perfect_transmutation_efficiency
The theorem asserts that transmuting any unstable nuclear configuration (ratio ≠ 1) to the stable configuration achieves exactly unit efficiency. Researchers modeling nuclear waste reduction via J-cost descent in Recognition Science would reference this result. The proof is a short tactic sequence that unfolds the efficiency definition, derives a nonzero cost from the instability hypothesis via nuclear_cost_zero_iff_stable, and simplifies using the zero-cost property of the stable state.
claimLet $cfg$ be a nuclear configuration with stability ratio $x ≠ 1$. Then the efficiency of transmuting $cfg$ to the stable configuration is exactly 1.
background
In the EN-006 module on Fission Product Transmutation, a NuclearConfig is a structure with positive real ratio $x$, where $x=1$ denotes a stable doubly-magic nucleus and $x≠1$ an unstable one. The J-cost is defined by nuclearCost $cfg$ := Jcost (cfg.ratio), which vanishes precisely when the ratio equals 1 according to nuclear_cost_zero_iff_stable. The stable_config is the canonical configuration with ratio 1, which has zero cost by stable_config_zero_cost. The module derives transmutation properties from the J-cost barrier structure, where efficiency measures the fractional cost reduction along a path to stability.
proof idea
The proof unfolds the definition of transmutation_efficiency. It first establishes that nuclearCost $cfg$ ≠ 0 by applying nuclear_cost_zero_iff_stable to the hypothesis cfg.ratio ≠ 1. Then it simplifies the expression using the zero cost of stable_config, yielding the result 1.
why it matters in Recognition Science
This theorem completes the derivation of perfect efficiency for transmutation to the stable state within the EN-006 framework. It feeds directly into the en006_certificate, which summarizes the status of all derived properties for fission product transmutation. The result relies on the J-cost structure where stable configurations are zero-cost attractors, aligning with the Recognition Science claim that optimal transmutation follows cost descent to doubly-magic nuclei.
scope and limits
- Does not apply when the initial configuration is already stable.
- Does not specify the sequence of physical steps required for transmutation.
- Does not address the energy barriers or reaction rates in actual nuclear processes.
- Does not extend to configurations outside the positive ratio domain.
formal statement (Lean)
215theorem perfect_transmutation_efficiency (cfg : NuclearConfig) (h_unstable : cfg.ratio ≠ 1) :
216 transmutation_efficiency cfg stable_config = 1 := by
proof body
Tactic-mode proof.
217 unfold transmutation_efficiency
218 have h0 : nuclearCost cfg ≠ 0 := by
219 intro h
220 exact h_unstable ((nuclear_cost_zero_iff_stable cfg).mp h)
221 simp [h0, stable_config_zero_cost]
222
223/-! ## §VII. Summary -/
224
225/-- The RS fission transmutation theorem.
226 Derives key properties of transmutation from J-cost structure. -/