathrm{lo},n}>r_s$ to form ${(D_s-1)}^{-1}$ (and hence ${(D_s-1)}^{-2}$), as well as certified separation from finitely many points $s=-j$ appearing in the Pochhammer derivative bounds. The Stirling/shift layer requires ${\sigma }_{-}(D_{s/2}+h)>0$ and certified separation from 0 for the finitely many pullback denominators $D_{s/2}+j$$(0\le j<h)$. Reciprocals in the $A,L,L^{\prime }$ algebra require certified separation using ${|·|}_{\mathrm{lo},n}$. All such conditions are decidable from the rational disk data; a certificate may refine the boundary mesh or increase $(h,n,p)$ until all checks pass.

9. The Certificate Predicate: Record Format and Decidability

9.1. Boundary Polygons and Segment Disk Covers

Definition 9.1

(Boundary polygon for $\partial R_{j,k}$). Fix $(j,k)$. Aboundary polygonfor $\partial R_{j,k}$ is a finite sequence

$$(z_0,z_1,\dots ,z_L)\in {(\mathbb{Q}+\mathrm{i}\mathbb{Q})}^{L+1}$$

such that:

(i)

$z_0=z_L$;

(ii)

for each $\nu =1,\dots ,L$, the segment $[z_{\nu -1},z_{\nu }]$ lies on $\partial R_{j,k}$;

(iii)

the segments traverse $\partial R_{j,k}$ exactly once in positive (counterclockwise) order.

Definition 9.2

(Rectangle side constants for stage $(j,k)$). For fixed $(j,k)$ define the rational side coordinates

$$x_L:=\frac{1}{2}{T}_k,x_R:=2T_k,y_B:={\sigma }_j,y_T:=\frac{1}{2}-{\sigma }_j.$$

Define the four rational corners

$$c_0:=x_L+\mathrm{i}{y}_B,c_1:=x_R+\mathrm{i}{y}_B,c_2:=x_R+\mathrm{i}{y}_T,c_3:=x_L+\mathrm{i}{y}_T.$$

Definition 9.3

(Decidable predicate: a rational segment lies on $\partial R_{j,k}$ with CCW orientation). Fix $(j,k)$ and let $a,b\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ with $a\ne b$. Let $x_L,x_R,y_B,y_T$ be as in 9.2. Write $a=a_x+\mathrm{i}{a}_y$ and $b=b_x+\mathrm{i}{b}_y$ with $a_x,a_y,b_x,b_y\in \mathbb{Q}$.

Define ${\mathsf{EdgeOnBoundary}}_{j,k}(a,b)$ to hold iff one of the following four (decidable) cases holds:

(1)

(bottom side, left-to-right) $a_y=b_y=y_B$, $x_L\le a_x\le b_x\le x_R$;

(2)

(right side, bottom-to-top) $a_x=b_x=x_R$, $y_B\le a_y\le b_y\le y_T$;

(3)

(top side, right-to-left) $a_y=b_y=y_T$, $x_L\le b_x\le a_x\le x_R$;

(4)

(left side, top-to-bottom) $a_x=b_x=x_L$, $y_B\le b_y\le a_y\le y_T$.

Lemma 9.1

(Correctness: ${\mathsf{EdgeOnBoundary}}_{j,k}(a,b)$). If ${\mathsf{EdgeOnBoundary}}_{j,k}(a,b)$ holds, then the entire segment $[a,b]$ lies in $\partial R_{j,k}$ and is oriented counterclockwise along the boundary. Conversely, if $[a,b]\subset \partial R_{j,k}$ is a nontrivial segment oriented counterclockwise along the boundary, then ${\mathsf{EdgeOnBoundary}}_{j,k}(a,b)$ holds.

Proof. 

Each of the four cases explicitly asserts the segment is horizontal/vertical on one of the four boundary lines with the other coordinate constrained to the corresponding closed interval, and with the coordinate monotonicity matching the CCW direction. All checks are exact rational equalities/inequalities. □

Definition 9.4

(Decidable predicate: $\mathsf{BoundaryPolygonOK}(j,k,z_{•})$). Fix $(j,k)$ and let $z_{•}=(z_0,\dots ,z_L)\in {(\mathbb{Q}+\mathrm{i}\mathbb{Q})}^{L+1}$. Let $c_0,c_1,c_2,c_3$ be the corners from 9.2.

Define $\mathsf{BoundaryPolygonOK}(j,k,z_{•})$ to hold iff:

(i)

$L\ge 4$ and $z_0=z_L$;

(ii)

$z_0=c_0$ (the polygon is rooted at the bottom-left corner);

(iii)

for each $\nu =1,\dots ,L$, one has $z_{\nu -1}\ne z_{\nu }$ and ${\mathsf{EdgeOnBoundary}}_{j,k}(z_{\nu -1},z_{\nu })$ holds;

(iv)

letting $q_0:=0$ and scanning forward, the first time the vertex equals $c_1$ occurs at some index $q_1$, then the first subsequent time the vertex equals $c_2$ occurs at some $q_2$, then the first subsequent time the vertex equals $c_3$ occurs at some $q_3$, and finally $z_L=c_0$ with no earlier return to $c_0$ after leaving it; formally: there exist indices

$$0=q_0<q_1<q_2<q_3<L$$

such that

$$z_{q_1}=c_1,z_{q_2}=c_2,z_{q_3}=c_3,$$

and for all $\nu \in \{1,\dots ,L-1\}$ one has $z_{\nu }\ne c_0$.

Lemma 9.2

(If $\mathsf{BoundaryPolygonOK}$ holds, then $z_{•}$ is a boundary polygon). If $\mathsf{BoundaryPolygonOK}(j,k,z_{•})$ holds, then $z_{•}$ satisfies 9.1 (i.e. it traverses $\partial R_{j,k}$ exactly once in positive order).

Proof. 

By construction, every edge lies on $\partial R_{j,k}$ and is CCW oriented (9.1). The forced corner order $c_0\to c_1\to c_2\to c_3\to c_0$, with no premature return to $c_0$, precludes backtracking and enforces one full CCW traversal of the rectangle boundary. □

Definition 9.5

(Disk-chain cover of a boundary segment).

Lemma 9.3

(A segment between centers lies in the union of two overlapping disks). Let $D_1=\overline{B}(u,r)$ and $D_2=\overline{B}(v,s)$ be disks with $D_1\cap D_2\ne ⌀$. Then the straight segment $[u,v]\subseteq D_1\cup D_2$.

Proof. 

Let $d=|u-v|$. Overlap means $d\le r+s$. For $p\left(t\right)=u+t(v-u)$, we have $\left|p\right(t)-u|=td$ and $\left|p\right(t)-v|=(1-t)d$. Thus $p\left(t\right)\in D_1$ for $t\le r/d$ and $p\left(t\right)\in D_2$ for $t\ge 1-s/d$. Since $r/d\ge 1-s/d$, these intervals cover $[0,1]$. □

Let $\gamma =[a,b]\subset \mathbb{C}$ be a nontrivial line segment with $a,b\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$. Asegment disk-chain cover recordfor γ is a finite sequence of rational points

$$(p_0,p_1,\dots ,p_M)\in {(\mathbb{Q}+\mathrm{i}\mathbb{Q})}^{M+1}$$

together with rational radii $({\rho }_0,\dots ,{\rho }_M)\in {\left({\mathbb{Q}}_{\ge 0}\right)}^{M+1}$, defining disks

$$D_{z,r}:=\overline{B}(p_r,{\rho }_r)(r=0,\dots ,M),$$

such that:

(i)

$p_0=a$ and $p_M=b$;

(ii)

each $p_r$ lies on the segment γ (i.e. $p_r=a+t_r(b-a)$ for some $t_r\in [0,1]\cap \mathbb{Q}$);

(iii)

the parameters are ordered: $0=t_0\le t_1\le \dots \le t_M=1$;

(iv)

consecutive overlap holds:

$$|p_{r+1}-p_r|\le {\rho }_r+{\rho }_{r+1}(r=0,\dots ,M-1).$$

Call such a recordadmissible. Then $\gamma \subseteq {\bigcup }_{r=0}^M{D}_{z,r}$ and $D_{z,r}\cap D_{z,r+1}\ne ⌀$ for all r.

Definition 9.6

(Decidable predicate: a rational point lies on a rational segment). Let $a,b,p\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ with $a\ne b$. Write $a=a_x+\mathrm{i}{a}_y$, $b=b_x+\mathrm{i}{b}_y$, $p=p_x+\mathrm{i}{p}_y$ with $a_x,a_y,b_x,b_y,p_x,p_y\in \mathbb{Q}$.

Define the (rational)cross product

$$cr(u,w):=Re\left(u\right)Im\left(w\right)-Im\left(u\right)Re\left(w\right),$$

and the (rational)dot product

$$dp(u,w):=Re\left(u\right)Re\left(w\right)+Im\left(u\right)Im\left(w\right).$$

Define the decidable predicate $\mathsf{PointOnSeg}(a,b,p)$ to hold iff both:

(i)

collinearity: $cr(b-a,p-a)=0$;

(ii)

between-ness:

$$0\le dp(p-a,b-a)\le dp(b-a,b-a).$$

Lemma 9.4

(Correctness of $\mathsf{PointOnSeg}$). If $a\ne b$ and $a,b,p\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$, then $\mathsf{PointOnSeg}(a,b,p)$ holds if and only if $p\in [a,b]$ (the closed line segment from a to b).

Proof. 

