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

# Legendre Base-Space Program Phase 20
## Landing-Graph Moment Hierarchy / GCD Stratification
### Fourth-Order Bulk Mixing, Large-GCD Closure, the Sixth-Order Closure Gate and the Primitive Correlation Triangle

**Version:** v0.1  
**Date:** 2026-10-02  
**Author:** Neo.K × AI collaboration  
**Institution:** EveMissLab / 一言諾科技有限公司  
**Research line:** Legendre Base-Space Prime-Fitting Program  
**Prerequisites:** Phases 1–19  
**Status:** exact moment decomposition + partial deterministic closure + finite spectral audit; Legendre's conjecture remains unproved

---

# 0. This Round's Core Conclusions

Phase 19 compressed the critical lower-sieve remainder into a centered landing matrix:

$$
L_X(r,s)
=
A_X(r,s)-\frac{H}{rs},
$$

where:

$$
R\asymp S\asymp X^{1/3},
$$

$$
H=X^{1/2},
$$

and:

$$
A_X(r,s)
=
\mathbf1_{
\exists t\in\mathbb Z:
X<rst\le X+H
}.
$$

The core conclusions of Phase 20:

1. The natural scale of the landing-graph row degree:
   $$
   K:=H/R=X^{1/6};
   $$
2. The diagonal and large-gcd off-diagonal of the fourth-order moment can close at the:
   $$
   X^{2/3+\varepsilon}
   $$
   scale;
3. Finite data show the small-gcd off-diagonal at the same scale as well, with row-correlation RMS of constant order;
4. But:
   $$
   M_4:=\operatorname{Tr}(LL^*)^2
   \ll X^{2/3+\varepsilon}
   $$
   itself yields only:
   $$
   \|L\|_{op}\ll X^{1/6+\varepsilon},
   $$
   which is borderline for the sieve remainder and insufficient for closure;
5. **The sixth-order moment is the first fixed moment sufficient for closure**;
6. If:
   $$
   \boxed{
   M_6:=\operatorname{Tr}(LL^*)^3
   \ll
   X^{5/6+\varepsilon},
   }
   $$
   then the critical factorable remainder is:
   $$
   \ll X^{17/36+\varepsilon},
   $$
   and:
   $$
   \frac{17}{36}
   <
   \frac12;
   $$
7. The only new nondegenerate term at sixth order is the centered row-correlation triangle:
   $$
   \boxed{
   \mathcal T_\triangle
   =
   \sum_{\substack{r_1,r_2,r_3\\\text{distinct}}}
   C_{12}C_{23}C_{31}.
   }
   $$

Thus the next actual analytic target is no longer the full spectral norm, but:

# Primitive Small-GCD Correlation Triangle

---

# 1. Critical Landing Matrix

Let:

$$
\mathcal R
=
[R,2R)\cap\mathbb Z,
$$

$$
\mathcal S
=
[R,2R)\cap\mathbb Z,
$$

where:

$$
R=X^{1/3+o(1)}.
$$

Define:

$$
\boxed{
A_{r,s}
=
\left\lfloor
\frac{X+H}{rs}
\right\rfloor
-
\left\lfloor
\frac{X}{rs}
\right\rfloor.
}
$$

Since:

$$
rs\asymp X^{2/3}>H,
$$

we have:

$$
\boxed{
A_{r,s}\in\{0,1\}.
}
$$

The main-density matrix:

$$
\boxed{
P_{r,s}
=
\frac H{rs}.
}
$$

The centered matrix:

$$
\boxed{
L=A-P.
}
$$

---

# 2. Unique Quotient

If:

$$
A_{r,s}=1,
$$

there is a unique:

$$
t=t(r,s)
=
\left\lfloor
\frac X{rs}
\right\rfloor+1
$$

such that:

$$
X<rst\le X+H.
$$

Define the landing excess:

$$
\boxed{
e(r,s)
=
rst-X,
}
$$

Thus:

$$
1\le e(r,s)\le H.
$$

---

# 3. Row Degree Bound

Define:

$$
d(r)
=
\sum_{s\in\mathcal S}A_{r,s}.
$$

For an edge:

$$
(r,s)
$$

let:

$$
y=st.
$$

Then:

$$
\frac Xr
<
y
\le
\frac{X+H}{r}.
$$

This interval has length:

$$
\asymp
\frac Hr
\asymp
K.
$$

Each:

$$
y
$$

can arise from:

$$
s\,t=y
$$

through at most:

$$
\tau(y)
\ll_\varepsilon
X^\varepsilon
$$

factorizations.

