diff --git a/CompPoly.lean b/CompPoly.lean
index 6eefd4d9..480b8dfc 100644
--- a/CompPoly.lean
+++ b/CompPoly.lean
@@ -104,6 +104,7 @@ import CompPoly.Fields.KoalaBear.Basic
 import CompPoly.Fields.KoalaBear.Fast
 import CompPoly.Fields.Mersenne31
 import CompPoly.Fields.Mersenne31.Basic
+import CompPoly.Fields.Mersenne31.Circle
 import CompPoly.Fields.Mersenne31.Fast
 import CompPoly.Fields.PrattCertificate
 import CompPoly.Fields.Secp256k1
diff --git a/CompPoly/Fields/Mersenne31.lean b/CompPoly/Fields/Mersenne31.lean
index fb5dac89..2303d1e7 100644
--- a/CompPoly/Fields/Mersenne31.lean
+++ b/CompPoly/Fields/Mersenne31.lean
@@ -5,6 +5,7 @@ Authors: Quang Dao, Varun Thakore
 -/
 
 import CompPoly.Fields.Mersenne31.Basic
+import CompPoly.Fields.Mersenne31.Circle
 import CompPoly.Fields.Mersenne31.Fast
 
 /-!
@@ -12,5 +13,6 @@ import CompPoly.Fields.Mersenne31.Fast
 
   Facade module for the Mersenne31 field. It re-exports the canonical `ZMod` model
   from `CompPoly.Fields.Mersenne31.Basic` and the native-word implementation from
-  `CompPoly.Fields.Mersenne31.Fast`.
+  `CompPoly.Fields.Mersenne31.Fast`, plus the circle-domain skeleton from
+  `CompPoly.Fields.Mersenne31.Circle`.
 -/
diff --git a/CompPoly/Fields/Mersenne31/Circle.lean b/CompPoly/Fields/Mersenne31/Circle.lean
new file mode 100644
index 00000000..80aa0128
--- /dev/null
+++ b/CompPoly/Fields/Mersenne31/Circle.lean
@@ -0,0 +1,349 @@
+/-
+Copyright (c) 2026 CompPoly Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Adrien Lacombe
+-/
+
+import CompPoly.Fields.Mersenne31.Basic
+import Mathlib.Tactic.Ring
+
+/-!
+# Mersenne31 Circle Domains
+
+This module mirrors the structural circle-domain layer used by STWO over the
+Mersenne31 field. It records the circle equation, the STWO M31 circle generator,
+index arithmetic modulo the circle order, and the coset/domain shapes used by
+canonical circle domains.
+
+`CirclePointIndex.toPoint` currently interprets an index via `Point.nsmul` on the
+canonical natural representative `i.val`. A follow-on PR should prove generator
+order `2^31` and that `toPoint` is a group homomorphism on `CirclePointIndex`.