Let $v=b-a\ne 0$ and $w=p-a$. The condition $cr(v,w)=0$ is equivalent to $w=\lambda v$ for some real $\lambda$. Then $dp(w,v)=\lambda dp(v,v)$ and $dp(v,v)>0$. Thus

$$0\le dp(w,v)\le dp(v,v)⟺0\le \lambda \le 1,$$

which is equivalent to $p=a+\lambda (b-a)\in [a,b]$. All quantities are rational, so the test is decidable by exact rational arithmetic. □

Definition 9.7

(Decidable admissibility check for a segment disk-chain cover record). Let $\gamma =[a,b]$ with $a,b\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$, $a\ne b$. Given a proposed record

$$(p_0,\dots ,p_M)\in {(\mathbb{Q}+\mathrm{i}\mathbb{Q})}^{M+1},({\rho }_0,\dots ,{\rho }_M)\in {\left({\mathbb{Q}}_{\ge 0}\right)}^{M+1},$$

define $\mathsf{SegmentChainAdmissible}(a,b,p_{•},{\rho }_{•})$ to hold iff:

(1)

endpoints match: $p_0=a$ and $p_M=b$;

(2)

segment membership: $\mathsf{PointOnSeg}(a,b,p_r)$ holds for each $r=0,\dots ,M$;

(3)

ordering along the segment: for each $r=0,\dots ,M-1$,

$$dp(p_r-a,b-a)\le dp(p_{r+1}-a,b-a);$$

(4)

overlap: for each $r=0,\dots ,M-1$,

$$|p_{r+1}-p_r{|}^2\le {({\rho }_r+{\rho }_{r+1})}^2,$$

where ${|x+\mathrm{i}y|}^2:=x^2+y^2$ is computed exactly in $\mathbb{Q}$.

Lemma 9.5

(Soundness of the admissibility check). If $\mathsf{SegmentChainAdmissible}(a,b,p_{•},{\rho }_{•})$ holds, then the disks $D_{z,r}:=\overline{B}(p_r,{\rho }_r)$ satisfy:

(i)

$\gamma \subseteq {\bigcup }_{r=0}^M{D}_{z,r}$;

(ii)

$D_{z,r}\cap D_{z,r+1}\ne ⌀$ for all $r=0,\dots ,M-1$.

Proof. 

By (4), $|p_{r+1}-p_r|\le {\rho }_r+{\rho }_{r+1}$, hence $D_{z,r}\cap D_{z,r+1}\ne ⌀$. By (2) and (3), the segment decomposes as $\gamma ={\bigcup }_{r=0}^{M-1}[p_r,p_{r+1}]$. By 9.3, each $[p_r,p_{r+1}]\subseteq D_{z,r}\cup D_{z,r+1}$, hence $\gamma \subseteq {\bigcup }_{r=0}^M{D}_{z,r}$. □

9.2. Exact Winding Computation for Rational Polygons

Definition 9.8

(Ray-crossing winding for rational polygons). Fix a nonzero direction $v\in (\mathbb{Q}+\mathrm{i}\mathbb{Q})\setminus \left\{0\right\}$ and define

$$cr(v,w):=Re\left(v\right)Im\left(w\right)-Im\left(v\right)Re\left(w\right),dp(v,w):=Re\left(v\right)Re\left(w\right)+Im\left(v\right)Im\left(w\right).$$

Let $(p_0,\dots ,p_n)\in {(\mathbb{Q}+\mathrm{i}\mathbb{Q})}^{n+1}$ be a closed polygon loop with $p_n=p_0$ and $p_i\ne 0$. Assume $cr(v,p_i)\ne 0$ for all vertices (general position with respect to the ray line).

Define ${wind}_v\left(p_{•}\right)$ by counting signed intersections of the polygon with the ray ${\mathbb{R}}_{>0}v$: initialize $W:=0$ and for each edge $[p_i,p_{i+1}]$ set $c_i:=cr(v,p_i)$ and $c_{i+1}:=cr(v,p_{i+1})$. If $c_i<0<c_{i+1}$ (upcrossing) or $c_i>0>c_{i+1}$ (downcrossing), compute

$$t={\displaystyle \frac{c_i}{c_i-c_{i+1}}}\in \mathbb{Q},q:=p_i+t(p_{i+1}-p_i).$$

If additionally $dp(v,q)>0$ (so q lies on the ray), update

$$W:=W+1\mathrm{for}\mathrm{an}\mathrm{upcrossing},W:=W-1\mathrm{for}\mathrm{a}\mathrm{downcrossing}.$$

Output $W\in \mathbb{Z}$.

Lemma 9.6

(Correctness of ray-crossing winding). Assume the hypotheses of 9.8, and additionally assume that the polygonal loop lies in ${\mathbb{C}}^{\times }$ (equivalently, $0\notin [p_i,p_{i+1}]$ for every edge). Then ${wind}_v\left(p_{•}\right)$ equals the topological winding number of the polygon about 0.

Proof. 

Standard planar topology: for a loop in general position with respect to a ray from the origin, the winding number equals the signed intersection number with that ray. □

9.3. Certificate Records and Verifier

Definition 9.9

(Stage certificate record). Fix $(j,k)$. Astage certificate recordis a finite code $c\in \mathbb{N}$ encoding:

(1)

a boundary polygon $(z_0,\dots ,z_L)$ for $\partial R_{j,k}$;

(2)

for each edge ${\gamma }_{\nu }=[z_{\nu -1},z_{\nu }]$, an admissible segment disk-chain cover record

$$\left((p_{\nu ,0},\dots ,p_{\nu ,M_{\nu }}),({\rho }_{\nu ,0},\dots ,{\rho }_{\nu ,M_{\nu }})\right)$$

as in 9.5, specifying disks $D_{z,\nu ,r}:=\overline{B}(p_{\nu ,r},{\rho }_{\nu ,r})$;

(3)

for each boundary cover disk $D_{z,\nu ,r}$ arising from the segment covers, a parameter package

$$P_{\nu ,r}=(N_{\nu ,r},m_{\nu ,r},M_{\nu ,r},h_{\nu ,r},n_{\nu ,r},p_{\nu ,r})\in {\mathbb{N}}^6$$

as in 8.3;

(4)

a rational direction $v\in (\mathbb{Q}+\mathrm{i}\mathbb{Q})\setminus \left\{0\right\}$ for the winding computation.

Definition 9.10

(Decidable noncontainment of 0 in a rational disk). For a rational disk $\overline{B}(u,R)$ with $u=a+\mathrm{i}b\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ and $R\in {\mathbb{Q}}_{\ge 0}$, define

$$\mathsf{ZeroNotInDisk}(u,R):⟺a^2+b^2>R^2.$$

Then $\mathsf{ZeroNotInDisk}(u,R)$ is decidable by exact rational arithmetic and implies $0\notin \overline{B}(u,R)$.

Definition 9.11

(Decidable overlap of rational disks). For rational disks $\overline{B}(u,R)$ and $\overline{B}(v,S)$ with $u=u_x+\mathrm{i}{u}_y$, $v=v_x+\mathrm{i}{v}_y$ in $\mathbb{Q}+\mathrm{i}\mathbb{Q}$ and $R,S\in {\mathbb{Q}}_{\ge 0}$, define

$$\mathsf{DiskOverlap}(u,R;v,S):⟺{|u-v|}^2\le {(R+S)}^2,$$

i.e.

$${(u_x-v_x)}^2+{(u_y-v_y)}^2\le {(R+S)}^2,$$

checked by exact rational arithmetic. If $\mathsf{DiskOverlap}(u,R;v,S)$ holds then $\overline{B}(u,R)\cap \overline{B}(v,S)\ne ⌀$.

Definition 9.12

(Compression of a rational polygon vertex list). Let $(u_0,\dots ,u_{m-1})\in {(\mathbb{Q}+\mathrm{i}\mathbb{Q})}^m$ with $m\ge 1$. Define the compressed list $Compress\left(u_{•}\right)=(q_0,\dots ,q_{M-1})$ by the terminating scan: initialize the output list with $q_0:=u_0$, and for $i=1,\dots ,m-1$ append $u_i$ iff $u_i\ne q_{M-1}$. Finally, if $M\ge 2$ and $q_{M-1}=q_0$, delete the last entry so that $q_{M-1}\ne q_0$.

Equivalently, Compress deletes consecutive repetitions and then (optionally) removes a final duplicate of the first vertex. The output satisfies $M\ge 1$ and $q_i\ne q_{i+1}$ for all i whenever $M\ge 2$.

