    Final Assembly and Goldbach Handoff Packet

                             Denis Saltykov

                                  May 2026



Review request

Please check the global assembly and final handoff only. This is not a re-
quest to review the technical branch estimates themselves. The CKP/X10,
X16/Shiu/BRS, and GoodAWACK/E10 inputs are being reviewed in sepa-
rate targeted packets.
The exact question is:

     Assuming the stated terminal branch estimates, does the active
     proof tree correctly assemble them into

                         𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ),

     and does the prime-power handoff then give a genuine prime
     representation for all sufficiently large even 𝑁 ?

A positive review should confirm the final implication

                         𝐼1 + 𝐺2 =⇒ 𝐺1 =⇒ 𝐺0.


Conventions

All Goldbach sums in this packet use ordered positive pairs:

                     𝑅Λ (𝑁 ) :=                Λ(𝑛1 )Λ(𝑛2 ).
                                     ∑︁

                                  𝑛1 +𝑛2 =𝑁
                                   𝑛1 ,𝑛2 ≥1


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The genuine prime-prime weighted sum is

                      𝑅𝑝𝑝 (𝑁 ) :=                    log 𝑝1 log 𝑝2 ,
                                          ∑︁

                                       𝑝1 +𝑝2 =𝑁
                                      𝑝1 ,𝑝2 prime

again over ordered pairs. This is the convention used by B1, I1, G2, G1,
and G0H.
For even 𝑁 , the Goldbach singular series is normalized as
                       ∏︁ 𝑝 − 1                      ∏︁ (︂           1
                                                                             )︂
         S(𝑁 ) = 2𝐶2              ,        𝐶2 =              1−                   > 0.
                       𝑝|𝑁
                           𝑝−2                       𝑝>2
                                                                  (𝑝 − 1)2
                       𝑝>2

Hence S(𝑁 ) ≥ 2𝐶2 > 0 for every even 𝑁 .


Assumed branch inputs

The final assembly packet assumes the following branch inputs.

  1. B1 exact decomposition:

                             𝑅Λ (𝑁 ) =                𝑐ℬ 𝑅ℬ (𝑁 ).
                                                ∑︁

                                            ℬ∈B𝐽0


  2. B3/F3/F4 exact tagged terminal partition:

                             𝑅ℬ (𝑁 ) =                 𝑅ℬ,𝜏 (𝑁 ).
                                                ∑︁

                                            𝜏 ∈𝒯 (ℬ)

     The terminal tags are disjoint at the routing-history level.

  3. Edge estimate:
                                  𝑅Edge (𝑁 ) = 𝑜(𝑁 ).

  4. LongAP/Local projection:

                       𝑅LongAP (𝑁 ) = 𝑀LongAP (𝑁 ) + 𝑜(𝑁 ).

  5. CKP projection and error:

                         𝑅CKP (𝑁 ) = 𝑀CKP (𝑁 ) + 𝑜(𝑁 ).

                                            2
  6. GoodAWACK estimate:
                            𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

  7. Local/Main compatibility:
                          𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

This packet checks that these inputs are assembled correctly; it does not
reprove them.


Terminal assembly in I1

The exact terminal partition used by I1 is
              𝑅Λ (𝑁 ) = 𝑅Edge (𝑁 ) + 𝑅LongAP (𝑁 ) + 𝑅CKP (𝑁 )
                        + 𝑅GoodAWACK (𝑁 ) + 𝑅LocalDiag (𝑁 ).
The local/main terms are gathered as
                  𝑀local (𝑁 ) = 𝑀LongAP (𝑁 ) + 𝑀CKP (𝑁 )
                               + 𝑀LocalDiag (𝑁 ).

There is no fourth local summand. Auxiliary local-looking terms created
by controlled CRT absorption, fixed-divisor quotienting, or primitive local
slicing are tagged refinements of one of the three displayed classes, while
endpoint and smooth-boundary localizations are C1 Edge errors.
The important normalization rule is H4-admission:
                   local
                  𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜ℬ,𝜏 (𝑁 ).
Thus H4 does not accept arbitrary local-looking main terms. It accepts
only canonical local projections attached to a parent B1 block and a unique
routing tag.
In the proof-source layer this is formulated as a single-local-model normal-
ization lemma. The same operation
                           Λ(𝑛) ↦→ Λ𝑄 (𝑛 mod 𝑄)
is applied inside the original tagged Goldbach convolution for every admitted
local/main source. The branch checks are:

                                     3
  • LongAP/Local: F3P gives the positive local-coefficient predicate, and
    D1.2A expands the resulting AP/local algebra as Loc𝑄 𝑅ℬ,𝜏 .
  • CKP ℎ = 0: G8a.5 and B1LD identify the zero Fourier mode with the
    same Loc𝑄 𝑅ℬ,𝜏 ; ℎ ̸= 0 is CKP/X10, not local.
  • LocalDiag: the diagonal cell is admitted only when it is a canonical
    tagged projection; noncanonical degeneracies are routed elsewhere.
  • Controlled CRT, quotient, and slicing refinements: the parent B1 tag
    and the Λ𝑄 -replacement rule must be preserved inside one of the three
    displayed local classes.
  • Endpoint or smooth-boundary localizations: these are routed to C1 as
    𝑜(𝑁 ), not assembled as local main terms.

