# Sheafified Zero Obstructions and Local-to-Global Lifting
## From Rational-Rectangle Certificates to a Decision over the Entire Critical Strip

**English Title:** *Sheafified Zero Obstructions and Local-to-Global Lifting: From Rational-Rectangle Certificates to a Decision over the Entire Critical Strip*  
**Author:** Neo.K (Hsu Chuan-Wei)  
**Institution:** EveMissLab / A Word Promise Technology Co., Ltd.  
**Version:** v0.1 (Internal Research Draft)  
**Date:** 2026-07-24  
**Nature:** Complex Analysis / Sheaf-Theoretic Methods / Topological Degree / Local-Global Principle / RH Decision Domain Research  
**Prerequisite Documents:**
1. *From Centering to Equivariant Topology: Thinking Methods and Methodological Clusters for the Legitimate Decision of RH*
2. *Equivariant Zero Configuration Topology: RH Orbit-Type Stratification, Effective Divisor Semirings, and Positive Obstructions*  
**Status:** Internal draft; does not constitute a proof of the Riemann Hypothesis

---

## Important Declaration

This document is not a proof of the Riemann Hypothesis.

This document addresses the second-level problem left by the previous paper:

> Given that off-axis zeros can be represented as positive divisor obstructions, how can we localize these obstructions, form verifiable certificates, maintain compatibility in overlapping regions, and determine what local information is sufficient to lift to a conclusion over the entire critical strip?

This document establishes:

- A sheaf of locally finite effective divisors;
- An off-axis zero obstruction sheaf;
- A countable rational-rectangle basis;
- Winding number certificates on zero-free boundaries;
- A positive integer-valued additive valuation valid for finite rectangular partitions;
- An equivalence decision between local certificates and global RH;
- Height exhaustion, boundary regularization, and tail gaps;
- The duality relationship among local logarithms, winding cohomology classes, and zero divisors;
- Certificate specifications suitable for formalization and computational verification.

This document does not prove that any arbitrary non-trivial winding number must be zero. What is truly still lacking is a unified proof, derived from the independent analytic and arithmetic structure of \(\xi\), that all off-axis local certificates are zero.

---

# Abstract

Let
\[
F(z)=\xi\left(\frac12+z\right),
\]
and let the centered open critical strip be
\[
X=\left\{z\in\mathbb C:\left|\operatorname{Re}z\right|<\frac12\right\},
\]
with the critical axis being
\[
A=i\mathbb R.
\]
Previous research has represented the non-trivial zeros of \(F\) as a locally finite, \(G\)-invariant effective divisor \(D_F\) on \(X\), and defined the off-axis positive obstruction
\[
\mathfrak O(D_F)=D_F|_{X\setminus A}.
\]
RH is equivalent to this off-axis positive obstruction being zero.

This document organizes this obstruction into a sheaf over open sets. For any open set \(U\subseteq X\), we define
\[
\mathscr Z^+(U)=\operatorname{Div}_{\mathrm{lf}}^+(U),
\]
and the off-axis obstruction sheaf
\[
\mathscr O^+(U)=\operatorname{Div}_{\mathrm{lf}}^+(U\setminus A).
\]
The restriction maps are given by the open set restrictions of the divisors. Local finiteness and overlap compatibility ensure that the above assignments satisfy the sheaf axioms. Fixing the function \(F\) yields a global section
\[
\mathfrak o_F\in\Gamma(X,\mathscr O^+),
\]
and RH is equivalent to
\[
\mathfrak o_F=0.
\]

Since \(X\setminus A\) splits into left and right branches, and the zero divisor is preserved by the critical reflection
\[
j(z)=-\overline z
\]
, it suffices to decide over the right half critical strip
\[
X^+=\left\{z:0<\operatorname{Re}z<\frac12\right\}.
\]
This document takes a basis of rational rectangles in \(X^+\) whose closures are compactly contained in \(X^+\). For each regular rectangle \(R\) with a zero-free boundary, we define the winding number certificate
\[
\omega_R(F)
=
\frac{1}{2\pi i}\oint_{\partial R}\frac{F'(z)}{F(z)}\,dz.
\]
By the argument principle,
\[
\omega_R(F)=D_F(R)\in\mathbb N_0.
\]
Therefore, the winding number is not merely an abstract homological quantity with a sign, but a positive zero mass counted with multiplicity.

This document proves a countable local decision theorem:
\[
\mathrm{RH}
\iff
\omega_R(F)=0
\]
holds for all regular rational rectangles \(R\Subset X^+\). If off-axis zeros exist, one can use the discreteness of zeros to select a rational rectangle with a zero-free boundary around them, yielding a positive winding number certificate.

For an additive partition formed by finitely many rectangles, the winding numbers cancel out on common internal boundaries, thus forming a positive integer-valued valuation:
\[
\omega_{\bigcup_iR_i}(F)=\sum_i\omega_{R_i}(F).
\]
This document deliberately avoids overclaiming this structure as a complete cosheaf over all open sets; it is an additive certificate system valid on regular regions, finite partitions, and legitimate boundary chains.

This document further establishes a regular exhaustion:
\[
U_1\Subset U_2\Subset\cdots\Subset X^+,
\qquad
\bigcup_nU_n=X^+,
\]
where each \(\partial U_n\) avoids zeros. Thus,
\[
\mathrm{RH}
\iff
\omega_{U_n}(F)=0
\quad\forall n.
\]
However, any finite prefix
\[
\omega_{U_n}(F)=0
\quad(n\le N)
\]
cannot deduce global RH. To lift from finite verification, an additional tail exclusion theorem, a unified local theorem, or an analytic proof valid for all exhaustion levels is still required.

