Explain the Lean def `CoeffPermutationInvariant` in module `IndisputableMonolith.Cost.Ndim.Symmetry`. 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.

The supplied source contains the module IndisputableMonolith.Cost.Ndim.Symmetry with the declaration CoeffPermutationInvariant.

(1) In plain English, the declaration asserts that a coefficient vector α of length n remains unchanged when its indices are arbitrarily reordered: every permutation of the positions leaves the values identical.

(2) In Recognition Science the property encodes permutation symmetry of coefficient weights in N-dimensional cost structures, ensuring that the cost model treats all directions equivalently.

(3) The formal statement is read as: given natural number n and vector α : Vec n, the predicate holds precisely when ∀ σ : Equiv.Perm (Fin n), ∀ i : Fin n, α (σ i) = α i.

(4) Visible dependencies appear in the same module via coeff_perm_invariant_of_uniform, which derives the invariance from UniformWeights, and uniform_of_coeff_perm_invariant, which recovers uniform weights from the invariance when n > 0. The module imports Calibration but supplies no further certificates.

(5) The declaration defines only the invariance predicate; it does not prove existence of non-uniform coefficients, specific numerical values, or any link to J-cost, phi, or the forcing chain.