The Erdős–Straus conjecture states that, for every integer n ≥ 2, the rational number 4/n can be decomposed into three positive unit fractions:
The denominators need not be distinct. We may therefore impose x ≤ y ≤ z to remove permutation redundancy. The question is whether at least one such decomposition exists for every n, rather than only for selected integers or residue classes [1].
Clearing denominators gives the exact balance 4xyz = n(xy + xz + yz). A more useful structural representation retains the arithmetic of the residual two-term decomposition. Since x is the smallest denominator, 1/x < 4/n ≤ 3/x, so
For each candidate x, set a = 4x − n > 0 and b = nx. The remaining equation is a/b = 1/y + 1/z, equivalently
Define u = ay − b and v = az − b. Every ordered decomposition corresponds exactly to a positive factor pair satisfying
Conversely, these conditions reconstruct an exact positive-integer solution. Thus a fixed n defines a finite family of first-denominator branches, each containing a finite divisor-compatibility structure. No arbitrary cutoff on y or z is needed for this complete representation.
Following the arithmetic-first approach of Sketch No. 009, divisor sets, congruences, and compatibility relations are evaluated in their native arithmetic form. SAT then records their logical consequences. Let Xx select a first denominator, while Ux,u and Vx,v select positive divisors of (nx)2 for that branch. The finite encoding is summarized by six principal constraint families:
For example, an incompatible factor pair produces the clause ¬Ux,u ∨ ¬Vx,v, while branch selection requires
Together with the uniqueness and reverse branch implications, these constraints form Φn. Auxiliary variables normalize its clauses to an equisatisfiable 3SAT formula Ψn.
Let ℳn denote the set of ordered denominator triples satisfying the original equation. With complete domains and exact constraints, SAT(Ψn) holds if and only if ℳn ≠ ∅. Structural observables refer to reconstructed triples, with auxiliary Boolean assignments projected out.
Classical number-theoretic methods already exploit divisibility, residue classes, and parametrizations to construct or enumerate unit-fraction decompositions [1, 2]. The proposed CJM distinction concerns the mechanism of judgment applied after the finite structure has been normalized.
This finite question must be separated from the conjecture itself:
An exact UNSAT certificate for one complete instance would identify a counterexample. SAT results for any finite collection of integers establish only those cases. Whether a physical CJM could judge the entire unbounded family from its generating rule is a further hypothesis; finite 3SAT normalization alone does not supply that step.
Two integers can both admit decompositions while distributing them very differently across their first-denominator branches. Define Dn = ⌊3n/4⌋ − ⌊n/4⌋, the number of candidate branches, and use the branch-coverage fraction as one proposed Changbal Function:
This quantity is zero when no decomposition exists and otherwise measures how broadly admissibility is distributed. A candidate Changbal Region would be a reproducible regime in which branch coverage reorganizes sharply as n varies, even while all examined instances remain satisfiable.
Ordinary residue-class effects and divisor abundance provide essential arithmetic baselines. A future experiment should predefine its comparison against those baselines and test the same observable on an independent range. For physical discrimination, known SAT instances and independently certified UNSAT restricted instances provide controls; the latter are not counterexamples to the conjecture. Neither an isolated fluctuation nor a single accepted decomposition establishes a Changbal Jump. No such region is claimed to have been observed here.
Unlike the necessary-condition architecture of Sketch No. 009, the complete fixed-n structure here is equivalent to an actual solution condition. This follows from the finite bound on x and the exact factor-pair reconstruction of y and z. Infinity enters through the unbounded family of integers n, not through an unavoidable infinite denominator search for one fixed n.
This representation nevertheless has a substantial preparation cost. Generating divisor domains and compatibility relations may already perform much of the classical work; enumerating all fully admissible pairs would decide the instance before CJM receives it. Any proposed physical advantage must therefore account for arithmetic preparation, encoding size, physical discrimination, and readout together. The 3SAT interface by itself establishes neither a speedup nor a new decomposition method.
This sketch does not prove the Erdős–Straus conjecture, provide a counterexample, or claim that the unbounded statement has been reduced to one ordinary finite 3SAT formula. It proposes a structural interpretation of exact decomposition compatibility and the subsequent physical CJM question.
With complete domains, SAT reconstructs a genuine decomposition for the selected n; a rigorously certified UNSAT result excludes every decomposition for that integer. With truncated domains, UNSAT excludes only the retained candidates. If exact constraints are relaxed, a SAT model need not reconstruct a valid solution.
Nor does this sketch demonstrate a physical mechanism for atemporal discrimination or establish an advantage over existing arithmetic methods. Its purpose is to preserve the native denominator and divisor relations, express their joint compatibility through SAT and 3SAT, and ask whether the resulting whole structure can support a physically distinct mode of admissibility judgment.