> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# SRBRD-11
## Square-Root Completion / Sieve-Parity Interface
### Fixed-Factor Truncation, Root-Complete Prime Certification, Bulk–Frontier Geometry and Parity-Barrier Audit

**Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program  
**Branch:** SRBRD — Symmetric Root-Bounded Relation Domain  
**Round:** SRBRD-11  
**Date:** 2026-09-30  
**Version:** v0.1  
**Authors:** Neo.K; Aletheia (GPT-5.6 Sol)  
**Type:** Exact Structural Audit / External Sieve Interface / Barrier Localization  
**Prerequisites:** SRBRD-00 through SRBRD-10  
**Status:** The exact boundary between the fixed-cutoff sieve core and dynamic root-completion has been established; Goldbach remains unproved

---

# 0. Why This Round Pauses the Calculation of Yet Another Moment Order

SRBRD-10 has located the current uniform gap as:

$$
\boxed{
\text{positive-moment CRT phase-hit discrepancy}.
}
$$

But if this gap is merely another representation of a classical sieve difficulty, mechanically continuing to increase:

- moment order；
- hypercell order；
- forest / cycle correction；

may only approach a known barrier again.

Thus this round conducts a necessary source audit:

> Where exactly is SRBRD's root-bounded relation the same as a classical sieve, and where is it genuinely different?

The central answer obtained in this round is:

$$
\boxed{
\text{The SRBRD bulk at a fixed cutoff }z
\text{ reduces exactly to a residue sieve;}
}
$$

However:

$$
\boxed{
\text{prime certification requires endpoint-dependent square-root completion.}
}
$$

The gap between the two corresponds precisely to:

> Small-factor-free does not mean prime.

This is closely related to the parity barrier in sieve theory.

---

# 1. Fixed Cutoff Divisor Domain

For:

$$
x>1,
$$

and an integer cutoff:

$$
z\ge2,
$$

Define:

$$
\boxed{
\mathcal D_z(x)
=
\left\{
r\in\mathbb P:
r\le z,
\quad
r\mid x
\right\}.
}
$$

If:

$$
x
$$

is odd, one can also consider only:

$$
r\ge3.
$$

---

# 2. z-Rough Number

Define:

$$
\boxed{
\operatorname{Rough}_z(x)
:
\Longleftrightarrow
\mathcal D_z(x)=\varnothing.
}
$$

That is:

> $x$ has no prime factor not exceeding $z$.

Note:

$$
\operatorname{Rough}_z(x)
$$

is not equivalent to:

$$
x\in\mathbb P
$$

unless the cutoff has reached square-root completeness.

---

# 3. Root-Completion Lemma

## Theorem SRBRD-11-T01

If:

$$
x\ge2,
$$

and:

$$
z^2\ge x,
$$

then:

$$
\boxed{
\operatorname{Rough}_z(x)
\Longleftrightarrow
x\in\mathbb P.
}
$$

---

# 4. T01 Proof

If $x$ is prime, of course:

$$
\mathcal D_z(x)=\varnothing
$$

as long as we count only the proper-factor range.

Conversely, if $x$ is composite,

there exists a prime:

$$
r
$$

such that:

$$
r^2\le x.
$$

From:

$$
x\le z^2,
$$

we obtain:

$$
r\le z.
$$

Thus:

$$
r\in\mathcal D_z(x),
$$

contradicting roughness.

---

# 5. Incomplete Cutoff Leaves a Second-Division State

If:

$$
\operatorname{Rough}_z(x)
$$

However:

$$
x
$$

is composite,

then:

$$
\operatorname{lpf}(x)>z.
$$

Therefore:

$$
\boxed{
x>
z^2.
}
$$

More precisely:

If:

$$
x=ab
$$

and:

$$
a=\operatorname{lpf}(x),
$$

then:

$$
a>z,
$$

while:

$$
b\ge a>z.
$$

Thus:

$$
\boxed{
x
=
ab,
\qquad
a,b>z.
}
$$

This is the exact form of the second division not yet being closed.

