Update

.codex-gitconfig

@@ -0,0 +1,10 @@
+[safe]
+	directory = C:/dev/OpenGA/.lake/packages/aesop
+	directory = C:/dev/OpenGA/.lake/packages/batteries
+	directory = C:/dev/OpenGA/.lake/packages/Cli
+	directory = C:/dev/OpenGA/.lake/packages/importGraph
+	directory = C:/dev/OpenGA/.lake/packages/LeanSearchClient
+	directory = C:/dev/OpenGA/.lake/packages/mathlib
+	directory = C:/dev/OpenGA/.lake/packages/plausible
+	directory = C:/dev/OpenGA/.lake/packages/proofwidgets
+	directory = C:/dev/OpenGA/.lake/packages/Qq

Riemannian.lean

@@ -4,6 +4,7 @@ import Riemannian.Curvature
 import Riemannian.Foundations.Notation
 import Riemannian.Foundations.Tactic
 import Riemannian.Gradient
+import Riemannian.LoopSpace
 import Riemannian.Metric
 import Riemannian.SecondFundamentalForm
 import Riemannian.TangentBundle.Smoothness
@@ -49,6 +50,8 @@ AltRegularity                ← consumer
     $|A|^2$, mean curvature.
   * `Gradient.lean`   — manifold gradient via Riesz duality, gradient norm
     squared.
+  * `LoopSpace.lean` — based and free loop-space primitives over a
+    topological space, parameterized by `[0, 1]`.
   * `BumpFunction.lean` — scalar / radial / manifold bumps + tangent
     vector field extension (`OpenGALib.BumpFunction`).
 
@@ -100,6 +103,12 @@ intermediate identities) are internal and may change without notice.
   * `Riemannian.manifoldGradientNormSq`
   * `Riemannian.manifoldGradient_riesz`
 
+**Loop space** (`LoopSpace.lean`):
+  * `Riemannian.LoopInterval`
+  * `Riemannian.BasedLoop`, `Riemannian.LoopSpace`
+  * `Riemannian.FreeLoop`, `Riemannian.FreeLoopSpace`
+  * `Riemannian.BasedLoop.const`, `Riemannian.FreeLoop.const`
+
 **Bump functions** (`BumpFunction.lean`):
   * `OpenGALib.BumpFunction.expDamping`
   * `OpenGALib.BumpFunction.smoothStep`

Riemannian/LoopSpace.lean

@@ -0,0 +1,121 @@
+import Mathlib.Topology.Basic
+import Mathlib.Data.Real.Basic
+import Mathlib.Tactic
+
+/-!
+# Riemannian.LoopSpace
+
+Basic loop-space primitives.
+
+This file defines:
+
+* `Riemannian.BasedLoop X base`: continuous loops in `X` based at `base`;
+* `Riemannian.FreeLoop X`: continuous loops in `X` with no chosen basepoint.
+
+Both are represented as continuous maps from the closed interval `[0, 1]`.
+This is intentionally topological rather than Riemannian: manifolds and
+Riemannian manifolds can consume the same definitions through their underlying
+topology.
+-/
+
+namespace Riemannian
+
+/-- The closed unit interval `[0, 1]`, used as the parameter space for loops. -/
+abbrev LoopInterval : Type := Set.Icc (0 : ℝ) 1
+
+namespace LoopInterval
+
+/-- The initial point `0` of the loop interval. -/
+def zero : LoopInterval :=
+  ⟨0, by constructor <;> norm_num⟩
+
+/-- The terminal point `1` of the loop interval. -/
+def one : LoopInterval :=
+  ⟨1, by constructor <;> norm_num⟩
+
+@[simp] theorem zero_val : ((zero : LoopInterval) : ℝ) = 0 := rfl
+
+@[simp] theorem one_val : ((one : LoopInterval) : ℝ) = 1 := rfl
+
+end LoopInterval
+
+/-- The **based loop space** at `base`: continuous maps
+`γ : [0, 1] → X` satisfying `γ(0) = base` and `γ(1) = base`. -/
+structure BasedLoop (X : Type*) [TopologicalSpace X] (base : X) where
+  /-- The underlying map from the unit interval. -/
+  toFun : LoopInterval → X
+  /-- The loop is continuous. -/
+  continuous_toFun : Continuous toFun
+  /-- The loop starts at the basepoint. -/
+  map_zero : toFun LoopInterval.zero = base
+  /-- The loop ends at the basepoint. -/
+  map_one : toFun LoopInterval.one = base
+
+/-- Synonym emphasizing the type as a space of based loops. -/
+abbrev LoopSpace (X : Type*) [TopologicalSpace X] (base : X) : Type _ :=
+  BasedLoop X base
+
+namespace BasedLoop
+
+variable {X : Type*} [TopologicalSpace X] {base : X}
+
+instance : CoeFun (BasedLoop X base) (fun _ => LoopInterval → X) :=
+  ⟨BasedLoop.toFun⟩
+
+@[simp] theorem apply_zero (γ : BasedLoop X base) :
+    γ LoopInterval.zero = base :=
+  γ.map_zero
+
+@[simp] theorem apply_one (γ : BasedLoop X base) :
+    γ LoopInterval.one = base :=
+  γ.map_one
+
+/-- The constant based loop at `base`. -/
+def const (base : X) : BasedLoop X base where
+  toFun := fun _ => base
+  continuous_toFun := continuous_const
+  map_zero := rfl
+  map_one := rfl
+
+instance instNonempty (base : X) : Nonempty (BasedLoop X base) :=
+  ⟨const base⟩
+
+end BasedLoop
+
+/-- The **free loop space** of `X`: continuous maps `γ : [0, 1] → X`
+whose endpoints agree. No basepoint is fixed. -/
+structure FreeLoop (X : Type*) [TopologicalSpace X] where
+  /-- The underlying map from the unit interval. -/
+  toFun : LoopInterval → X
+  /-- The loop is continuous. -/
+  continuous_toFun : Continuous toFun
+  /-- The endpoints agree. -/
+  isLoop : toFun LoopInterval.zero = toFun LoopInterval.one
+
+/-- Synonym emphasizing the type as a space of free loops. -/
+abbrev FreeLoopSpace (X : Type*) [TopologicalSpace X] : Type _ :=
+  FreeLoop X
+
+namespace FreeLoop
+
+variable {X : Type*} [TopologicalSpace X]
+
+instance : CoeFun (FreeLoop X) (fun _ => LoopInterval → X) :=
+  ⟨FreeLoop.toFun⟩
+
+@[simp] theorem apply_zero_eq_apply_one (γ : FreeLoop X) :
+    γ LoopInterval.zero = γ LoopInterval.one :=
+  γ.isLoop
+
+/-- The constant free loop at a point. -/
+def const (x : X) : FreeLoop X where
+  toFun := fun _ => x
+  continuous_toFun := continuous_const
+  isLoop := rfl
+
+noncomputable instance instNonempty [h : Nonempty X] : Nonempty (FreeLoop X) :=
+  ⟨const (Classical.choice h)⟩
+
+end FreeLoop
+
+end Riemannian