A gapped generalization of Kingman’s subadditive ergodic theorem

The paper’s Theorem 3.1 states exactly the gapped almost-subadditive result the model aims to prove and gives a complete four-step proof using a gapped Steele-type construction. The proof establishes f = liminf fn/n is T-invariant (Step 1), produces many good k with controlled ratios via Birkhoff (Step 2), shows the bad times have vanishing density (Step 3), and constructs gapped intervals to bound the limsup by a truncated version of f (Step 4), yielding existence of the limit almost surely . By contrast, the model’s outline misapplies the gapped inequality on blocks with remainder r=0 (not permitted by the paper’s n,m∈N convention), asserts a non-justified squeeze limsup ≤ liminf + ε without the paper’s essential truncation max{f,−ε−1}, and uses conditional expectations E(gn|I_a) in a way not connected to a valid lower bound or the final Steele partitioning argument. Hence the paper is correct and the model’s proof is flawed/incomplete.