+-/
+
+namespace Mersenne31
+namespace Circle
+
+/-- The canonical Mersenne31 field used by the circle-domain skeleton. -/
+abbrev Field := Basic.Field
+
+/-- Predicate for points on the circle `x^2 + y^2 = 1`. -/
+def OnCircle (x y : Field) : Prop :=
+  x ^ 2 + y ^ 2 = 1
+
+/-- A point on the Mersenne31 circle. -/
+structure Point where
+  x : Field
+  y : Field
+  onCircle : OnCircle x y
+
+namespace Point
+
+@[ext]
+theorem ext {p q : Point} (hx : p.x = q.x) (hy : p.y = q.y) : p = q := by
+  cases p
+  cases q
+  simp_all
+
+/-- The identity point of the circle group. -/
+def zero : Point where
+  x := 1
+  y := 0
+  onCircle := by
+    simp [OnCircle]
+
+/-- The circle-group inverse, equal to complex conjugation. -/
+def conjugate (p : Point) : Point where
+  x := p.x
+  y := -p.y
+  onCircle := by
+    simpa [OnCircle, pow_two] using p.onCircle
+
+/-- The antipodal point. Reserved for follow-on circle-group lemmas. -/
+def antipode (p : Point) : Point where
+  x := -p.x
+  y := -p.y
+  onCircle := by
+    simpa [OnCircle, pow_two] using p.onCircle
+
+/-- Circle-group addition, written as multiplication of complex coordinates. -/
+def add (p q : Point) : Point where
+  x := p.x * q.x - p.y * q.y
+  y := p.x * q.y + p.y * q.x
+  onCircle := by
+    dsimp [OnCircle]
+    calc
+      (p.x * q.x - p.y * q.y) ^ 2 + (p.x * q.y + p.y * q.x) ^ 2 =
+          (p.x ^ 2 + p.y ^ 2) * (q.x ^ 2 + q.y ^ 2) := by
+        ring
+      _ = 1 := by
+        rw [p.onCircle, q.onCircle]
+        ring
+
+instance : Zero Point := ⟨zero⟩
+
+instance : Neg Point := ⟨conjugate⟩
+
+instance : Add Point := ⟨add⟩
+
+/-- Repeated circle-group addition. -/
+def nsmul (p : Point) : Nat → Point
+  | 0 => 0
+  | n + 1 => nsmul p n + p
+
+@[simp]
+theorem zero_x : (0 : Point).x = 1 := rfl
+
+@[simp]
+theorem zero_y : (0 : Point).y = 0 := rfl
+
+@[simp]
+theorem conjugate_x (p : Point) : (-p).x = p.x := rfl
+
+@[simp]
+theorem conjugate_y (p : Point) : (-p).y = -p.y := rfl
+
+@[simp]
+theorem zero_add (p : Point) : (0 : Point) + p = p := by
+  obtain ⟨px, py, hp⟩ := p
+  apply Point.ext
+  · change (1 : Field) * px - (0 : Field) * py = px
+    ring
+  · change (1 : Field) * py + (0 : Field) * px = py
+    ring
+
+end Point
+
+/-- STWO's Mersenne31 circle generator x-coordinate. -/
+def generatorX : Field := 2
+
+/-- STWO's Mersenne31 circle generator y-coordinate. -/
+def generatorY : Field := 1268011823
+
+/-- STWO's Mersenne31 circle generator lies on `x^2 + y^2 = 1`. -/
+theorem generator_onCircle : OnCircle generatorX generatorY := by
+  change ((2 : Field) ^ 2 + (1268011823 : Field) ^ 2 = 1)
+  decide
+
+/-- STWO's generator for the Mersenne31 circle group. -/
+def generator : Point where
+  x := generatorX
+  y := generatorY
+  onCircle := generator_onCircle
+
+/-- The log order of STWO's Mersenne31 circle group. -/
+@[reducible]
+def logOrder : Nat := 31
+
+/-- The order of the Mersenne31 circle-index group, `2^31`. -/
+@[reducible]
+def order : Nat := 2 ^ logOrder
+
+/-- Integer index for multiples of the Mersenne31 circle generator. -/
+abbrev CirclePointIndex := ZMod order
+
+namespace CirclePointIndex
+
+/-- The distinguished generator index. -/
+def generator : CirclePointIndex := 1
+
+/-- Subgroup generator index for the subgroup of order `2^logSize`. For
+`logSize ≤ logOrder` this is the canonical generator of that subgroup; callers
+that rely on the bound (`Coset.new`, `odds`, `halfOdds`) carry it themselves. -/
+def subgroupGen (logSize : Nat) : CirclePointIndex :=
+  (2 ^ (logOrder - logSize) : Nat)
+
+/-- Interpret an index as a repeated multiple of the STWO circle generator.