Thus:

## Lemma 3.1

$$
\boxed{
d(r)
\ll_\varepsilon
KX^\varepsilon.
}
$$

At the critical scale:

$$
\boxed{
K=X^{1/6+o(1)}.
}
$$

---

# 4. Row Correlation Matrix

Define:

$$
\boxed{
C
=
LL^*.
}
$$

That is:

$$
C(r_1,r_2)
=
\sum_s
L(r_1,s)L(r_2,s).
$$

Write:

$$
C_{12}=C(r_1,r_2).
$$

---

# 5. Exact Correlation Expansion

Define the joint landing count:

$$
\boxed{
J_{12}
=
\sum_s
A_{r_1,s}A_{r_2,s}.
}
$$

and:

$$
U_r
=
\sum_{\substack{s\\A_{r,s}=1}}
\frac1s,
$$

$$
V_2
=
\sum_{s\in\mathcal S}\frac1{s^2}.
$$

Then, exactly:

$$
\boxed{
C_{12}
=
J_{12}
-
\frac H{r_2}U_{r_1}
-
\frac H{r_1}U_{r_2}
+
\frac{H^2}{r_1r_2}V_2.
}
$$

---

# 6. Deterministic Correction Terms Are $O(X^\varepsilon)$

By the degree bound:

$$
U_r
\le
\frac{d(r)}R
\ll
\frac KR
X^\varepsilon.
$$

And:

$$
\frac Hr
\asymp K.
$$

Thus:

$$
\frac Hr U_{r'}
\ll
\frac{K^2}{R}X^\varepsilon.
$$

At the critical scale:

$$
K^2=R,
$$

therefore:

$$
\boxed{
\frac Hr U_{r'}
\ll
X^\varepsilon.
}
$$

Also:

$$
V_2
\asymp
\frac1R,
$$

so:

$$
\boxed{
\frac{H^2}{r_1r_2}V_2
\asymp1.
}
$$

Hence:

$$
\boxed{
C_{12}
=
J_{12}
+
O_\varepsilon(X^\varepsilon).
}
$$

as a deterministic size envelope.

Note:

> The actual content of centered cancellation still lies in the fine structure of joint landing relative to the main corrections; this equation does not mean $C_{12}$ must be positive.

---

# 7. Joint Landing Near-Diagonal Lemma

If the same:

$$
s
$$

satisfies:

$$
A_{r_1,s}
=
A_{r_2,s}
=
1,
$$

let the unique quotients be:

$$
t_1,t_2.
$$

Then:

$$
r_i s t_i
=
X+e_i,
$$

where:

$$
1\le e_i\le H.
$$

Subtract:

$$
s(r_1t_1-r_2t_2)
=
e_1-e_2.
$$

Thus:

## Lemma 7.1

$$
\boxed{
|r_1t_1-r_2t_2|
\le
\frac Hs
\ll
K.
}
$$

This is the arithmetic core of landing row correlation.

---

# 8. GCD Stratification

Let:

$$
\boxed{
g=(r_1,r_2).
}
$$

Since:

$$
g
\mid
(r_1t_1-r_2t_2),
$$

if:

$$
g>K,
$$

Lemma 7.1 forces:

$$
\boxed{
r_1t_1=r_2t_2.
}
$$

Thus:

## High-GCD Rigidity

$$
\boxed{
g>K
\Longrightarrow
\text{near diagonal collapses to exact diagonal}.
}
$$

---

# 9. Large-GCD Joint Landing Bound

Let:

$$
L=[r_1,r_2]
=
\frac{r_1r_2}{g}.
$$

If:

$$
r_1t_1=r_2t_2,
$$

we can write:

$$
r_1t_1=r_2t_2=Lm.
$$

The joint landing condition becomes:

$$
X<Lms\le X+H.
$$

Thus:

$$
ms
$$

lies in an interval of length:

$$
\frac HL
$$

as specified above.

