The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves

We introduce a framework that converts the search for maximal gaps between integers coprime to a primorial p#—and for maximal deserts between twin-pair centers—into a finite enumeration over the phases of a fixed layer sieve. An atlas is the set of admissible (phase, covering assignment) pairs; we prove that the gap (resp. desert) positions of each length are exactly the CRT lift classes of the atlas entries, with a smallest-prime refinement giving an explicit bijection. Consequences are proved without computation: a characterization of the Jacobsthal function g(p#) and of the maximal twin-desert width W(p) as the largest feasible atlas width; first occurrences as minimal CRT lifts; a mirror involution pairing all positions; a capacity pre-sieve; and a complete defect stratification of phases. A Symmetry Theorem characterizes the palindromic structure of gap kill-patterns in terms of center divisibility. All values g(p#) and W(p) are certified by independent sieve cross-checks for the Jacobsthal values g(p#), 13 ≤ p ≤ 37, and for the twin-desert widths W(p), 13 ≤ p ≤ 31; W(37) ≥ 462 is certified with maximality provisional.

A central distinction is maintained throughout: the Jacobsthal function g(p#) is a property of the primorial period and can exceed 2(p−2); the location-sensitive prime gap bound 2(⌊√x⌋ − 2) is a separate conjecture bounding the prime gap surrounding each position x ≥ 25, verified for every x ≤ 41² by direct computation, and capable of being smaller than the global Jacobsthal value g(p#) at some positions x. The atlas reduces this conjecture to an explicit finite statement at each prime and locates precisely the open difficulty. The certified tables accompany submissions of new integer sequences to the OEIS. The enumeration cost grows with covering-set size; the present implementation reaches its practical limit at p = 41, well inside the range of record computations—the contribution is structural (positions, twin deserts, mirror pairing, the Symmetry Theorem), not record-setting.