Finally, this document treats
\[
\frac{1}{2\pi i}\frac{F'}F\,dz
\]
as an integral-period closed form on the zero complement space, whose period class is identical to the \(H^1\) class induced by the map
\[
F/|F|:X\setminus Z(F)\to S^1
\]
. The local winding number certificate is precisely the pairing of this class on a boundary 1-cycle, while the zero divisor is its positive mass representation under planar duality. This provides a unified interface among the divisor sheaf, boundary certificates, and topological obstructions.

**Keywords:** Riemann Hypothesis, Sheaf, Locally Finite Divisor, Off-Axis Obstruction, Winding Number, Argument Principle, Rational Rectangle, Local-to-Global Lifting, Exhaustion, Logarithmic Cohomology

---

# 1. Research Problem

## 1.1 The Gap Left by the Previous Paper

Previous research established:

\[
\mathrm{RH}
\iff
\mathfrak O(D_F)=0,
\]

where

\[
\mathfrak O(D_F)
=
D_F|_{X\setminus A}.
\]

But this remains a global condition.

To investigate further, we must answer:

1. How can we read the off-axis obstruction in a finite region?
2. How do local obstructions restrict to smaller regions?
3. When is local data from different regions compatible?
4. Can local certificates be glued into a global certificate?
5. Does there exist a countable sufficient decision family?
6. What exactly is missing between finite-height verification and global RH?

## 1.2 The Core Stance of This Document

This document adopts:

\[
\boxed{
\text{Divisor data is managed by sheaves, and boundary counting is managed by positive valuations.}
}
\]

The reasons are:

- Zero divisors can naturally restrict and glue, forming a true sheaf;
- Winding numbers depend on the region and boundary regularity, possessing additivity over finite legitimate partitions;
- Directly claiming winding numbers as a complete cosheaf over general open sets easily overlooks boundary zeros, orientations, and region types.

Therefore, this document distinguishes between:

\[
\text{Local geometric data}
\quad\text{and}\quad
\text{Local certificate data}.
\]

---

# 2. Basic Spaces and Symmetry Reduction

## 2.1 Open Critical Strip

Definition:

\[
X
=
\left\{
z\in\mathbb C:
-\frac12<\operatorname{Re}z<\frac12
\right\}.
\]

Critical axis:

\[
A=i\mathbb R.
\]

Off-axis space:

\[
X^\times=X\setminus A.
\]

## 2.2 Left and Right Branches

Definition:

\[
X^+
=
\left\{
z:
0<\operatorname{Re}z<\frac12
\right\},
\]

\[
X^-
=
\left\{
z:
-\frac12<\operatorname{Re}z<0
\right\}.
\]

Then:

\[
X^\times=X^+\sqcup X^-.
\]

## 2.3 Critical Reflection

Definition:

\[
j(z)=-\overline z.
\]

Then:

\[
j(X^+)=X^-,
\qquad
j(X^-)=X^+.
\]

By the \(j\)-invariance of the zero divisor:

\[
D_F|_{X^-}
=
j_*\left(D_F|_{X^+}\right).
\]

Therefore:

### Proposition 2.1

\[
D_F|_{X^\times}=0
\iff
D_F|_{X^+}=0.
\]

That is to say, it suffices to exclude zeros in the right half critical strip.

---

# 3. Sheaf of Locally Finite Effective Divisors

## 3.1 Divisors on Open Sets

For any open set \(U\subseteq X\), define:

\[
\mathscr Z^+(U)
=
\operatorname{Div}_{\mathrm{lf}}^+(U).
\]

Its elements are:

\[
D=\sum_{\rho\in U}m_\rho[\rho],
\qquad
m_\rho\in\mathbb N_0,
\]

and for any compact set \(K\Subset U\), only finitely many support points lie in \(K\).

## 3.2 Restriction Maps

If:

\[
V\subseteq U,
\]

define:

\[
\operatorname{res}_{U,V}:
\mathscr Z^+(U)\to\mathscr Z^+(V),
\]

\[
D\mapsto D|_V.
\]

It satisfies:

\[
\operatorname{res}_{U,U}=\operatorname{id},
\]

and:

\[
\operatorname{res}_{V,W}\circ\operatorname{res}_{U,V}
=
\operatorname{res}_{U,W}.
\]

## 3.3 Local Uniqueness

If \(D_1,D_2\in\mathscr Z^+(U)\), and for an open cover

\[
U=\bigcup_iU_i
\]

we have:

\[
D_1|_{U_i}=D_2|_{U_i},
\]

then:

\[
D_1=D_2.
\]

Because the multiplicity of each point can be read in any \(U_i\) containing that point.

## 3.4 Gluing

Suppose:

\[
D_i\in\mathscr Z^+(U_i)
\]

and on all overlaps:

\[
D_i|_{U_i\cap U_j}
=
D_j|_{U_i\cap U_j}.
\]

Then we can define pointwise:

\[
m_\rho=m_\rho(D_i)
\]

where we choose any \(U_i\) containing \(\rho\).

Compatibility ensures this is well-defined.

For any \(\rho\in U\), choose some \(U_i\ni\rho\). By the local finiteness of \(D_i\), there exists a relatively compact neighborhood of \(\rho\) in which the support is finite. Therefore, the glued divisor remains locally finite.

