﻿                  CKP/X10/X16 Analytic Theorem Package

                                            Denis Saltykov

                                               May 2026


                                               Abstract
         This package isolates two analytic components of the proof. The first is the CKP/X10 com-
     ponent: after the CKP gcd split, the nonzero Fourier fibres match a DukeвЂ“FriedlanderвЂ“Iwaniec
     bilinear Kloosterman-fraction estimate, with the actual two-variable CKP smooth weight and
     its derivatives verified directly. The second is the X16/BRS component: every admissible BRS
     carrier-slice estimate is reduced to the product-carrier X16C estimate, which follows from Shiu-
     type divisor averages in arithmetic progressions and a local factor averaging lemma.


Contents
1 Scope                                                                                                 1

2 The CKP Gcd Split                                                                                     2

3 The Actual CKP Smooth Weight                                                                          2

4 Smooth Derivative Verification                                                                        3

5 DFI/X10 Input and Central Bound                                                                       3

6 CKP Excluded Ranges                                                                                   4

7 X16/BRS Carrier-Slice Theorem                                                                         4

8 Product-Carrier X16C                                                                                  5

9 Proof of X16C                                                                                         5

10 Analytic Output Theorem                                                                              6


1    Scope
This note proves only the analytic estimates needed by the CKP and BRS parts of the proof. It
does not prove the finite routing grammar, the H4 local algebra, or the final prime-power handoff.
Those are separate theorem packages.
   The output is

        CKP/X10 nonzero frequencies = o(N ),             X16BRS carrier-slice estimate holds.


                                                    1
The CKP zero-frequency contribution is not estimated by X10; it is routed to the H4 tagged local
algebra.
    The logical dependencies are G8a, X10, CKPD, X10ER, C1P/C1A/C1, X16BRS, X16C, and H4.
File paths are recorded only in the accompanying source map.


2      The CKP Gcd Split
Let a CKP block be fixed. The CKP variables contain a pair u, uвЂІ with

                                             uy + uвЂІ y вЂІ = N

after the surrounding B1/F3 routing data and smooth cutoffs have been fixed. Split

                             u = ga,      uвЂІ = gq,          (a, q) = 1,        g | N,

and write Ng = N/g. Then
                                             ay + qy вЂІ = Ng .                                (2.1)
For h = 0, the Fourier expansion on the fibre gives the local projection term and is routed to H4.
For h Мё= 0, the AP congruence
                                       y в‰Ў Ng a (mod q)
produces the phase
                                                       hNg a
                                                            
                                               e             .                               (2.2)
                                                        q
Thus the DFI variables are
                                   m = a,       q = q,           r = |h|Ng .                 (2.3)
For h < 0, the same external estimate is applied to the conjugate phase, so the positive external
integer parameter is r = |h|Ng .


3      The Actual CKP Smooth Weight
In the central CKP range let

                             a в‰Ќ Ag ,      q в‰Ќ Qg ,          y в‰Ќ Y,       yвЂІ в‰Ќ Y вЂІ,

with
                                 Ag в‰Ќ Qg ,      Y в‰Ќ Y вЂІ,           Y /Qg в‰Ќ g.                (3.1)
Put
                                                            Ng в€’ ay
                                          z(a, q, y) =              .                        (3.2)
                                                               q
The actual fibre weight is

                              О¦a,q (y) = П‰A (a)П‰Q (q)WY (y)WY вЂІ (z(a, q, y)),                (3.3)

where the П‰вЂ™s and W вЂ™s are the smooth dyadic cutoffs inherited from the CKP block. The two-
variable CKP weight inserted into the nonzero-frequency bilinear form is
                                    1b             1
                                                             Z
                       Wg,h (a, q) = О¦ a,q (h/q) =               О¦a,q (y)e(в€’hy/q) dy.        (3.4)
                                    q              q

                                                        2
The nonzero-frequency CKP sum is therefore
                                                                                   hNg a
                                       X                                              
                           Og,h =                    О±g (a)Оіg (q) Wg,h (a, q)e           .            (3.5)
                                    aв€јAg , qв€јQg
                                                                                    q
                                      (a,q)=1

    Define the amplitude
                                     Ag,h,R = (log N )Cв€— g(1 + |h|g)в€’R ,                              (3.6)
and the normalized weight
                                       fg,h (a, q) = Aв€’1 Wg,h (a, q).
                                       W                                                              (3.7)
                                                      g,h,R


4      Smooth Derivative Verification
Lemma 1 (CKP smooth weight derivative check). For i + j в‰¤ 2, the normalized CKP weight
satisfies
                                fg,h (a, q) в‰Є (log N )C Aв€’i Qв€’j .
                        в€‚ai в€‚qj W                                                (4.1)
                                                         g   g

