X16/Shiu/BRS Carrier-Slice Estimate
Focused External Review Packet

revised packet after first assessment

1

Review request

Please check the divisor-sum step used in the BRS singular TC1 branch.
This is not a request to verify the whole Goldbach proof. The requested
review is the following local question:
Does the stated Shiu/AP divisor-correlation argument prove the
X16C core carrier-slice estimate, and hence the BRS carrier-slice
estimate, with the claimed logarithmic losses and without circular
dependence on C1 or TTH?
The point of the packet is to make this carrier-slice step reviewable independently of the larger proof architecture.

2

Minimal context

The proof has a singular TC1 branch in which a terminal B1-origin coarea
test could concentrate on a short additive image of a marked carrier. The
BRS lemma says that this situation is not a genuine terminal GoodAWACK
residual: either it is already routed to an existing tag, or the short-image
subcell is a strict Edge contribution. The only analytic input needed for this
conclusion is the X16 carrier-slice estimate.
The local chain being reviewed is
BRS + X16BRS + X16C =⇒ TTH.
1

Here TTH is the structural consequence that every unrouted active B1-origin
coarea test has near-global length
𝐻 ≥ 𝑋(log 𝑋)−𝐵 .
That near-global range is then used elsewhere with the Davenport/AP input.
This packet asks only about the X16/Shiu/BRS part.

3

BRS carrier-slice statement

Let ℬ be a fixed-depth typed B1 dyadic block. Let 𝐶 be a B1 carrier of
dyadic height 𝑋𝐶 , and let 𝐼 be an additive interval. Put
𝑌16 := max{|𝐼 ∩ Z|, 𝑋𝐶 (log 𝑁 )−𝐵16 }.
The required carrier-slice estimate is
Massℬ (𝐶 ∈ 𝐼) ≪ 𝑁 (log 𝑁 )𝐶16

𝑌16
+ 𝑁 1−𝜌16 (log 𝑁 )𝐶16 .
𝑋𝐶

(X16-BRS)

Here 𝐶16 = 𝐶16 (𝐽0 ), 𝜌16 = 𝜌16 (𝐽0 ) > 0, and in the active proof-source file
one may take, after harmless enlargement,
𝐶16 = 100𝐽02 + 100,

𝜌16 =

1
106 𝐽04

.

The allowed BRS carriers are:
1. grouped product carriers;
2. Goldbach complementary carriers 𝑁 − 𝑃 ;
3. quotient carriers 𝑠 from a recorded equation 𝐿 = 𝑑𝑠;
4. controlled divisor quotients 𝐿/𝑑0 with 𝑑0 ≤ (log 𝑁 )𝐶 .
The reduction from these four carriers to grouped product carriers is in
X16BRS. Quotient carriers require the divisor to have already been tagged
or controlled before BRS is invoked.

2

4

X16C core analytic model

For a grouped product carrier 𝑃 with height 𝑋𝑃 , the B1 equation is reduced
to a fixed-depth divisor-correlation model. If 𝑃 = 𝑝, if 𝑢 is the complementary product on the same B1 side, and if the opposite side is forced to have
product
𝑄 = 𝑁 − 𝑝𝑢,
then the tuple mass is bounded by
(log 𝑁 )𝑂𝐽0 (1)

∑︁

𝜏𝐾1 (𝑝)

∑︁

𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0 ,

(1)

𝑢≍𝑈

#
𝑝∈𝐼16

#
with 𝐾𝑖 ≤ 2𝐽0 , 𝑋𝑃 𝑈 ≍ 𝑁 , and |𝐼16
| ≍ 𝑌16 . The target estimate for the
double sum in (1) is

∑︁
#
𝑝∈𝐼16

𝜏𝐾1 (𝑝)

∑︁

𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0
(2)

𝑢≍𝑈

≪ 𝑌16 𝑈 (log 𝑁 )

𝑂𝐽0 (1)

+𝑁

1−𝜌16

(log 𝑁 )

𝑂𝐽0 (1)

.

Since 𝑋𝑃 𝑈 ≍ 𝑁 , (2) gives (X16-BRS) for product carriers. The complementary, quotient, and controlled quotient carriers are then reduced to this
case in X16BRS.
This correlation is the key point. The proof does not use the false shortcut
∑︀
of bounding only 𝑝∈𝐼 𝜏𝐾 (𝑝). It keeps the moving complementary divisor
factor 𝜏𝐾3 (𝑁 − 𝑝𝑢).