+
+Uses the canonical natural representative of `i`; see the module docstring for
+the planned homomorphism proof. -/
+def toPoint (i : CirclePointIndex) : Point :=
+  Point.nsmul Circle.generator i.val
+
+@[simp]
+theorem toPoint_zero : toPoint 0 = 0 := by
+  unfold toPoint
+  simp [Point.nsmul]
+
+@[simp]
+theorem toPoint_generator : toPoint CirclePointIndex.generator = Circle.generator := by
+  unfold toPoint CirclePointIndex.generator Circle.generator
+  change Point.nsmul Circle.generator (0 + 1) = Circle.generator
+  simp [Point.nsmul, Point.zero_add]
+
+@[simp]
+theorem subgroupGen_zero : subgroupGen 0 = 0 := by
+  change ((2 ^ (logOrder - 0) : Nat) : ZMod order) = 0
+  simp [order]
+
+@[simp]
+theorem subgroupGen_logOrder : subgroupGen logOrder = generator := by
+  simp [subgroupGen, generator]
+
+end CirclePointIndex
+
+/-- A coset of circle indices with a fixed additive step. -/
+structure Coset where
+  initialIndex : CirclePointIndex
+  stepSize : CirclePointIndex
+  logSize : Nat
+  logSize_le_logOrder : logSize ≤ logOrder
+
+namespace Coset
+
+/-- Create a coset with STWO's subgroup step for `logSize`. -/
+def new (initialIndex : CirclePointIndex) (logSize : Nat) (hlogSize : logSize ≤ logOrder) :
+    Coset where
+  initialIndex := initialIndex
+  stepSize := CirclePointIndex.subgroupGen logSize
+  logSize := logSize
+  logSize_le_logOrder := hlogSize
+
+/-- The additive subgroup of size `2^logSize`. -/
+def subgroup (logSize : Nat) (hlogSize : logSize ≤ logOrder) : Coset :=
+  new 0 logSize hlogSize
+
+/-- The STWO coset `G_{2n} + <G_n>`. -/
+def odds (logSize : Nat) (hlogSize : logSize + 1 ≤ logOrder) : Coset :=
+  new (CirclePointIndex.subgroupGen (logSize + 1)) logSize
+    (Nat.le_trans (Nat.le_succ logSize) hlogSize)
+
+/-- The STWO coset `G_{4n} + <G_n>`, whose conjugate completes `odds (logSize + 1)`. -/
+def halfOdds (logSize : Nat) (hlogSize : logSize + 2 ≤ logOrder) : Coset :=
+  new (CirclePointIndex.subgroupGen (logSize + 2)) logSize
+    (Nat.le_trans (Nat.le_add_right logSize 2) hlogSize)
+
+/-- Number of indices in the coset. -/
+def size (c : Coset) : Nat :=
+  2 ^ c.logSize
+
+/-- The `i`th index in the coset order. -/
+def indexAt (c : Coset) (i : Nat) : CirclePointIndex :=
+  c.initialIndex + c.stepSize * (i : CirclePointIndex)
+
+/-- The circle point at the `i`th coset index. -/
+def pointAt (c : Coset) (i : Nat) : Point :=
+  CirclePointIndex.toPoint (c.indexAt i)
+
+/-- The conjugate coset `-initial - <step>`. -/
+def conjugate (c : Coset) : Coset where
+  initialIndex := -c.initialIndex
+  stepSize := -c.stepSize
+  logSize := c.logSize
+  logSize_le_logOrder := c.logSize_le_logOrder
+
+@[simp]
+theorem indexAt_zero (c : Coset) : c.indexAt 0 = c.initialIndex := by
+  simp [indexAt]
+
+@[simp]
+theorem indexAt_succ (c : Coset) (i : Nat) :
+    c.indexAt (i + 1) = c.indexAt i + c.stepSize := by
+  simp [indexAt, Nat.cast_add, Nat.cast_one]
+  ring
+
+@[simp]
+theorem conjugate_logSize (c : Coset) : c.conjugate.logSize = c.logSize := rfl
+
+@[simp]
+theorem conjugate_initialIndex (c : Coset) :