Thus there exists a unique:

\[
D\in\mathscr Z^+(U)
\]

such that:

\[
D|_{U_i}=D_i.
\]

### Theorem 3.1

\[
U\longmapsto\mathscr Z^+(U)
\]

is a sheaf on \(X\) taking values in commutative monoids.

---

# 4. Off-Axis Obstruction Sheaf

## 4.1 Definition

For any open set \(U\subseteq X\), define:

\[
\mathscr O^+(U)
=
\operatorname{Div}_{\mathrm{lf}}^+(U\setminus A).
\]

Since \(U\setminus A\) is an open set, the results of the previous section directly yield:

### Theorem 4.1

\[
U\longmapsto\mathscr O^+(U)
\]

is a sheaf of commutative monoids on \(X\).

## 4.2 Obstruction Section of a Fixed Function

Let:

\[
D_F=\operatorname{div}_0(F)|_X.
\]

For each \(U\subseteq X\), define:

\[
\mathfrak o_F(U)
=
D_F|_{U\setminus A}.
\]

These local divisors are compatible under restriction, thus forming a global section:

\[
\mathfrak o_F
\in
\Gamma(X,\mathscr O^+).
\]

## 4.3 Sheaf-Theoretic Form of RH

### Theorem 4.2

The following propositions are equivalent:

1. RH holds;
2. \(\mathfrak o_F=0\);
3. For all open sets \(U\subseteq X\), \(\mathfrak o_F(U)=0\);
4. For all \(p\in X\), the stalk of the obstruction section \((\mathfrak o_F)_p=0\);
5. For all \(p\in X^+\), \((\mathfrak o_F)_p=0\).

### Explanation

This is the locality principle of sheaves:

\[
\text{Global section is zero}
\iff
\text{Every stalk is zero}.
\]

However, this is merely a local linguistic reconstruction of RH and does not provide an independent reason for the stalks to be zero.

---

# 5. Rational-Rectangle Basis

## 5.1 Basis Definition

Let:

\[
\mathcal B_{\mathbb Q}^+
\]

be the set of all open rectangles

\[
R=(a,b)\times(c,d)
\]

, where:

\[
a,b,c,d\in\mathbb Q,
\]

\[
0<a<b<\frac12,
\qquad
c<d.
\]

Identify the complex number \(z=x+iy\) with \((x,y)\).

The closure of each \(R\) satisfies:

\[
\overline R\Subset X^+.
\]

## 5.2 Countability

Since \(\mathbb Q^4\) is countable,

\[
\mathcal B_{\mathbb Q}^+
\]

is a countable set.

## 5.3 Basis Property

For any \(p\in X^+\) and any open neighborhood \(U\ni p\), there exists:

\[
R\in\mathcal B_{\mathbb Q}^+
\]

such that:

\[
p\in R,
\qquad
\overline R\subset U.
\]

Thus it is a countable relatively compact basis for \(X^+\).

## 5.4 Why Use Relatively Compact Rectangles

The relatively compact condition simultaneously avoids:

- The critical axis \(A\);
- The right boundary of the critical strip;
- Infinity.

Therefore, there are only finitely many zeros in each rectangle, and the argument principle can be legitimately applied.

---

# 6. Regular Rectangles and Winding Number Certificates

## 6.1 Regularity

For:

\[
R\in\mathcal B_{\mathbb Q}^+,
\]

If:

\[
F(z)\ne0
\qquad
\forall z\in\partial R,
\]

then \(R\) is called an \(F\)-regular rectangle.

Denote all regular rectangles by:

\[
\mathcal B_{\mathbb Q,F}^{\mathrm{reg},+}.
\]

## 6.2 Boundary Phase Map

For a regular rectangle, define:

\[
\phi_R:
\partial R\to S^1,
\]

\[
\phi_R(z)=\frac{F(z)}{|F(z)|}.
\]

Its topological degree is:

\[
\omega_R(F)
=
\deg(\phi_R).
\]

## 6.3 Integral Form

Fixing the positive orientation of \(\partial R\), we have:

\[
\omega_R(F)
=
\frac{1}{2\pi i}
\oint_{\partial R}
\frac{F'(z)}{F(z)}\,dz.
\]

## 6.4 Positivity

By the argument principle, since \(F\) is an entire function:

\[
\omega_R(F)
=
\sum_{\rho\in R}
m_\rho
=
D_F(R).
\]

Therefore:

\[
\omega_R(F)\in\mathbb N_0.
\]

The positivity here relies on:

1. The boundary taking a positive orientation;
2. \(F\) having no poles in the interior;
3. Zeros being counted with positive multiplicity.

## 6.5 Certificate Content

A legitimate rectangle certificate contains at least:

\[
\mathsf{Cert}(R,F)
=
\left(
R,
\mathsf{BoundaryFree},
\omega_R,
\mathsf{Multiplicity},
\mathsf{Dependencies}
\right).
\]

where:

- \(R\): Rectangle endpoints;
- \(\mathsf{BoundaryFree}\): Proof that the boundary is zero-free;
- \(\omega_R\): Integer winding number;
- \(\mathsf{Multiplicity}\): Declaration of counting by multiplicity;
- \(\mathsf{Dependencies}\): Analytic theorems, numerical bounds, and axiomatic dependencies used.

If the proof of a zero-free boundary is missing, the winding number certificate is incomplete.

