CKP/X10 Smooth-Weight Matching Packet
targeted external-review packet
May 30, 2026

Review request

Please check the CKP branch only. The question is whether the CKP nonzero
frequency contribution produced by the internal reduction ts the hypotheses
of the stated DukeFriedlanderIwaniec Kloosterman-fraction input, called
X10 in the proof ledger, after the smooth-weight derivative verication in
CKPD.
This is not a request to review the full Goldbach proof.
Target claim
Target claim. The CKP terminal contribution satises

RCKP (N ) = MCKP (N ) + o(N ),
where MCKP (N ) is the canonical local projection admitted by H4.
The nonzero-frequency CKP contribution is o(N ) by X10. The zerofrequency part is local and is passed to H4 through B1LD.
Logical chain

G1a + G2a + G3a + CKP D + G4a/X10 + X10ER + B1LD =⇒ G8a.
More explicitly:
1. G1a gives the exact gcd splitting u = ga, u′ = gq .
2. G2a performs the smooth AP/Fourier expansion.
3. G3a converts the nonzero-frequency CKP terms to bilinear Kloostermanfraction sums with two-variable smooth weights Wg,h (a, q).
1

4. CKPD proves the derivative bounds required to insert these weights
into X10.
5. G4a applies X10 on the central balanced nonzero-frequency range.
6. X10ER records that high-frequency, small-conductor, large-g , boundary, and short-volume CKP ranges are routed to C1A/C1 or H4 before
X10 is invoked.
7. G8a combines the nonzero-frequency estimate with the zero-frequency
local term.
Expanded block descriptions

The symbols in the chain mean the following local assertions.
G1a: gcd splitting

Starting from a tagged CKP atom of the schematic form
X
α(u)α′ (u′ )β(y)β ′ (y ′ )WU (u)WU ′ (u′ )WY (y)WY ′ (y ′ ),
uy+u′ y ′ =N

G1a splits

u′ = gq,

u = ga,

(a, q) = 1.

The equation becomes

ay + qy ′ = Ng ,

Ng = N/g.

Layers with g ∤ N are empty. Large or unbalanced g -layers are routed outside
X10. The central CKP branch retains balanced layers with inherited divisorbounded coecient sequences.
G2a: smooth AP/Fourier expansion

For xed g, a, q , the equation

ay + qy ′ = Ng
is reduced to the congruence

y ≡ Ng a

(mod q).
2

G2a applies a smooth AP/Fourier expansion to the y -bre. The frequency
h = 0 produces the local term. Frequencies h ̸= 0 produce oscillatory terms
with phase


hNg a
e
.
q
G3a: weighted Kloosterman-fraction form

G3a keeps the Fourier bre as a genuine two-variable smooth weight rather
than separating it articially. The nonzero-frequency layer has the form


X
hNg a
Og,h =
αg (a)γg (q)Wg,h (a, q)e
.
q
a∼Ag , q∼Qg
(a,q)=1

This is the exact signed phase sent to X10. The external DFI integer parameter is
r = |h|Ng .
For h < 0, we apply the same estimate to the conjugate phase, so the positive
external integer parameter is still r = |h|Ng .
X10/DFI theorem statement used in this packet

The external theorem invoked by X10 is used in the following dyadic smoothweight form. Let M, Q ≥ 1, r ≥ 1, and let αm , βq be arbitrary complex
sequences supported on m ≍ M , q ≍ Q. Let F (m, q) be supported on the
same dyadic box, satisfy |F (m, q)| ≤ 1, and obey
i j
∂m
∂q F (m, q) ≪ η i+j M −i Q−j ,

0 ≤ i, j ≤ 2.

(DFI-wt)

Then, for every ε > 0,

X
m≍M, q≍Q
(m,q)=1



rm
αm βq F (m, q)e
q



≪ε η 2 ∥α∥2 ∥β∥2 (r + M Q)3/8 (M + Q)11/48+ε .

The CKP application has η ≤
polylogarithmic loss budget.

(log N )C ,

3

so the η 2

(DFI-X10)
factor is part of the

CKPD: exact weight, normalization, and derivative check

CKPD proves that the actual nonseparated Fourier bre weight satises the
DFI/X10 smooth-weight hypotheses. The central layer has

a ≍ Ag ,

q ≍ Qg ,

y′ ≍ Y ′,

y ≍ Y,

with central support

Ag ≍ Qg ,

Y ≍ Y ′,

Y
≍ g.
Qg

(CS)

The remaining bre variable is

z(a, q, y) =

Ng − ay
.
q

Let ωA , ωQ , WY , WY ′ be the smooth dyadic cutos from the xed CKP tag,
with
(r)

(r)

ωA ≪r A−r
g ,

ωQ ≪r Q−r
g ,

(r)

(r)

WY ′ ≪r (Y ′ )−r .

