THE FIVE-FOLD CAPACITY RECORD

Abstract

THE FIVE-FOLD CAPACITY RECORD Paper X of the Five-Fold Dark Series Gates on the returning door words \(14\), \(41\), \(11111\), and \(44444\) Chloe Jane Tully Independent researcher · New South Wales, Australia correspondence on request · September 2026 https://orcid.org/0009-0007-5661-7332 10.5281/zenodo.23006153 chloetully5@gmail.com ABSTRACT Paper IX named the door language \(L^*=\{1,4\}^*\) and the capacity record of a walk. This paper computes that record on the four shortest nonempty returning door words: the out-and-back words \(14\) and \(41\), and the two full cycles \(11111\) and \(44444\). Capacity is re-asked at every letter. A syntactic return is not an automatic double conversion. On an out-and-back the same edge is gated twice; spending the last empty site on the first letter can close the door for the second. On a five-cycle five distinct doors are gated once each; a single zero stops the conversion sequence and does not open the complementary edge. In the single-cell collapse the gate is emptiness of at least one end. Hall deficiency remains a different predicate. This paper adds no edge. KEYWORDS capacity record; returning door word; \(14\); \(41\); \(11111\); \(44444\); emptiness gate; \(C_5\); Paper X; seal \(5\)–\(41\)–\(72\)–\(\varphi\)