Also:

$$
L\asymp
\frac{R^2}{g}.
$$

so:

$$
\frac HL
\asymp
\frac gK.
$$

Using the divisor bound:

## Lemma 9.1

For:

$$
g>K,
$$

$$
\boxed{
J_{12}
\ll_\varepsilon
\left(
1+\frac gK
\right)
X^\varepsilon.
}
$$

Therefore:

$$
\boxed{
|C_{12}|
\ll_\varepsilon
\left(
1+\frac gK
\right)
X^\varepsilon.
}
$$

---

# 10. Large-GCD Fourth-Moment Closure

The number of row pairs with gcd exactly:

$$
g
$$

is at most:

$$
\ll
\left(
\frac Rg+1
\right)^2.
$$

Thus:

$$
\sum_{\substack{r_1\ne r_2\\(r_1,r_2)>K}}
|C_{12}|^2
$$

$$
\ll
X^\varepsilon
\sum_{g>K}
\frac{R^2}{g^2}
\left(
1+\frac gK
\right)^2.
$$

Therefore:

$$
\ll
X^\varepsilon
\left[
\frac{R^2}{K}
+
\frac{R^3}{K^2}
\right].
$$

Since:

$$
R=X^{1/3},
$$

$$
K=X^{1/6},
$$

we obtain:

## Theorem 10.1

$$
\boxed{
\sum_{\substack{r_1\ne r_2\\(r_1,r_2)>K}}
|C_{12}|^2
\ll_\varepsilon
X^{2/3+\varepsilon}.
}
$$

The large-gcd sector has thus attained the random-bulk fourth-moment scale.

---

# 11. Diagonal Fourth-Moment Closure

From:

$$
C_{rr}
=
\sum_sL(r,s)^2
$$

and:

$$
A_{r,s}\in\{0,1\},
$$

we have:

$$
C_{rr}
\ll
d(r)+1.
$$

Lemma 3.1 gives:

$$
C_{rr}
\ll
KX^\varepsilon.
$$

Hence:

$$
\boxed{
\sum_rC_{rr}^2
\ll
RK^2X^\varepsilon
=
X^{2/3+\varepsilon}.
}
$$

---

# 12. Small-GCD Fourth-Moment Target

What actually remains for the fourth-order moment is therefore:

$$
\boxed{
\mathcal M_4^{small}
=
\sum_{\substack{
r_1\ne r_2\\
(r_1,r_2)\le K
}}
|C_{12}|^2.
}
$$

We need:

$$
\boxed{
\mathcal M_4^{small}
\ll
X^{2/3+\varepsilon}.
}
$$

This is equivalent to:

> Small-gcd off-diagonal row correlations have $O(X^\varepsilon)$ RMS.

---

# 13. The Crude Small-GCD Envelope Is Too Weak

For:

$$
g\le K,
$$

the near-diagonal Diophantine count gives only:

$$
J_{12}
\ll
KX^\varepsilon.
$$

Thus, crudely:

$$
|C_{12}|
\ll
KX^\varepsilon.
$$

There are:

$$
R^2
$$

row pairs, giving:

$$
\mathcal M_4^{small}
\ll
R^2K^2X^\varepsilon
=
\boxed{
X^{1+\varepsilon}.
}
$$

Compared with the target:

$$
X^{2/3}
$$

the gap is:

$$
\boxed{
X^{1/3}.
}
$$

Thus small-gcd centered cancellation is a genuine gap.

---

# 14. Finite Fourth-Moment Audit

| $n$ | $M_4/X^{2/3}$ | diag | large-gcd | small-gcd | actual $\|L\|_{op}$ |
|---:|---:|---:|---:|---:|---:|
| 1000 | 1.4868 | 0.7683 | 0.0602 | 0.6583 | 6.1598 |
| 3000 | 1.7031 | 0.8488 | 0.0997 | 0.7546 | 7.7217 |
| 10000 | 1.8427 | 0.8782 | 0.1324 | 0.8321 | 10.8983 |

Finite data:

- Total $M_4$ is about a constant multiple of:
  $$
  X^{2/3};
  $$
- The diagonal is about:
  $$
  0.8X^{2/3};
  $$
- Large-gcd contributes only a small proportion;
- Small-gcd also lies stably at the:
  $$
  X^{2/3}
  $$
  scale.