5

X16C checking table

Lemma X16C records the Shiu/AP route as the following review checklist:
• carrier fixing: a fixed product 𝑝 = 𝑃 (𝑎𝑖 : 𝑖 ∈ 𝑆) has at most 𝜏𝐾1 (𝑝)
factorizations;
• same-side complement: the complementary product 𝑢 has at most
𝜏𝐾2 (𝑢) factorizations;
• opposite side: the Goldbach complement 𝑁 −𝑝𝑢 is kept as 𝜏𝐾3 (𝑁 −𝑝𝑢),
not averaged away;
3

• fixed 𝑝: 𝑁 − 𝑝𝑢 is treated in one residue class modulo 𝑝;
• fixed 𝑢: 𝑁 − 𝑢𝑝 is treated in one residue class modulo 𝑢 in the largecarrier orientation;
• Cauchy–Schwarz is followed by Shiu/AP for squared divisor weights;
•

𝑢∼𝑈 𝜏𝐾 (𝑢)

∑︀

ment;

2 is controlled by the standard fixed-divisor second mo-

• non-coprime AP classes are handled by separating local gcd factors;
• CRT and quotient restrictions are tagged and cost only polylogarithmic loss;
• if 𝑌16 𝑈 ≤ 𝑁 1−𝜌16 , the residual small-volume case is handled by the
trivial divisor bound.
Thus Shiu/AP is applied only after the relevant arithmetic progression and
carrier orientation have been fixed.

6

Shiu/AP input

The analytic input is Shiu’s Brun–Titchmarsh theorem for multiplicative
functions in arithmetic progressions:
P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J. Reine
Angew. Math. 313 (1980), 161–170, DOI 10.1515/crll.1980.313.161.
The fixed divisor-function second moment used in the Cauchy–Schwarz step
is cited as:
G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory,
Graduate Studies in Mathematics 163, American Mathematical Society, 3rd
ed., 2015, Ch. II.5, Theorem 5.
The proof uses the standard fixed-divisor-function corollary: for fixed 𝐾, 𝐴,
𝑓 (𝑛) = 𝜏𝐾 (𝑛)𝐴 is admissible in AP intervals, including the case 𝑓 (𝑛) =
𝜏𝐾 (𝑛)2 . The precise form used in X16C is the following.
For fixed 𝐾, 𝐴 ≥ 1 and 0 < 𝛿 < 1/10, put 𝑓 (𝑛) = 𝜏𝐾 (𝑛)𝐴 . If 𝐽 ⊂ [1, 𝑁 ] is
an interval of length 𝐻, 𝑁 𝛿 ≤ 𝐻 ≤ 𝑁 , and 𝑞 ≤ 𝐻 1−𝛿 , then for every residue

4

class 𝑎 mod 𝑞
𝑓 (𝑛) ≪

∑︁
𝑛∈𝐽
𝑛≡𝑎 (mod 𝑞)

(︂

𝐻
+ 1 (log 𝑁 )𝐶SH ℰ𝑞,𝑎 .
𝑞
)︂

(SH)

Lemma (Local factor averaging). The factor ℰ𝑞,𝑎 is a local noncoprime-class cost supported on primes dividing (𝑎, 𝑞). It is harmless
on average in the following precise sense. For every X16 carrier interval
#
#
𝐼16
⊂ [𝑋/2, 3𝑋] with |𝐼16
| ≫ 𝑋(log 𝑁 )−𝐵16 ,
∑︁

1/2

#
𝜏𝐾0 (𝑐)𝐴 ℰ𝑐,𝑁 ≪ |𝐼16
|(log 𝑁 )𝐶loc .

(LFA)

#
𝑐∈𝐼16

The same estimate holds over a full dyadic interval 𝑐 ≍ 𝑋. This dyadic
version is used in Case 2 when 𝑐 = 𝑢.
This is the only place where non-coprime AP classes enter the X16C estimate.
The proof of (LFA) is as follows. For a non-coprime class write 𝑔 = (𝑎, 𝑞),
𝑎 = 𝑔𝑎1 , 𝑞 = 𝑔𝑞1 , with (𝑎1 , 𝑞1 ) = 1. The summand becomes 𝑓 (𝑔𝑛1 ) in a
coprime class modulo 𝑞1 , and submultiplicativity gives 𝑓 (𝑔𝑛1 ) ≪ 𝑓 (𝑔)𝑓 (𝑛1 ).
Thus the local cost is bounded by a fixed divisor power of 𝑔 = (𝑎, 𝑞) and
by a harmless Euler factor over primes dividing 𝑞. In X16C the relevant
residue is 𝑎 = 𝑁 , so the cost is dominated by a fixed divisor power of (𝑐, 𝑁 ).
Averaging the multiplicative function
𝑐 ↦→ 𝜏𝐾0 (𝑐)𝐴 𝜏𝑀 ((𝑐, 𝑁 ))𝐵
over the carrier interval gives (LFA) by Shiu’s ordinary interval theorem; if
the carrier scale is polylogarithmic, the same bound is trivial after increasing
the logarithmic exponent.
The requested check is whether this Shiu/AP input applies to the sums in
(1) in the two range splits below.