WY ≪r Y −r ,

The smooth bre is

Φa,q (y) = ωA (a)ωQ (q)WY (y)WY ′ (z(a, q, y)).

(Phi)

For h ̸= 0, the actual two-variable weight entering the Kloosterman-fraction
sum is


Z
1
hy
Wg,h (a, q) =
Φa,q (y)e −
dy.
(W)
q R
q
Thus the nonzero-frequency layer is

Og,h =



X
a∼Ag , q∼Qg
(a,q)=1

hNg a
αg (a)γg,h (q)Wg,h (a, q)e
q


.

(CKP-X10)

The amplitude used to normalize the DFI weight is

Ag,h,R = (log N )C∗ g(1 + |h|g)−R ,

(Amp)

and the normalized DFI weight is

fg,h (a, q) = A−1 Wg,h (a, q).
W
g,h,R
4

(Wtilde)

The factor Ag,h,R is kept outside the DFI weight and is included in the nal
g, h-summation below.
The derivative check uses the exact dependence z(a, q, y) = (Ng − ay)/q ,
not a separated surrogate. On the central support,

y
∂a z = − ,
q

z
∂q z = − .
q

Since y ≍ Y , z ≍ Y ′ , Ag ≍ Qg , and Y ≍ Y ′ , each a-derivative of WY ′ (z)
contributes
Y
y
≪ A−1
WY′ ′ (z) ≪ (Y ′ )−1
g ,
q
Qg
and each q -derivative contributes O(Q−1
g ). Repeated and mixed derivatives
follow by Faa di Bruno, giving for xed i, j
−j
∂ai ∂qj Φa,q (y) ≪ A−i
g Qg 1y≍Y .

(Phi-der)

Dierentiating the Fourier integral in (W), derivatives falling on Φa,q give the
−j
−1 give powers of Q−1 , and
expected A−i
g
g Qg factors, derivatives falling on q
derivatives falling onR the phase cost at most powers of 1+|h|g . If |h|g ≤ 1, the
trivial estimate q −1 |Φa,q (y)|dy ≪ Y /Qg ≍ g , and its dierentiated version,
gives the desired bound because 1 + |h|g ≍ 1. If |h|g > 1, integration by
parts in y gives, for any xed B, i, j ,
−j
−B+i+j
∂ai ∂qj Wg,h (a, q) ≪ A−i
.
g Qg g(1 + |h|g)

(W-der)

In the central DFI range |h|g ≤ (log N )BHF , choose R in (Amp) larger than
the derivative order and the summation losses. Then

fg,h (a, q) ≪ 1,
W

−j
fg,h (a, q) ≪ (log N )C A−i
∂ai ∂qj W
g Qg

This is the smooth-weight hypothesis required by X10.

(0 ≤ i, j ≤ 2).
(DFI-wt)

G4a/X10: DFI estimate

G4a invokes the X10 external input, a DFI bilinear Kloosterman-fraction
estimate in the dyadic form


X
rm
αm βq F (m, q)e
q
m≍M, q≍Q
(m,q)=1

5

with the substitution

m = a,

r = |h|Ng ,

M = Ag ,

Q = Qg .

For h < 0, this is the same application to the conjugate phase. Boundary,
high-frequency, small-conductor, and large-g ranges are excluded from the
X10 application and routed by the existing X10ER, C1A/C1, G2a, and G8a
interfaces.
B1LD and G8a: zero frequency and recombination

The h = 0 term is not estimated by X10. B1LD identies the niteconvolution local densities inherited from B1 with the local model used by
H4. G8a then combines:
1. zero-frequency local projection;
2. central nonzero-frequency cancellation by G3a/G4a/X10/CKPD;
3. boundary, high-frequency, small-conductor, and large-g routing through
X10ER and C1A/C1;
to prove

RCKP (N ) = MCKP (N ) + o(N ).
Central DFI loss accounting

In the central balanced range write

Ag ≍ Qg ≍ Sg ,

Sg =

N 1/2+O(η0 )
.
g

The coecient norms are
C
∥αg ∥2 ≪ A1/2
g (log N ) ,

C
∥γg,h ∥2 ≪ Q1/2
g (log N ) .

