Explain the Lean theorem `uniform_of_coeff_perm_invariant` 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.

Here is an explanation of the Lean theorem uniform_of_coeff_perm_invariant from the Recognition Science canon.

1. Plain English Translation

The theorem states that if an $n$-dimensional system has a set of coefficients (or weights) that remain unchanged regardless of how you swap or shuffle their coordinate indices, and the system has at least one dimension ($n > 0$), then all the coefficients must be identical to one another.

2. Why it Matters in Recognition Science

Recognition Science (RS) is fundamentally a parameter-free formalization of physics. In multi-dimensional cost frameworks, attaching arbitrary weights to different coordinate axes introduces free parameters (empirical "dials" that must be fit). By proving that permutation symmetry forces uniform weights, this theorem mathematically guarantees that an isotropic space cannot harbor hidden parameters. It enforces parameter-freedom directly from a baseline structural symmetry.

3. How to Read the Formal Statement

• {n : ℕ}: An implicit natural number representing the dimension of the space.
• (hn : 0 < n): The hypothesis that the dimension is strictly greater than 0. This guarantees the space is not empty.
• {α : Vec n}: An implicit vector $\alpha$ containing $n$ coefficients.
• (hperm : CoeffPermutationInvariant α): The core premise that $\alpha$ satisfies CoeffPermutationInvariant, meaning $\alpha(\sigma(i)) = \alpha(i)$ for any valid index $i$ and any permutation $\sigma$.
• UniformWeights α: The conclusion that $\alpha$ possesses uniform weights (all elements share the same value).

4. Visible Dependencies and Proof Structure

Within the supplied source, the theorem explicitly depends on the predicate CoeffPermutationInvariant.

The proof mechanism is short and constructive:

1. Because $n > 0$ (hn), the system has a valid "base" index i0 = 0.
2. For any arbitrary index i, Lean constructs a specific permutation that only swaps i0 and i (Equiv.swap i0 i).
3. By feeding this specific swap permutation into the hperm symmetry rule evaluated at i0, the proof derives $\alpha(i) = \alpha(i_0)$. Since this holds for any index $i$, all weights evaluate to the exact same base weight.

5. What this Declaration Does Not Prove

• That space is isotropic: It does not prove that an RS system must have permutation symmetry; it is a conditional theorem showing that if it possesses the symmetry, its weights are uniform.
• The dimension of space: It applies universally to any integer dimension $n > 0$. The RS proof that spatial dimension $D = 3$ is handled elsewhere in the framework (e.g., via topological linking).
• The numerical value of the weights: It proves that all weights are identical, but it does not evaluate their actual scalar magnitude.