| _version_ | 1866901482337468416 |
|---|---|
| author | Ball, Alan |
| author_facet | Ball, Alan |
| contents | <p>This paper proves a single structural result: if a directed temporal transition is to be represented intrinsically within an integer-valued state space; without relying on an external observer, label convention, or sign rule; then the minimal possible alphabet is necessarily balanced ternary.</p> <p> </p> <p>We formalise this as three constraints on any candidate state space: (i) a neutral ground state, (ii) the existence of a directed unit transition away from that ground state, and (iii) closure under the inverse transition for all states reachable from the ground state by a single step. Under these constraints, the binary set {0,1} fails (direction becomes a matter of extrinsic convention), while the set S = {-1,0,+1 } is uniquely forced (up to rescaling) and is additively symmetric.</p> <p> </p> <p>No physical interpretation is assumed or required. The paper isolates the minimal algebraic substrate for representing a first directed distinction; what this alphabet generates under further structural operations is left explicitly open.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18806015 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Balanced Ternary by Necessity Ball, Alan balanced ternary minimal alphabet intrinsic direction state space closure inverse transition foundational logic Philosophy Mathematical logic <p>This paper proves a single structural result: if a directed temporal transition is to be represented intrinsically within an integer-valued state space; without relying on an external observer, label convention, or sign rule; then the minimal possible alphabet is necessarily balanced ternary.</p> <p> </p> <p>We formalise this as three constraints on any candidate state space: (i) a neutral ground state, (ii) the existence of a directed unit transition away from that ground state, and (iii) closure under the inverse transition for all states reachable from the ground state by a single step. Under these constraints, the binary set {0,1} fails (direction becomes a matter of extrinsic convention), while the set S = {-1,0,+1 } is uniquely forced (up to rescaling) and is additively symmetric.</p> <p> </p> <p>No physical interpretation is assumed or required. The paper isolates the minimal algebraic substrate for representing a first directed distinction; what this alphabet generates under further structural operations is left explicitly open.</p> |
| title | Balanced Ternary by Necessity |
| topic | balanced ternary minimal alphabet intrinsic direction state space closure inverse transition foundational logic Philosophy Mathematical logic |
| url | https://doi.org/10.5281/zenodo.18806015 |