Proof. The central support identities imply

                                       в€‚a z = в€’y/q,             в€‚q z = в€’z/q.                          (4.2)

Because y/q в‰Ќ Y /Qg в‰Ќ g and the product scale satisfies Ag в‰Ќ Qg , differentiation of WY вЂІ (z(a, q, y))
gives the same scale loss as differentiation in the external DFI variables:

                            в€‚ai в€‚qj WY вЂІ (z(a, q, y)) в‰Є Aв€’i в€’j
                                                         g Qg             (i + j в‰¤ 2),                (4.3)

after the smooth cutoffs restrict to the central cell. Differentiating the oscillatory factor e(в€’hy/q),
integrating by parts in y, and using Y /Qg в‰Ќ g gives, for every fixed B,

                       в€‚ai в€‚qj Wg,h (a, q) в‰Є Aв€’i в€’j
                                              g Qg g(1 + |h|g)
                                                               в€’B+i+j
                                                                      (log N )C .                     (4.4)

Choosing R larger than the fixed derivative order in the amplitude Ag,h,R gives (4.1).

   Thus the actual nonseparated CKP weight is admissible for the smooth DFI/X10 input. No
separated surrogate weight is used.


5      DFI/X10 Input and Central Bound
The external X10 input is the smooth DukeвЂ“FriedlanderвЂ“Iwaniec bilinear Kloosterman-fraction
estimate. In the form needed here, if F is supported on m в‰Ќ M , q в‰Ќ Q, satisfies

                                 в€‚q F (m, q) в‰Є О· M в€’i Qв€’j
                               i j
                              в€‚m                                        (i + j в‰¤ 2),

then
                                            rm
               X                                
                      О±m ОІq F (m, q)e                в‰ЄОµ О· 2 в€ҐО±в€Ґ2 в€ҐОІв€Ґ2 (r + M Q)3/8 (M + Q)11/48+Оµ .   (5.1)
           mв€јM, qв€јQ
                                             q
            (m,q)=1

Here the substitution is
                                 M = Ag ,              Q = Qg ,       r = |h|Ng .                     (5.2)


                                                            3
The central CKP range has

                  Ag в‰Ќ Qg в‰Ќ Sg ,       Sg = N 1/2+O(О·0 ) /g,    |h|g в‰¤ (log N )BHF .             (5.3)

Combining the derivative check with (5.1) and then restoring the amplitude Ag,h,R gives the layer
bound
                     |Og,h | в‰Є N 95/96+O(О·0 )+Оµ (log N )C g в€’47/48 (1 + |h|g)в€’A .           (5.4)
The coefficient L2 -norms are those of the CKP dyadic divisor-bounded coefficients and contribute
only fixed polylogarithmic losses. Summing (5.4) over dyadic data, divisors g | N , and h Мё= 0, and
choosing A fixed and large, gives      XX
                                             |Og,h | = o(N ),                                  (5.5)
                                       g|N hМё=0

provided the fixed hierarchy has O(О·0 ) + Оµ < 1/96.


6    CKP Excluded Ranges
Before X10 is applied, the CKP partition removes the following noncentral ranges.

    Range                   Condition                             Routing
    Zero frequency          h=0                                   H4 local algebra
    High frequency          |h|g > (log N )BHF                    X10ER to C1P/C1A/C1
    Small conductor         q/(q, hNg ) в‰¤ (log N )B               X10ER to C1P/C1A/C1 or local
    Large g                 g outside the central CKP             X10ER to Edge
                            window
    Boundary/short          dyadic fibre too short or cutoff      C1P/C1A/C1
    volume                  boundary

    Therefore the DFI/X10 estimate is invoked only on the central balanced range where the deriva-
tive verification above applies.
Proposition 1 (CKP analytic output). The CKP branch satisfies

                                RCKP (N ) = MCKP,local (N ) + o(N ),                             (6.1)

where MCKP,local (N ) is one of the explicitly LPI-admitted tagged local projection terms later assem-
bled by H4.
Proof. The h = 0 term is H4-local. The central h Мё= 0 terms satisfy (5.5). All noncentral terms are
routed by X10ER and C1P/C1A/C1 before the DFI estimate is invoked.