---

# 6. Connection to the Original Intuition

The even number:

$$
N=2m
$$

first gives the compulsory division:

$$
N\xrightarrow{/2}m.
$$

For the endpoint:

$$
x=m\pm d,
$$

fixed cutoff：

$$
z
$$

excludes only division actions with:

$$
r\le z
$$

as displayed above.

If:

$$
x>z^2,
$$

even if all these actions have been excluded,

there may still exist:

$$
\boxed{
x=ab,
\quad
a,b>z.
}
$$

Thus prime certification actually requires:

$$
\boxed{
z\ge\sqrt x.
}
$$

---

# 7. Symmetric Bulk for a Fixed Cutoff

Let:

$$
N=2m,
$$

and:

$$
z^2\le m.
$$

Define:

$$
\boxed{
\Omega_m^{\mathrm{bulk}}(z)
=
\left\{
d\in\Omega_m:
d\le m-z^2
\right\}.
}
$$

For:

$$
d\in\Omega_m^{bulk}(z),
$$

left endpoint：

$$
m-d\ge z^2.
$$

right endpoint：

$$
m+d\ge m\ge z^2.
$$

---

# 8. Bulk Root-Automatic Theorem

## Theorem SRBRD-11-T02

If:

$$
d\in\Omega_m^{bulk}(z),
$$

and:

$$
3\le r\le z
$$

is prime,

then:

$$
\boxed{
r^2\le m-d
}
$$

and:

$$
\boxed{
r^2\le m+d.
}
$$

Thus, in the bulk:

> The root condition is entirely automatic for every $r\le z$.

---

# 9. Bulk Left Obstruction = Residue Class

From T02:

$$
r
$$

produces an obstruction at the left endpoint if and only if:

$$
r\mid(m-d).
$$

That is:

## Theorem SRBRD-11-T03

$$
\boxed{
d\in O_{L,r}
\Longleftrightarrow
d\equiv m\pmod r
}
$$

For:

$$
d\in\Omega_m^{bulk}(z),
\quad
r\le z.
$$

---

# 10. Bulk Right Obstruction = Residue Class

Similarly:

## Theorem SRBRD-11-T04

$$
\boxed{
d\in O_{R,r}
\Longleftrightarrow
d\equiv -m\pmod r
}
$$

For:

$$
d\in\Omega_m^{bulk}(z),
\quad
r\le z.
$$

---

# 11. Collapse for Center Divisors

If:

$$
r\mid m,
$$

then:

$$
m\equiv-m\equiv0\pmod r.
$$

Thus, in the bulk, the obstruction residues collapse from two to one:

$$
\boxed{
d\equiv0\pmod r.
}
$$

If:

$$
r\nmid m,
$$

then:

$$
\boxed{
d\equiv\pm m\pmod r
}
$$

are two distinct residue classes.

---

# 12. Bulk Sieve Survivor

Define:

$$
\boxed{
\mathcal S_z(m)
=
\left\{
d\in\Omega_m^{bulk}(z):
d\not\equiv\pm m\pmod r
\text{ for every odd prime }r\le z
\right\},
}
$$

where:

$$
r\mid m
$$

when this holds,

$$
\pm m
$$

are treated as the same residue.

---

# 13. Bulk Sieve Equivalence

## Theorem SRBRD-11-T05

For:

$$
d\in\Omega_m^{bulk}(z),
$$

we have:

$$
\boxed{
d\in\mathcal S_z(m)
}
$$

if and only if:

$$
\boxed{
m-d
\text{ and }
m+d
\text{ are both }z\text{-rough}.
}
$$

Thus the fixed-cutoff SRBRD bulk reduces exactly to a:

$$
\boxed{
\text{two-linear-form residue sieve}.
}
$$

---

# 14. Classical Sieve Core

In the bulk,

for each:

$$
r\le z,
$$

the offset needs only to avoid:

$$
\boxed{
1\text{ or }2
}
$$

residue classes modulo:

$$
r.
$$

Therefore:

$$
\mathcal S_z(m)
$$

is a very standard finite sieve object.

