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CJM Problem Sketch No. 003

The Goldbach Conjecture as a Structural Admissibility Landscape

Conceptual approach only. No proof or solution is claimed.

Keunsoo Yoon
Independent Research Group (Seoul, Republic of Korea)
austiny@gatech.edu, austiny@snu.ac.kr

Aug 16, 2026

Open Question.
Does a problem ฮฆ admit a solution if and only if it is structurally admissible under atemporal resonance?

โ… . The Problem

Goldbach's conjecture states that every even integer n โ‰ฅ 4 can be written as the sum of two primes:

n = p + q,   where p and q are prime.

The conjecture has been computationally verified over enormous finite ranges, yet no proof establishes it for all even integers. The deeper question is therefore not only whether one particular n has a decomposition, but how the full family of admissible prime-pair decompositions changes as n itself varies.

โ…ก. Structural Reinterpretation

For a finite horizon N, consider the ordered family of even integers

4, 6, 8, ..., N.

Each even integer n defines a layer Ln. Within that layer, candidate states are pairs (p, n โˆ’ p) with p โ‰ค n โˆ’ p. A state is admissible only when both components are prime.

Let

Mn = { (p, q) : p + q = n, and p, q are prime }

denote the full admissible model space of layer Ln. Goldbach's conjecture is equivalent to the statement that Mn is nonempty for every even n โ‰ฅ 4. CJM, however, asks not only whether Mn is empty, but how its internal structure changes across neighboring layers.

โ…ข. SAT โ†’ 3SAT Encoding

For each candidate value p โ‰ค n/2, let Xn,p represent selection of the decomposition (p, n โˆ’ p). The finite encoding is summarized by five principal constraint families:

  1. Xn,p is admissible only when both p and n โˆ’ p are prime.
  2. every selected decomposition must satisfy p + (n โˆ’ p) = n within its corresponding layer Ln.
  3. Cn is true if and only if at least one admissible decomposition exists in layer Ln.
  4. admissible selections within each layer must remain consistent with the finite candidate set defined for that even value n.
  5. for the finite horizon N, global coverage requires Cn to hold for every even n with 4 โ‰ค n โ‰ค N.

Each constraint family expands across the relevant candidate values, decompositions, and layers of the finite instance. The resulting Boolean structure is expressed as SAT and normalized through the common CJM 3SAT gate.

Goldbach Structure โ†’ SAT โ†’ 3SAT โ†’ CJM

โ…ฃ. The CJM Question

A conventional SAT interpretation reduces each layer to a binary question:

Mn โ‰  โˆ…   or   Mn = โˆ….

This discards most of the structure. Two even integers may both be SAT while possessing very different admissible model spaces.

CJM therefore introduces a structural observable, written abstractly as a Changbal Function:

CG(n) = F(Mn),

where F may measure properties such as the number of admissible prime pairs, their spread around n/2, their concentration, distributional entropy, or other structural features preserved across the layered representation.

Thus the primary object is no longer a sequence of SAT/UNSAT bits, but a changing admissibility landscape:

n โ†’ Mn โ†’ CG(n).

The classical question "Does every even integer up to N admit a prime-pair decomposition?" is therefore reformulated by CJM as:

As n varies across {4, 6, ..., N}, how does the admissible model space Mn โ€” not merely its emptiness โ€” evolve, and where does that evolution become nonlinear?

โ…ค. Expected Changbal Region

Heuristically, |Mn| grows roughly with n / (log n)ยฒ on average, though this trend fluctuates locally and is not itself the object of interest here. A single even integer with unusually many or unusually few decompositions is not, by itself, a Changbal Jump โ€” such fluctuations are expected from ordinary variation in the distribution of primes.

A stronger candidate is a Changbal Region: a range of neighboring even integers in which CG(n) departs sharply and persistently from its surrounding behavior, indicating a nonlinear reorganization of the admissible model space.

Importantly, such a region need not coincide with a SAT โ†’ UNSAT transition. Every layer in the region may remain satisfiable while the internal structure of Mn changes abruptly. This distinction separates the CJM question from a simple sequence of satisfiability tests. Within such a region, a sufficiently sharp structural transition may be identified as a Changbal Jump.

Problem Family โ†’ 3SAT โ†’ Model Space โ†’ Changbal Function โ†’ Changbal Region โ†’ Changbal Jump

โ…ฅ. Limitation

This sketch does not prove Goldbach's conjecture, does not establish that every Mn is nonempty, and does not claim that a Changbal Region has already been observed. Its purpose is to define a finite experimental framework in which Goldbach's conjecture can be examined not merely as a repeated existence test, but as a changing structural landscape viewed through the common CJM representation, in the same spirit as the Recamรกn and Collatz sketches.