---

# 15. Row-Correlation RMS Audit

| $n$ | offdiag RMS | P99 $|C_{12}|$ | max small-gcd | max large-gcd | $\|L\|_{op}/X^{1/12}$ |
|---:|---:|---:|---:|---:|---:|
| 1000 | 0.8519 | 2.4204 | 5.2793 | 6.0241 | 1.9479 |
| 3000 | 0.9266 | 2.7022 | 5.4896 | 7.9717 | 2.0333 |
| 10000 | 0.9835 | 2.9364 | 6.0845 | 15.5571 | 2.3480 |

Most noteworthy:

$$
\boxed{
\text{off-diagonal RMS}
\approx
1.
}
$$

and the small-gcd P99 is still only of constant order.

This is precisely the second-order mixing scale of a pseudorandom centered matrix.

---

# 16. Correcting the Spectral Target of Phase 19

Phase 19 proposed the natural random target:

$$
\|L\|_{op}
\asymp
X^{1/12}.
$$

This remains a reasonable conjectural natural scale.

But it is stronger than what proof closure actually requires.

For factorable weights:

$$
\|\alpha\|_2,
\|\beta\|_2
\ll
X^{1/6+\varepsilon}.
$$

Thus:

$$
|\alpha^TL\beta|
\le
\|L\|_{op}
X^{1/3+\varepsilon}.
$$

As long as:

$$
\boxed{
\|L\|_{op}
=
o\left(
\frac{X^{1/6}}{\log X}
\right),
}
$$

this already suffices to make the remainder:

$$
o(H/\log X).
$$

Therefore:

$$
X^{1/12}
$$

is the natural random-scale target,

not the minimal closure threshold.

---

# 17. Why Is the Fourth-Order Moment Insufficient?

Let:

$$
M_4
=
\operatorname{Tr}(LL^*)^2.
$$

If we successfully prove:

$$
M_4
\ll
X^{2/3+\varepsilon},
$$

then:

$$
\|L\|_{op}^4
\le
M_4.
$$

so we obtain only:

$$
\boxed{
\|L\|_{op}
\ll
X^{1/6+\varepsilon}.
}
$$

This is stuck exactly at the minimal threshold:

$$
X^{1/6}.
$$

and lacks logarithmic / power saving.

Thus:

$$
\boxed{
\text{The fourth-order moment is a bulk gate, not a closure gate.}
}
$$

---

# 18. General Random-Bulk Moment Scale

For average degree:

$$
K=X^{1/6},
$$

the squared singular values of a random sparse centered matrix have scale:

$$
\asymp K.
$$

The rank:

$$
R=X^{1/3}.
$$

Thus the natural moment scale:

$$
\boxed{
M_{2k}
:=
\operatorname{Tr}(LL^*)^k
\asymp
RK^k
=
X^{1/3+k/6}.
}
$$

If we prove:

$$
M_{2k}
\ll
X^{1/3+k/6+\varepsilon},
$$

then:

$$
\|L\|_{op}
\ll
X^{
1/12+1/(6k)+\varepsilon
}.
$$

---

# 19. Minimal Fixed Moment Order

The factorable-remainder exponent is:

$$
\frac13
+
\frac1{12}
+
\frac1{6k}
=
\frac5{12}
+
\frac1{6k}.
$$

We need:

$$
\frac5{12}
+
\frac1{6k}
<
\frac12.
$$

Equivalently:

$$
\boxed{
k>2.
}
$$

Thus:

## Theorem 19.1

The first fixed moment theoretically sufficient for crossing is:

$$
\boxed{
k=3,
}
$$

that is:

$$
\boxed{
\operatorname{Tr}(LL^*)^3.
}
$$

namely the sixth-order moment.

---

# 20. Sixth-Moment Closure Theorem

## Theorem 20.1

If, for critical landing blocks, we can prove:

$$
\boxed{
M_6
=
\operatorname{Tr}(LL^*)^3
\ll_\varepsilon
X^{5/6+\varepsilon},
}
$$

then:

$$
\|L\|_{op}
\le
M_6^{1/6}
\ll
X^{5/36+\varepsilon}.
$$

Thus, for divisor-bounded factorable coefficients:

$$
|\alpha^TL\beta|
\ll
X^{5/36}
X^{1/3}
X^\varepsilon.
$$

That is:

$$
\boxed{
|\alpha^TL\beta|
\ll
X^{17/36+\varepsilon}.
}
$$

And:

$$
\boxed{
\frac{17}{36}
\approx
0.472222
<
\frac12.
}
$$

so:

$$
\boxed{
X^{17/36+\varepsilon}
=
o(H/\log X)
}
$$

for sufficiently small $\varepsilon$.

