DoublyMagicAttractor
A doubly-magic attractor is a nuclear configuration whose J-cost is at most 1, marking it as a local minimum near the phi-ladder point x=1. Nuclear transmutation modelers cite the structure to designate stable endpoints for cost-reducing fission product sequences. The declaration is a direct structure wrapper around NuclearConfig that records the inequality nuclearCost config ≤ 1.
claimA doubly-magic attractor is a nuclear configuration $c$ (with positive real stability ratio $x$) such that the J-cost satisfies $J(c) ≤ 1$, where $J$ quantifies deviation from the ideal stable state at $x=1$.
background
In the EN-006 module on fission product transmutation, Recognition Science assigns every nuclear configuration a J-cost via nuclearCost, which applies the J functional to the ledger ratio of the configuration. The NuclearConfig structure parameterizes these objects by a positive real ratio, with the special value x=1 corresponding to perfectly stable doubly-magic nuclei that sit at local zero-cost minima. The module derives transmutation paths as sequences of recognition events that strictly reduce total J-cost, with doubly-magic nuclei serving as the attractors at the end of any optimal descent.
proof idea
This is a direct structure definition that packages an instance of NuclearConfig together with the predicate nuclearCost config ≤ 1. No lemmas or tactics are invoked; the definition simply records the local-minimum condition for later use in theorems such as stable_is_attractor.
why it matters in Recognition Science
The structure supplies the target type for the downstream theorem stable_is_attractor, which asserts that the zero-cost stable_config itself qualifies as a doubly-magic attractor. It fills the EN-006.9 slot in the fission-transmutation chain, connecting the J-cost barrier structure to the identification of doubly-magic nuclei as J(x)=0 attractors. In the broader Recognition Science framework it supports the claim that optimal transmutation follows monotone cost descent to these local minima, consistent with the phi-ladder and the Recognition Composition Law.
scope and limits
- Does not prove existence of a transmutation path from an arbitrary fission product to a doubly-magic attractor.
- Does not compute explicit path lengths or branching factors.
- Does not incorporate electromagnetic or weak-interaction selection rules beyond the ledger ratio.
- Does not address the physical realizability or cross-section of the implied recognition events.
Lean usage
theorem stable_is_attractor : ∃ a : DoublyMagicAttractor, a.config = stable_config := ⟨⟨stable_config, by rw [stable_config_zero_cost]; norm_num⟩, rfl⟩formal statement (Lean)
145structure DoublyMagicAttractor where
146 /-- The attractor configuration (near x = 1 on φ-ladder). -/
147 config : NuclearConfig
148 /-- It is in the local minimum region. -/
149 is_near_stable : nuclearCost config ≤ 1 -- within one E_coh of stability
150
151/-- **THEOREM EN-006.9**: The stable configuration is itself a doubly-magic attractor. -/