+    c.conjugate.initialIndex = -c.initialIndex := rfl
+
+@[simp]
+theorem conjugate_stepSize (c : Coset) : c.conjugate.stepSize = -c.stepSize := rfl
+
+@[simp]
+theorem conjugate_indexAt (c : Coset) (i : Nat) :
+    c.conjugate.indexAt i = -c.indexAt i := by
+  simp [indexAt, conjugate]
+  ring
+
+end Coset
+
+/-- A valid STWO circle domain: a half coset followed by its conjugate. -/
+structure CircleDomain where
+  halfCoset : Coset
+
+namespace CircleDomain
+
+/-- Construct a circle domain from the half coset. -/
+def new (halfCoset : Coset) : CircleDomain where
+  halfCoset := halfCoset
+
+/-- Domain log size. A domain contains a half coset and its conjugate. -/
+def logSize (D : CircleDomain) : Nat :=
+  D.halfCoset.logSize + 1
+
+/-- Number of indices in the domain. -/
+def size (D : CircleDomain) : Nat :=
+  2 ^ D.logSize
+
+/-- The `i`th domain index: first the half coset, then the conjugate coset. -/
+def indexAt (D : CircleDomain) (i : Nat) : CirclePointIndex :=
+  if i < D.halfCoset.size then
+    D.halfCoset.indexAt i
+  else
+    D.halfCoset.conjugate.indexAt (i - D.halfCoset.size)
+
+/-- The circle point at the `i`th domain index. -/
+def pointAt (D : CircleDomain) (i : Nat) : Point :=
+  CirclePointIndex.toPoint (D.indexAt i)
+
+@[simp]
+theorem size_eq_two_mul_halfSize (D : CircleDomain) :
+    D.size = 2 * D.halfCoset.size := by
+  simp [size, logSize, Coset.size, pow_succ, Nat.mul_comm]
+
+@[simp]
+theorem indexAt_left (D : CircleDomain) (i : Nat) (hi : i < D.halfCoset.size) :
+    D.indexAt i = D.halfCoset.indexAt i := by
+  simp [indexAt, hi]
+
+@[simp]
+theorem indexAt_right (D : CircleDomain) (i : Nat) :
+    D.indexAt (D.halfCoset.size + i) = -D.halfCoset.indexAt i := by
+  have hnot : ¬ D.halfCoset.size + i < D.halfCoset.size := by
+    exact Nat.not_lt.mpr (Nat.le_add_right _ _)
+  simp [indexAt, hnot, Coset.conjugate_indexAt]
+
+end CircleDomain
+
+/-- A canonical STWO coset `G_{2n} + <G_n>` for `1 ≤ logSize < logOrder`. -/
+structure CanonicCoset where
+  logSize : Nat
+  one_le_logSize : 1 ≤ logSize
+  logSize_succ_le_logOrder : logSize + 1 ≤ logOrder
+
+namespace CanonicCoset
+
+/-- The full canonical coset `G_{2n} + <G_n>`. -/
+def coset (c : CanonicCoset) : Coset :=
+  Coset.odds c.logSize c.logSize_succ_le_logOrder
+
+/-- The half coset used to form the canonical circle domain. -/
+def halfCoset (c : CanonicCoset) : Coset :=
+  Coset.halfOdds (c.logSize - 1) (by
+    have hEq : c.logSize - 1 + 2 = c.logSize + 1 := by
+      have hOne : 1 ≤ c.logSize := c.one_le_logSize
+      omega
+    rw [hEq]
+    exact c.logSize_succ_le_logOrder)
+
+/-- The canonical circle domain with the same log size. -/
+def circleDomain (c : CanonicCoset) : CircleDomain :=
+  CircleDomain.new c.halfCoset
+
+@[simp]
+theorem circleDomain_logSize (c : CanonicCoset) : c.circleDomain.logSize = c.logSize := by
+  unfold circleDomain CircleDomain.new CircleDomain.logSize halfCoset Coset.halfOdds Coset.new
+  dsimp
+  have hOne : 1 ≤ c.logSize := c.one_le_logSize
+  omega
+
+@[simp]