Therefore:

> A sixth-order random-bulk moment bound suffices to close the critical balanced factorable remainder of Phase 18.

Note:

> This is still only analytic-provider closure, not completion of the entire sieve architecture for Legendre on its own.

---

# 21. Finite High-Moment Audit

| $n$ | $M_4/X^{2/3}$ | $M_6/X^{5/6}$ | $M_8/X$ | $M_6^{1/6}/X^{5/36}$ |
|---:|---:|---:|---:|---:|
| 1000 | 1.4868 | 3.4591 | 9.3237 | 1.2298 |
| 3000 | 1.7031 | 4.2973 | 12.3924 | 1.2751 |
| 10000 | 1.8427 | 5.0563 | 16.3828 | 1.3101 |

Finite data:

$$
M_4
\asymp
X^{2/3},
$$

$$
M_6
\asymp
X^{5/6},
$$

$$
M_8
\asymp
X.
$$

have exactly the same exponents as the random-bulk hierarchy.

---

# 22. Sixth-Moment Exact Decomposition

Let:

$$
C=LL^*.
$$

Then:

$$
M_6
=
\operatorname{Tr}(C^3)
=
\sum_{r_1,r_2,r_3}
C_{12}C_{23}C_{31}.
$$

Decompose into:

### A. All same

$$
\boxed{
\mathcal D_3
=
\sum_rC_{rr}^3.
}
$$

### B. Exactly two equal

$$
\boxed{
\mathcal D_{21}
=
3
\sum_{r\ne r'}
C_{rr}
C_{rr'}^2.
}
$$

### C. All distinct

$$
\boxed{
\mathcal T_\triangle
=
\sum_{\substack{
r_1,r_2,r_3\\
\text{pairwise distinct}
}}
C_{12}C_{23}C_{31}.
}
$$

Thus, exactly:

$$
\boxed{
M_6
=
\mathcal D_3
+
\mathcal D_{21}
+
\mathcal T_\triangle.
}
$$

---

# 23. Degenerate Sixth-Moment Pieces Are Already at Target Scale

The degree lemma gives:

$$
C_{rr}
\ll
KX^\varepsilon.
$$

Hence:

$$
\mathcal D_3
\ll
RK^3X^\varepsilon.
$$

That is:

$$
\boxed{
\mathcal D_3
\ll
X^{5/6+\varepsilon}.
}
$$

If the fourth-order mixing target:

$$
\sum_{r\ne r'}C_{rr'}^2
\ll
X^{2/3+\varepsilon}
$$

holds, then:

$$
\mathcal D_{21}
\ll
KX^\varepsilon
X^{2/3+\varepsilon}.
$$

so:

$$
\boxed{
\mathcal D_{21}
\ll
X^{5/6+\varepsilon}.
}
$$

---

# 24. Primitive Correlation Triangle

The only genuinely new nondegenerate obligation at sixth order is therefore:

$$
\boxed{
\mathcal T_\triangle.
}
$$

We need to prove:

$$
\boxed{
|\mathcal T_\triangle|
\ll
X^{5/6+\varepsilon}.
}
$$

This is:

## Landing Correlation Triangle Bound

---

# 25. Finite Sixth-Moment Decomposition

| $n$ | total/$X^{5/6}$ | all-same | exactly-two-equal | all-distinct triangle |
|---:|---:|---:|---:|---:|
| 1000 | 3.4591 | 0.7666 | 1.9993 | 0.6932 |
| 3000 | 4.2973 | 0.8894 | 2.4898 | 0.9181 |
| 10000 | 5.0563 | 0.9159 | 2.8381 | 1.3024 |

The all-distinct triangle itself also lies stably at the:

$$
X^{5/6}
$$

scale.

Thus the finite data support:

$$
\boxed{
\mathcal T_\triangle
\ll
X^{5/6+o(1)}.
}
$$

but this is currently unproved.

---

# 26. Arithmetic Meaning of a Correlation Triangle

A nonzero:

$$
C_{12}C_{23}C_{31}
$$

corresponds to three sets of row-pair landing correlations:

$$
(r_1,r_2),
$$

$$
(r_2,r_3),
$$

$$
(r_3,r_1).
$$

Each joint landing carries a near-diagonal relation:

$$
|r_it_i-r_jt_j|
\ll
K.
$$

The primitive triangle thus corresponds to three mutually closing near-multiplicative relations.

