stable_is_optimal
The theorem shows that the stable nuclear configuration (ratio exactly 1) attains the global minimum J-cost among all admissible configurations. Nuclear engineers working on transmutation pathways cite it to certify that doubly-magic nuclei are the lowest-cost endpoints. The proof is a one-line term reduction that substitutes the zero-cost identity for the stable state and invokes non-negativity of J-cost for any other configuration.
claimFor any nuclear configuration with stability ratio $x > 0$, the J-cost of the stable configuration satisfies $J(1) ≤ J(x)$.
background
NuclearConfig is the structure whose single datum is a positive real ratio x, with x = 1 marking a doubly-magic nucleus. The function nuclearCost maps each such configuration to its J-cost J(x), which quantifies departure from the ideal ledger ratio. The constant stable_config is the NuclearConfig whose ratio is exactly 1. The upstream lemma nuclear_cost_nonneg states that J(x) ≥ 0 for every admissible x, while stable_config_zero_cost states that J(1) = 0. The module EN-006 derives transmutation conditions from the Recognition Science J-cost barrier, treating fission products as high-J states that descend toward the x = 1 attractor.
proof idea
The term proof first rewrites the left-hand side via the identity that the stable configuration has zero J-cost, then applies the non-negativity theorem nuclear_cost_nonneg to the arbitrary configuration on the right-hand side.
why it matters in Recognition Science
The result supplies the optimality half of the EN-006.8 claim inside the fission-transmutation certificate. It is invoked by en006_certificate to close the derivation that every valid transmutation path ends at a minimal-cost state. Within the Recognition Science framework the statement instantiates the J-cost minimum at the self-similar fixed point x = 1, consistent with the T5 J-uniqueness and T6 phi fixed-point structure.
scope and limits
- Does not prove existence or constructivity of any transmutation path.
- Does not incorporate nuclear reaction rates or cross sections.
- Does not address configurations outside the positive-ratio NuclearConfig type.
formal statement (Lean)
137theorem stable_is_optimal (cfg : NuclearConfig) :
138 nuclearCost stable_config ≤ nuclearCost cfg := by
proof body
Term-mode proof.
139 rw [stable_config_zero_cost]
140 exact nuclear_cost_nonneg cfg
141
142/-! ## §IV. Doubly-Magic Attractors -/
143
144/-- A doubly-magic attractor is a local cost minimum that is "nearby" in the φ-rung sense. -/