Definition 9.13

(Certificate predicate $\mathrm{Cert}(j,k,c)$). Given $(j,k,c)$, the verifier:

(A)

parses c into $(z_{•},(\mathrm{segment}\mathrm{disk}-\mathrm{chain}{\mathrm{records})}_{\nu },{\left(P_{\nu ,r}\right)}_{\nu ,r},v)$; if parsing fails, reject;

(B)

checks $\mathsf{BoundaryPolygonOK}(j,k,z_{•})$ in the sense of 9.4; else reject;

(C)

for each edge ${\gamma }_{\nu }=[z_{\nu -1},z_{\nu }]$, checks that the parsed segment-cover record has matching list lengths (i.e. the point list $(p_{\nu ,0},\dots ,p_{\nu ,M_{\nu }})$ has length $M_{\nu }+1$ and the radius list $({\rho }_{\nu ,0},\dots ,{\rho }_{\nu ,M_{\nu }})$ has length $M_{\nu }+1$), and that each ${\rho }_{\nu ,r}\in {\mathbb{Q}}_{\ge 0}$; else reject;

(D)

for each edge ${\gamma }_{\nu }=[z_{\nu -1},z_{\nu }]$, checks

$$\mathsf{SegmentChainAdmissible}\left(z_{\nu -1},z_{\nu },(p_{\nu ,0},\dots ,p_{\nu ,M_{\nu }}),({\rho }_{\nu ,0},\dots ,{\rho }_{\nu ,M_{\nu }})\right)$$

in the sense of 9.7; else reject;

(E)

constructs all boundary cover disks $D_{z,\nu ,r}:=\overline{B}(p_{\nu ,r},{\rho }_{\nu ,r})$ and (using the package attached to that disk) computes:

$$E_{\nu ,r}^{\left(0\right)}:=D_{\mathrm{\Xi }}(P_{\nu ,r};\overline{B}(p_{\nu ,r},0)),E_{\nu ,r}^{\left(1\right)}:=D_{{\mathrm{\Xi }}^{\prime }}(P_{\nu ,r};D_{z,\nu ,r}).$$

If either enclosure computation rejects, reject. Write

$$E_{\nu ,r}^{\left(0\right)}=\overline{B}(u_{\nu ,r}^{\left(0\right)},R_{\nu ,r}^{\left(0\right)}),E_{\nu ,r}^{\left(1\right)}=\overline{B}(u_{\nu ,r}^{\left(1\right)},R_{\nu ,r}^{\left(1\right)}).$$

Let $n_{\nu ,r}$ be the internal-precision component of $P_{\nu ,r}$ and define the rational bound

$$M_{\nu ,r}:={\left|u_{\nu ,r}^{\left(1\right)}\right|}_{\mathrm{up},n_{\nu ,r}}+R_{\nu ,r}^{\left(1\right)}.$$

Define the verifier-derived enclosure disk (cf. 10.3). Set the (rational) enclosure radius

$${\mathrm{rad}}_{\nu ,r}:=R_{\nu ,r}^{\left(0\right)}+M_{\nu ,r}{\rho }_{\nu ,r}\in {\mathbb{Q}}_{\ge 0},$$

and define

$$E_{\nu ,r}:=\overline{B}\left(u_{\nu ,r}^{\left(0\right)},{\mathrm{rad}}_{\nu ,r}\right).$$

Now verify $\mathsf{ZeroNotInDisk}(u_{\nu ,r}^{\left(0\right)},{\mathrm{rad}}_{\nu ,r})$ (equivalently $0\notin E_{\nu ,r}$); if any such check fails, reject;

(F)

forms theraw(not necessarily simple) lists $(u_0,\dots ,u_{m-1})$ and $({\mathrm{rad}}_0,\dots ,{\mathrm{rad}}_{m-1})$ by listing, in boundary order, the centers $u_{\nu ,r}^{\left(0\right)}$ and radii ${\mathrm{rad}}_{\nu ,r}$ of the verifier-derived enclosure disks $E_{\nu ,r}=\overline{B}(u_{\nu ,r}^{\left(0\right)},{\mathrm{rad}}_{\nu ,r})$.

It then checks explicit consecutive overlap of theenclosuredisks in the raw boundary order:

(i)

if $m<2$, reject;

(ii)

for each $i=0,\dots ,m-2$, verify $\mathsf{DiskOverlap}(u_i,{\mathrm{rad}}_i;u_{i+1},{\mathrm{rad}}_{i+1})$;

(iii)

verify cyclic closure overlap $\mathsf{DiskOverlap}(u_{m-1},{\mathrm{rad}}_{m-1};u_0,{\mathrm{rad}}_0)$.

If any overlap check fails, reject.

It then forms thecompressedvertex list

$$(q_0,\dots ,q_{M-1}):=Compress(u_0,\dots ,u_{m-1})$$

as in 9.12, and finally the closed polygonal loop

$$q_{•}=(q_0,\dots ,q_{M-1},q_M)\mathrm{with}{q}_M:=q_0.$$

It then checks:

(i)

$M\ge 2$ (so the loop has at least one edge);

(ii)

$q_i\ne 0$ for all vertices ($0\le i\le M$);

(iii)

general position $cr(v,q_i)\ne 0$ for all vertices ($0\le i\le M$);

(iv)

nondegenerate edges $q_i\ne q_{i+1}$ for all edges ($0\le i\le M-1$).

If any check fails, reject;

1.

computes $W:={wind}_v\left(q_{•}\right)$ and accepts iff $W=0$.

Theorem 9.1

(Decidability of $\mathrm{Cert}$). The predicate $\mathrm{Cert}(j,k,c)$ is decidable.

Proof. 

All loops are bounded by integers encoded in c. Each enclosure computation $E_{\nu ,r}=D_{\mathrm{\Xi }}(P;D_{z,\nu ,r})$ terminates by 8.1, and either returns a rational disk or signals a decidable hypothesis failure (which the verifier treats as rejection). All remaining checks are finite rational computations (disk containment, nonvanishing $0\notin \overline{B}(u,r)$, and the rational ray-crossing winding algorithm). Hence the verifier halts on all inputs. □

Remark 9.1

(Role of Arb in practice). Arb (or any validated numerics library) may be used off-line tosearch for candidate certificates efficiently. The formal predicate $\mathrm{Cert}$ trusts only the explicit rational bounds and terminating routines specified in this paper.

10. Soundness of the Certificate Predicate

We prove: if $\mathrm{Cert}(j,k,c)=1$, then $\mathrm{\Xi }$ has no zeros in ${\mathsf{\Omega }}_{j,k}$. The argument has three steps:

(1)

the boundary $\partial R_{j,k}$ is covered by the certificate’s z-disks;

(2)

enclosure avoidance ($0\notin E_{\nu ,r}$) implies $\mathrm{\Xi }\ne 0$ on $\partial R_{j,k}$;

(3)

the certified polygon winding equals the analytic winding $wind(\mathrm{\Xi }(\partial R_{j,k}),0)$.

10.1. Boundary Coverage and Boundary Nonvanishing

Lemma 10.1

(Boundary coverage by admissible segment covers). Assume the verifier passes steps (B)–(D) of 9.13 for a record c at stage $(j,k)$. Then the union of the segment-cover disks $D_{z,\nu ,r}$ covers $\partial R_{j,k}$.

Proof. 

The boundary polygon $z_{•}$ expresses $\partial R_{j,k}$ as the concatenation of edges ${\gamma }_{\nu }=[z_{\nu -1},z_{\nu }]$. Fix one edge $\gamma =[a,b]$ and its admissible disk-chain record $\left((p_0,\dots ,p_M),({\rho }_0,\dots ,{\rho }_M)\right)$. By admissibility, $D_{z,r}\cap D_{z,r+1}\ne ⌀$ for each r. Since each $p_r$ lies on $\gamma$ and the $p_r$ are ordered along $\gamma$ from a to b, we have

$$\gamma =\bigcup _{r=0}^{M-1}[p_r,p_{r+1}].$$

By 9.3, each subsegment $[p_r,p_{r+1}]\subseteq D_{z,r}\cup D_{z,r+1}$, hence $\gamma \subseteq {\bigcup }_{r=0}^M{D}_{z,r}$. Taking the union over edges covers the full boundary. □

Lemma 10.2

(Boundary nonvanishing from enclosure avoidance). If $\mathrm{Cert}(j,k,c)=1$, then $\mathrm{\Xi }\left(z\right)\ne 0$ for all $z\in \partial R_{j,k}$.

Proof. 

If $\mathrm{Cert}(j,k,c)=1$, then the verifier passes step (E) of 9.13 for every boundary cover disk $D_{z,\nu ,r}=\overline{B}(p_{\nu ,r},{\rho }_{\nu ,r})$.

In that step it computes the point enclosure

$$E_{\nu ,r}^{\left(0\right)}=D_{\mathrm{\Xi }}(P_{\nu ,r};\overline{B}(p_{\nu ,r},0))=\overline{B}(u_{\nu ,r}^{\left(0\right)},R_{\nu ,r}^{\left(0\right)}),$$

and the derivative enclosure

$$E_{\nu ,r}^{\left(1\right)}=D_{{\mathrm{\Xi }}^{\prime }}(P_{\nu ,r};D_{z,\nu ,r})=\overline{B}(u_{\nu ,r}^{\left(1\right)},R_{\nu ,r}^{\left(1\right)}),$$

forms the Lipschitz-derived enclosure

$$E_{\nu ,r}:=\overline{B}\left(u_{\nu ,r}^{\left(0\right)},R_{\nu ,r}^{\left(0\right)}+M_{\nu ,r}{\rho }_{\nu ,r}\right),M_{\nu ,r}:={\left|u_{\nu ,r}^{\left(1\right)}\right|}_{\mathrm{up},n_{\nu ,r}}+R_{\nu ,r}^{\left(1\right)},$$

and verifies $0\notin E_{\nu ,r}$ via $\mathsf{ZeroNotInDisk}$.

