Find a natural bijective proof of the Rogers–Ramanujan identity

Find a genuinely natural bijective proof of the Rogers–Ramanujan identity that avoids reliance on the Garsia–Milne involution principle.

Background

The Rogers–Ramanujan identity equates the number of partitions of an integer into parts differing pairwise by at least two with the number of partitions whose parts are congruent to 1 or 4 modulo 5. Garsia and Milne established a bijective proof using their involution principle, which constructs an explicit matching through auxiliary sets and mappings.

The paper explains that the involution-principle proof is highly general but typically provides little insight into the specific combinatorial structures involved. Consequently, the unresolved problem is to produce a genuinely natural, structurally informative bijection for the Rogers–Ramanujan identity rather than one obtained through the involution principle.