Cost Ndim Connections X Flat Connection

The flat connection

In differential geometry, a connection is a rule for comparing vectors at nearby points, and a flat connection is the special case where the comparison is path-independent: transport a vector around any closed loop and it returns unchanged. The declaration xFlatConnection in the framework's machine-checked library of formal theorems defines the zero connection on an n-dimensional space. Its defining formula is simply that the connection coefficients are all zero, which is the most direct way to satisfy the flatness condition. The theorem xFlatConnection_apply confirms this by showing that applying the connection to any indices yields zero.

The declaration's purpose is to serve as a baseline in a comparison with a second connection, called tPulledConnection. The second connection arises from changing coordinates from t to x via the relation t = log x, a standard logarithmic transformation. In the t-coordinates, the space is flat by construction; pulling the connection back to x-coordinates introduces a diagonal term of the form Γⁱᵢᵢ = -1/xᵢ. The theorem tPulledConnection_diag states exactly this diagonal coefficient, while tPulledConnection_offDiag proves all off-diagonal coefficients vanish.

The framework then asks when these two connections are projectively equivalent, meaning they define the same straight lines up to reparametrization. The theorem projectivelyEquivalent_one_dim proves that in one dimension, the pulled-back connection is projectively equivalent to the zero connection. In two or more dimensions, however, the theorem not_projectivelyEquivalentToZeroAt_tPulledConnection shows this equivalence fails, provided no coordinate xᵢ is zero. This dichotomy is the substantive content: the one-dimensional case is special, and the framework's higher-dimensional structure is genuinely different from the flat baseline.

In Recognition Science, this distinction matters because the framework derives spatial dimensions from a recognition cycle, and this connection result is part of the geometric scaffolding. The declaration does not itself assert anything about physics, curvature, or the number of dimensions; it is a definitional building block. Its role is to fix a reference point so that the framework can state, in a machine-checked way, that the logarithmic-coordinate connection is not projectively flat in higher dimensions.

/-- The flat affine connection in `x`-coordinates. -/
def xFlatConnection {n : ℕ} (_x : Vec n) (_i _j _k : Fin n) : ℝ := 0

@[simp] theorem xFlatConnection_apply {n : ℕ} (x : Vec n) (i j k : Fin n) :
    xFlatConnection x i j k = 0 := rfl

theorem projectivelyEquivalent_one_dim {x : Vec 1} :
    ProjectivelyEquivalentToZeroAt (tPulledConnection x) := by
  refine ⟨fun _ => -((x 0)⁻¹) / 2, ?_⟩
  intro i j k
  fin_cases i
  fin_cases j
  fin_cases k
  simp [delta, tPulledConnection]

theorem not_projectivelyEquivalentToZeroAt_tPulledConnection {n : ℕ}
    (hn : 2 ≤ n) (x : Vec n) (hx : ∀ i : Fin n, x i ≠ 0) :
    ¬ ProjectivelyEquivalentToZeroAt (tPulledConnection x) := by
  let i0 : Fin n := ⟨0, lt_of_lt_of_le (by decide : 0 < 2) hn⟩
  let i1 : Fin n := ⟨1, lt_of_lt_of_le (by decide : 1 < 2) hn⟩
  have hi01 : i0 ≠ i1 := by
    simp [i0, i1]
  intro hproj
  rcases hproj with ⟨ψ, hψ⟩
  have hpsi1 : ψ i1 = 0 := by
    have h := hψ i0 i0 i1
    simpa [eq_comm, delta, tPulledConnection, hi01] using h
  have hdiag : tPulledConnection x i1 i1 i1 = delta i1 i1 * ψ i1 + delta i1 i1 * ψ i1 := by
    simpa using hψ i1 i1 i1
  have hxinv_zero : (x i1)⁻¹ = 0 := by
    have h' : -(x i1)⁻¹ = 0 := by
      simpa [delta, tPulledConnection, hpsi1] using hdiag
    exact neg_eq_zero.mp h'
  exact (inv_ne_zero (hx i1)) hxinv_zero

What this page does not claim

The declaration does not assert any physical law or property of space. The theorem does not prove that the pulled-back connection is flat in any dimension; it only addresses projective equivalence to the zero connection. The result does not apply when any coordinate xᵢ is zero, a condition stated in the theorem's hypothesis.

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/Connections.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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