Using the branch estimates and the H4 identity,
             𝑅Λ (𝑁 ) = 𝑀local (𝑁 ) + 𝑜(𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).
This is Theorem I1.
The global summability of the displayed 𝑜(𝑁 ) terms is recorded in GEB:
strict Edge savings, CKP derivative losses, TC1 Davenport/AP losses,
X16/BRS carrier-slice losses, and local boundary terms are checked after
the polylogarithmic terminal summations. GEB handles terminal branch
summability; G2 separately handles the prime-power error.


No double counting in H4

The no-double-counting mechanism is tag-based:

  1. B1 gives an exact sum over parent blocks ℬ.
  2. F3/F4 give an exact partition of each parent block into routing tags
     𝜏.
  3. H4 sums local terms indexed by (ℬ, 𝜏 ), not by their visual algebraic
     shape.
  4. Two local-looking expressions that coincide algebraically but come
     from distinct tags remain distinct complementary summands of the
     exact partition.

                                    4
Therefore LocalDiag, LongAP/Local, and CKP zero-frequency contribu-
tions are not double counted when they are passed through H4.


Prime-power removal in G2

The von Mangoldt function is supported on primes and prime powers:

                                    log 𝑝,       𝑛 = 𝑝𝑘 , 𝑘 ≥ 1,
                               {︃
                      Λ(𝑛) =
                                    0,           otherwise.

The difference 𝑅Λ (𝑁 ) − 𝑅𝑝𝑝 (𝑁 ) is the nonnegative contribution of ordered
pairs in which at least one coordinate is a nontrivial prime power 𝑝𝑘 , 𝑘 ≥ 2.
The number of such prime powers up to 𝑁 is 𝑂(𝑁 1/2 ), and for each selected
coordinate the other coordinate is uniquely determined. Since Λ(𝑛) ≤ log 𝑁 ,
the total contribution is

               𝑅Λ (𝑁 ) − 𝑅𝑝𝑝 (𝑁 ) = 𝑂(𝑁 1/2 (log 𝑁 )2 ) = 𝑜(𝑁 ).

Combining this with I1 gives

                         𝑅𝑝𝑝 (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).


Final positivity in G1/G0H

Since S(𝑁 ) ≥ 2𝐶2 > 0 for even 𝑁 , the 𝑜(𝑁 ) error is eventually smaller than
𝐶2 𝑁 . Hence for all sufficiently large even 𝑁 ,

                            𝑅𝑝𝑝 (𝑁 ) ≥ 𝐶2 𝑁 > 0.

Every summand in 𝑅𝑝𝑝 (𝑁 ) is nonnegative and is strictly positive exactly
for an ordered prime representation 𝑁 = 𝑝1 + 𝑝2 . Therefore 𝑅𝑝𝑝 (𝑁 ) > 0
implies the existence of at least one genuine prime pair.
The result is the sufficiently-large binary Goldbach statement. No finite
verification for small even 𝑁 is included in this packet.




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What a positive check would confirm

     Assuming the stated branch estimates and external inputs, I1
     assembles the terminal tagged partition into 𝑅Λ (𝑁 ) = S(𝑁 )𝑁 +
     𝑜(𝑁 ), G2 removes the nontrivial prime-power support at 𝑜(𝑁 )
     cost, and G1/G0H correctly convert positivity of 𝑅𝑝𝑝 (𝑁 ) into a
     genuine prime representation for all sufficiently large even 𝑁 .


Failure modes to look for

A negative check should identify the first concrete failure, for example:

  1. B1 is not exact or introduces an unrecorded truncation error;

  2. F3/F4 terminal routing is not an exact disjoint tagged partition;

  3. a terminal class is omitted from I1;

  4. a branch error is summed over too many descendants and no longer
     remains 𝑜(𝑁 );

  5. H4 double-counts or loses a local/main term;

  6. the singular-series normalization does not match the ordered 𝑅𝑝𝑝 con-
     vention;

  7. the prime-power removal misses a non-prime Λ-support case;

  8. positivity of 𝑅𝑝𝑝 is used before nontrivial prime powers are removed.




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