collocation_minimizes_cost
plain-language theorem explainer
Collocation at distance zero yields strictly lower J-cost than any positive distance on the phi ladder. Derivations of brain holography from first principles cite this to force local caching. The proof reduces immediately to the strict increase of J-cost on successive phi powers.
Claim. For every natural number $d$ with $0 < d$, the J-cost at $phi^0$ is strictly smaller than the J-cost at $phi^d$.
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 argument. J-cost is the derived cost of a multiplicative recognizer comparator on positive ratios, equivalently the cost of any recognition event. The constant phi is the unique positive solution to $r^2 = r + 1$ and generates the phi-ladder used for rung assignments.
proof idea
The proof is a one-line wrapper that applies the upstream theorem establishing J-cost of phi to the m is less than to the n whenever m is less than n, instantiated at m equal to zero and n equal to d.
why it matters
It supplies the key inequality used in the brain holography fully forced theorem and the local cache forcing certificate. The certificate assembles this inequality with J-cost zero at the origin and the Fibonacci derivation of phi. This step realizes the T5 J-uniqueness implication that farther allocations incur higher cost within the forcing chain from T0 to T8.
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