SRBRD's:

- hypercells；
- moments；
- CRT intersections；

can all be regarded in this region as reorganizations of sieve inclusion-exclusion / residue intersection.

---

# 15. Root Frontier

Define:

$$
\boxed{
\Omega_m^{\mathrm{front}}(z)
=
\Omega_m
\setminus
\Omega_m^{bulk}(z).
}
$$

That is:

$$
d>m-z^2.
$$

In the frontier:

$$
m-d<z^2.
$$

Thus the left endpoint's root domain starts to be shorter than:

$$
[3,z]
$$

as displayed above.

Only here does SRBRD's characteristic moving root truncation appear.

---

# 16. Fixed Cutoff Cannot Complete a Wide Symmetric Bulk

If the bulk contains some:

$$
d>0,
$$

then:

$$
z^2<m.
$$

Since:

$$
d\le m-z^2.
$$

but the right endpoint satisfies:

$$
m+d
>
m
>
z^2.
$$

Thus:

$$
\boxed{
z
<
\sqrt{m+d}.
}
$$

therefore the fixed cutoff $z$ is not root-complete for the right endpoint.

---

# 17. Bulk-vs-Completion Theorem

## Theorem SRBRD-11-T06

If:

$$
d>0
$$

and:

$$
d\in\Omega_m^{bulk}(z),
$$

then:

$$
\boxed{
z^2<m+d.
}
$$

Thus:

$$
\boxed{
\operatorname{Rough}_z(m+d)
}
$$

cannot, by itself, imply:

$$
m+d\in\mathbb P.
$$

---

# 18. No Wide Fixed-Cutoff Prime-Certification Zone

If one wants the same fixed:

$$
z
$$

to be root-complete for both endpoints throughout:

$$
\Omega_m^{bulk}(z)
$$

as displayed above,

then, for the maximum right endpoint:

$$
2m-z^2
$$

we have:

$$
z^2\ge2m-z^2.
$$

That is:

$$
\boxed{
z^2\ge m.
}
$$

but a nontrivial bulk radius requires:

$$
z^2<m.
$$

Thus:

## Theorem SRBRD-11-T07

$$
\boxed{
\text{No fixed cutoff simultaneously has positive bulk radius and root completeness for both endpoints.}
}
$$

boundary：

$$
z^2=m
$$

at this boundary, the bulk degenerates to the vicinity of:

$$
d=0
$$

as displayed above.

---

# 19. This Is the Necessity of a Dynamic Root Cutoff

Goldbach prime-pair certification cannot be completed with a single fixed:

$$
z
$$

over the entire offset domain.

An actual exact criterion must move the cutoff with the endpoint:

$$
\boxed{
z_L(d)
=
\sqrt{m-d},
}
$$

$$
\boxed{
z_R(d)
=
\sqrt{m+d}.
}
$$

The formal core instead uses:

$$
r^2\le m\pm d
$$

to avoid real square roots.

This is one place where SRBRD genuinely differs structurally from a classical fixed sieve.

---

# 20. False Survivors

Define a fixed-cutoff false survivor:

$$
\boxed{
d\in
\mathcal S_z(m)
\setminus
\operatorname{Res}(m).
}
$$

Then at least one endpoint:

$$
x=m\pm d
$$

is a:

$$
z\text{-rough composite}.
$$

Thus:

$$
\boxed{
x=ab,
\quad
a,b>z.
}
$$

These states are precisely the gap between a fixed-cutoff sieve and primality.

---

# 21. Second-Division Interpretation

For a false-survivor endpoint:

$$
x,
$$

All:

$$
r\le z
$$

division actions fail.

