Cost Ndim Symmetry Coeff Permutation Invariant

Permutation symmetry

Permutation symmetry is the property that a mathematical expression does not change when its inputs are rearranged. In the Recognition Science framework, the declaration CoeffPermutationInvariant formalizes this for coefficient weights: a vector of coefficients is permutation invariant when swapping any two positions leaves every coefficient unchanged. The definition states that for any permutation σ of the index set, applying σ to the index and reading the coefficient gives the same value as reading the original index. In plain terms, the coefficients do not care which slot they occupy.

The framework proves two directions of this property. First, if all coefficients are equal (a condition called uniform weights), then permutation invariance follows immediately: equal numbers cannot change under rearrangement. Second, and more substantively, if coefficients are permutation invariant and the dimension n is positive, then all coefficients must be equal. The proof constructs the common value by picking one index and showing every other coefficient equals it, using a swap permutation that exchanges the chosen index with any other. Together these two theorems establish that permutation invariance and uniformity are equivalent for positive dimensions.

This equivalence matters because it connects a symmetry condition to a concrete structural conclusion. In the framework's n-dimensional cost analysis, permutation invariance is the formal way to say the cost function treats all coordinate directions alike. The theorems show that this symmetry assumption is not a loose qualitative idea but a precise condition that forces the coefficients to collapse to a single shared value. The result holds for any positive integer dimension n, from one dimension upward, and the proof is machine-checked in the framework's library of formal theorems.

The declaration does not claim that this equivalence holds in dimension zero, where the positivity condition is absent. It also does not assert that permutation invariance is the only symmetry of interest, nor does it say anything about what the common coefficient value must be. The theorems establish the equivalence between two conditions; they do not determine the numerical value of the uniform coefficient, which remains a separate question in the framework's cost analysis.

theorem coeff_perm_invariant_of_uniform {n : ℕ} {α : Vec n}
    (hU : UniformWeights α) :
    CoeffPermutationInvariant α := by
  rcases hU with ⟨a, ha⟩
  intro σ i
  simp [ha]

theorem uniform_of_coeff_perm_invariant {n : ℕ} (hn : 0 < n) {α : Vec n}
    (hperm : CoeffPermutationInvariant α) :
    UniformWeights α := by
  let i0 : Fin n := ⟨0, hn⟩
  refine ⟨α i0, ?_⟩
  intro i
  have h := hperm (Equiv.swap i0 i) i0
  simpa using h

What this page does not claim

The equivalence does not hold without the positivity condition on dimension n. The theorems do not determine the numerical value of the uniform coefficient. Permutation invariance is not claimed to be the only symmetry of interest in the cost framework.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/Ndim/Symmetry.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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