Quantifying Distinguishable Worlds in Everett's Multiverse: Quantum Darwinism and Finite-Memory Records of Changing von Neumann Machines

Everettian quantum mechanics describes alternative histories within one evolving quantum state, while quantum Darwinism relates their accessibility to information retained in environmental records. I examine how finite record memory constrains the number of distinguishable histories and the organisation of their conditional ancestry. When d + 1 histories must each be decoded from a d-dimensional fragment with success d/(d + 1), the established discrimination optimum forces a regular-simplex geometry. Its conditional mixtures then determine exact ancestral edge costs and a joint criterion for the permissible topology, depth, total variation and largest step. In the three-history binary example, equally weighted labels decoded with success 2/3 require 61.6% greater minimum conditional variation in a qubit than in an optimally designed three-dimensional record. For six histories, minimising total variation and minimising the largest step select different genealogies. Quantitative stability estimates show that the additional constraint persists under sufficiently small departures from ideal decoding and weights. At this threshold, trace-distance contraction prevents a fixed channel acting only on an unrestricted optimum from compressing it into the smaller record at unchanged accuracy, although an explicit controlled preparation realises the alternative encoding. A von Neumann-inspired application distinguishes hereditary function, present configurations and retained mutation histories, and a comparison of finite and infinite counts separates these quantities from cosmological multiplicity. The resulting theory makes the limitations of environmental memory consequential for readable branching while keeping conditional variation distinct from physical rates of change. Its bounds apply to specified records and histories without assigning an absolute number of worlds since the universe’s beginning.