---

# 7. Countable Local Decision Theorem

## 7.1 Theorem

### Theorem 7.1

The following propositions are equivalent:

1. RH holds;
2. \(D_F|_{X^+}=0\);
3. For all \(F\)-regular rational rectangles
   \[
   R\in\mathcal B_{\mathbb Q,F}^{\mathrm{reg},+},
   \]
   we have
   \[
   \omega_R(F)=0.
   \]

## 7.2 Proof

### \(1\Rightarrow2\)

If RH holds, all zeros lie on \(A\), so the right half-strip has no zeros.

### \(2\Rightarrow3\)

If the right half-strip has no zeros, any \(R\Subset X^+\) has no zeros in its interior, so:

\[
\omega_R(F)=0.
\]

### \(3\Rightarrow2\)

Assume for contradiction that there exists a zero \(\rho\) in the right half-strip.

Since the zero set is discrete, there exists an open disk \(B(\rho,\delta)\Subset X^+\) such that the closed disk contains no zeros other than \(\rho\).

Using the rational-rectangle basis, select:

\[
\rho\in R,
\qquad
\overline R\subset B(\rho,\delta).
\]

Then \(\partial R\) contains no zeros, thus \(R\) is regular, and:

\[
\omega_R(F)
=
m_\rho
>0,
\]

which contradicts the assumption.

Proof complete.

## 7.3 Significance

RH has been transformed into a condition where a **countable family of certificates are all zero**.

However, "countable" does not mean "finite".

Nor does it imply that an infinite proposition can be completed in finite time through one-by-one numerical verification.

---

# 8. From Divisor Sheaves to Certificate Valuations

## 8.1 Region Class

Let:

\[
\mathsf{Reg}_F^+
\]

denote all bounded regions \(U\Subset X^+\) satisfying the following conditions:

1. The boundary consists of finitely many piecewise \(C^1\) Jordan curves;
2. The boundary takes a positive orientation;
3. \(F\) has no zeros on \(\partial U\).

## 8.2 Certificate Valuation

Define:

\[
\nu_F(U)
=
\frac{1}{2\pi i}
\oint_{\partial U}\frac{F'}F\,dz.
\]

By the argument principle:

\[
\nu_F(U)=D_F(U)\in\mathbb N_0.
\]

## 8.3 Monotonicity

If:

\[
U\subseteq V,
\]

and:

\[
D_F(V\setminus U)\ge0,
\]

then:

\[
\nu_F(U)\le\nu_F(V).
\]

For positive divisors, this is the monotonicity of zero mass.

## 8.4 Disjoint Additivity

If \(U,V\in\mathsf{Reg}_F^+\) and their closures are disjoint, then:

\[
\nu_F(U\sqcup V)
=
\nu_F(U)+\nu_F(V).
\]

## 8.5 Finite Partition Additivity

Suppose \(U\) is partitioned by finitely many regions \(U_1,\ldots,U_n\), satisfying:

- Their interiors are pairwise disjoint;
- The closure of their union equals \(\overline U\);
- All outer boundaries and internal partition edges do not pass through zeros;
- Each common internal edge has opposite orientations in adjacent regions.

Then:

\[
\sum_{k=1}^n
\oint_{\partial U_k}\frac{F'}F\,dz
=
\oint_{\partial U}\frac{F'}F\,dz.
\]

The integrals on common internal boundaries cancel each other out, therefore:

\[
\boxed{
\nu_F(U)
=
\sum_{k=1}^n\nu_F(U_k).
}
\]

## 8.6 Why It Is Called a Valuation Rather Than a Complete Cosheaf

The above structure possesses finite additivity, but it still relies on:

- The regularity of region boundaries;
- Zero-free boundaries;
- Legitimate partitions;
- Orientation consistency.

For an arbitrary open cover, there is no automatically well-defined operation to "directly push forward local winding numbers".

Therefore, this document calls:

\[
U\mapsto\nu_F(U)
\]

a positive integer-valued region valuation or certificate valuation, rather than unconditionally claiming it is a cosheaf over the category of general open sets.

---

# 9. Gluing of Local Certificates

## 9.1 Gluing Zero Certificates

If a regular region \(U\) admits a finite regular partition:

\[
U=\bigcup_{k=1}^nU_k,
\]

and:

\[
\nu_F(U_k)=0
\qquad
\forall k,
\]

then:

\[
\nu_F(U)=0.
\]

## 9.2 Positivity Provides Reverse Decomposition

Since:

\[
\nu_F(U_k)\in\mathbb N_0,
\]

If:

\[
\nu_F(U)
=
\sum_k\nu_F(U_k)
=
0,
\]

then:

\[
\nu_F(U_k)=0
\qquad
\forall k.
\]

If positive and negative coefficients were allowed, this reverse conclusion would not hold.

Thus, positivity ensures:

\[
\text{Global zero certificate}
\iff
\text{Every finite block is zero}.
\]

## 9.3 Relationship with the Divisor Sheaf

On region \(U\):

\[
\nu_F(U)
=
\left|
\mathfrak o_F(U)
\right|,
\]

where the right side represents the total multiplicity of the off-axis effective divisor within \(U\).

The divisor sheaf preserves:

- Exact positions;
- Multiplicities;
- Orbit types.

The certificate valuation only preserves:

- The total positive mass within the region.

Therefore, the certificate valuation is a compressed projection of the divisor sheaf.