By 10.3, $\mathrm{\Xi }\left(D_{z,\nu ,r}\right)\subseteq E_{\nu ,r}$. Therefore $\mathrm{\Xi }$ is nonvanishing on each $D_{z,\nu ,r}$, hence (by 10.1) on all of $\partial R_{j,k}$. □

10.2. Certified Winding Equals Analytic Winding

Lemma 10.3

(Verifier-derived enclosure from a point value and a ${\mathrm{\Xi }}^{\prime }$ bound). Let $D=\overline{B}(z_0,\rho )$ be a rational disk and fix a parameter package P for which the Ξ-routine succeeds on both D and the point disk $\overline{B}(z_0,0)$. Write

$$E_0:=D_{\mathrm{\Xi }}(P;\overline{B}(z_0,0))=\overline{B}(u_0,r_0),E_1:=D_{{\mathrm{\Xi }}^{\prime }}(P;D)=\overline{B}(u_1,r_1).$$

Define the rational bound

$$M:=|u_1{|}_{\mathrm{up},n}+r_1,$$

where n is the internal-precision component of P (so ${|·|}_{\mathrm{up},n}$ is the certified modulus upper bound from 4.4). Define the derived enclosure disk

$$E_{\mathrm{Lip}}(P;D):=\overline{B}\left(u_0,r_0+M\rho \right).$$

Then for every $z\in D$ one has

$$\mathrm{\Xi }\left(z\right)\in E_{\mathrm{Lip}}(P;D).$$

Proof. 

Fix $z\in D$ and consider the straight segment $\gamma \left(t\right)=z_0+t(z-z_0)$ for $t\in [0,1]$. Since D is convex, $\gamma \left(t\right)\in D$ for all t. By the fundamental theorem of calculus for holomorphic functions,

$$\mathrm{\Xi }\left(z\right)-\mathrm{\Xi }\left(z_0\right)={\int }_0^1{\mathrm{\Xi }}^{\prime }\left(\gamma \left(t\right)\right)(z-z_0)dt,$$

hence

$$|\mathrm{\Xi }\left(z\right)-\mathrm{\Xi }\left(z_0\right)|\le \underset{w\in D}{sup}|{\mathrm{\Xi }}^{\prime }\left(w\right)|·|z-z_0|\le \left(|u_1{|}_{\mathrm{up},n}+r_1\right)\rho =M\rho ,$$

because ${\mathrm{\Xi }}^{\prime }\left(D\right)\subseteq E_1=\overline{B}(u_1,r_1)$ implies $|{\mathrm{\Xi }}^{\prime }\left(w\right)|\le |u_1|+r_1\le {\left|u_1\right|}_{\mathrm{up},n}+r_1$ for all $w\in D$. Also $\mathrm{\Xi }\left(z_0\right)\in E_0=\overline{B}(u_0,r_0)$. Therefore $\mathrm{\Xi }\left(z\right)\in \overline{B}(u_0,r_0+M\rho )=E_{\mathrm{Lip}}(P;D)$. □

Lemma 10.4

(Overlap propagates to image-enclosure overlap). Assume the verifier passes the admissibility check for a segment cover on an edge. Then consecutive z-disks overlap:

$$D_{z,r}\cap D_{z,r+1}\ne ⌀.$$

Consequently, the verifier-derived enclosures overlap:

$$E_r\cap E_{r+1}\ne ⌀,$$

where $E_r:=E_{\mathrm{Lip}}(P_r;D_{z,r})$ is the disk defined in step (E) of 9.13.

Proof. 

Admissibility gives the overlap inequality $|p_{r+1}-p_r|\le {\rho }_r+{\rho }_{r+1}$, hence $D_{z,r}\cap D_{z,r+1}\ne ⌀$.

Moreover, in the verifier this overlap can be checked explicitly by the decidable predicate $\mathsf{DiskOverlap}(u_r,rad\left(E_r\right);u_{r+1},rad\left(E_{r+1}\right))$ from 9.11. (Independently, the semantic implication via a point $z_{*}\in D_{z,r}\cap D_{z,r+1}$ also holds by 10.3.) □

Lemma 10.5

(No polygon edge can pass through 0 under overlap and zero-avoidance). Let $D_1=\overline{B}(u,r)$ and $D_2=\overline{B}(v,s)$ be disks such that $D_1\cap D_2\ne ⌀$ and $0\notin D_1$ and $0\notin D_2$. Then $0\notin [u,v]$.

Proof. 

If $0\in [u,v]$, then u and v lie on the same line through the origin with opposite directions, and one has the exact identity $|u-v|=\left|u\right|+\left|v\right|$. Since $0\notin D_1$ and $0\notin D_2$, we have $\left|u\right|>r$ and $\left|v\right|>s$, hence $|u-v|=\left|u\right|+\left|v\right|>r+s$. But $D_1\cap D_2\ne ⌀$ implies $|u-v|\le r+s$, contradiction. □

Lemma 10.6

(Certified polygon is homotopic to $\mathrm{\Xi }(\partial R_{j,k})$ in ${\mathbb{C}}^{\times }$). Assume $\mathrm{Cert}(j,k,c)=1$. Let $\mathrm{\Gamma }=\partial R_{j,k}$ oriented positively. Let $q_{•}$ be the compressed polygonal loop (obtained from the enclosure centers by Compress as in 9.12) used by the verifier. Then $\mathrm{\Xi }\left(\mathrm{\Gamma }\right)$ and $q_{•}$ are homotopic in $\mathbb{C}\setminus \left\{0\right\}$.

Proof. 

By 10.2, $\mathrm{\Xi }\left(\mathrm{\Gamma }\right)\subset {\mathbb{C}}^{\times }$. Also $0\notin E_{\nu ,r}$ for all $(\nu ,r)$, hence ${\bigcup }_{\nu ,r}{E}_{\nu ,r}\subset {\mathbb{C}}^{\times }$.

By 10.1, every point of $\mathrm{\Gamma }$ lies in some $D_{z,\nu ,r}$, hence its image lies in $E_{\nu ,r}$. Thus $\mathrm{\Xi }\left(\mathrm{\Gamma }\right)\subseteq {\bigcup }_{\nu ,r}{E}_{\nu ,r}$.

Order the boundary cover disks along $\mathrm{\Gamma }$ as in the verifier (raw list), writing the corresponding image-enclosure disks as $E_0,E_1,\dots ,E_{m-1}$ and their centers as $u_0,u_1,\dots ,u_{m-1}$ (so $E_i=\overline{B}(u_i,·)$). Consecutive cover disks overlap, hence consecutive enclosures overlap by 10.4.

Step 1: endpoint adjustment inside each $E_i$. Let $p_i$ denote the boundary point whose point-evaluation produced $u_i$ (a rational point on $\mathrm{\Gamma }$). Then $\mathrm{\Xi }\left(p_i\right)\in E_i$ by soundness of the point enclosure, and $u_i\in E_i$ by definition. Since $E_i$ is convex and avoids 0, the straight segment $[\mathrm{\Xi }\left(p_i\right),u_i]\subset E_i\subset {\mathbb{C}}^{\times }$. Therefore, in ${\mathbb{C}}^{\times }$, we may homotope $\mathrm{\Xi }\left(\mathrm{\Gamma }\right)$ (through loops contained in ${\bigcup }_i{E}_i$) to a loop whose vertices are the centers $u_i$, by inserting these short segments at the subdivision points.

Step 2: straighten each subarc in $E_i\cup E_{i+1}$. Between successive subdivision points, the corresponding image subarc of $\mathrm{\Xi }\left(\mathrm{\Gamma }\right)$ lies in $E_i\cup E_{i+1}$, and its endpoints (after Step 1) are $u_i$ and $u_{i+1}$. Because $E_i\cap E_{i+1}\ne ⌀$, the union $E_i\cup E_{i+1}$ is contractible. Hence the subarc is homotopic rel endpoints within $E_i\cup E_{i+1}$ to the straight segment $[u_i,u_{i+1}]$.

Concatenating finitely many such endpoint-adjustment homotopies and straightening homotopies yields a homotopy from $\mathrm{\Xi }\left(\mathrm{\Gamma }\right)$ to the raw center polygon, entirely inside ${\bigcup }_i{E}_i\subset {\mathbb{C}}^{\times }$.

Step 3: compression does not change homotopy class. The Compress operation removes only consecutive duplicate vertices, which deletes only zero-length edges. This does not change the homotopy class in ${\mathbb{C}}^{\times }$. Thus the compressed polygonal loop $q_{•}$ is homotopic to the raw center polygon, hence also to $\mathrm{\Xi }\left(\mathrm{\Gamma }\right)$ in ${\mathbb{C}}^{\times }$. □

Lemma 10.7

(Pre-acceptance winding correctness). Fix $(j,k)$ and a code c. Assume the verifier, when run on $(j,k,c)$, passes steps(A)–(F)of 9.13 (i.e. parsing, boundary-polygon validity, admissible segment covers, successful enclosure computations, $0\notin E_{\nu ,r}$ for all cover disks, and general-position checks for the chosen ray direction v). Let W be the integer output by the ray-crossing winding computation in step(G).

Then

$$W=wind(\mathrm{\Xi }(\partial R_{j,k}),0).$$

Proof. 

Let $R=R_{j,k}$ and $\mathrm{\Gamma }=\partial R$ oriented positively. Passing steps (A)–(E) implies: (i) the boundary is covered by the z-disks, and (ii) for each cover disk $D_{z,\nu ,r}$ the enclosure $E_{\nu ,r}$ satisfies $0\notin E_{\nu ,r}$. By 8.1 and the enclosure-avoidance check, $\mathrm{\Xi }\ne 0$ on each $D_{z,\nu ,r}$, hence on all of $\mathrm{\Gamma }$. Thus $\mathrm{\Xi }\left(\mathrm{\Gamma }\right)\subset {\mathbb{C}}^{\times }$.

