cachedAccessCost
plain-language theorem explainer
cachedAccessCost defines the total access cost for n items as the frequency-weighted sum of ε for cached accesses and full dist otherwise, plus αK maintenance overhead. Researchers modeling recognition hierarchies and local caching in information-theoretic physics cite this as the objective function underlying the Local Cache Theorem. The definition is a direct algebraic term with a conditional sum and linear overhead term, encoding the caching trade-off without reduction.
Claim. The cached access cost for $n$ items is $C = ∑_{i=1}^n (f_i · ε if cached_i else f_i · d_i) + α K$, where $f_i$ and $d_i$ are the frequency and distance of item $i$, cached is the decidable predicate selecting cached items, $ε$ is the cached access distance, $α$ is the per-item maintenance coefficient, and $K$ is the number of cached items.
background
The module supplies the machine-verified core of the Inevitability of Local Minds paper, centered on the Local Cache Theorem and φ-Optimal Hierarchy. Upstream results define cost via multiplicative recognizers as derivedCost of the comparator and via observer forcing as Jcost of the recognition state. Constants.K supplies the dimensionless bridge ratio $K = φ^{1/2}$. The definition supplies the explicit objective used to state benefit under non-uniform frequency, distance spread, and positive maintenance.
proof idea
The definition is a direct term-mode expression: the sum over Fin n of the conditional (freq i * ε if cached i else freq i * dist i), followed by the linear maintenance term α * K. No lemmas or tactics are invoked; it is the primitive cost model.
why it matters
This supplies the objective function for the local_cache_benefit result that caching strictly reduces total cost under (A1-A3). It anchors the information argument that local caching is forced in recognition systems, linking to J-cost gradients and the fibonacci partition that forces φ. The definition fills the cost-model step in the module's hierarchy before the φ-optimal recurrence.
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