access_cost_pos_of_nonzero
plain-language theorem explainer
Nonzero natural distance d yields strictly positive J-cost on the phi-power ladder. Workers on the GCIC local cache forcing argument cite it to establish that any remote access carries positive cost, closing the zero-cost case. The proof rewrites the claim to the origin case and invokes the strict monotonicity of Jcost on phi powers.
Claim. Let $d$ be a positive natural number. Then $0 < Jcost(φ^d)$, where $Jcost$ is the J-cost function and $φ$ the golden ratio fixed point.
background
The module proves that J-cost minimization on connected graphs forces hierarchical local caching, closing Gap G1 in the brain holography proof. Jcost is the cost function with Jcost(φ^0) = 0; distances are measured on the phi-ladder via powers of φ. Upstream results include access_cost_zero_at_origin stating Jcost(φ^0) = 0 and Jcost_phi_pow_strictMono asserting Jcost(φ^m) < Jcost(φ^n) whenever m < n.
proof idea
The proof is a one-line wrapper. It rewrites the left-hand side via access_cost_zero_at_origin to reduce to the origin case, then applies Jcost_phi_pow_strictMono directly to the hypothesis 0 < d.
why it matters
This result supports collocation_minimizes_cost and the local_cache_forcing_certificate by showing remote storage always incurs positive cost. It fills the nonzero-distance step in the J-cost forcing chain and connects to the Recognition Science T5 J-uniqueness property on the phi-ladder (T6). No open questions are touched.
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