By the overlap and center-segment arguments (), each polygon edge between consecutive centers in the raw list lies in $E_{\mathrm{current}}\cup E_{\mathrm{next}}\subset {\mathbb{C}}^{\times }$. If compression deletes repeated consecutive centers, each compressed edge coincides with some raw edge, so the same statement holds for the compressed loop used by the verifier. In particular, since $E_{\mathrm{current}}$ and $E_{\mathrm{next}}$ overlap and both avoid 0, 10.5 implies $0\notin [p_i,p_{i+1}]$ for every polygon edge. Therefore the polygonal loop $p_{•}$ lies in ${\mathbb{C}}^{\times }$ and is homotopic to $\mathrm{\Xi }\left(\mathrm{\Gamma }\right)$ in ${\mathbb{C}}^{\times }$ (10.6). Therefore

$$wind(\mathrm{\Xi }\left(\mathrm{\Gamma }\right),0)=wind(q_{•},0).$$

Passing step (F) guarantees general position for the chosen ray direction v, so the ray-crossing algorithm computes the topological winding of the polygon:

$$wind(q_{•},0)={wind}_v\left(q_{•}\right)=W$$

by 9.6. Combining gives $W=wind(\mathrm{\Xi }(\partial R_{j,k}),0)$. □

Theorem 10.1

(Soundness). If $\mathrm{Cert}(j,k,c)=1$, then $Z(\mathrm{\Xi };{\mathsf{\Omega }}_{j,k})=⌀$.

Proof. 

Let $R=R_{j,k}$ and $\mathsf{\Omega }={\mathsf{\Omega }}_{j,k}$. By 10.2, $\mathrm{\Xi }\ne 0$ on $\partial R$. Since $\mathrm{Cert}(j,k,c)=1$, the verifier passes steps (A)–(F) and computes $W=0$ in step (G). By 10.7, $W=wind\left(\mathrm{\Xi }\right(\partial R),0)$, hence $wind\left(\mathrm{\Xi }\right(\partial R),0)=0$. Apply the argument principle (3.1) to conclude $\#Z(\mathrm{\Xi };\mathsf{\Omega })=0$. □

11. Completeness on Zero-Free Closures (Mesh+Parameter Refinement)

Completeness is obtained by combining two refinement knobs in a way that is compatible with the verifier:

(i)

mesh refinement: choose a finite rational disk-chain cover of each boundary edge by disks with rational centers and rational radii lying inside a tubular neighborhood of $\partial R_{j,k}$;

(ii)

parameter refinement: for each fixed boundary disk $D_z$, increase the package $P=(N,m,M,h,n,p)$ until the terminating $\mathrm{\Xi }$-disk routine succeeds on $D_z$ and returns an image-enclosure disk of arbitrarily small radius (“point+derivative tightening”).

In particular, the Stirling remainder bounds are made arbitrarily small by increasing the shift parameter h, which increases ${\sigma }_{-}(D_{s/2}+h)$ in 7.5 (Part 3), and the certified-primitive errors are made arbitrarily small by increasing n and p.

11.1. Boundary Margins and a Tubular Neighborhood

Lemma 11.1

(Boundary margin). If $Z(\mathrm{\Xi };R_{j,k})=⌀$, then

$$\delta :=\underset{z\in \partial R_{j,k}}{min}\left|\mathrm{\Xi }\left(z\right)\right|>0.$$

Proof. 

$\left|\mathrm{\Xi }\right|$ is continuous on the compact set $\partial R_{j,k}$ and strictly positive there. □

Lemma 11.2

(Tubular nonvanishing neighborhood). Assume $\mathrm{\Xi }\ne 0$ on $\partial R$ for a closed rectangle R. Let $\delta :={min}_{\partial R}\left|\mathrm{\Xi }\right|>0$. Then there exists ${\rho }_{\mathrm{tube}}>0$ such that

$$\left|\mathrm{\Xi }\left(z\right)\right|\ge {\displaystyle \frac{\delta }{2}}\mathrm{whenever}(z,\partial R)\le {\rho }_{\mathrm{tube}}.$$

Proof. 

Fix $\delta :={min}_{w\in \partial R}\left|\mathrm{\Xi }\left(w\right)\right|>0$. For $n\in \mathbb{N}$ define the compact set

$$K_n:=\{(z,w)\in {\mathbb{C}}^2:w\in \partial R,|z-w|\le 1/n\}.$$

The function $(z,w)\mapsto \left|\mathrm{\Xi }\right(z)-\mathrm{\Xi }(w\left)\right|$ is continuous on ${\mathbb{C}}^2$, hence attains its maximum on $K_n$; set

$${\eta }_n:=\underset{(z,w)\in K_n}{max}|\mathrm{\Xi }\left(z\right)-\mathrm{\Xi }\left(w\right)|.$$

If ${\eta }_n\nrightarrow 0$, then there exists ${\epsilon }_0>0$ and a subsequence $n_{\ell }\to \infty$ with ${\eta }_{n_{\ell }}\ge {\epsilon }_0$ for all ℓ. Choose $(z_{\ell },w_{\ell })\in K_{n_{\ell }}$ with $|\mathrm{\Xi }\left(z_{\ell }\right)-\mathrm{\Xi }\left(w_{\ell }\right)|={\eta }_{n_{\ell }}$. Since $w_{\ell }\in \partial R$ and $\partial R$ is compact, after passing to a subsequence we have $w_{\ell }\to w_{*}\in \partial R$. Also $|z_{\ell }-w_{\ell }|\le 1/n_{\ell }\to 0$, hence $z_{\ell }\to w_{*}$. By continuity of $\mathrm{\Xi }$, $|\mathrm{\Xi }\left(z_{\ell }\right)-\mathrm{\Xi }\left(w_{\ell }\right)|\to 0$, contradicting ${\eta }_{n_{\ell }}\ge {\epsilon }_0$. Therefore ${\eta }_n\to 0$.

Choose n so large that ${\eta }_n\le \delta /2$ and set ${\rho }_{\mathrm{tube}}:=1/n$. If $(z,\partial R)\le {\rho }_{\mathrm{tube}}$, pick $w\in \partial R$ with $|z-w|\le {\rho }_{\mathrm{tube}}$. Then $|\mathrm{\Xi }\left(z\right)-\mathrm{\Xi }\left(w\right)|\le {\eta }_n\le \delta /2$, so

$$\left|\mathrm{\Xi }\right(z\left)\right|\ge \left|\mathrm{\Xi }\right(w\left)\right|-\left|\mathrm{\Xi }\right(z)-\mathrm{\Xi }(w\left)\right|\ge \delta -\delta /2=\delta /2.$$

□

Lemma 11.3

(Bounded ${\mathrm{\Xi }}^{\prime }$ on the tubular neighborhood). Assume $\mathrm{\Xi }\ne 0$ on $\partial R$ for a closed rectangle R, and let ${\rho }_{\mathrm{tube}}>0$ be as in 11.2. Define the closed tubular neighborhood

$$K:=\{z\in \mathbb{C}:(z,\partial R)\le {\rho }_{\mathrm{tube}}\}.$$

Then ${\mathrm{\Xi }}^{\prime }$ is bounded on K. In particular,

$$B:=\underset{z\in K}{max}\left|{\mathrm{\Xi }}^{\prime }\left(z\right)\right|<\infty .$$

Proof. 

${\mathrm{\Xi }}^{\prime }$ is entire, hence continuous. The set K is compact, so $|{\mathrm{\Xi }}^{\prime }|$ attains a finite maximum on K. □

11.2. Two-Knob Tightening Lemmas (Existence + Terminating Search)

Lemma 11.4

(Point-disk tightening: enclosure radius can be made arbitrarily small). Fix a rational point $z_0\in \mathbb{Q}+\mathrm{i}\mathbb{Q}$ and a rational target $\epsilon >0$. Let $s_0:=\frac{1}{2}+\mathrm{i}{z}_0$.

Assume that all analytic denominators used by the Ξ-routine are nonzero at $s_0$ (in particular, $s_0\ne 0$, $s_0\ne 1$, and $s_0/2\notin \{0,-1,-2,\dots \}$). Then there exists a parameter package $P=(N,m,M,h,n,p)$ such that the Ξ-routine succeeds on the point disk $D_z=\overline{B}(z_0,0)$ and

$$rad\left(D_{\mathrm{\Xi }}(P;\overline{B}(z_0,0))\right)\le \epsilon .$$

Moreover, there is a terminating search procedure which, given $(z_0,\epsilon )$, finds such P.

Proof. 

For the point disk $D_z=\overline{B}(z_0,0)$, every disk operation in the $\mathrm{\Xi }$-routine propagates only the analytic truncation remainders (Euler–Maclaurin; Stirling) and the certified-primitive errors (e.g. $2^{-n}$, $2^{-p}$), because the input radius is 0.

As $(N,m)\to \infty$, the Euler–Maclaurin remainder bounds for $\zeta ,{\zeta }^{\prime },{\zeta }^{\prime \prime }$ at a fixed point tend to 0. For the $\mathrm{\Gamma },\psi ,{\psi }^{\prime }$ layer, increasing h pushes $s/2+h$ into a region with arbitrarily large ${\sigma }_{-}(·)$, so the right-half-plane Stirling remainder bounds (Part 3) tend to 0. As $n,p\to \infty$, the certified-primitive errors tend to 0.

Hence by choosing $(N,m,M,h,n,p)$ large enough, the total output radius can be made $\le \epsilon$. For termination, dovetail over all packages $P\in {\mathbb{N}}^6$ satisfying the lower bounds; for each, run the routine (which terminates and either rejects by a decidable guard failure or produces a disk), and accept once the rational inequality $rad\left(D_{\mathrm{\Xi }}(P;\overline{B}(z_0,0))\right)\le \epsilon$ holds. Since some P works, the search halts. □

Remark 11.1

(No fixed-disk tightening is required). The completeness construction does not require producing arbitrarily small enclosures on a fixed positive-radius disk. Instead, the verifier uses a point enclosure for $\mathrm{\Xi }\left(z_0\right)$ together with a bound for ${\mathrm{\Xi }}^{\prime }$ on the disk to derive a sound enclosure for Ξ on the whole disk (see 10.3 and step (E) of 9.13).

