---

Ⅰ. The Problem

The diagonal Ramsey number R(k, k) is the smallest integer N such that every red-blue coloring of the edges of the complete graph K_N necessarily contains a monochromatic complete subgraph K_k.

For example,

Already the next diagonal case remains unresolved:

More fundamentally, the growth of R(k,k) as k → ∞ remains unknown. Classical probabilistic arguments give an exponential lower bound, while recent work has improved the long-standing exponential upper bound to a form

For CJM, however, the central question is not merely the computation of another finite Ramsey number. The deeper target is the unbounded family of thresholds

and the global structure traced by these thresholds as k increases without bound.

Ⅱ. Structural Reinterpretation

For fixed integers k and N, define

when there exists a red-blue coloring of K_N containing no monochromatic K_k, and

when no such coloring exists.

Thus each k generates a structural transition from an admissible avoidance region to an unavoidable Ramsey region.

The Ramsey number itself is precisely the boundary:

As k ranges without bound, these individual transitions form an unbounded two-dimensional threshold structure over (k,N).

The infinite problem is therefore not whether each individual threshold exists— Ramsey's theorem already guarantees that—but whether the global geometry and growth law of the entire threshold boundary can be structurally distinguished without enumerating every value sequentially.

Ⅲ. SAT → 3SAT Encoding

For fixed (k,N), assign one Boolean variable

to every edge {i,j} of K_N, where 1 represents red and 0 represents blue.

For every set S of k vertices, let E(S) denote the edges of the corresponding K_k. Two principal constraints are imposed:

1. At least one edge of E(S) must be blue,
   ∨_e∈E(S) ¬x_e,
   preventing an all-red K_k.
2. At least one edge of E(S) must be red,
   ∨_e∈E(S) x_e,
   preventing an all-blue K_k.

Applying these constraints to every k-vertex subset produces a finite Boolean formula Φ_k,N.

Clauses longer than three literals may be converted through auxiliary variables into the common CJM 3SAT representation.

The infinite family is not compressed into one ordinary finite 3SAT formula. Rather, a finite rule generates Φ_k,N for arbitrary (k,N), producing an unbounded SAT/UNSAT landscape.

Ⅳ. The CJM Question

For any fixed (k,N), physical CJM may first be posed as the discrimination

The transition point between these two states determines R(k,k). But determining one Ramsey number is only the local problem.

The intended CJM question concerns the unbounded threshold family itself. For a proposed asymptotic bound B(k), consider the global statement

Its negation states that the proposed boundary is violated arbitrarily far into the sequence.

Under the CJM hypothesis, P ≡ NP^J, the question is whether such an unbounded structural statement may be discriminated through the finite rule generating the Ramsey landscape rather than by sequentially solving k = 3, 4, 5, 6, ... one case at a time.

Can the finite 3SAT rule generating the Ramsey threshold landscape permit physical CJM to discriminate the global growth structure of R(k,k) over an unbounded domain?

Ⅴ. Expected Changbal Region

Ramsey numbers already contain a natural structural transition. For fixed k, increasing N produces

exactly when N reaches R(k,k).

Thus the Ramsey problem may generate not merely one Changbal point, but an entire sequence of Changbal transitions:

As k increases, these points trace an unbounded Ramsey Threshold Boundary.

A useful representation is the (k, log N) plane. Since diagonal Ramsey numbers grow exponentially, an asymptotic relation of the form

would appear approximately as a linear boundary whose normalized position satisfies

The known lower and upper bounds still leave a substantial asymptotic region unresolved; even the existence of a single limiting exponential base remains unknown.

A candidate Changbal Region would therefore be sought as a reproducible nonlinear physical response associated not only with individual SAT/UNSAT transitions, but with the structural organization of this unbounded boundary.

Ⅵ. Limitation

This sketch does not determine any unknown Ramsey number and does not claim that the infinite Ramsey family has been converted into one finite 3SAT instance.

For each finite (k,N), the corresponding avoidance problem admits a finite SAT representation and therefore a finite 3SAT normalization. The collection of all such instances is nevertheless unbounded.

The additional CJM hypothesis is that a finitely specified rule may define an infinite structural object whose global threshold behavior can be discriminated atemporally without sequential enumeration of all of its finite members. Thus 3SAT supplies the local structural language; the unbounded Ramsey surface supplies the infinite problem; and CJM supplies the proposed global discrimination.