---

# 10. Rational Rectangle Grids and Finite Certificate Complexes

## 10.1 Rectangle Grid

Take finite sets of rational coordinates:

\[
0<x_0<x_1<\cdots<x_m<\frac12,
\]

\[
y_0<y_1<\cdots<y_n.
\]

Generate a rectangular grid:

\[
R_{ij}
=
(x_{i-1},x_i)\times(y_{j-1},y_j).
\]

## 10.2 Regular Grid

If \(F\) has no zeros on all grid lines, this grid is called \(F\)-regular.

## 10.3 Cell Certificates

Each cell has:

\[
\omega_{ij}
=
\nu_F(R_{ij})
\in\mathbb N_0.
\]

## 10.4 Block Certificates

For any finite block \(K\) composed of cells:

\[
\nu_F(K)
=
\sum_{R_{ij}\subset K}
\omega_{ij}.
\]

## 10.5 Certificate Complex

We can treat:

- Vertices as rational grid points;
- Edges as directed grid lines;
- 2-cells as rectangles;
- The boundary operator as recording the directed boundary of each rectangle;
- The integral of \(F'/F\) as a linear functional on 1-chains.

Internal boundary cancellation can be written as:

\[
\partial\left(\sum_iR_i\right)
=
\sum_i\partial R_i.
\]

This is the chain-level reason why rectangle certificates are composable.

## 10.6 Engineering Value

Finite certificate complexes are suitable for:

- Exact interval arithmetic;
- Distributed verification;
- Certificate caching;
- Re-subdivision of failed regions;
- Finite combinatorial formalization in Lean;
- Separating numerical verification from topological correctness.

---

# 11. Boundary Zeros and Regularization

## 11.1 Boundary Issues

If:

\[
F(z_0)=0,
\qquad
z_0\in\partial U,
\]

then:

\[
F/|F|
\]

is undefined at \(z_0\), and:

\[
\frac{F'}F
\]

has a pole at that point.

Therefore, the original boundary cannot be directly used for a winding number certificate.

## 11.2 Perturbed Boundaries

By the discreteness of zeros, one can apply arbitrarily small shifts to finite rectangle boundaries so that the new boundaries avoid all zeros.

But one must record:

- The direction of the shift;
- Which zeros are included or excluded before and after the shift;
- The relationship between the new region and the original region.

## 11.3 Inner and Outer Approximations

For a region \(U\) that might pass through zeros, one can select:

\[
U^-_\epsilon\Subset U\Subset U^+_\epsilon
\]

and both boundaries are zero-free.

If there are no zeros in the intermediate layer, then:

\[
\nu_F(U^-_\epsilon)
=
\nu_F(U^+_\epsilon).
\]

This common value can serve as the stable certificate for \(U\).

## 11.4 Not Using Undeclared Half-Multiplicity Rules

Some contour integral conventions assign half-multiplicity to boundary zeros.

This document does not take this convention as a default.

Unless indented contours, orientations, and limit rules are explicitly established, all certificates require zero-free boundaries.

---

# 12. Regular Exhaustion

## 12.1 Exhaustion Definition

A sequence of regions:

\[
U_1\Subset U_2\Subset\cdots\Subset X^+
\]

If:

\[
\bigcup_{n=1}^\infty U_n=X^+,
\]

then it is called an exhaustion of \(X^+\).

## 12.2 Rectangular Exhaustion

We can take:

\[
U_n
=
\left\{
z:
\varepsilon_n<\operatorname{Re}z<\frac12-\delta_n,
\quad
|\operatorname{Im}z|<T_n
\right\},
\]

where:

\[
\varepsilon_n\downarrow0,
\qquad
\delta_n\downarrow0,
\qquad
T_n\uparrow\infty.
\]

## 12.3 Existence of Regular Exhaustion

The zero set is countable and discrete.

We can choose \(\varepsilon_n,\delta_n,T_n\) such that all rectangle boundaries avoid zeros while maintaining the exhaustion property.

Because at each step we only need to avoid finitely or countably many bad coordinate values, and a dense complement still exists in the real numbers.

## 12.4 Exhaustion Decision Theorem

### Theorem 12.1

For any \(F\)-regular exhaustion \((U_n)\), the following are equivalent:

1. RH holds;
2. \(D_F|_{X^+}=0\);
3. 
   \[
   \nu_F(U_n)=0
   \qquad
   \forall n.
   \]

### Proof

If the right half-strip has no zeros, each \(U_n\) counts to zero.

Conversely, if there exists a zero \(\rho\) in the right half-strip, by the exhaustion property, there exists \(N\) such that:

\[
\rho\in U_N.
\]

Therefore:

\[
\nu_F(U_N)\ge m_\rho>0.
\]

Proof complete.

---

# 13. Finite Verification and Global Gaps

## 13.1 Finite Prefixes

If we only know:

\[
\nu_F(U_n)=0
\qquad
1\le n\le N,
\]

we can only deduce:

\[
D_F(U_N)=0.
\]

We cannot deduce:

\[
D_F(X^+)=0.
\]

## 13.2 The Missing Tail Proposition

To lift a finite prefix to RH, we at least also need:

\[
D_F(X^+\setminus U_N)=0.
\]

This is the tail exclusion proposition.

