local_cache_forcing_certificate
plain-language theorem explainer
The local cache forcing certificate assembles four properties of the J-cost function evaluated at powers of the golden ratio: strict increase with exponent, zero value at the origin, strict positivity for positive distances, and uniqueness of phi as the positive solution to the quadratic r^2 = r + 1. Researchers working on the brain holography argument would cite this result to close Gap G1. The proof is a one-line term that directly invokes the four supporting lemmas on monotonicity and the Fibonacci characterization.
Claim. Let $J$ be the J-cost function and let $phi$ be the golden ratio. Then $J(phi^m) < J(phi^n)$ whenever $m < n$ are natural numbers, $J(phi^0) = 0$, $J(phi^0) < J(phi^d)$ for every natural number $d > 0$, and every positive real $r$ satisfying $r^2 = r + 1$ equals $phi$.
background
The module establishes that J-cost minimization on connected graphs forces hierarchical local caching. J-cost is the function satisfying the Recognition Composition Law, with the explicit form $J(x) = (x + x^{-1})/2 - 1$. The four upstream lemmas supply the required pieces: access_cost_zero_at_origin states that Jcost (phi ^ 0) = 0; collocation_minimizes_cost proves Jcost (phi ^ 0) < Jcost (phi ^ d) for d > 0; Jcost_phi_pow_strictMono gives the strict increase J(phi^m) < J(phi^n) for m < n; and phi_from_fibonacci_ratio shows that any positive r obeying the quadratic recurrence equals phi.
proof idea
The proof is a term-mode construction that packages the four lemmas into a single conjunction via the exact constructor. It applies Jcost_phi_pow_strictMono for the first conjunct, access_cost_zero_at_origin for the second, collocation_minimizes_cost for the third, and phi_from_fibonacci_ratio for the fourth. No further tactics or reductions appear.
why it matters
This is the master certificate of the LocalCacheForcing module and directly supports the claim that J-cost minimization forces local caching, thereby closing Gap G1 in the brain holography proof. It encodes the cost-growth and uniqueness properties that align with T6 (phi forced as self-similar fixed point) and the eight-tick octave structure. No downstream theorems are recorded yet.
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