1. Plain English Statement
CoeffPermutationInvariant defines what it means for an $N$-dimensional vector of coefficients to be perfectly symmetric. It states that if you take a list of values and shuffle their index positions in any possible way, the value at any given index remains identically the same. The only way this can be mathematically true is if every element in the list is exactly equal to the others.
2. Role in Recognition Science In Recognition Science, physics emerges without fitted parameters. When scaling cost functions to $N$ dimensions, the framework requires structural isotropy—no spatial or logical dimension can be privileged with a unique weight by fiat. This MODEL defines the exact permutation symmetry required to prevent arbitrary dimensional biasing, serving as the formal condition that forces dimensional weights to be identically uniform.
3. Reading the Formal Statement
def CoeffPermutationInvariant {n : ℕ} (α : Vec n) : Prop :=
∀ σ : Equiv.Perm (Fin n), ∀ i : Fin n, α (σ i) = α i
def ... : Prop: This declares a mathematical proposition (a true/false statement).{n : ℕ} (α : Vec n): The property applies to any vector $\alpha$ of length $n$.∀ σ : Equiv.Perm (Fin n): For every possible mathematical permutation (shuffle) $\sigma$ of the $n$ indices...∀ i : Fin n: ...and for every specific index $i$...α (σ i) = α i: ...the value of the vector at the permuted position $\sigma(i)$ equals the value at the original position $i$.
4. Visible Dependencies and Certificates This MODEL serves as the central hub for two bidirectional certificates in the supplied source:
- coeff_perm_invariant_of_uniform (THEOREM): Proves the trivial direction—if weights are already known to be uniform, they satisfy this permutation invariance.
- uniform_of_coeff_perm_invariant (THEOREM): Proves the crucial forcing direction—if a vector possesses this permutation symmetry (and $n > 0$), the weights are strictly forced to be uniform.
5. Limitations (What it does not prove) This declaration is purely a definitional predicate (MODEL). It does not prove that nature must adopt this symmetry globally (that requires upstream forcing theorems). Additionally, it dictates only that the coefficients are equal to one another; it does not dictate their absolute magnitude or require them to be normalized (e.g., forcing them to sum to $1$), which requires separate calibration constraints.