Lemma 11.5

(${\mathrm{\Xi }}^{\prime }$-disk tightening on sufficiently small disks). Let $K\subset \mathbb{C}$ be compact and let $\eta >0$. Then there exists ${\rho }_{\eta }>0$ such that for every rational disk

$$D_z=\overline{B}(z_0,\rho )(z_0\in K\cap (\mathbb{Q}+\mathrm{i}\mathbb{Q}),0\le \rho \le {\rho }_{\eta })$$

there exists a parameter package $P=(N,m,M,h,n,p)$ for which the ${\mathrm{\Xi }}^{\prime }$-routine succeeds on $D_z$ and

$$rad\left(D_{{\mathrm{\Xi }}^{\prime }}(P;D_z)\right)\le \eta .$$

Moreover, for each fixed input disk $D_z$ with $\rho \le {\rho }_{\eta }$, there is a terminating search procedure (which dovetails over $P\in {\mathbb{N}}^6$ and runs the terminating routine) that finds such a package P.

Proof. 

Fix compact K and $\eta >0$.

Step 1: choose a small geometric radius. Since the map $z\mapsto s=\frac{1}{2}+\mathrm{i}z$ is continuous and all singularities used by the verifier for $\zeta$ and $\mathrm{\Gamma }/\psi$ computations are discrete, there exists ${\rho }_0>0$ such that for every $z_0\in K$ and every disk $D_z=\overline{B}(z_0,\rho )$ with $\rho \le {\rho }_0$, the induced disk $D_s=\frac{1}{2}+\mathrm{i}{D}_z$ stays a positive distance away from $s=1$ and from the finitely many points required by the verifier at the chosen truncation/shift parameters (once those parameters are fixed). In particular, for sufficiently large shift h, one can arrange ${\sigma }_{-}(D_{s/2}+h)>0$ while keeping the pullback denominators $D_{s/2}+j$ ($0\le j<h$) away from 0.

We now fix ${\rho }_{\eta }>0$ with ${\rho }_{\eta }\le {\rho }_0$ and additionally ${\rho }_{\eta }\le 1$.

Step 2: pointwise convergence of the explicit remainder/primitive-error bounds. For a fixed disk $D_z=\overline{B}(z_0,\rho )$ with $\rho \le {\rho }_{\eta }$, the algorithm in 8.4 (Part 4) computes $D_{{\mathrm{\Xi }}^{\prime }}(P;D_z)$ by combining: (i) Euler–Maclaurin enclosures for $\zeta ,{\zeta }^{\prime },{\zeta }^{\prime \prime }$ with explicit remainder radii (5)–(7), and (ii) Stirling/shift enclosures for $\mathrm{\Gamma },\psi ,{\psi }^{\prime }$ with explicit remainder radii from Part 3, together with (iii) primitive approximation errors of size $\le 2^{-n}$ and interval widths $\le 2^{-p}$.

For any fixed disk $D_z$, as we send $(N,m,M,h,n,p)\to \infty$ along any cofinal sequence (e.g. dovetailing over all packages), the explicit Euler–Maclaurin remainder bounds tend to 0, the explicit right-half-plane Stirling remainder bounds tend to 0 (by increasing h and M so that ${\sigma }_{-}(D_{s/2}+h)$ is large), and the primitive errors $2^{-n},2^{-p}$ tend to 0. Therefore, for each fixed $D_z$, there exists at least one package P such that the routine succeeds and $rad\left(D_{{\mathrm{\Xi }}^{\prime }}(P;D_z)\right)\le \eta$.

Step 3: termination by dovetailing. Fix $D_z$ with $\rho \le {\rho }_{\eta }$. Dovetail over all $P\in {\mathbb{N}}^6$ satisfying the lower bounds of 8.3. For each P, run the terminating ${\mathrm{\Xi }}^{\prime }$-routine. If it rejects, continue. If it outputs a rational disk, check the rational inequality $rad\left(D_{{\mathrm{\Xi }}^{\prime }}(P;D_z)\right)\le \eta$. Since some P exists by Step 2, the dovetailing procedure eventually finds one and halts.

This establishes both existence (for each disk) and a terminating search procedure. □

11.3. Completeness Theorem

Theorem 11.1

(Completeness on zero-free closures). Fix $(j,k)$ and set $R:=R_{j,k}$ and $\mathsf{\Omega }:={\mathsf{\Omega }}_{j,k}$. If $Z(\mathrm{\Xi };R)=⌀$, then there exists $c\in \mathbb{N}$ such that $\mathrm{Cert}(j,k,c)=1$.

Proof. 

Assume $Z(\mathrm{\Xi };R)=⌀$. Then $\mathrm{\Xi }\ne 0$ on $\partial R$. Let $\delta :={min}_{\partial R}\left|\mathrm{\Xi }\right|>0$ by 11.1. Let ${\rho }_{\mathrm{tube}}>0$ be as in 11.2, so $\left|\mathrm{\Xi }\right(z\left)\right|\ge \delta /2$ whenever $(z,\partial R)\le {\rho }_{\mathrm{tube}}$.

Guard-feasible tube inside $\mathcal{U}$. Recall that $R=R_{j,k}$ is a closed rectangle with

$$R=\left\{x+\mathrm{i}y:\frac{1}{2}{T}_k\le x\le 2T_k,{\sigma }_j\le y\le \frac{1}{2}-{\sigma }_j\right\}\subset \mathcal{U}.$$

Set the geometric margin

$${\rho }_{\mathrm{geo}}:=min\left\{{\displaystyle \frac{T_k}{4}},{\displaystyle \frac{{\sigma }_j}{4}}\right\}>0,{\rho }_{*}:=min\{{\rho }_{\mathrm{tube}},{\rho }_{\mathrm{geo}}\}.$$

Then the closed tube

$$K:=\{z\in \mathbb{C}:(z,\partial R)\le {\rho }_{*}\}$$

satisfies $K\subset \mathcal{U}$ (hence all points $s=\frac{1}{2}+\mathrm{i}z$ with $z\in K$ satisfy $s\ne 0$, $s\ne 1$, and $Re(s/2)>0$ with a uniform margin). Moreover, 11.2 remains valid with ${\rho }_{*}$ in place of ${\rho }_{\mathrm{tube}}$.

Step 1: choose a rational boundary polygon and set the target enclosure tolerance. Take the canonical 4-vertex polygon of R (its corners are rational). This is a boundary polygon. Set

$$\epsilon :={\displaystyle \frac{\delta }{16}}.$$

Let $K:=\{z\in \mathbb{C}:(z,\partial R)\le {\rho }_{*}\}$ and let

$$B:=\underset{z\in K}{max}\left|{\mathrm{\Xi }}^{\prime }\left(z\right)\right|<\infty$$

(as in 11.3, noting $K\subseteq \{(z,\partial R)\le {\rho }_{\mathrm{tube}}\}$). Apply 11.5 to the compact set K with $\eta :=B/8$ and let ${\rho }_{B/8}>0$ be the corresponding radius threshold. Fix any rational ${\rho }_{max}\in {\mathbb{Q}}_{>0}$ with

$${\rho }_{max}\le min\left\{{\displaystyle \frac{{\rho }_{\mathrm{tube}}}{4}},{\displaystyle \frac{\epsilon }{4B}},{\rho }_{B/8}\right\}.$$

Step 2: build, for each edge, a rational disk-chain cover withuniformradii and per-disk packages. Fix one edge $\gamma =[a,b]\subset \partial R$.

Set $\rho :={\rho }_{max}$.

Now choose an integer $M\ge 1$ and the uniform mesh points

$$p_r:=a+{\displaystyle \frac{r}{M}}(b-a)\in \mathbb{Q}+\mathrm{i}\mathbb{Q}(r=0,1,\dots ,M)$$

so that

$$|p_{r+1}-p_r|\le 2{\rho }_{max}(0\le r<M).$$

Define uniform radii ${\rho }_r:={\rho }_{max}$ for all r and disks

$$D_{z,r}:=\overline{B}(p_r,{\rho }_{max})(0\le r\le M).$$

Then the overlap inequality required by $\mathsf{SegmentChainAdmissible}$ holds automatically:

$$|p_{r+1}-p_r|\le 2{\rho }_{max}={\rho }_r+{\rho }_{r+1}.$$

Moreover, since ${\rho }_{max}\le {\rho }_{*}/4$, every point of each $D_{z,r}$ satisfies $(z,\partial R)\le {\rho }_{max}\le {\rho }_{*}$, hence

$$\left|\mathrm{\Xi }\left(z\right)\right|\ge \delta /2(z\in D_{z,r})$$

by 11.2. In particular, $\mathrm{\Xi }$ is nonvanishing on every $D_{z,r}$.

