learning_compounds
plain-language theorem explainer
Learning events with identical experience traces, attention and spacing but more repetitions have decay rates at least as large as those with fewer repetitions. Memory modelers in Recognition Science cite this to show repetition compounds retention by slowing free-energy decay. The proof uses phi-power monotonicity, bonus equality, rate inequality via multiplication lemmas, cost equality and nonnegativity to reach the result by linear arithmetic.
Claim. For learning events $e_1$ and $e_2$ sharing the same experience trace, attention value and spacing interval, with the repetition count of $e_1$ exceeding that of $e_2$ and positive trace strengths, the decay rates satisfy $dr_1 >= dr_2$, where each decay rate is the negative product of the learning rate and the memory cost of the shared trace.
background
Memory is treated as a thermodynamic system in which retention competes with free-energy decay. A LearningEvent packages a LedgerMemoryTrace experience with bounded attention in [0,1], a repetition count and a spacing value. The learning rate is phi raised to the negative repetition count, multiplied by attention and by one plus the spaced bonus; the spaced bonus equals the log of one plus spacing over the working memory window, divided by log phi. The memory cost sums a complexity term times J-cost of strength, a log time term scaled by breath cycle, and an interference term from ledger balance, all discounted emotionally. Nonnegativity of memory cost is a separate theorem. This result depends on the reflexivity lemma le_refl from ArithmeticFromLogic and on the module's positivity results for costs and bonuses.
proof idea
After binding the decay expressions, the proof proves the phi-power inequality phi to the negative r1 less than or equal to phi to the negative r2 by rpow_le_rpow_of_exponent_le using phi greater than one and the negated repetition ordering. Spaced bonus equality follows by definition unfolding and the spacing hypothesis. The learning rate inequality is obtained by two mul_le_mul_of_nonneg_right steps using the power result, common attention and nonnegativity of the bonus. Memory costs are identified by the experience equality. Nonnegativity is supplied by memory_cost_nonneg with le_refl. The conclusion follows by simplification and nlinarith.
why it matters
This theorem belongs to the fully proven set listed in the MemoryLedger module documentation. It is referenced by the memory_ledger_status definition that records all completed thermodynamic memory results. In the Recognition Science framework it supports the thermodynamic view of memory, connecting to the J-uniqueness fixed point and the phi self-similar point in the forcing chain. It closes the learning-compounding piece without residual scaffolding.
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