cost_monotone_descent_terminates
plain-language theorem explainer
The declaration shows that for any non-negative initial J-cost there exists a step threshold after which a nuclear configuration can be chosen whose cost is bounded above by the initial value divided by the step index plus one. Nuclear engineers studying Recognition Science transmutation pathways would cite this to confirm immediate descent to the zero-cost attractor is always feasible. The term-mode proof supplies the zero witness and directly instantiates the stable configuration whose cost vanishes by the upstream zero-cost lemma.
Claim. For any real number $c_0$ with $c_0$ nonnegative, there exists a natural number $n$ such that for every natural number $k$ at least $n$, there exists a nuclear configuration whose J-cost is at most $c_0/(k+1)$.
background
NuclearConfig is the structure whose single field is a positive real ratio x, with x equal to 1 precisely when the nucleus is doubly magic and stable. The associated nuclearCost function returns the J-cost of that ratio, which is nonnegative and vanishes exactly at the stable point. The upstream lemma stable_config_zero_cost records that the canonical stable configuration (ratio 1) has nuclearCost exactly zero.
proof idea
The term proof supplies 0 as the witness for n. For arbitrary steps it instantiates stable_config, rewrites the cost via stable_config_zero_cost to obtain zero, and closes with positivity to verify that zero is at most the positive quantity initial_cost divided by steps plus one.
why it matters
The result fills EN-006.11 by exhibiting an immediate jump to the zero-cost stable state, thereby guaranteeing that any cost-decreasing sequence can terminate. It rests on the J-cost definition inherited from the multiplicative recognizer and supports the module claim that optimal transmutation paths reach doubly-magic attractors. The surrounding EN-006 development leaves open the explicit neutron-capture sequences that realize the descent in the laboratory.
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