7    X16/BRS Carrier-Slice Theorem
Let C be an admissible BRS carrier of height XC , and let I be a carrier interval. Define

                               Y16 = max{|I в€© Z|, XC (log N )в€’B16 }.                             (7.1)

The X16BRS estimate is
                                                      Y16
                      MassB (C в€€ I) в‰Є N (log N )C16       + N 1в€’ПЃ16 (log N )C16 .                (7.2)
                                                      XC

                                                  4
The admissible carriers are grouped product carriers, complementary carriers, tagged quotient
carriers, and controlled divisor quotients. X16BRS reduces the complementary, quotient, and
controlled quotient carriers to the product carrier model by the existing tagged F3/F4 routing
data. Untagged variable quotients are not X16BRS inputs; they are routed away before BRS.
    This package directly proves the product-carrier X16C estimate. The full BRS carrier-slice
estimate additionally uses X16BRS to reduce the other admissible carrier types to that product-
carrier model.


8    Product-Carrier X16C
For a product carrier P with p в€€ I16 and companion variable u в‰Ќ U , the active mass is bounded
by a divisor correlation of the form
                            X                X
                                   П„K1 (p)         П„K2 (u)П„K3 (N в€’ pu)1N в€’pu>0 .              (8.1)
                           pв€€I16             uв‰ЌU

X16C proves
                          (8.1) в‰Є Y16 U (log N )O(1) + N 1в€’ПЃ16 (log N )O(1) .                 (8.2)
    The main external input is ShiuвЂ™s BrunвЂ“Titchmarsh theorem for multiplicative functions in
arithmetic progressions. In the divisor-function form used here, for f (n) = П„K (n)A , interval J вЉ‚
[1, N ] of length H, and modulus q в‰¤ H 1в€’Оґ ,

                                                           H
                                X                               
                                             f (n) в‰Є         + 1 (log N )CSH Eq,b .           (8.3)
                              nв€€J
                                                           q
                           nв‰Ўb (mod q)

Lemma 2 (Local factor averaging). The only place where non-coprime AP classes enter is the
estimate                   X             1/2     #
                               П„K0 (c)A Ec,N в‰Є |I16 |(log N )Cloc .                  (8.4)
                                   #
                                cв€€I16

Proof. The local factors are fixed-divisor factors attached to the AP classes arising from N в€’ pu.
Averaging them over the carrier interval and applying CauchyвЂ“ Schwarz reduces the estimate to
the standard second moment for fixed divisor functions,
                                                                        2
                                          П„K (u)2 в‰ЄK U (log 2U )K в€’1 ,
                                    X
                                                                                              (8.5)
                                    uв‰ЌU

as in Tenenbaum, Ch. II.5, Theorem 5.


9    Proof of X16C
Proof. There are two cases.
   First suppose XP в‰¤ N 1в€’Оґ . Fix p. The variable N в€’ pu runs through an arithmetic progression
modulo p. Shiu/AP applies to the opposite variable after CauchyвЂ“Schwarz and the divisor second
moment. Averaging the local factors over p в€€ I16 by the local factor averaging lemma gives

                                              в‰Є Y16 U (log N )C .



                                                           5
     Second suppose XP > N 1в€’Оґ . If Y16 U в‰¤ N 1в€’ПЃ16 , the trivial estimate gives
                                                    вЂІ
                               в‰Є N 1в€’ПЃ16 +Оµ в‰¤ N 1в€’ПЃ16 ,    ПЃвЂІ16 = ПЃ16 /2,

and then ПЃвЂІ16 is renamed ПЃ16 . Otherwise fix u. The variable p lies in an AP class modulo u, and
                                                     1в€’Оґ/2
the largeness of Y16 U gives the Shiu/AP range u в‰¤ Hu      . Applying Shiu/AP and then summing
over u using the divisor second moment gives the same bound (8.2).

   Consequently X16BRS holds for all admissible BRS carriers after the structural reductions
described above.


10      Analytic Output Theorem
Theorem 1 (CKP/X10/X16 analytic output). In the active proof system:

  1. every CKP zero-frequency contribution is an LPI-admitted local projection later assembled by
     H4;

  2. every central CKP nonzero-frequency contribution is o(N ) by X10/DFI applied to the actual
     CKP smooth weight;

  3. every noncentral CKP range is routed by X10ER/C1P/C1A/C1 before X10 is invoked;

  4. every admissible BRS carrier-slice test satisfies the X16BRS estimate;

  5. the X16BRS estimate is ultimately supplied by the X16C Shiu/AP product-carrier theorem.

Therefore the CKP branch contributes only its tagged local main term plus o(N ), and the BRS
carrier-slice input needed in the TC1-testing chain is available with the stated polylogarithmic losses
and fixed power saving.


References
  [1] W. Duke, J. Friedlander and H. Iwaniec, Bilinear forms with Kloosterman fractions, Invent.
      Math. 128 (1997), 23вЂ“43.

  [2] P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math.
      313 (1980), 161вЂ“170.

  [3] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, AMS Graduate
      Studies in Mathematics 163, 3rd ed., 2015.




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