learning_rate
plain-language theorem explainer
learning_rate defines the update strength for a memory event as phi to the power of negative repetitions, multiplied by attention and one plus the spacing bonus. Physicists modeling thermodynamic memory retention cite it when proving non-negativity or compounding under repeated exposure. The definition is a direct product expression with no additional lemmas required.
Claim. For an event with repetition count $n$, attention weight $a$ satisfying $0 ≤ a ≤ 1$, and spacing interval $s$, the rate is $φ^{-n} a (1 + b(s))$ where $b(s) = log(1 + s/w) / log φ$ and $w$ is the working memory window.
background
MemoryLedger treats retention as a thermodynamic balance between J-cost decay and recognition reinforcement. A LearningEvent packages a LedgerMemoryTrace with bounded attention, repetition count, and spacing. The spaced_bonus computes a logarithmic increment normalized to the golden ratio base phi, drawing from the self-similar fixed point in the forcing chain. Upstream, the module establishes positivity for base decay and emotional adjustments before introducing this rate.
proof idea
The definition is a one-line algebraic product: exponential decay in repetitions scaled by attention and the spaced bonus. It directly feeds the non-negativity proof via multiplication of non-negative terms and the compounding theorem via monotonicity in repetitions.
why it matters
This supplies the explicit rate expression used by the learning_compounds theorem to establish that additional repetitions increase the rate under matched conditions, and by learning_rate_nonneg to confirm non-negativity. It translates the Recognition Science phi-ladder and eight-tick octave into a concrete memory update, supporting the claim that all thermodynamic memory theorems are proven. It touches no open scaffolding.
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