The DFI estimate is applied with

m = a,
Because

n = q,

r = |h|Ng ,

M = Ag ,

|h|Ng
≍ |h|g ≤ (log N )BHF ,
Ag Qg
6

Q = Qg .

the DFI factor (r + M Q)3/8 contributes only a polylogarithmic loss beyond
(Ag Qg )3/8 . Including the amplitude factor (Amp), one obtains

|Og,h | ≪ (log N )C g(1 + |h|g)−A ∥αg ∥2 ∥γg,h ∥2 (Sg2 )3/8 (2Sg )11/48+ε .
Thus

|Og,h | ≪ (log N )C g(1 + |h|g)−A Sg1+3/4+11/48+ε ,

and since 1 + 3/4 + 11/48 = 95/48,

|Og,h | ≪ N 95/96+O(η0 )+ε (log N )C g −47/48 (1 + |h|g)−A .

(Layer)

The h-sum is absolutely convergent for xed large A:
X
(1 + |h|g)−A ≪ 1.
h̸=0

Also g | N by the G1a splitting, hence the number of g -layers is divisorbounded:
#{g : g | N } ≪ε N ε .
Therefore

XX

|Og,h | ≪ N 95/96+O(η0 )+o(1) = o(N ),

g|N h̸=0

after choosing η0 and the DFI ε so that

O(η0 ) + ε + o(1) <

1
.
96

Excluded-range decomposition

The DFI theorem is invoked only on the central nonzero-frequency range.
The complete CKP nonzero-frequency decomposition is

CKPh̸=0 = CentralDFI⊔HighFreq⊔SmallConductor⊔LargeG⊔Boundary/Short.
The routing is:
 CentralDFI: central balance, |h|g ≤ (log N )BHF , and non-small conductor; routed to X10 after CKPD.
 HighFreq: |h|g > (log N )BHF ; routed by G2a/X10ER to C1A/C1.
 SmallConductor: q/(q, hNg ) ≤ (log N )B ; routed to C1A/C1.
7

 LargeG: g > N η0 or outside CKP balance; routed by G1a/G8a/X10ER
to C1A/C1.
 Boundary/Short: boundary support or short volume; routed to C1A/C1
or H4 as appropriate.
 Zero frequency: h = 0; routed to the B1LD/H4 local term, not X10.
Thus this packet directly veries the central DFI matching. The full CKP
conclusion additionally uses the listed X10ER/C1A/C1 routing of excluded
ranges and the B1LD/H4 treatment of the zero frequency.
Precise point to verify

The central technical assertion is that

Wg,h (a, q),
as produced by G3a, satises the smoothness, support, derivative, modulus,
coprimality, and parameter restrictions required by the X10 Kloostermanfraction theorem statement.
Please especially check:
1. whether the variables a, q in G3a match the variables in X10;
2. whether all arithmetic moduli and coprimality restrictions are compatible;
3. whether CKPD dierentiates the actual weight entering X10, not a
simplied surrogate;
4. whether summation over g, h and dyadic parameters preserves the
claimed o(N ) saving;
5. whether the zero-frequency term is correctly excluded from the X10
estimate and routed to H4.
Minimal source map

The detailed proof-source les are:
1. ../../../manuscript_md/appendices/appendix_c_ckp_x10.md
8

2. ../../../Lemmas/ckp_x10_smooth_weight_derivative_appendix_ltx.md
3. ../../../External/x_10_verification_ltx.md
4. ../../../Lemmas/g_8_a_ltx.md
The full technical chain is:

G1a,

G2a,

G3a,

CKP D,

G4a,

G8a,

B1LD,

X10.

Checklist
X10 hypotheses

 The theorem statement used as X10 is stated precisely enough for this
packet.
 The CKP variables a, q match the X10 variables.
 The summation ranges satisfy the X10 size restrictions.
 The moduli and coprimality assumptions match X10.
 The nonzero-frequency parameter h is in the required range.
Smooth weights

 CKPD dierentiates the actual weight Wg,h (a, q).
 The support of Wg,h is compatible with X10.
 All rst and higher derivative bounds required by X10 are proved.
 Boundary cutos do not create uncontrolled derivative losses.
 Summing over dyadic partitions preserves the claimed saving.
Local term

 The zero-frequency term is not incorrectly estimated by X10.
 The zero-frequency term is correctly identied as a canonical local
term.
 B1LD/H4 compatibility is sucient for the CKP local contribution.
9

Expected outcome

A positive review would say:
The CKP nonzero-frequency reduction ts the stated X10 hypotheses after CKPD, and no missing derivative/modulus/range
condition is visible.
A negative review should identify the rst failed hypothesis or range.

10

