efficiency_bounded
Transmutation efficiency, defined as the relative J-cost reduction between two nuclear configurations, lies in the closed unit interval whenever the final configuration has J-cost no larger than the initial. Nuclear engineers modeling waste transmutation pathways cite the bound to certify feasible cost-descent sequences. The proof splits on whether the initial cost vanishes and then applies non-negativity of J-cost together with division inequalities.
claimLet $J$ be the J-cost of a nuclear configuration. If $J(f) ≤ J(i)$ for configurations $i$ and $f$, then $0 ≤ η ≤ 1$ where $η = 1$ whenever $J(i) = 0$ and $η = (J(i) - J(f))/J(i)$ otherwise.
background
In the EN-006 module a NuclearConfig is a structure carrying a positive real stability ratio x, with x = 1 marking the stable doubly-magic nuclei. The associated nuclearCost is the J-cost J(x) that measures departure from this ideal state; non-negativity of nuclearCost follows from the corresponding property of the J function. Transmutation efficiency between an initial and final configuration is the relative cost reduction: identically 1 when the initial state is already stable, otherwise the normalized drop (initial cost minus final cost) divided by initial cost.
proof idea
The term proof unfolds the definition of transmutation_efficiency and splits on the predicate nuclearCost initial = 0. The zero-cost case returns 1, which lies in [0,1] by direct arithmetic. In the positive-cost case it invokes nuclear_cost_nonneg on both configurations to obtain non-negative numerator and denominator, then applies div_nonneg and the division lemma div_le_one together with linarith to conclude the upper bound.
why it matters in Recognition Science
The theorem supplies the elementary interval bound required by the EN-006 derivation of fission-product transmutation. It is invoked by the en006_certificate that aggregates all derived statements for the module. Within Recognition Science the result underwrites the interpretation of transmutation as monotonic descent on the J-cost landscape toward the x = 1 attractor, consistent with the J-uniqueness and cost-monotone-descent steps of the forcing chain.
scope and limits
- Does not apply when final J-cost exceeds initial J-cost.
- Does not assert existence of any transmutation path between given configurations.
- Does not incorporate nuclear reaction rates or cross sections.
- Does not bound efficiency for multi-step paths beyond the single-step case.
formal statement (Lean)
200theorem efficiency_bounded (initial final : NuclearConfig)
201 (h : nuclearCost final ≤ nuclearCost initial) :
202 0 ≤ transmutation_efficiency initial final ∧
203 transmutation_efficiency initial final ≤ 1 := by
proof body
Term-mode proof.
204 unfold transmutation_efficiency
205 split_ifs with h0
206 · constructor <;> norm_num
207 · constructor
208 · apply div_nonneg
209 · linarith [nuclear_cost_nonneg final]
210 · exact nuclear_cost_nonneg initial
211 · rw [div_le_one (lt_of_le_of_ne (nuclear_cost_nonneg initial) (Ne.symm h0))]
212 linarith [nuclear_cost_nonneg final]
213
214/-- **THEOREM EN-006.14**: Perfect transmutation (to stable state) has 100% efficiency. -/