But compositeness still comes from:

$$
x=ab,
$$

$$
a,b>z.
$$

Thus:

> A fixed-cutoff sieve excludes only the first layer of small divisions;
> Root completion excludes all possible second-layer factorization witnesses.

This directly corresponds to the originally proposed:

$$
\boxed{
\text{scope of the second division}.
}
$$

---

# 22. Almost-Prime Bridge

If:

$$
z
$$

is chosen sufficiently large,

a composite number that is:

$$
z\text{-rough}
$$

and:

$$
x\le X
$$

cannot have too many prime factors.

For example, if:

$$
z^3>X,
$$

then:

$$
x
$$

cannot have three prime factors all:

$$
>z
$$

as displayed above.

Thus:

$$
\boxed{
z\text{-rough}
+
z^3>X
}
$$

forces:

$$
x
$$

to be:

- prime；
- Or a product of two primes.

This is an elementary factor-count bridge.

---

# 23. Relation to Prime + $P_2$

This explains why:

> Excluding sufficiently many small factors

naturally reaches:

$$
P_2
$$

first, rather than directly reaching primes.

In the external sieve literature, Chen's theorem is an important result at precisely this level:

> A sufficiently large even integer can be represented as a prime plus a number with at most two prime factors.

This is not a theorem of this document, but an external benchmark.

---

# 24. External Sieve-Parity Audit

Classical sieve theory has the:

$$
\boxed{
\text{parity problem}.
}
$$

Roughly speaking:

> Certain kinds of local divisibility / Type-I sieve information alone have difficulty distinguishing primes from almost-prime states with different prime-factor parity.

Binary Goldbach is one of the typical settings for the parity problem.

Therefore:

$$
\boxed{
\text{SRBRD bulk residue data alone should not be expected to break through Goldbach automatically.}
}
$$

---

# 25. Important Qualification

This does not mean:

$$
\boxed{
\text{SRBRD has been proved equivalent to a traditional sieve and therefore cannot succeed.}
}
$$

For SRBRD also has structures absent from the classical fixed-cutoff bulk:

1. dynamic root boundary；
2. left/right asymmetric truncation；
3. centered product：
   $$
   m^2-d^2;
   $$
4. factor/cofactor hypergraph；
5. exact cycle danger complexes；
6. occupancy / moment dual constraints。

The actual question is:

> Do these extra structures provide enough Type-II / bilinear-like information to escape the parity barrier?

This remains unknown.

---

# 26. Source-Class Interpretation

SRBRD information can be divided into:

## Sieve-Core Information

- residues：
  $$
  d\equiv\pm m\pmod r;
  $$
- fixed-cutoff survivor；
- finite intersection counts；
- Type-I-like local divisibility data。

## Root-Dynamic Information

- endpoint-specific：
  $$
  r^2\le m\pm d;
  $$
- left/right moving frontier。

## Cofactor Coupling

- 
  $$
  m-d=ar;
  $$
- 
  $$
  m+d=bs;
  $$
- 
  $$
  2m=ar+bs.
  $$

## Centered Product Coupling

$$
\boxed{
(m-d)(m+d)
=
m^2-d^2.
}
$$

The last two categories are more likely to be sources of parity escape.

---

# 27. A Formal Parity-Risk Flag

It is recommended that metadata be added to future theorems:

```yaml
parity_risk:
  local_residue_only: high
  fixed_cutoff_only: high
  dynamic_root_used: medium
  cofactor_bilinear_used: lower
  centered_product_cross_side_used: lower
```

This is not a mathematical theorem.

It is a research-method audit:

> To prevent us from mistaking the same sieve information for a new source after changing its language.

---

# 28. Large Sieve Interface

Large-sieve-type theorems can give upper bounds on:

> Sets of integers avoiding specified residue classes

as displayed above.