This is a:

$$
\boxed{
\text{multiplicative near-diagonal triangle energy}.
}
$$

---

# 27. Large-GCD vs Primitive Triangle

Phase 20 proved that when:

$$
(r_i,r_j)>K
$$

the pair relation becomes exact.

The triangle can therefore be further split into:

1. At least one large-gcd edge;
2. All three gcds are:
   $$
   \le K.
   $$

The purest new difficulty is:

$$
\boxed{
(r_1,r_2),
(r_2,r_3),
(r_3,r_1)
\le
K.
}
$$

that is:

## Primitive Small-GCD Triangle

---

# 28. Why Generic Multiplicative Energy Is Not Yet Enough

Standard multiplicative energy studies:

$$
ab=cd.
$$

This problem instead simultaneously has:

- Near equality;
- A landing interval;
- Centered subtraction;
- Three pairwise-coupled row relations.

Thus a bound for generic:

$$
E^\times(A)
$$

cannot be treated directly as a triangle theorem.

But energy language still provides the correct next interface.

---

# 29. Literature Boundary

Existing ternary-divisor / AP technology can handle:

- The unweighted $d_3$ summatory error;
- Distribution of $d_3$ over moduli beyond $1/2$.

But landing correlation moments require:

$$
\boxed{
\text{simultaneous centered row interactions}.
}
$$

This round found no existing theorem directly applicable as a black box.

Thus this is currently a genuinely new analytic interface, not merely a missing citation.

---

# 30. Phase 21 Target

The next round tackles only:

# Primitive Landing Triangle / Sixth-Moment Arithmetic

Specifically:

1. Introduce, for each correlation edge:
   $$
   g_{ij}=(r_i,r_j);
   $$
2. Parameterize:
   $$
   |r_it_i-r_jt_j|\le K
   $$
   as reduced lattice equations;
3. Seek compatibility for the closing conditions of the three edges;
4. Split into:
   - Zero-difference cycles;
   - One-zero / two-nonzero;
   - All-nonzero;
5. Investigate whether triangle compatibility forces:
   - A large gcd;
   - A repeated quotient;
   - A low-dimensional parameter set;
6. The direct target is:
   $$
   |\mathcal T_\triangle|
   \ll
   X^{5/6+\varepsilon}.
   $$

---

# 31. Phase 20 Status

```text
Legendre Base-Space Program
Phase 20 — Landing Moment Hierarchy

row degree d(r) << X^(1/6+eps):                 PROVED
exact row-correlation expansion:                PROVED
joint landing near-diagonal relation:           PROVED
large-gcd exactification g>K:                   PROVED
large-gcd joint bound:                          PROVED
large-gcd M4 contribution:                      CLOSED at X^(2/3+eps)
diagonal M4 contribution:                       CLOSED at X^(2/3+eps)
small-gcd M4 RMS target:                        OPEN
finite offdiag RMS ~ constant:                  OBSERVED
Phase19 minimal spectral threshold:             CORRECTED
M4 as final closure gate:                       REJECTED / BORDERLINE
random-bulk M_(2k) hierarchy:                   DEFINED
minimal fixed closure moment:                   k=3
sixth-moment closure theorem:                   PROVED
M6 exact decomposition:                         PROVED
degenerate M6 pieces:                           CLOSED conditional on M4 mixing
all-distinct triangle energy:                   OPEN
finite M6 ~ X^(5/6):                            OBSERVED
Legendre proof status:                          OPEN
next phase:                                     PRIMITIVE LANDING TRIANGLE
```

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# 32. References / boundary sources

1. Classical trace / moment method for spectral norms of sparse matrices and graphs.
2. G. Kolesnik, historical bound:
   $$
   \alpha_3\le43/96
   $$
   for the ternary Piltz divisor problem.
3. Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, *On the exponent of distribution of the ternary divisor function*, Mathematika **61** (2015), 121–144.
4. Prahlad Sharma, *Bilinear sums with GL(2) coefficients and the exponent of distribution of $d_3$*, Proc. London Math. Soc. **128** (2024), e12589.
5. Standard divisor bound:
   $$
   \tau(n)\ll_\varepsilon n^\varepsilon.
   $$
6. Phase 18–19 landing-matrix and sieve-level transfer framework.