7
Let

Bilinear estimate
𝑆=

∑︁
#
𝑝∈𝐼16

𝜏𝐾1 (𝑝)

∑︁

𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0 .

𝑢≍𝑈

5

Case 1: 𝑋𝑃 ≤ 𝑁 1−𝛿
#
Fix 𝑝 ∈ 𝐼16
. The positive values 𝑁 − 𝑝𝑢, with 𝑢 ≍ 𝑈 , belong to the residue
class 𝑁 mod 𝑝. They are contained in an interval 𝐽𝑝 ⊂ [1, 𝑁 ], and we enlarge
monotonically to length 𝐻𝑝 = 𝑁 . Since 𝑋𝑃 𝑈 ≍ 𝑁 ,

𝐻𝑝
𝑁
+1≍
+ 1 ≍ 𝑈.
𝑝
𝑝
The modulus condition is explicit:
𝑝 ≤ 𝑁 1−𝛿 = 𝐻𝑝1−𝛿 ≤ 𝐻𝑝1−𝛿/2 .
Cauchy–Schwarz gives
∑︁

𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0

𝑢≍𝑈

)︃1/2 (︃

(︃

≤

∑︁

𝜏𝐾2 (𝑢)2

)︃1/2
∑︁

𝑢≍𝑈

𝜏𝐾3 (𝑁 − 𝑝𝑢)2 1𝑁 −𝑝𝑢>0

.

𝑢≍𝑈

The first factor uses the standard second moment
∑︁

2

𝜏𝐾 (𝑢)2 ≪𝐾 𝑈 (log 2𝑈 )𝐾 −1 .

𝑢≍𝑈
2 in the residue
The second factor is controlled by (SH), applied to 𝑓 = 𝜏𝐾
3
class 𝑁 mod 𝑝:

∑︁

𝜏𝐾3 (𝑁 − 𝑝𝑢)2 1𝑁 −𝑝𝑢>0 ≪ 𝑈 (log 𝑁 )𝑂𝐽0 (1) ℰ𝑝,𝑁 .

𝑢≍𝑈

Therefore, for fixed 𝑝,
∑︁

1/2

𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0 ≪ 𝑈 (log 𝑁 )𝑂𝐽0 (1) ℰ𝑝,𝑁 .

𝑢≍𝑈

Now (LFA) gives
∑︁

1/2

𝜏𝐾1 (𝑝)ℰ𝑝,𝑁 ≪ 𝑌16 (log 𝑁 )𝑂𝐽0 (1) .

#
𝑝∈𝐼16

Consequently

𝑆 ≪ 𝑌16 𝑈 (log 𝑁 )𝑂𝐽0 (1) .
6

Case 2: 𝑋𝑃 > 𝑁 1−𝛿
Then 𝑈 ≪ 𝑁 𝛿 . If

𝑌16 𝑈 ≤ 𝑁 1−𝜌16 ,

the trivial divisor bound gives the required power-saving term:
𝑆 ≪ 𝑁 𝜀 𝑌16 𝑈 (log 𝑁 )𝑂𝐽0 (1) ≤ 𝑁 1−𝜌16 +𝜀 (log 𝑁 )𝑂𝐽0 (1) .
Taking 𝜀 = 𝜌16 /2, and then relabelling the resulting positive saving after the
initial harmless shrinkage of 𝜌16 , gives the second term in (2). Explicitly,
with 𝜌′16 = 𝜌16 /2,
′
𝑁 1−𝜌16 +𝜀 ≤ 𝑁 1−𝜌16 ,
and then 𝜌′16 is renamed as 𝜌16 .
#
, the values 𝑁 − 𝑢𝑝
Assume now that 𝑌16 𝑈 > 𝑁 1−𝜌16 . Fix 𝑢 ≍ 𝑈 . As 𝑝 ∈ 𝐼16
lie in the residue class 𝑁 mod 𝑢 and in an interval 𝐽𝑢 ⊂ [1, 𝑁 ] of length

𝐻𝑢 ≍ 𝑢𝑌16 .
Since 𝑢 ≍ 𝑈 , the non-small-volume assumption gives
𝐻𝑢 ≫ 𝑈 𝑌16 > 𝑁 1−𝜌16 .
The required Shiu modulus condition follows from
𝛿 < (1 − 𝜌16 )(1 − 𝛿/2).