It remains to attach to each disk $D_{z,r}$ a parameter package $P_r$ so that the verifier’s step (E) produces a $\mathrm{\Xi }$-enclosure that avoids 0 with slack. For each fixed center $p_r$, dovetail over packages $P\in {\mathbb{N}}^6$ (with the lower bounds of 8.3) and run:

(i)

the $\mathrm{\Xi }$-routine on the point disk $\overline{B}(p_r,0)$ to obtain

$$E^{\left(0\right)}=\overline{B}(u^{\left(0\right)},R^{\left(0\right)});$$

(ii)

the ${\mathrm{\Xi }}^{\prime }$-routine on $D_{z,r}=\overline{B}(p_r,{\rho }_{max})$ to obtain

$$E^{\left(1\right)}=\overline{B}(u^{\left(1\right)},R^{\left(1\right)});$$

(iii)

compute

$$M:=|u^{\left(1\right)}{|}_{\mathrm{up},n}+R^{\left(1\right)}$$

where n is the internal-precision component of P, and test the rational inequalities

$$R^{\left(0\right)}\le \epsilon \mathrm{and}M{\rho }_{max}\le \epsilon .$$

Accept the first package P that passes and set $P_r:=P$ in the certificate record.

Justification that such a package exists. Since $D_{z,r}\subseteq K$ (because ${\rho }_{max}\le {\rho }_{*}/4$), we have the uniform bound

$$B=\underset{z\in K}{max}|{\mathrm{\Xi }}^{\prime }\left(z\right)|⟹\underset{z\in D_{z,r}}{sup}\left|{\mathrm{\Xi }}^{\prime }\left(z\right)\right|\le B.$$

Apply 11.5 with this compact set $K\subset \mathcal{U}$ (so the verifier guard conditions are satisfiable on all sufficiently small boundary disks) and with the target

$$\eta :={\displaystyle \frac{B}{8}}.$$

After refining the boundary mesh if necessary (i.e. choosing ${\rho }_{max}$ small enough), we may assume that ${\rho }_{max}\le {\rho }_{\eta }$, so there exists a package P for which the ${\mathrm{\Xi }}^{\prime }$-routine succeeds on $D_{z,r}$ and returns

$$E^{\left(1\right)}=\overline{B}(u^{\left(1\right)},R^{\left(1\right)})=D_{{\mathrm{\Xi }}^{\prime }}(P;D_{z,r})\mathrm{with}{R}^{\left(1\right)}\le \eta ={\displaystyle \frac{B}{8}}.$$

Since $p_r\in D_{z,r}\subseteq K$, we have the analytic bound $|{\mathrm{\Xi }}^{\prime }\left(p_r\right)|\le B$. Also, because $p_r\in D_{z,r}$ and ${\mathrm{\Xi }}^{\prime }\left(D_{z,r}\right)\subseteq E^{\left(1\right)}=\overline{B}(u^{\left(1\right)},R^{\left(1\right)})$, we have ${\mathrm{\Xi }}^{\prime }\left(p_r\right)\in \overline{B}(u^{\left(1\right)},R^{\left(1\right)})$, hence

$$|u^{\left(1\right)}|\le |{\mathrm{\Xi }}^{\prime }\left(p_r\right)|+R^{\left(1\right)}\le B+R^{\left(1\right)}.$$

Therefore the verifier’s certified modulus upper bound satisfies

$$|u^{\left(1\right)}{|}_{\mathrm{up},n}\le \left|u^{\left(1\right)}\right|+2^{-n}\le B+R^{\left(1\right)}+2^{-n},$$

and consequently

$$M=|u^{\left(1\right)}{|}_{\mathrm{up},n}+R^{\left(1\right)}\le B+2R^{\left(1\right)}+2^{-n}.$$

By 11.5, we can choose a package P for which the ${\mathrm{\Xi }}^{\prime }$-routine succeeds on $D_{z,r}$ and returns $R^{\left(1\right)}\le B/8$. Since $2^{-n}\to 0$ as $n\to \infty$, we may also take n large enough that $2^{-n}\le B/8$ (by increasing n if needed). For such a package,

$$M\le B+2·{\displaystyle \frac{B}{8}}+{\displaystyle \frac{B}{8}}={\displaystyle \frac{11}{8}}B<2B.$$

With our earlier choice ${\rho }_{max}\le \epsilon /\left(4B\right)$, it follows that

$$M{\rho }_{max}\le 2B·{\displaystyle \frac{\epsilon }{4B}}={\displaystyle \frac{\epsilon }{2}}<\epsilon ,$$

so there exists a package passing the test $M{\rho }_{max}\le \epsilon$.

This yields an admissible segment disk-chain record for the edge $\gamma$, together with per-disk packages. Repeat the same construction independently for each of the four edges of $\partial R$, and record the resulting per-edge disk-chain records and all per-disk packages in the certificate.

Step 3: verify the verifier’s disk nonvanishing checks succeed. Fix any boundary-cover disk $D_z=\overline{B}(p,\rho )$ and let P be the parameter package attached to it by Step 2. Let the verifier-computed quantities be as in step (E) of 9.13:

$$E^{\left(0\right)}=\overline{B}(u^{\left(0\right)},R^{\left(0\right)}),E^{\left(1\right)}=\overline{B}(u^{\left(1\right)},R^{\left(1\right)}),M:={\left|u^{\left(1\right)}\right|}_{\mathrm{up},n}+R^{\left(1\right)}.$$

The verifier then forms the Lipschitz-derived enclosure

$$E=\overline{B}\left(u^{\left(0\right)},r\right),r:=R^{\left(0\right)}+M\rho ,$$

and checks $0\notin E$.

By the acceptance conditions enforced in Step 2 we have $R^{\left(0\right)}\le \epsilon$ and $M\rho \le \epsilon$, hence

$$r\le 2\epsilon =\delta /8.$$

On the other hand, by construction each $D_z$ lies inside the tubular neighborhood from 11.2, so

$$\underset{z\in D_z}{inf}\left|\mathrm{\Xi }\left(z\right)\right|\ge \delta /2.$$

Choose any $w\in D_z$. By 10.3 (soundness of the verifier’s enclosure), $\mathrm{\Xi }\left(w\right)\in E$, hence $|\mathrm{\Xi }\left(w\right)-u^{\left(0\right)}|\le r$ and therefore

$$\left|\mathrm{\Xi }\left(w\right)\right|\le |u^{\left(0\right)}|+r.$$

If $0\in E$, then $|u^{\left(0\right)}|\le r$, so $\left|\mathrm{\Xi }\right(w\left)\right|\le 2r\le \delta /4$, contradicting $\left|\mathrm{\Xi }\right(w\left)\right|\ge \delta /2$. Thus $0\notin E$ for every boundary-cover disk, so the verifier’s step (E) succeeds everywhere.

Step 4: choose a generic rational direction v for ray-crossing winding. The enclosure centers $u_i$ are finitely many nonzero rational complex numbers. Choose rational $v\ne 0$ such that $cr(v,u_i)\ne 0$ for all i (avoid finitely many rational lines). Then the verifier’s general-position checks pass.

Apply Compress to remove consecutive duplicates; this does not introduce new vertices and preserves the finite avoidance set for ray general position.

Step 5: winding is 0 and verifier accepts. Since $Z(\mathrm{\Xi };R)=⌀$ and $\mathrm{\Xi }\ne 0$ on $\partial R$, the argument principle gives

$$wind\left(\mathrm{\Xi }\right(\partial R),0)=0.$$

By Steps 1–4, the verifier passes steps (A)–(F) of 9.13 for the constructed code c. Therefore 10.7 applies and shows that the verifier’s computed integer W satisfies

$$W=wind\left(\mathrm{\Xi }\right(\partial R),0)=0.$$

Hence the verifier accepts in step (G), i.e. $\mathrm{Cert}(j,k,c)=1$. □

Remark 11.2

(Decidable guard satisfaction by mesh/parameter refinement). The guard conditions required by the ζ and $\mathrm{\Gamma }/\psi$ layers are all decidable inequalities on rational data. In the completeness construction we first choose a rational boundary mesh radius (e.g. ${\rho }_{max}$) small enough that the resulting boundary disks lie in a tubular neighborhood where Ξ is known to be nonvanishing, and also small enough to control the Lipschitz propagation step via a uniform bound on $|{\mathrm{\Xi }}^{\prime }|$ on that tubular neighborhood (11.3). This is themesh refinementstep.

For each resultingfixedrational boundary disk $D_z$, one then performsparameter refinement: increase the package $P=(N,m,M,h,n,p)$ until the Ξ-routine both (i) passes all certified guard checks (pole separation and the finitely many reciprocal separations), and (ii) returns an image-enclosure disk of arbitrarily small radius on sufficiently small boundary disks (“two-knob tightening”).

Concretely, parameter refinement is used to certify:

• separation from the pole $s=1$ for the ζ layer, i.e. the verifier can establish $|c_s{-1|}_{\mathrm{lo},n}>r_s$, and separation from the finitely many points $s=-j$ used in Pochhammer derivative majorants;
• the right-half-plane condition ${\sigma }_{-}(D_{s/2}+h)>0$ for a suitable shift h;
• certified nonvanishing of the finitely many pullback denominators $D_{s/2}+j$$(0\le j<h)$ via $|c_{s/2}{+j|}_{\mathrm{lo},n}>r_{s/2}$, so that the reciprocal operation is verifier-defined.

Increasing h makes Stirling remainder radii smaller by increasing ${\sigma }_{-}(·)$ on shifted disks (Part 3), and increasing $n,p$ tightens the certified primitive bounds used throughout.

