access_cost_zero_at_origin
plain-language theorem explainer
J-cost vanishes at the phi-ladder origin. Cache-forcing arguments in graph recognition cite this to set the free-access baseline for collocated storage. The proof reduces immediately to the unit lemma for Jcost via simplification.
Claim. $J(x) = 0$ at $x = 1$, where $J(x) = (x-1)^2/(2x)$ is the J-cost of a positive ratio $x$.
background
The Local Cache Forcing module shows that J-cost minimization on connected graphs forces hierarchical local caching, closing Gap G1 in the brain holography proof. J-cost is the function $J(x) = (x-1)^2/(2x)$ induced by the multiplicative recognizer comparator. The upstream lemma Jcost_unit0 states that Jcost(1) = 0, which supplies the zero point here.
proof idea
One-line wrapper that applies the lemma Jcost_unit0 from the Cost module by simplification.
why it matters
Anchors the local cache forcing certificate by providing the zero-cost baseline for collocated access. It feeds directly into access_cost_pos_of_nonzero and caching_is_forced to establish that remote patterns incur strictly higher total cost. The result supports the master certificate that collocation minimizes cost, consistent with the Recognition Composition Law and the self-similar fixed point phi.
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