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Ⅰ. The Problem

For a positive integer n, define its radical as the product of its distinct prime factors:

Repeated prime powers therefore contribute only once. For example,

The ABC conjecture concerns coprime positive integers satisfying

For every ε > 0, the conjecture predicts that only finitely many such triples satisfy

Equivalently, for every ε > 0 there exists a constant K_ε such that

for all coprime triples satisfying a + b = c.

For CJM, the target is therefore not merely to locate another unusually strong ABC triple inside a finite numerical interval. The central question is whether, for each fixed positive ε, exceptional triples eventually disappear or continue to reappear arbitrarily far into the integer domain.

Ⅱ. Structural Reinterpretation

For a coprime ABC triple, define its quality

For a fixed ε > 0, define an exceptional ABC state by

whenever

The ABC conjecture can then be viewed as the claim that for every fixed positive ε, this exceptional state has only finite support.

The two competing global structures may be written schematically as

Thus the conjecture is reinterpreted as a global extinction-versus-persistence problem. The local arithmetic rule is finite, but the statement that exceptional behavior eventually terminates is inherently unbounded.

Ⅲ. SAT → 3SAT Encoding

For a fixed finite encoding width, fixed positive rational ε = r / s, and fixed numerical search region, an ABC instance may be represented through Boolean arithmetic constraints.

The finite encoding may be summarized by five principal conditions:

1. Boolean variables encode positive integers a, b, c, constrained by a + b = c.
2. Finite Boolean constraints impose the coprimality condition gcd(a,b,c) = 1.
3. Prime-factor support variables encode the distinct prime divisors of abc, allowing construction of R = rad(abc).
4. For ε = r / s, the exceptional inequality may be written without real-number arithmetic as
   c^s > R^s+r.
5. Finite bounds may restrict the search region, for example
   B < c ≤ N.

The resulting bounded Boolean structure can be expressed as SAT and normalized through the common CJM 3SAT gate.

This reduction has a precise but limited meaning. It can determine whether an exceptional ABC triple exists inside a chosen finite region. It cannot determine whether exceptional triples will never appear again beyond every finite bound.

Therefore, unlike a fixed Ramsey instance, the global ABC state is not itself represented by one finite SAT formula.

Ⅳ. The CJM Question

For fixed ε and finite bounds B < c ≤ N, a SAT instance can ask whether at least one exceptional triple exists in that finite interval.

But even if the answer is UNSAT for an arbitrarily large finite N, the possibility remains that another exceptional triple occurs beyond it.

The intended CJM output therefore concerns a fundamentally different state:

Here TERMINATING means that there exists some finite B_ε beyond which no ε-exceptional ABC triple exists. UNBOUNDED means that exceptional triples continue to occur beyond every finite bound.

Unlike the Ramsey encoding, neither global state is generally reducible to a single finite SAT instance. In particular, TERMINATING is itself an unbounded statement: no finite search interval can certify that an exceptional triple never reappears beyond it.

The proposed role of CJM is therefore not merely to solve a larger SAT instance. It is to ask whether a finite local rule may support an atemporal judgment about the global termination or persistence of the infinite structure generated by that rule.

Under the CJM hypothesis, P ≡ NP^J, this introduces a stronger gap than in the Ramsey case:

Can a finitely specified ABC rule, locally representable through 3SAT, permit physical CJM to distinguish eventual extinction from unbounded recurrence when no finite SAT instance can certify termination itself?

Ⅴ. Expected Changbal Region

The quality parameter

provides a natural local structural observable. For fixed ε, exceptional triples satisfy

Finite computation may identify isolated high-quality triples or finite intervals in which no such triples occur. Neither observation, however, determines whether the exceptional region has globally terminated.

A candidate Changbal Region would therefore not correspond simply to the presence or absence of an exceptional triple at one finite scale. It would have to correspond to a reproducible physical distinction between two global structural regimes:

This makes the ABC setting more demanding than a finite threshold problem. The conjectured transition is not located at one known finite coordinate; it concerns whether a terminal boundary exists at all for each ε.

Ⅵ. Limitation

This sketch does not prove the ABC conjecture and does not claim that the conjecture, or its quantifier structure, has been reduced to one ordinary finite 3SAT formula.

For fixed finite bounds and fixed rational ε > 0, local ABC relations can be represented by finite Boolean constraints and normalized to 3SAT.

This finite representation can establish only bounded existence or bounded nonexistence. It cannot certify the global statement that exceptional triples never occur again beyond some unknown finite boundary.

The additional CJM hypothesis is therefore stronger than the claim that difficult finite SAT instances may be discriminated physically. It asks whether a finitely specified local rule may define an infinite structural object whose termination or recurrence can itself be judged atemporally.

Thus 3SAT supplies the local radical-balance language; the ABC quantifiers create the finite-to-infinite gap; and CJM supplies the proposed global extinction-versus-persistence discrimination.