strict_transmutation_progress
For any nuclear configuration whose stability ratio differs from 1, a direct transmutation step reaches the stable configuration while strictly lowering J-cost. Nuclear engineers working on waste transmutation pathways cite the result to guarantee monotonic descent. The term proof invokes the positivity of cost for unstable nuclei, the zero cost of the stable endpoint, and constructs the step record with two linarith discharges.
claimLet $cfg$ be a nuclear configuration with stability ratio $x ≠ 1$. Then there exists a transmutation step whose initial configuration is $cfg$ and whose final configuration is the stable (doubly-magic) nucleus such that the J-cost of the final state is strictly smaller than the J-cost of the initial state.
background
NuclearConfig is the structure carrying a positive real ratio $x$ that equals 1 precisely when the nucleus is doubly magic and stable. nuclearCost is the function that returns Jcost of this ratio, serving as the instability measure in the EN-006 module. TransmutationStep is the record type whose reduces_cost field records that the final J-cost is at most the initial J-cost. The module derives fission-product transmutation from the J-cost barrier structure, with stable_config defined as the ratio-1 nucleus whose cost is identically zero.
proof idea
The term proof first obtains a positive initial cost via transmutation_cost_pos applied to the instability hypothesis. It then obtains the zero cost of stable_config via stable_config_zero_cost. The refine tactic builds the TransmutationStep record with initial $cfg$ and final stable_config, after which linarith discharges both the non-strict reduction inequality and the strict inequality required by the existential.
why it matters in Recognition Science
The declaration supplies the strict-progress half of the EN-006.12 claim inside the fission-transmutation module. It is invoked by the en006_certificate that records the full derived status of the EN-006 suite. Within Recognition Science it confirms that every unstable configuration admits a cost-decreasing step to the nearest zero-cost attractor, consistent with the optimal-path existence asserted in the module documentation.
scope and limits
- Does not establish physical realizability of the constructed step.
- Does not produce an explicit sequence of intermediate nuclei.
- Does not bound the total number of steps to stability.
- Does not incorporate decay rates or cross sections.
formal statement (Lean)
181theorem strict_transmutation_progress
182 (cfg : NuclearConfig) (h_unstable : cfg.ratio ≠ 1) :
183 ∃ step : TransmutationStep,
184 step.initial = cfg ∧
185 nuclearCost step.final < nuclearCost step.initial := by
proof body
Term-mode proof.
186 have hcost_pos := transmutation_cost_pos cfg h_unstable
187 have hscz : nuclearCost stable_config = 0 := stable_config_zero_cost
188 refine ⟨⟨cfg, stable_config, ?_⟩, rfl, ?_⟩
189 · linarith [hscz.le, hcost_pos]
190 · linarith [hscz.le, hcost_pos]
191
192/-! ## §VI. RS Transmutation Efficiency -/
193
194/-- The transmutation efficiency: ratio of cost reduction to initial cost. -/