A Hubble-Volume Mass Ledger in FRW Cosmology: An Internal-Transfer No-Go Theorem and the Required Accretion History

In a spatially flat Friedmann–Robertson–Walker (FRW) spacetime, the mass inside the Hubble radius obeys 𝑀= 𝑐3/(2𝐺𝐻)iden- tically. Dynamics enter only when an accretion law is imposed. Motivated by the Babichev–Dokuchaev–Eroshenko relativistic perfect-fluid result 𝑀 ∝𝑀2 (𝜌+𝑝), we test a quadratic mass ledger. We show that the constant-coefficient law 𝑀= A𝑀2 forces the total inertial mass density 𝜌tot +𝑝tot to be constant. An internal exchange 𝑄between matter and a dark component cancels from total conservation and cannot change the resulting logarithmic expansion history. Once normalized to the observed present expansion derivative, this history undershoots the standard recombination value of 𝐻2/𝐻2 0 by nearly eight orders of magnitude and cannot contain the usual radiation era, matter–radiation equality, sound horizon, or big-bang nucleosynthesis. We give two exact corollaries: a vacuum-like dark component requires steady-state matter creation, while standard dilution without interaction requires a compensating negative dark inertial density. Dropping only the assumption that Ais constant gives the unique history A(𝑡)= 8𝜋𝐺2 𝑐3 (𝜌𝑚 + 4 3 𝜌𝑟 +𝐾), which scales as 𝑎−4, then 𝑎−3, and approaches a constant floor. The theorem is an FRW constraint independent of any black-hole- interior interpretation.