Reversible Multi–Radix Logic and the Collapse of Irreversible1 Ledger Security: A Hamiltonian Framework for Zero–Entropy2 State Evolution3

Abstract

Classical ledger systems, including proof-of-work (PoW) and proof-of-stake (PoS) blockchains;derive their security from the assumption that irreversible computation incurs a thermodynamic cost. This assumption, rooted in Landauer’s principle, implies that reversing orre-writing global state requires expenditure of significant physical energy, and therefore canbe made economically infeasible.In this paper, we introduce the RHEA–Λ Gate Family: a reversible multi–radix (2–3–5)logic primitive with a triangular, measure-preserving topology that embeds directly intoHamiltonian phase-space flows. Each gate includes an intrinsic symbolic (glyph/entropy)register enabling perfect, lossless history retention without information erasure. Whencomposed into circuits, Λ–gates form fully reversible, entropy-preserving state-transitionoperators capable of implementing arbitrary classical computations at asymptotically zeroenergy in adiabatic regimes.We show that any ledger whose security relies on computational irreversibility becomesvulnerable in a computational substrate that supports (i) strictly reversible evolution, (ii)zero–entropy symbolic memory, and (iii) multi-radix reversible hashing. In such substrates,the economic barrier that protects ledger history vanishes: all PoW functions become ther-modynamically free, PoS penalties become reversible, and Merkle-tree hashing no longerprovides unidirectional security. We formalize this result as an impossibility theorem forirreversible-cost security models, and we construct a reversible ledger architecture whose cor-rectness is maintained through Hamiltonian invariants rather than dissipative computationalcost.The Λ framework thereby provides both (a) a constructive alternative to irreversible ledgermechanisms and (b) the first proof that classical reversible computation, when extendedto higher radices with symbolic memory, nullifies the energy-based assumptions underlyingmodern blockchain security.