access_cost_increases_with_distance
plain-language theorem explainer
Farther distances on the phi-ladder incur strictly higher J-cost. Researchers modeling holographic memory or recognition-based caching cite this to show remote access is penalized under J-cost minimization. The proof is a direct one-line application of the strict monotonicity theorem for J-cost on phi-powers.
Claim. For natural numbers $d_1 < d_2$, the J-cost of the state at rung $d_1$ satisfies $Jcost(phi^{d_1}) < Jcost(phi^{d_2})$, where J-cost is the derived cost of a multiplicative recognizer comparator.
background
J-cost is the cost function induced by a multiplicative recognizer: the derived cost of its comparator on positive ratios. For any recognition event the cost equals the J-cost of its state. The module proves that J-cost minimization on connected graphs forces hierarchical local caching, closing Gap G1 in the brain holography argument. The upstream result Jcost_phi_pow_strictMono states that J(phi^m) < J(phi^n) whenever m < n, obtained by composing Jcost_strictMono_on_Ici_one with phi_pow_strictMono.
proof idea
One-line wrapper that applies Jcost_phi_pow_strictMono directly to the hypothesis d1 < d2.
why it matters
This theorem supplies the monotonicity step needed for collocation_minimizes_cost and caching_is_forced, the core results that close Gap G1. It sits inside the access-cost properties section of the Local Cache Forcing paper and rests on the Recognition Composition Law together with the phi fixed-point property. No open scaffolding remains here.
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