## 13.3 Three Legitimate Lifting Modes

### Mode 1: Full Exhaustion Proof

Prove separately for all \(n\):

\[
\nu_F(U_n)=0.
\]

### Mode 2: Unified Local Theorem

Establish a theorem valid for all regular rectangles:

\[
R\Subset X^+
\Longrightarrow
\nu_F(R)=0.
\]

### Mode 3: Finite Verification Plus Tail Theorem

First prove that a finite region has no zeros, then prove:

\[
|\operatorname{Im}z|>T_0
\Longrightarrow
F(z)\ne0
\quad
\text{in }X^+.
\]

## 13.4 Illegitimate Lifting

The following inferences are all invalid:

\[
\text{Verified to a very high height}
\Longrightarrow
\text{Holds for all heights},
\]

\[
\nu_F(U_n)=0
\text{ for many }n
\Longrightarrow
\nu_F(U_n)=0
\text{ for all }n,
\]

\[
\lim_{n\to\infty}
\frac{\nu_F(U_n)}{\text{Total number of zeros}}
=0
\Longrightarrow
\nu_F(U_n)\equiv0.
\]

Density zero does not equal non-existence.

---

# 14. Local Logarithms and Winding Cohomology Classes

## 14.1 Zero Complement Space

Let:

\[
Y_F=X^+\setminus Z(F).
\]

On \(Y_F\):

\[
F:Y_F\to\mathbb C^\times.
\]

## 14.2 Circle-Valued Map

Define:

\[
u_F
=
\frac{F}{|F|}
:
Y_F\to S^1.
\]

It induces the cohomology class:

\[
[u_F]
\in
H^1(Y_F;\mathbb Z).
\]

## 14.3 Logarithmic Derivative

On \(Y_F\), define:

\[
\eta_F
=
\frac{1}{2\pi i}\frac{F'}F\,dz.
\]

It is a closed form and has integer periods:

\[
\int_\gamma\eta_F
\in\mathbb Z
\]

for any closed loop \(\gamma\subset Y_F\).

## 14.4 Periods and Winding Numbers

We have:

\[
\int_\gamma\eta_F
=
\deg(u_F|_\gamma).
\]

If \(\gamma=\partial U\), then:

\[
\int_{\partial U}\eta_F
=
D_F(U).
\]

## 14.5 Local Logarithms

On a zero-free simply connected open set \(V\subset Y_F\), there exists a holomorphic function \(L_V\) such that:

\[
e^{L_V}=F.
\]

On overlaps:

\[
L_V-L_W
\in
2\pi i\mathbb Z.
\]

These integer jumps form a Čech 1-cocycle, representing the same topological class as \([u_F]\).

## 14.6 Relationship with Zero Divisors

The winding class takes values on infinitesimal circles enclosing zeros:

\[
m_\rho.
\]

Therefore, the zero divisor can be reconstructed from all local periods:

\[
D_F
=
\sum_\rho
\left(
\int_{\gamma_\rho}\eta_F
\right)
[\rho].
\]

where \(\gamma_\rho\) is a small positively oriented circle enclosing only \(\rho\).

## 14.7 Topological Unification in This Document

From this we obtain:

\[
\boxed{
\text{Divisor sheaf}
\longleftrightarrow
\text{Local periods}
\longleftrightarrow
\text{Boundary winding certificates}
}
\]

However, this correspondence still merely encodes existing zeros into topological data.

It does not prove that these local periods are zero.

---

# 15. Three Levels of Local-to-Global

## 15.1 Level 1: Locality of Sheaves

\[
\mathfrak o_F=0
\iff
(\mathfrak o_F)_p=0
\quad
\forall p.
\]

This is a formal local-to-global equivalence.

## 15.2 Level 2: Certificate Additivity

In a finite regular partition:

\[
\nu_F(U)
=
\sum_i\nu_F(U_i).
\]

This is a computational local-to-global principle for finite regions.

## 15.3 Level 3: Non-Compact Space Lifting

From:

\[
\nu_F(U_n)=0
\quad
\forall n
\]

deducing a global zero obstruction relies on the exhaustion property.

If we only grasp finitely many \(n\), there remains a gap at infinity.

## 15.4 Where the True Difficulty Lies

Sheaf theory itself tells us:

> If every local obstruction has been proven to be zero, then the global obstruction is zero.

But the core difficulty of RH is precisely:

\[
\text{How do we independently prove that each local off-axis obstruction is zero?}
\]

Thus, sheaf theory manages the proof; it does not create the missing analytic-arithmetic theorems.

---

# 16. Information Compression of Local Certificates

## 16.1 Complete Divisors

\[
D_F|_R
\]

preserves:

- The position of each zero;
- The multiplicity of each zero;
- Orbit types;
- Relative geometry.

## 16.2 Winding Number Certificates

\[
\omega_R(F)
\]

only preserves:

- The total multiplicity within the region.

Therefore:

\[
D_F|_R
\longmapsto
\omega_R(F)
\]

is a strong coarsening.

## 16.3 Decision Completeness

For a single rectangle:

\[
\omega_R(F)=0
\iff
D_F(R)=0
\]

because the divisor is effective.

So regarding "whether zeros exist within the region", the winding number certificate is complete.

But it is incomplete for the following questions:

- Where are the zeros;
- Do they form 4-element orbits;
- Distances between zeros;
- Are they multiple zeros;
- How do they move with parameters.

## 16.4 Multi-Scale Certificate Families

Using a family of nested rectangles:

\[
R_1\supset R_2\supset\cdots,
\]

one can progressively locate zeros.

If:

\[
\omega_{R_n}(F)>0
\]

and the diameter tends to zero, then the support can be localized to a unique limit point.

This provides an interface for numerical isolation and formalized zero certificates.

---

# 17. Certificate Failure Modes

## 17.1 Zero-Free Boundary Unproven

Only calculating an integral approximation without proving:

\[
\inf_{z\in\partial R}|F(z)|>0,
\]

cannot guarantee winding number stability.

