The Energy Cost of Heredity: thermodynamic boundaries of the Eigen error threshold, the limits of adaptive memory and the architecture of the replicator

How much free energy does a self-copying system need in order to retain hereditary information, and within what limits is energy able to acquire it? This paper answers these questions with a closed-form theory and numerical experiments. Main results: (1) for Eigen's quasispecies model with Landauer-priced accuracy, a closed-form formula of the dominance boundary of the master sequence E_c(L, σ) is obtained, together with its two-term asymptotics of relative error not exceeding 2·10⁻⁴% (L ≥ 50); the price of the transition from survival to dominance is strictly linear in genome length: 1.35 kT per monomer at σ = 10. (2) In the equilibrium regime energy does not remove the hard limit on genome length: at prebiotic chemistry (Δ_max = 2–5 kT) a genome of ribozyme scale is ruled out. (3) Kinetic proofreading removes the limit, paying a premium of ≈ 7 over the equilibrium estimate and a factor of 3.3 above the Landauer minimum. (4) An agent-based model of protocells in a fluctuating environment shows that heritability is set by copying accuracy (H agrees with 1 − 2ν + 2ν²); under a limited resource, dissipation grows with the mutation rate (correlation +0.7…+0.9) and anticorrelates with heritability (−0.7) — the "dissipation versus heredity" contradiction is measured, not declared. (5) The accuracy price of a matrix-free autocatalytic cycle coincides with that of the template one: the advantage of the template is a property of competition (the economics of parasites), not of chemistry. (6) An optimal distribution of proofreading cycles over an inhomogeneous genome (water-filling) lowers the budget to 61%; compartmentalization suppresses the unlinked parasite as a threshold filter (p_m* ≈ 0.1), and the threshold tracks the percolation geometry of the migration network (p_m* ≈ 0.2 p_c in all five geometries). (7) The parasite's price is derived from an explicit reaction network (r = k/M); the suppression threshold becomes a topological requirement k ≥ 0.52M. (8) Three pressures are tied together by a theorem on the minimal budget: budgetary pressures (accuracy, environment) add up, the parasitic one multiplies. (9) The residual-error budget per genome is length-independent while the required accuracy per position grows only logarithmically — hence the L ln L form of the budget; in a nonstationary environment the heredity window collapses in phenotype variability (σ_x* ∝ τ^(−1/2)), not in accuracy. Control checks on synthetic texts and administrative Linear B series show no false positives. This deposit contains two parallel language versions with identical content: EN — The Energy Cost of Heredity (20 pp.); RU — Энергетическая цена наследственности (20 стр.).