+theorem circleDomain_size (c : CanonicCoset) : c.circleDomain.size = 2 ^ c.logSize := by
+  simp [CircleDomain.size]
+
+end CanonicCoset
+
+end Circle
+end Mersenne31
diff --git a/tests/CompPolyTests.lean b/tests/CompPolyTests.lean
index 0321cad5..0f05a7f9 100644
--- a/tests/CompPolyTests.lean
+++ b/tests/CompPolyTests.lean
@@ -24,6 +24,7 @@ import CompPolyTests.Data.MvPolynomial.Notation
 import CompPolyTests.Fields.Binary.AdditiveNTT.NovelPolynomialBasis
 import CompPolyTests.Fields.Binary.BF128Ghash.Prelude
 import CompPolyTests.Fields.KoalaBear.Fast
+import CompPolyTests.Fields.Mersenne31.Circle
 import CompPolyTests.Fields.PrattCertificate
 import CompPolyTests.LinearAlgebra.Dense
 import CompPolyTests.Multilinear.Equiv
diff --git a/tests/CompPolyTests/Fields/Mersenne31/Circle.lean b/tests/CompPolyTests/Fields/Mersenne31/Circle.lean
new file mode 100644
index 00000000..075438b6
--- /dev/null
+++ b/tests/CompPolyTests/Fields/Mersenne31/Circle.lean
@@ -0,0 +1,76 @@
+/-
+Copyright (c) 2026 CompPoly Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Adrien Lacombe
+-/
+
+import CompPoly.Fields.Mersenne31.Circle
+
+/-!
+# Mersenne31 Circle Domain Tests
+
+Regression checks for the STWO-style Mersenne31 circle-domain skeleton.
+-/
+
+namespace Mersenne31.Circle
+
+example : OnCircle generatorX generatorY := generator_onCircle
+
+example : generator.x = 2 := rfl
+
+example : generator.y = 1268011823 := rfl
+
+example : OnCircle (generator + generator).x (generator + generator).y :=
+  (generator + generator).onCircle
+
+example : OnCircle (Point.antipode generator).x (Point.antipode generator).y :=
+  (Point.antipode generator).onCircle
+
+#guard logOrder = 31
+#guard order = 2147483648
+
+example : CirclePointIndex.toPoint 0 = 0 := by
+  simp
+
+example : CirclePointIndex.toPoint CirclePointIndex.generator = generator := by
+  simp
+
+example : CirclePointIndex.subgroupGen 0 = 0 := by
+  simp
+
+example : CirclePointIndex.subgroupGen logOrder = CirclePointIndex.generator := by
+  simp
+
+def smallHalfCoset : Coset :=
+  Coset.halfOdds 3 (by decide)
+
+#guard smallHalfCoset.size = 8
+
+example : smallHalfCoset.indexAt 0 = smallHalfCoset.initialIndex := by
+  simp [smallHalfCoset]
+
+example (i : Nat) : smallHalfCoset.conjugate.indexAt i = -smallHalfCoset.indexAt i := by
+  simp [smallHalfCoset]
+
+def smallDomain : CircleDomain :=
+  CircleDomain.new smallHalfCoset
+
+#guard smallDomain.logSize = 4
+#guard smallDomain.size = 16
+
+example (i : Nat) :
+    smallDomain.indexAt (smallHalfCoset.size + i) = -smallHalfCoset.indexAt i := by
+  change smallDomain.indexAt (smallDomain.halfCoset.size + i) = -smallDomain.halfCoset.indexAt i
+  exact CircleDomain.indexAt_right smallDomain i
+
+def smallCanonicCoset : CanonicCoset where
+  logSize := 4
+  one_le_logSize := by decide
+  logSize_succ_le_logOrder := by decide
+
+#guard smallCanonicCoset.coset.logSize = 4
+#guard smallCanonicCoset.halfCoset.logSize = 3
+#guard smallCanonicCoset.circleDomain.logSize = 4
+#guard smallCanonicCoset.circleDomain.size = 16
+
+end Mersenne31.Circle