This may be used for:

- phase-hit discrepancy；
- residual candidate count；
- CRT lattice packing。

But SRBRD ultimately needs a:

$$
\boxed{
\text{lower bound establishing that the residual is nonempty}.
}
$$

Thus large-sieve upper control alone is still not Goldbach closure.

---

# 29. Larger Sieve Interface

Gallagher's larger sieve considers:

> How large can a set in an interval be if it occupies few residue classes modulo many prime powers?

This has a formal interface with SRBRD's:

- sparse CRT hypercells；
- lattice hit sets；

as displayed above.

But the larger sieve is also an:

$$
\boxed{
\text{upper-bound technology}.
}
$$

and cannot directly replace a prime-pair lower bound.

---

# 30. Why the Sieve Comparison Is Useful

It tells us:

After SRBRD-11, one should not continue pursuing only:

$$
\boxed{
\text{more refined residue-density estimates}.
}
$$

because this is very likely to remain within the parity-sensitive sieve core.

The next target more worth attacking is:

$$
\boxed{
\text{cofactor coupling / bilinear structure}.
}
$$

---

# 31. Cofactor Pair Equation

If:

$$
r
$$

acts on the left,

$$
s
$$

acts on the right,

one can write:

$$
m-d=ar,
$$

$$
m+d=bs.
$$

Adding:

$$
\boxed{
ar+bs=2m.
}
$$

Subtracting:

$$
\boxed{
bs-ar=2d.
}
$$

where:

$$
a\ge r,
\quad
b\ge s
$$

follows from the root condition.

This is not merely a residue statement.

It is a bilinear/cofactor coupling.

---

# 32. Root Conditions in Cofactor Form

If:

$$
m-d=ar,
$$

and:

$$
r^2\le m-d,
$$

then:

$$
\boxed{
a\ge r.
}
$$

Similarly:

$$
\boxed{
b\ge s.
}
$$

Thus the obstruction relation can be rewritten as:

$$
\boxed{
ar+bs=2m,
\qquad
a\ge r,
\quad
b\ge s,
}
$$

with odd prime：

$$
r,s.
$$

If an endpoint has multiple witness primes,

multiple cofactor constraints are obtained.

---

# 33. Why This May Be Type-II-Like

A local sieve asks only:

$$
r\mid(m-d).
$$

The cofactor form instead retains, in:

$$
m-d=ar
$$

the variable:

$$
\boxed{
a
}
$$

and couples it to the opposite side:

$$
bs
$$

through:

$$
ar+bs=2m
$$

as displayed above.

That is, it simultaneously retains:

- divisor；
- quotient；
- additive relation。

This has an additional multiplicative variable beyond merely recording residue classes.

---

# 34. Candidate Escape Principle

If SRBRD can ultimately break through the local sieve barrier,

it is most likely not by using:

$$
\boxed{
\text{more one-sided residue classes}.
}
$$

but by using:

$$
\boxed{
\text{three-way divisor–cofactor–opposite-side coupling}.
}
$$

This reconnects with the user's original intuition about a second division:

> It is not enough to know whether division is possible; the cofactor left after division must also be retained.

---

# 35. Root-Complete vs Cofactor-Complete

A distinction can be made:

### Root-Complete

For an endpoint:

$$
x
$$

check all prime divisors satisfying:

$$
r^2\le x
$$

as displayed above.

### Cofactor-Complete

If:

$$
r\mid x,
$$

also retain:

$$
\boxed{
a=x/r.
}
$$

and:

$$
a\ge r.
$$

SRBRD-00 through 10 mainly concern the root / incidence side.

SRBRD-12 should formally promote:

$$
\boxed{
\text{cofactor state}
}
$$

to a primary variable.

---

# 36. New No-Go Criterion

If, in the future, a proposed theorem:

1. Uses only:
   $$
   d\bmod r;
   $$
2. Uses only a fixed cutoff;
3. Does not read the cofactor;
4. Does not use a cross-side bilinear relation;

then, even if its notation is very new,

it should first be regarded as a:

$$
\boxed{
\text{sieve-core reformulation}
}
$$

rather than automatically counted as a source escape.