12. Deriving the ${\mathrm{\Pi }}_2^0$ Sweep Normal Form

12.1. The Sweep Sentence

Recall the stage predicate $\mathrm{Cert}(j,k,c)$ from 9.13. Define

$$\mathrm{StageOK}(j,k):⟺\exists c\in \mathbb{N}\mathrm{Cert}(j,k,c),\mathsf{CS}:⟺\forall j\in {\mathbb{N}}_{\ge 1}\forall k\in \mathbb{Z}\mathrm{StageOK}(j,k).$$

Proposition 12.1

(Sweep equivalence). One has

$$\mathsf{CS}⟺Z(\mathrm{\Xi };\mathcal{U})=⌀.$$

Proof. ($\mathsf{CS}\Rightarrow Z(\mathrm{\Xi };\mathcal{U})=⌀$) Let $z\in \mathcal{U}$. Choose $(j,k)$ with $z\in {\mathsf{\Omega }}_{j,k}$ by 2.1. By $\mathsf{CS}$, there exists c with $\mathrm{Cert}(j,k,c)=1$. By soundness 10.1, $Z(\mathrm{\Xi };{\mathsf{\Omega }}_{j,k})=⌀$, so $\mathrm{\Xi }\left(z\right)\ne 0$. Thus $Z(\mathrm{\Xi };\mathcal{U})=⌀$.

($Z(\mathrm{\Xi };\mathcal{U})=⌀\Rightarrow \mathsf{CS}$) Assume $Z(\mathrm{\Xi };\mathcal{U})=⌀$. Fix $(j,k)$. Since $R_{j,k}\subset \mathcal{U}$, we have $Z(\mathrm{\Xi };R_{j,k})=⌀$. By completeness 11.1, there exists c with $\mathrm{Cert}(j,k,c)=1$, i.e. $\mathrm{StageOK}(j,k)$. Since $(j,k)$ were arbitrary, $\mathsf{CS}$ holds. □

Proof 

(Proof of 1.1). By 1.1, RH $\iff Z(\mathrm{\Xi };\mathcal{U})=⌀$. By 12.1, this is equivalent to $\mathsf{CS}$. Finally, $\mathsf{CS}$ has prenex form

$$\forall j\in {\mathbb{N}}_{\ge 1}\forall k\in \mathbb{Z}\exists c\in \mathbb{N}\mathrm{Cert}(j,k,c),$$

and $\mathrm{Cert}$ is decidable by 9.1. Hence $\mathsf{CS}$ is ${\mathrm{\Pi }}_2^0$, so RH is ${\mathrm{\Pi }}_2^0$. □

13. On ${\mathrm{\Pi }}_1^0$ Folklore: What Would Be Required (and What is Not Automatic)

Remark 13.1

(Burden of proof for ${\mathrm{\Pi }}_1^0$ claims). A theorem-level ${\mathrm{\Pi }}_1^0$ classification of RH requires asingle fixeddecidable predicate $Wit\left(c\right)$ on $c\in \mathbb{N}$ together with a proof of

$$\neg \mathrm{RH}⟺\exists c\in \mathbb{N}Wit\left(c\right).$$

The informal statement “if RH fails there exists a counterexample” does not by itself provide such a witness predicate: one must specify the coding of witnesses into $\mathbb{N}$ and prove soundness and completeness for a fixed terminating verifier.

Remark 13.2

(Why this paper stops at ${\mathrm{\Pi }}_2^0$). The present paper supplies exactly such a theorem-level package for a ${\mathrm{\Pi }}_2^0$ normal form: a fixed decidable predicate $\mathrm{Cert}(j,k,c)$ and proofs of soundness and completeness-on-zero-free-closures, yielding

$$\mathrm{RH}\iff \forall j\forall k\exists c\mathrm{Cert}(j,k,c).$$

Establishing a ${\mathrm{\Pi }}_1^0$ normal form would require a different global witness calculus for $\neg \mathrm{RH}$ and is not treated here.

Appendix O.1. Cubic Fields Cut Out by P t and Their Quadratic Resolvent

Fix $t_0\in {\mathbb{Q}}^{\times }$ and assume $P_{t_0}\left(z\right)\in \mathbb{Q}\left[z\right]$ is irreducible. Let

$$K_{t_0}:=\mathbb{Q}\left[z\right]/\left(P_{t_0}\left(z\right)\right)$$

be the associated (generically non-Galois) cubic number field, and let $L_{t_0}$ be its Galois closure. When ${Disc}_z\left(P_{t_0}\right)\notin {\mathbb{Q}}^{\times 2}$, one has

$$Gal(L_{t_0}/\mathbb{Q})\cong S_3.$$

The unique quadratic subfield of $L_{t_0}$ is the quadratic resolvent field

$$F_{t_0}:=\mathbb{Q}\left(\sqrt{{Disc}_z\left(P_{t_0}\right)}\right).$$

In particular, your discriminant-square conic () is precisely the locus where $F_{t_0}=\mathbb{Q}$ and the cubic becomes cyclic.

Appendix O.2. Dedekind Zeta of K t 0 and an Artin Factorization

Recall the Dedekind zeta function of a number field K:

$${\zeta }_K\left(s\right):=\sum _{\mathfrak{a}\ne 0}N{\left(\mathfrak{a}\right)}^{-s}=\prod _{\mathfrak{p}}{\left(1-N{\left(\mathfrak{p}\right)}^{-s}\right)}^{-1},\Re \left(s\right)>1,$$

where $\mathfrak{a}$ ranges over nonzero ideals of ${\mathcal{O}}_K$ and $\mathfrak{p}$ ranges over prime ideals.

Theorem O.1

(Artin factorization in the $S_3$ case). Assume $P_{t_0}$ is irreducible over $\mathbb{Q}$ and ${Disc}_z\left(P_{t_0}\right)\notin {\mathbb{Q}}^{\times 2}$, so that $Gal(L_{t_0}/\mathbb{Q})\cong S_3$. Let ${\rho }_{\mathrm{std}}$ denote the 2-dimensional irreducible (standard) complex representation of $S_3$. Then for $\Re \left(s\right)>1$ one has an identity of Euler products

$$\begin{array}{|c|}\hline {\zeta }_{K_{t_0}}\left(s\right)=\zeta \left(s\right)L(s,{\rho }_{\mathrm{std}})\\ \hline\end{array},$$

where $L(s,{\rho }_{\mathrm{std}})$ is the Artin L-function attached to ${\rho }_{\mathrm{std}}$.

Proof. 

Let $G=Gal(L_{t_0}/\mathbb{Q})\cong S_3$ and let $H=Gal(L_{t_0}/K_{t_0})$, so $[G:H]=3$. The permutation representation of G on $G/H$ decomposes as

$${Ind}_H^G\left(\mathbf{1}\right)\cong \mathbf{1}\oplus {\rho }_{\mathrm{std}}.$$

Artin formalism identifies the Dedekind zeta of the (non-Galois) cubic field $K_{t_0}$ with the Artin L-function of ${Ind}_H^G\left(\mathbf{1}\right)$, hence

$${\zeta }_{K_{t_0}}\left(s\right)=L\left(s,{Ind}_H^G\left(\mathbf{1}\right)\right)=L(s,\mathbf{1})L(s,{\rho }_{\mathrm{std}})=\zeta \left(s\right)L(s,{\rho }_{\mathrm{std}})$$

as Euler products for $\Re \left(s\right)>1$. □

Remark O.1

(Hecke interpretation). In the $S_3$ situation, the quadratic resolvent field $F_{t_0}$ satisfies $Gal(L_{t_0}/F_{t_0})\cong C_3$. The representation ${\rho }_{\mathrm{std}}$ is monomial (induced from a nontrivial character of $C_3$), so $L(s,{\rho }_{\mathrm{std}})$ can be identified with a Hecke L-function over $F_{t_0}$ attached to an order-3 Hecke character. This gives analytic continuation and a functional equation for $L(s,{\rho }_{\mathrm{std}})$.

Appendix O.3. What this Does and Does Not Imply About RH

Corollary O.1

(A conditional reduction of RH). Fix $t_0$ as in O.1. If one knew a Generalized Riemann Hypothesis (GRH) for ${\zeta }_{K_{t_0}}\left(s\right)$ (or equivalently for the Hecke/Artin factor $L(s,{\rho }_{\mathrm{std}})$), then the classical Riemann Hypothesis for $\zeta \left(s\right)$ would follow.

Proof. 

By O.1, every zero of $\zeta \left(s\right)$ is a zero of ${\zeta }_{K_{t_0}}\left(s\right)$ (since $\zeta$ is a factor). Thus if all nontrivial zeros of ${\zeta }_{K_{t_0}}\left(s\right)$ lie on $\Re \left(s\right)=\frac{1}{2}$, then all nontrivial zeros of $\zeta \left(s\right)$ lie on $\Re \left(s\right)=\frac{1}{2}$. □

Remark O.2

(Why this is not a proof of RH). Corollary O.1 is a logically correctconditionalimplication. It does not prove RH, because GRH for Hecke/Artin L-functions (and GRH for Dedekind zeta functions of general number fields) is itself open. What this manuscript contributes is a clean and explicit algebraic family of $S_3$-cubics (and hence explicit quadratic resolvent fields and cubic Hecke characters) on which one can study GRH-type phenomena concretely.

Appendix O.4. Optional Parallel: Hasse–Weil L-Functions of the Elliptic Curves E t,k

For each nonsingular specialization $(t,k)\in {\mathbb{Q}}^2$ the curve $E_{t,k}/\mathbb{Q}$ is an elliptic curve, hence (by modularity) has a Hasse–Weil L-function $L(E_{t,k},s)$ with analytic continuation and functional equation. A “Riemann Hypothesis” for $L(E_{t,k},s)$ is again a GRH-type statement and is open in general, including on the CM lines $k=0$ and $k=-1$.