emotional_forgets_slower
plain-language theorem explainer
Memory traces with higher emotional weight retain strictly more strength after any positive number of forgetting steps than otherwise identical traces. Thermodynamic models of retention would cite this to quantify how affect modulates decay. The proof unfolds the forgetting map, invokes the emotional cost reduction lemma to obtain a strictly lower rate, then chains positivity, rate inequality, and the strict monotonicity of the exponential to reach the goal via linarith.
Claim. Let $T_1$ and $T_2$ be memory traces sharing the same complexity, strength $s>0$, encoding tick, and ledger balance, yet with emotional weight of $T_1$ strictly exceeding that of $T_2$. For positive integer $n$ and time $t$ meeting the non-perfect-retention condition, the remaining strength after $n$ forgetting steps satisfies $s·exp(-r_1 n w)>s·exp(-r_2 n w)$, where the decay rates $r_i$ are proportional to the respective memory costs at time $t$ and higher emotional weight yields strictly lower cost.
background
A memory trace records complexity, emotional weight (in [0,1]), encoding tick, strength (in [0,1]), and ledger balance. The module casts retention versus free-energy decay as a solvable thermodynamic problem inside Recognition Science. Upstream lemmas supply the J-cost of recognition events and the positivity of base decay rate and breath cycle; the key sibling result emotional_reduces_cost shows that larger emotional weight strictly lowers instantaneous memory cost under the shared-parameter hypotheses.
proof idea
Unfold apply_forgetting and forgetting_rate. Apply emotional_reduces_cost to obtain memory_cost(T1,t)<memory_cost(T2,t). Multiply by positive base_decay_rate and divide by positive breath_cycle to obtain strictly lower rate. Scale the negative exponent by positive n and working_memory_window to obtain a strictly larger (less negative) argument. Invoke exp_lt_exp to enlarge the exponential factor. Multiply by identical positive strengths and close with linarith.
why it matters
The theorem completes the fully-proven thermodynamic memory suite listed in memory_ledger_status. It supplies the quantitative link between emotional weight and slower free-energy decay, consistent with Recognition Science treatment of cost as J-cost and retention as controlled by the phi-ladder decay dynamics. No scaffolding remains; the result closes the discrete forgetting model for the ledger.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.