---

# 37. Dynamic Root Still Matters

This does not mean discarding SRBRD-03 through 10.

By contrast:

Dynamic roots make:

- The bulk gradually shrink;
- The frontier gradually expand;
- Large witnesses become right-only;
- Occupancy order decrease;
- The danger complex become sparse.

These can be combined with a cofactor method.

The genuinely new main route should be:

$$
\boxed{
\text{Dynamic Root Geometry}
+
\text{Cofactor Bilinear Coupling}.
}
$$

---

# 38. Bulk–Frontier Hybrid Strategy

In the future:

$$
\Omega_m
$$

can be partitioned into:

$$
\Omega_m^{bulk}(z)
\bigsqcup
\Omega_m^{front}(z).
$$

### Bulk

Use:

- sieve / large-sieve / moment envelope；
- But remain alert to the parity barrier.

### Frontier

Use:

- root truncation；
- occupancy cap；
- exact finite closure；
- one-sided large witnesses。

### Coupling

Use:

$$
ar+bs=2m
$$

to reconnect the information in the two regions.

---

# 39. External Benchmark: Chen-Type Endpoint

An important benchmark of known sieve methods is:

$$
\boxed{
\text{prime}+P_2.
}
$$

Thus, if SRBRD merely re-derives:

> A prime on one side and at most two prime factors on the other,

it has still not moved beyond the known sieve frontier.

A genuinely new result must at least be able to exclude the final:

$$
P_2
$$

false-survivor layer,

or find a different exact source to complete this step.

---

# 40. Parity-Escape Test

For future SRBRD theorems, it is recommended to add:

```yaml
ParityEscapeAudit:
  uses_only_local_divisibility: false?
  preserves_cofactor: true?
  uses_cross_side_product: true?
  uses_bilinear_relation: true?
  distinguishes_prime_from_rough_P2: true?
```

Only when the final item is actually supported by a theorem

could one claim:

$$
\boxed{
\text{parity escape evidence}.
}
$$

---

# 41. Formal Theorem Queue

```yaml
SRBRD_11:
  T01: rough_plus_square_completion_iff_prime
  T02: bulk_small_prime_root_condition_automatic
  T03: bulk_left_obstruction_iff_plus_residue
  T04: bulk_right_obstruction_iff_minus_residue
  T05: bulk_survivor_iff_both_endpoints_z_rough
  T06: positive_bulk_radius_implies_right_not_root_complete
  T07: no_positive_radius_fixed_cutoff_double_root_completion
  T08: rough_composite_implies_two_large_factors
  T09: z_cubed_gt_X_implies_z_rough_is_prime_or_P2
```

---

# 42. External Literature Audit

This round uses the external literature only for:

$$
\boxed{
\text{barrier comparison / source audit}.
}
$$

It does not covertly treat external theorems as results already proved in SRBRD.

External cross-checks:

1. P. Gallagher, **A larger sieve**, *Acta Arithmetica* 18 (1971), 77–81.
2. Classical large / larger sieve frameworks control sets by residue-class occupancy or avoidance.
3. The sieve literature explicitly records the parity problem for binary Goldbach.
4. Chen-type methods can prove that a sufficiently large even integer is:
   $$
   p+P_2.
   $$

---

# 43. Research Verdict

This round is not a:

$$
\boxed{
\text{SRBRD-11 PROOF PASS}.
}
$$

but the more important:

$$
\boxed{
\mathrm{SRBRD\mbox{-}11\ SIEVE\mbox{-}PARITY\ BOUNDARY\ IDENTIFIED}.
}
$$

We now know:

### The Parts That Are No Longer the Problem

- finite representation；
- root-bounded primality equivalence；
- CRT exactness；
- finite closure；
- moment polynomial certificates。

### The Actually Dangerous Parts

If a uniform proof relies only on:

- local residues；
- fixed small-factor cutoff；
- phase density；

it is very likely to remain within classical parity-sensitive sieve information.