## 17.2 Missed Counts from Numerical Phase Jumps

Discrete sampling might cross phase changes exceeding \(\pi\), causing missed winding counts.

One must provide:

- Derivative bounds;
- Step size bounds;
- Interval envelopes;
- Adaptive subdivision.

## 17.3 Floating-Point Values Near Zero

If \(|F|\) is extremely small on the boundary, ordinary floating-point calculations cannot rule out true zeros.

## 17.4 Double Counting from Overlapping Regions

If rectangles overlap internally:

\[
\nu_F(U\cup V)
\ne
\nu_F(U)+\nu_F(V)
\]

one should instead use inclusion-exclusion or switch to a legitimate partition.

## 17.5 Internal Boundary Orientations Not Cancelled

When gluing grids, internal edges must appear with opposite orientations.

## 17.6 Finite Covers Misidentified as Global Covers

No bounded finite grid can cover the entire \(X^+\).

## 17.7 Misinterpreting Local Logarithm Existence as Global Logarithm Existence

Choosing a logarithm branch in each small region does not imply the existence of a single global logarithm on the zero complement space.

Its obstruction is precisely:

\[
[u_F]\in H^1(Y_F;\mathbb Z).
\]

---

# 18. Formalization Specifications

## 18.1 Sheaf Module

Suggested to establish:

```text
LocallyFiniteEffectiveDivisor
DivisorRestriction
EffectiveDivisorSheaf
OffAxisObstructionSheaf
GlobalOffAxisSection
StalkVanishing
```

Core theorems:

```text
divisor_sheaf_locality
divisor_sheaf_gluing
off_axis_global_zero_iff_stalk_zero
right_half_zero_iff_full_off_axis_zero
```

## 18.2 Rectangle Module

```text
RationalRectangle
RelativelyCompactRectangle
BoundaryZeroFree
RegularRectangle
PositiveOrientation
```

## 18.3 Winding Number Module

```text
PhaseMap
WindingCertificate
ArgumentPrincipleCertificate
CertificateNonnegative
```

Core theorem:

\[
\omega_R(F)=D_F(R).
\]

## 18.4 Partition Module

```text
RectangleComplex
CompatibleOrientation
InternalEdgeCancellation
CertificateAdditivity
```

## 18.5 Exhaustion Module

```text
RegularExhaustion
ExhaustionCoversRightHalfStrip
AllExhaustionCertificatesZero
```

Core theorem:

\[
\left(
\forall n,\ \nu_F(U_n)=0
\right)
\iff
D_F|_{X^+}=0.
\]

## 18.6 Trust Boundaries

Numerical certificates must separate the following:

- Analytic theorems;
- Exact integer conclusions;
- Interval arithmetic;
- External function libraries;
- Floating-point hardware;
- Verified programs;
- Unverified preprocessing.

---

# 19. Certificate Data Format

Each local certificate can be recorded using the following structure:

```yaml
certificate_id:
function_id:
region:
  x_min:
  x_max:
  y_min:
  y_max:
orientation: positive
boundary_free:
  method:
  lower_bound_abs_F:
  precision:
winding:
  integer_value:
  method:
multiplicity_policy: counted
dependencies:
  theorems:
  axioms:
  software:
parent_partition:
children:
status:
```

## 19.1 Required Fields

- `region`: Must be exactly reconstructible;
- `boundary_free`: Cannot just say "numerically looks non-zero";
- `integer_value`: Must have a proof of integer stability;
- `dependencies`: List computational and axiomatic dependencies;
- `parent_partition`: Supports gluing and avoids double counting.

## 19.2 Global Certificate Checklist

A full exhaustion certificate requires:

```text
ExhaustionDefinition
BoundaryRegularityForEveryLevel
ZeroWindingForEveryLevel
CoverageProof
SymmetryReductionProof
DependencyAudit
```

It cannot merely be a finite-length numerical table.

---

# 20. Main Results of This Document

This document completes the following structures.

## 20.1 True Sheaves

Locally finite effective divisors and off-axis effective divisors form sheaves.

## 20.2 Countable Decision Family

RH is equivalent to the winding numbers of all regular rational rectangles being zero.

## 20.3 Positive Additive Certificates

Under finite legitimate partitions, winding number certificates are additive, and positivity prevents cancellation.

## 20.4 Regular Exhaustion

RH is equivalent to all certificates of any regular exhaustion being zero.

## 20.5 Topological Unification

Divisor multiplicities, boundary winding numbers, and \(H^1\) period classes are different representations of the same zero obstruction.

---

# 21. What This Document Has Not Accomplished

This document has not proven:

\[
\omega_R(F)=0
\]

holds for any arbitrary rectangle containing the possibility of unknown zeros.

This document also has not established:

- A full-height zero-free region;
- Prime-side positivity;
- Admissible test function separation;
- Explicit formula contradictions;
- A proof of RH.

Therefore, the substantive position of this paper is:

\[
\boxed{
\text{Deconstructing "the non-existence of global off-axis zeros" into a countable, positive-valued, and glueable local certificate problem.}
}
\]

---

# 22. Next Stage: Equivariant Arithmetic Separation

This document outputs to the next paper:

\[
\left(
\mathscr O^+,
\mathfrak o_F,
\mathcal B_{\mathbb Q,F}^{\mathrm{reg},+},
\nu_F,
[u_F],
(U_n)
\right).
\]

The core question of the next paper is:

> If a certain rectangle certificate \(\nu_F(R)>0\), can we construct a test function near its support that can enter the \(\zeta\) explicit formula, and cause the off-axis positive mass to produce a sign that cannot be cancelled by on-axis zeros, Gamma terms, and the prime side?