(3)

Indeed, 𝑢 ≍ 𝑈 ≪ 𝑁 𝛿 , while 𝐻𝑢 ≫ 𝑁 1−𝜌16 ; for large 𝑁 , (3) implies
𝑢 ≤ 𝐻𝑢1−𝛿/2 .
For the displayed choices 𝛿 = 1/(20𝐽02 ) and 𝜌16 = 1/(106 𝐽04 ), (3) holds for
every 𝐽0 ≥ 1.
Cauchy–Schwarz in the 𝑝-variable gives
∑︁

𝜏𝐾1 (𝑝)𝜏𝐾3 (𝑁 − 𝑢𝑝)1𝑁 −𝑢𝑝>0

#
𝑝∈𝐼16

⎞1/2 ⎛
⎞1/2
∑︁
∑︁
⎜
⎟
⎜
⎟
≤⎝
𝜏𝐾1 (𝑝)2 ⎠ ⎝
𝜏𝐾3 (𝑁 − 𝑢𝑝)2 1𝑁 −𝑢𝑝>0 ⎠ .
⎛

#
𝑝∈𝐼16

#
𝑝∈𝐼16

7

The first factor is

1/2

≪ 𝑌16 (log 𝑁 )𝑂𝐽0 (1) ,

using the 𝑞 = 1 divisor-function interval estimate and the floor 𝑌16 ≥
𝑋𝑃 (log 𝑁 )−𝐵16 . The second factor is controlled by (SH), applied modulo
𝑢 to the residue class 𝑁 mod 𝑢:
∑︁

𝜏𝐾3 (𝑁 − 𝑢𝑝)2 1𝑁 −𝑢𝑝>0 ≪ 𝑌16 (log 𝑁 )𝑂𝐽0 (1) ℰ𝑢,𝑁 .

#
𝑝∈𝐼16

Thus
∑︁

1/2

𝜏𝐾1 (𝑝)𝜏𝐾3 (𝑁 − 𝑢𝑝)1𝑁 −𝑢𝑝>0 ≪ 𝑌16 (log 𝑁 )𝑂𝐽0 (1) ℰ𝑢,𝑁 .

#
𝑝∈𝐼16

Summing this bound with weight 𝜏𝐾2 (𝑢) and using the dyadic form of (LFA)
gives
𝑆 ≪ 𝑌16 𝑈 (log 𝑁 )𝑂𝐽0 (1) + 𝑁 1−𝜌16 (log 𝑁 )𝑂𝐽0 (1) .
This proves (2), hence X16C.

8

Noncircularity and parameters

The X16 floor is a monotone upper-bound device, not an appeal to TTH.
If the actual interval is shorter than 𝑋𝐶 (log 𝑁 )−𝐵16 , it is enlarged only for
the purpose of an upper bound. The added contribution is exactly of size
𝑁 (log 𝑁 )𝐶16

𝑋𝐶 (log 𝑁 )−𝐵16
= 𝑁 (log 𝑁 )𝐶16 −𝐵16 ,
𝑋𝐶

up to fixed C1/B1 coefficient losses, and the parameter register chooses
𝐵16 > 𝐶0 + 𝐶1 + 𝐶16 + 20.
After BRS is proved, TTH chooses 𝐵𝜅 large enough to dominate
𝐵16 + 𝐶0 + 𝐶1 + 𝐶16 + 𝜌−1
16 + 20.
Thus X16C is used before TTH and should not depend on TTH.
This packet directly verifies the product-carrier X16C estimate. The full
BRS carrier-slice estimate additionally uses X16BRS to reduce complementary, quotient, and controlled quotient carriers to this product-carrier model.
8

9

Expected positive outcome

A positive review can be short:
The BRS carrier-slice estimate follows from the stated Shiu/AP
divisor-correlation argument. The AP moduli, divisor weights, local factors, Cauchy–Schwarz loss, and X16 floor are compatible
with the claimed constants, and the argument is not circular in C1
or TTH.

10

Failure modes to flag

Please flag the first point at which any of the following fails:
1. a BRS carrier is not covered by the X16BRS reduction;
2. the divisor-correlation majorant (1) does not dominate the actual B1
tuple mass;
3. Shiu’s theorem does not apply to the AP modulus/interval range used
in one of the cases;
4. the non-coprime AP residue classes create local factors not absorbed
by the stated averaging;
5. the Cauchy–Schwarz square-root step loses more than recorded;
6. the trivial saving branch does not cover the complementary largecarrier range;
7. the proof of X16C uses C1 or TTH circularly.

9

