Recamán’s sequence begins with a0 = 0. At step n, one first attempts to move backward by n. If the result is nonnegative and has not appeared earlier, that value is selected; otherwise, the sequence moves forward by n:
| an−1 − n | if an−1 − n ≥ 0 and has not been visited, |
| an−1 + n | otherwise. |
The sequence is simple to generate, yet its long-term coverage remains irregular. Some integers appear quickly, while others remain absent for extremely long intervals. The central question is therefore not merely how to compute the next term, but how the full pattern of permitted and forbidden visits is organized.
The usual view treats Recamán’s sequence as a temporal recursion:
The CJM approach instead represents the first N stages as one finite layered path structure. Each layer Ln corresponds to a step n, and each possible integer position v becomes a candidate state. From v at layer n−1, two potential transitions are considered:
The subtraction edge is admissible only when v − n is nonnegative and has not appeared in any previous layer. Otherwise, the addition edge becomes compulsory. The sequence is therefore reinterpreted as a globally constrained path rather than as a list produced one term at a time.
Let Pn,v mean that the sequence occupies value v at step n, and let Hn,v mean that v has been visited by step n. The encoding imposes four basic conditions:
The visit history is updated by
These Boolean constraints are first expressed as SAT and then normalized through the common CJM gate:
For a target integer T and a finite horizon N, add the condition
CJM then examines whether the complete 3SAT structure remains admissible. The question is no longer merely, “At which step does T appear?” but:
Is the appearance of T structurally admissible within the first N layers under the full Recamán rule?
The same framework can test whether an interval {0, …, K} is completely covered, whether a target remains absent, or whether a modified Recamán-type path satisfying additional global conditions exists.
By varying N, T, or the coverage range K, one may observe changes in clause interaction, path rigidity, visit density, and admissibility. A single first appearance is not by itself a Changbal Jump. A stronger candidate would be a repeatable critical region in which many target values or related path variants undergo a collective structural transition.
This sketch does not prove that every integer appears in Recamán’s sequence, nor does it establish a nonclassical computational advantage. Its purpose is narrower: to demonstrate how a temporal recurrence can be translated into a finite 3SAT path structure and examined through CJM as a problem of global structural admissibility.