Formally, we need to establish:

\[
\nu_F(R)>0
\Longrightarrow
\exists h_R\in\mathcal H_{\mathrm{adm}}
\quad
\mathcal Q_F(h_R)<0,
\]

and then independently prove from the arithmetic side:

\[
\mathcal Q_F(h)\ge0
\qquad
\forall h\in\mathcal H_{\mathrm{adm}}.
\]

Only then is it possible to turn a local certificate into a substantive contradiction.

---

# 23. Conclusion

This paper further sheafifies the off-axis positive obstruction from the previous paper.

Fixing the function \(F\) yields a global section of the off-axis obstruction sheaf:

\[
\mathfrak o_F
\in
\Gamma(X,\mathscr O^+).
\]

The sheaf-theoretic form of RH is:

\[
\boxed{
\mathrm{RH}
\iff
\mathfrak o_F=0.
}
\]

The countable rectangle form of RH is:

\[
\boxed{
\mathrm{RH}
\iff
\omega_R(F)=0
\quad
\forall
R\in\mathcal B_{\mathbb Q,F}^{\mathrm{reg},+}.
}
\]

The exhaustion form of RH is:

\[
\boxed{
\mathrm{RH}
\iff
\nu_F(U_n)=0
\quad
\forall n.
}
\]

These three forms each undertake different tasks:

- The sheaf-theoretic form is responsible for the consistency of local data;
- The rational rectangle form provides countable local certificates;
- The exhaustion form manages non-compact global quantifiers.

The most critical correction in this document is:

> **Local divisors form true sheaves; winding number certificates only form positive additive valuations on regular regions and legitimate finite partitions, and cannot be broadly termed as complete cosheaves before checking boundaries and orientations.**

This document also precisely delineates the boundary of local-to-global lifting:

\[
\text{All local certificates are zero}
\Longrightarrow
\text{Global obstruction is zero}
\]

is correct;

but:

\[
\text{Finitely many local certificates are zero}
\Longrightarrow
\text{Global obstruction is zero}
\]

is incorrect.

Therefore, the next step should not be to continue adding more equivalent forms of RH, but to start addressing the true substantive arrow:

\[
\boxed{
\text{Non-zero local off-axis certificate}
\Longrightarrow
\text{Arithmetically admissible separating test function}.
}
\]

This will be the first formal lifting problem transitioning from the topological decision framework into the analytic-arithmetic proof content.

---

# Appendix A: Main Symbols

| Symbol | Meaning |
|---|---|
| \(X\) | Centered open critical strip |
| \(A\) | Critical axis \(i\mathbb R\) |
| \(X^\pm\) | Left and right half critical strips |
| \(\mathscr Z^+\) | Sheaf of locally finite effective divisors |
| \(\mathscr O^+\) | Sheaf of off-axis effective divisors |
| \(\mathfrak o_F\) | Global off-axis obstruction section of \(F\) |
| \(\mathcal B_{\mathbb Q}^+\) | Rational-rectangle basis for the right half-strip |
| \(\omega_R(F)\) | Rectangle boundary winding number certificate |
| \(\nu_F(U)\) | Zero-counting valuation for regular regions |
| \(U_n\) | Regular exhaustion |
| \(\eta_F\) | \((2\pi i)^{-1}(F'/F)dz\) |
| \([u_F]\) | \(H^1\) class of the circle-valued phase map |

---

# Appendix B: Logical Strength Table

| Result | Logical Status |
|---|---|
| \(\mathscr Z^+\) is a sheaf | General structure theorem |
| \(\mathscr O^+\) is a sheaf | General structure theorem |
| \(\omega_R(F)=D_F(R)\) | Argument principle |
| Rectangle certificates are finitely additive | Boundary chain cancellation |
| RH \(\iff\mathfrak o_F=0\) | Equivalent restatement |
| RH \(\iff\) All regular rational rectangle certificates are zero | Countable local decision |
| RH \(\iff\) All certificates of a regular exhaustion are zero | Non-compact exhaustion decision |
| Any specific unknown rectangle certificate is zero | Yet to be proven |
| All rectangle certificates are zero | Equivalent to RH, yet to be proven |

---

# Appendix C: Local-to-Global Failure Checklist

1. Boundary zeros not excluded;
2. Local regions do not cover the global domain;
3. Only finite height;
4. Internal overlap of blocks;
5. Inconsistent orientations on common boundaries;
6. Tail at infinity uncontrolled;
7. Vicinity of the critical axis not exhausted with \(\varepsilon\downarrow0\);
8. Mistaking density zero for the empty set;
9. Mistaking numerical near-zero for exact zero;
10. Mistaking local logarithm branches for a global logarithm.

---

# Appendix D: Version Boundaries

v0.1 has completed:

- Effective divisor sheaf;
- Off-axis obstruction sheaf;
- Countable basis of rational rectangles;
- Regular boundaries and winding number certificates;
- Finite partition additivity;
- Regular exhaustion theorem;
- Local logarithms and \(H^1\) classes;
- Formalization and certificate formats;
- Local-to-global failure classification.

v0.1 has not yet completed:

- Lean 4 implementation;
- Exact interval arithmetic verifier;
- Complete formalization for general curved regions;
- Test function lifting;
- Sign control in explicit formulas;
- Any proof of RH.