Cost Ndim Symmetry Uniform Of Coeff Perm Invariant

Permutation symmetry

In the mathematics of cost functions, a vector of weights assigns a relative importance to each coordinate. A permutation symmetry says the weights do not care which coordinate they sit on: swap any two indices, and the weight values stay exactly the same. The framework's machine-checked library of formal theorems proves that this symmetry is equivalent to uniformity: if a positive number of coordinates all have the same weight, the weights are permutation-invariant, and conversely, if the weights are permutation-invariant, they are all equal to one another.

The forward direction is immediate from the definition. The reverse direction is the substantive result: uniform_of_coeff_perm_invariant establishes that a permutation-invariant weight vector over a positive number of coordinates must be constant. The proof picks one coordinate as a reference, then uses a transposition (a swap of two coordinates) to show every other coordinate carries the same weight as the reference. The theorem holds for any finite dimension n greater than zero, and it requires no continuity, no differentiability, and no regularity condition beyond the symmetry itself.

In Recognition Science, this result matters because the framework's cost function J(x) = (x + 1/x)/2 - 1 is derived from conditions that include a forced composition law. When the framework extends the cost to multiple dimensions, the permutation symmetry is one of the plain conditions that helps pin down the form of the multidimensional cost. The theorem does not by itself derive the full multidimensional cost; it only characterizes what permutation symmetry forces about the weights.

The classical picture is familiar from any symmetric problem: if a system looks the same after relabeling its parts, the parts must be treated equally. The theorem makes that intuition precise for finite weight vectors, and it does so without any smoothness assumption. A reader who knows only that a cost function is symmetric under coordinate swaps can already conclude that no coordinate is special.

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

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 theorem does not derive the full multidimensional cost function, only the uniformity of weights under permutation symmetry. The theorem does not apply to zero coordinates, since the proof requires a positive dimension n. The theorem does not require or establish any continuity or smoothness property of the cost function.

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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