Explain the Lean structure `DarkMatterHaloCert` in module `IndisputableMonolith.Physics.DarkMatterHaloProfileFromRS`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

(1) Plain English

DarkMatterHaloCert is a Lean record (structure) that packages three proved facts about dark-matter halo density profiles in Recognition Science (RS):

• There exist exactly five canonical regimes (NFW inner, NFW outer, Einasto, isothermal sphere, truncation edge).
• Each regime is assigned a density that is a positive real number of the form 1 / phi^k for rung k.
• These densities are strictly decreasing as k increases.

The structure is instantiated by the concrete certificate darkMatterHaloCert.

(2) Why it matters in Recognition Science

RS derives all constants and structural features from the single J-cost functional equation. This certificate shows that the five-regime halo taxonomy and the phi-ladder density scaling are forced by the same arithmetic (phi-powers) that produces the other RS constants. It supplies a named, machine-checked object that downstream modules can depend on when linking halo profiles to the recognition ladder.

(3) How to read the formal statement

structure DarkMatterHaloCert where
  five_regimes : Fintype.card HaloRegime = 5
  density_always_pos : ∀ k, 0 < densityRung k
  density_strictly_decreasing : ∀ k, densityRung (k + 1) < densityRung k

• HaloRegime is an inductive type with five constructors.
• densityRung k := 1 / phi ^ k (noncomputable definition).
• The three fields are the bundled theorems.

(4) Visible dependencies or certificates in the supplied source

• haloRegime_count proves the cardinality is 5.
• density_pos proves positivity for every k.
• density_strictDecr proves strict decrease.
• darkMatterHaloCert is the concrete instance that assembles the three proofs.
• densityRung and HaloRegime are the supporting definitions; phi is imported from IndisputableMonolith.Constants.

(5) What this declaration does not prove

It does not derive the choice of the five specific regimes from the J-cost equation or the forcing chain; it does not prove that these profiles match observed rotation curves; and it does not connect the halo certificate to the master RealityCertificate or spacetime emergence theorems.