equilibrium_remember_prob
plain-language theorem explainer
equilibrium_remember_prob supplies the equilibrium retention probability for any memory trace inside a recognition system. It evaluates the logistic form of the Boltzmann factor using the trace's J-cost at the current tick and the system's recognition temperature. Researchers analyzing temperature-dependent memory retention in Recognition Science cite this definition when proving high-temperature disorder or low-temperature bistability. The definition is a direct one-line algebraic wrapper around memory_cost and the exponential ratio.
Claim. Let $J$ be the memory cost of the trace at tick $t$ and let $T_R$ be the recognition temperature of the system. The equilibrium retention probability is then $p = e^{-J/T_R}/(e^{-J/T_R}+1)$.
background
The MemoryLedger module models memory traces as thermodynamic objects whose retention balances J-cost against thermal noise. LedgerMemoryTrace is the structure that records a trace's complexity, emotional weight (bounded in [0,1]), encoding tick, strength (bounded in [0,1]), and ledger balance. memory_cost computes the effective J-cost from complexity times Jcost(strength), logarithmic time decay, interference from ledger imbalance, and an emotional discount factor. RecognitionSystem is the record containing the recognition temperature TR > 0, which sets the scale of fluctuations; its beta is the inverse 1/TR. This definition sits inside the module's thermodynamic treatment of retention versus free-energy decay and directly uses the temperature concept from BoltzmannDistribution.
proof idea
This is a one-line wrapper that binds J to memory_cost trace t and then returns the logistic expression exp(-J / sys.TR) / (exp(-J / sys.TR) + 1).
why it matters
The definition is the central object for all temperature-dependent memory theorems in the module. It is invoked by high_temp_maximizes_entropy to establish that p approaches 0.5 as TR grows, maximizing entropy, and by low_temp_bistable to show that p collapses toward 0 or 1 as TR shrinks. It realizes the Recognition Science link between J-cost minimization and thermal occupation probabilities, consistent with the RCL and the phi-ladder structure of costs. The module states that all thermodynamic memory theorems are now fully proven.
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