The Legendre conjecture states that, for every positive integer n, at least one prime lies strictly between two consecutive squares:
Equivalently, if π(x) counts the primes not exceeding x, then π((n + 1)2) − π(n2) ≥ 1 for every n ≥ 1. Extensive finite verification supports the conjecture [1], but verifying many intervals does not establish the statement for all integers.
For fixed n, the open interval contains exactly 2n integers. Write each candidate as m = n2 + j, with
Every composite candidate has a prime divisor q ≤ √m < n + 1, hence q ≤ n. Conversely, any candidate divisible by a prime q ≤ n is composite, since m > n ≥ q. Each such prime therefore defines an exact exclusion set
Let ℳn be the set of positions surviving all these exclusions:
These surviving positions correspond exactly to the primes in the interval. For n = 1, the exclusion family is empty and the two candidates are 2 and 3. The conjecture is thus the assertion that the composite exclusions never achieve complete coverage: ℳn ≠ ∅ for every n ≥ 1.
The arithmetic-first representation evaluates the exclusion relations in their native modular form. For fixed n, let Xj indicate that position j is selected as a prime witness. The finite encoding is summarized by four principal constraint families:
Together these clauses form a finite formula Φn whose models select exactly one surviving prime position. The long existence clause is split using auxiliary variables, and the shorter clauses are normalized to the common equisatisfiable 3SAT representation Ψn.
Auxiliary assignments are projected out when examining the surviving position space. This explicit encoding uses 2n witness variables before auxiliaries; it is not a claim of a representation polynomial in the binary input length log n. Nor does reduction to 3SAT make this special instance family NP-complete.
Existing sieves already combine modular exclusions, including through parallel and bitwise operations [1]. The proposed CJM question concerns whether the complete normalized structure could support a distinct physical mechanism of admissibility judgment.
For one fixed n, this is a complete finite question. A satisfying model supplies a prime witness; an exact UNSAT certificate for the complete encoding establishes a prime-free square interval and therefore a counterexample. The global conjecture, however, has the unbounded form
Testing every interval up to any finite bound cannot certify this universal statement. Whether CJM could judge the entire family from its finite generating rule is an additional hypothesis, beyond the finite 3SAT representation itself.
The prime count |ℳn| records how many positions survive, but does not describe their arrangement. To isolate interval geometry, let Gn be the largest number of consecutive candidate positions containing no prime, including runs adjacent to either square boundary. Define one proposed Changbal Function by
This normalized quantity measures the largest contiguous region covered by composite exclusions. It equals 1 precisely when the whole interval is prime-free. Two intervals with the same prime count can nevertheless have different values because their surviving primes occupy different positions.
A candidate Changbal Region would require a reproducible change in this geometry across a range of intervals, beyond ordinary prime-density and residue-class effects. A future experiment should predefine this observable and compare it with randomized position controls preserving the interval length, prime count, and exclusions through a predeclared small-prime cutoff below n. An independent interval range should then test whether the observed difference persists.
Physical discrimination additionally requires known SAT controls and certified UNSAT restricted instances, such as candidate subsets containing only verified composites. Those restricted controls are not counterexamples to Legendre's conjecture. An isolated large gap, a thinning prime density, or a sharp physical response alone does not establish a Changbal Jump. No region or physical discrimination result is claimed to have been observed here.
This construction preserves the ordinary sieve's arithmetic information. Once all exclusion sets are explicitly available, the finite existence question is resolved by checking whether their union covers the interval. In the displayed SAT encoding, unit clauses eliminate composite positions; any remaining position supplies a model. General-purpose combinatorial search is therefore unnecessary for this representation.
The proposed distinction is consequently physical, not a new primality criterion or a harder Boolean reformulation. A CJM experiment must account for prime-list preparation, exclusion generation, encoding, physical realization, and readout, and compare the whole procedure with a classical sieve. Supplying an already computed survivor table would decide the finite arithmetic question before CJM receives it.
Prime-density heuristics suggest about n/log n primes in a square interval for large n [1]. This is a heuristic scale, not a uniform guarantee for every interval. Average abundance cannot rule out an exceptional interval with complete composite coverage; the global CJM question concerns precisely that universal noncoverage requirement.
This sketch does not prove the Legendre conjecture, identify a counterexample, establish a new prime-gap bound, or claim that the unbounded family has been compressed into one ordinary finite 3SAT formula.
With the complete candidate interval and all prime exclusions through n, SAT yields an actual prime and rigorously certified UNSAT excludes every prime in that interval. If only some candidate positions are retained, UNSAT concerns that restricted domain. If only some valid divisor exclusions are imposed, SAT may retain composites; UNSAT still excludes the full interval provided every original candidate position is retained.
Neither structural observables nor 3SAT normalization alone demonstrate a physical CJM mechanism or a computational advantage. The purpose is to express Legendre's prime-interval question as an exact modular coverage structure, preserve the distinction between finite certification and universal validity, and formulate the subsequent question of atemporal admissibility discrimination.