### The Next Source Target

$$
\boxed{
\text{cofactor-preserving cross-side bilinear structure}.
}
$$

---

# 44. Recommendations for the Next Round

# SRBRD-12
## Cofactor-State / Bilinear Division Closure

Formally promote the obstruction state from:

$$
(r,\sigma)
$$

to:

$$
\boxed{
(r,a,\sigma)
}
$$

where:

$$
m+\sigma d=ar,
\qquad
a\ge r.
$$

and study the two-sided relation:

$$
\boxed{
ar+bs=2m.
}
$$

Core questions:

1. How many cofactor pairs are there for fixed $r,s$?
2. How does the root condition become:
   $$
   a\ge r,\ b\ge s;
   $$
3. cofactor parity / gcd constraints；
4. How do multiple factor witnesses at the same $d$ share cofactors?
5. Does a bilinear capacity upper bound exist?
6. Can one distinguish:
   - prime；
   - rough semiprime；
   - higher composite；
7. Does it actually produce parity-escape information?

This returns directly to the original question:

$$
\boxed{
\text{After being able to divide again, what remains after the division?}
}
$$

---

# Appendix A: Bulk Sieve Contract

```yaml
cutoff:
  z:
    integer: true
    z^2 <= m

bulk:
  d:
    d <= m-z^2

for_prime_r_le_z:
  left_root_condition:
    automatic: true
  right_root_condition:
    automatic: true

obstruction:
  left:
    d == m mod r
  right:
    d == -m mod r

survivor:
  both_endpoints_z_rough: true
```

---

# Appendix B: Root Completion Contract

```yaml
Rough_z(x):
  no_prime_factor_le_z: true

if:
  z^2 >= x

then:
  Rough_z(x) iff Prime(x)

false_survivor:
  rough_and_composite:
    implies:
      lpf(x) > z
      x = a*b
      a > z
      b > z
```

---

# Appendix C: Bulk–Completion Conflict

```yaml
nontrivial_bulk:
  exists_d_positive:
    requires:
      z^2 < m

double_endpoint_completion_over_bulk:
  requires:
    z^2 >= max_right_endpoint
    max_right_endpoint:
      2m-z^2

therefore:
  requires:
    z^2 >= m

conclusion:
  positive_radius_bulk_and_uniform_double_root_completion:
    impossible: true
```

---

# Appendix D: External Barrier Notes

```yaml
classical_sieve:
  bulk_relation:
    residue_avoidance: true

parity_problem:
  relevance:
    binary_Goldbach: high

known_benchmark:
  Chen_type:
    sufficiently_large_even:
      representation:
        prime_plus_P2: true

SRBRD_escape_candidate:
  - dynamic_root_boundary
  - cofactor_state
  - cross_side_bilinear_relation
  - centered_product_coupling
```

---

# Appendix E: Commit Decision

```yaml
research_round: SRBRD-11

decision:
  fixed_cutoff_bulk_equals_residue_sieve: true
  root_completion_requires_dynamic_cutoff: true
  parity_risk_identified: true
  cofactor_source_promoted_for_next_round: true
  uniform_goldbach_closure: false

main_barrier:
  local_sieve_information_cannot_be_assumed_to_distinguish_prime_from_rough_almost_prime

next:
  SRBRD-12_CofactorState_Bilinear_Division_Closure
```

---

# References

- P. Gallagher, “A larger sieve,” *Acta Arithmetica* 18 (1971), 77–81, DOI 10.4064/aa-18-1-77-81.
- Encyclopedia of Mathematics, “Sieve method.”
- Terence Tao, “The parity problem obstruction for the binary Goldbach problem with bounded error” (2014).
- Terence Tao, “254A, Supplement 5: The linear sieve and Chen’s theorem” (2015).