Proof of Sequential Memory Execution (PoSME)

This document defines Proof of Sequential Memory Execution (PoSME), a cryptographic primitive combining mutable arena state, data- dependent pointer-chase addressing, and per-block causal hash binding in a single step function. A Prover executes K sequential steps over a mutable N-block arena. Each step reads d blocks at addresses determined by the previous read's result (pointer chasing), writes one block with spatial neighborhood entanglement (incorporating A\[w-1\] and A\[w+1\]), and advances a transcript chain. The construction provides three properties: (1) unconditional sequential time enforcement anchored in physics-bounded latency floors, (2) forgery prevention via causal hashes (reduces to collision resistance of H), and (3) TMTO resistance scaling as $1/\alpha$ with spatial entanglement, where $\alpha$ is the adversary's storage fraction. Verification requires O(Q * d^R * log N) hash evaluations with no arena allocation. No trusted setup is required.