The declaration DarkMatterHaloCert appears in module IndisputableMonolith.Physics.DarkMatterHaloProfileFromRS.
(1) In plain English it is a record that bundles three proved facts: the type HaloRegime has exactly five inhabitants (haloRegime_count), the function densityRung k = 1/phi^k is always positive (density_pos), and the sequence is strictly decreasing (density_strictDecr).
(2) In Recognition Science the certificate supplies a parameter-free structural skeleton for dark-matter halo density profiles. Each of the five regimes sits one rung lower on the phi-ladder, so the model inherits the same reciprocal symmetry and discreteness already forced by the J-cost functional equation elsewhere in the canon.
(3) The formal statement is read as a Lean structure (record type) whose three fields are themselves theorems: five_regimes asserts Fintype.card HaloRegime = 5, density_always_pos asserts ∀ k, 0 < densityRung k, and density_strictly_decreasing asserts ∀ k, densityRung (k+1) < densityRung k. The value darkMatterHaloCert (darkMatterHaloCert) is the concrete inhabitant that supplies the three proofs.
(4) Visible dependencies inside the supplied source are the inductive definition HaloRegime (five constructors), the theorem haloRegime_count, the definition densityRung, and the two supporting lemmas density_pos and density_strictDecr. The module imports only Mathlib and IndisputableMonolith.Constants.
(5) The declaration proves only the cardinality and the positivity/monotonicity properties of densityRung. It does not prove any mapping from the five HaloRegime constructors to named profiles (NFW, Einasto, etc.), nor any dynamical equations, observational fits, or falsification criteria.