diff --git a/Complexity.lean b/Complexity.lean
index 2e559d5461..194cc232ac 100644
--- a/Complexity.lean
+++ b/Complexity.lean
@@ -766,6 +766,7 @@ public import Complexity.Computability.Complexity.FineGrained.CountingEndpoint
 public import Complexity.Computability.Complexity.FineGrained.CountingParameterBounds
 public import Complexity.Computability.Complexity.FineGrained.CountingScale
 public import Complexity.Computability.Complexity.FineGrained.CountingSparsification
+public import Complexity.Computability.Complexity.FineGrained.CountingSparsifyContext
 public import Complexity.Computability.Complexity.FineGrained.CountingTreeBound
 public import Complexity.Computability.Complexity.FineGrained.ExhaustiveSearch
 public import Complexity.Computability.Complexity.FineGrained.Exponents
diff --git a/Complexity/Computability/Complexity/FineGrained/CountingSparsification.lean b/Complexity/Computability/Complexity/FineGrained/CountingSparsification.lean
index ffb89e22fa..47e717e4ae 100644
--- a/Complexity/Computability/Complexity/FineGrained/CountingSparsification.lean
+++ b/Complexity/Computability/Complexity/FineGrained/CountingSparsification.lean
@@ -89,6 +89,10 @@ parameter uniformity and the native subexponential counting endpoint remain sepa
   time-and-space transfer certificate for `CountingSparsify.search`.
 * `Cslib.Complexity.CountingSparsify.exists_computableInO_search_polySpace`: the same time envelope
   with independent polynomial workspace, from separately promised leaf time and workspace.
+* `Cslib.Complexity.CountingSparsify.stepWith`,
+  `Cslib.Complexity.CountingSparsify.iterate_stepWith_fst`,
+  `Cslib.Complexity.CountingSparsify.iterate_stepWith_countingThreshold`: the step with a captured
+  runtime context, which keeps the context and agrees with `step` at the context's base.
 
 ## References
 
@@ -1081,6 +1085,53 @@ theorem exists_computableInO_search_polySpace (hM2 : 2 ≤ M) {k B L_coeff : ℕ
     (polyBounded_pow ((polyBounded_nodeBound K L_coeff).add (.const 1)) b)
   exact ((hinit.add (hstep.add hleaf)).add (polyBounded_stateBound K L_coeff)).add hclock
 
+/-! ### Captured runtime context -/
+
+section ContextWrapper
+
+variable {P : Type*}
+
+/-- One step of the depth-first search of the counting-sparsification tree carrying an unchanged
+runtime context `p : P`, using threshold family `θ p` and context-aware leaf counter `g`
+(returning the occurring-variable count `ψ.modelCount` on sparse leaves). -/
+def stepWith (θ : P → ℕ → ℕ) (K : ℕ) (g : P × CNF ℕ → ℕ) (ps : P × State) : P × State :=
+  (ps.1,
+    if ps.2.2.1.isEmpty then ps.2
+    else if isSparse (θ ps.1) K (ps.2.2.1.headD []) then
+      leafUpdate ps.2 (g (ps.1, ps.2.2.1.headD []))
+    else
+      (ps.2.1, (countingNext (θ ps.1) K (ps.2.2.1.headD []) false ::
+        countingNext (θ ps.1) K (ps.2.2.1.headD []) true :: ps.2.2.1.tail, ps.2.2.2)))
+
+@[simp]
+theorem stepWith_fst (θ : P → ℕ → ℕ) (K : ℕ) (g : P × CNF ℕ → ℕ) (ps : P × State) :
+    (stepWith θ K g ps).1 = ps.1 := rfl
+
+@[simp]
+theorem iterate_stepWith_fst (θ : P → ℕ → ℕ) (K : ℕ) (g : P × CNF ℕ → ℕ) (r : ℕ)
+    (ps : P × State) : ((stepWith θ K g)^[r] ps).1 = ps.1 := by
+  induction r generalizing ps with
+  | zero => rfl
+  | succ r ih => rw [Function.iterate_succ_apply, ih, stepWith_fst]
+
+@[simp]
+theorem stepWith_countingThreshold (MOf : P → ℕ) (K : ℕ) (g : P × CNF ℕ → ℕ)
+    (p : P) (s : State) :
+    stepWith (fun q => countingThreshold (MOf q)) K g (p, s) =
+      (p, step (MOf p) K (fun ψ => g (p, ψ)) s) := rfl
+
+@[simp]
+theorem iterate_stepWith_countingThreshold (MOf : P → ℕ) (K : ℕ) (g : P × CNF ℕ → ℕ)
+    (r : ℕ) (p : P) (s : State) :
+    (stepWith (fun q => countingThreshold (MOf q)) K g)^[r] (p, s) =
+      (p, (step (MOf p) K (fun ψ => g (p, ψ)))^[r] s) := by
+  induction r generalizing s with
+  | zero => rfl
+  | succ r ih => rw [Function.iterate_succ_apply, stepWith_countingThreshold, ih,
+      Function.iterate_succ_apply]
+
+end ContextWrapper
+
 end CountingSparsify
 
 end Cslib.Complexity
diff --git a/Complexity/Computability/Complexity/FineGrained/CountingSparsifyContext.lean b/Complexity/Computability/Complexity/FineGrained/CountingSparsifyContext.lean
new file mode 100644
index 0000000000..c5cf5e56e3
--- /dev/null
+++ b/Complexity/Computability/Complexity/FineGrained/CountingSparsifyContext.lean
@@ -0,0 +1,228 @@
+/-
+Copyright (c) 2026 Samuel Schlesinger. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Samuel Schlesinger
+-/
+
+module
+-- upstream: candidate
+
+public import Complexity.Computability.Complexity.FineGrained.CountingSparsification
+import Complexity.Computability.Complexity.Problems.SAT.PolyTime
+import Complexity.Computability.Machines.Turing.MultiTape.Complexity.PolyTime.Basic
+import Complexity.Computability.Machines.Turing.MultiTape.Complexity.PolyTime.List
+import Complexity.Computability.Machines.Turing.MultiTape.Complexity.PolyTime.Nat
+import Complexity.Computability.Machines.Turing.MultiTape.Complexity.PolyTime.Nat.Binary
+import Complexity.Computability.Machines.Turing.MultiTape.Complexity.BigO.Plain
+
+/-!
+# The counting-sparsification step with a captured runtime context
+
+The counting-sparsification search (`Cslib.Complexity.CountingSparsify.step`) fixes its threshold
+base `M` and its leaf counter `g` before the input. A single machine for Proposition 6.4
+[OpenAI2026ETH] instead computes its parameters from the input: the threshold base from the
+sparsification parameter, and the leaf counter's own parameters from the declared number of
+variables (Section 6.4, printed pp. 43–44). This module carries a runtime context `c : C` beside
+the search state: `Cslib.Complexity.CountingSparsify.stepWith θ K g` reads its thresholds `θ c` and
+passes `c` to the leaf counter `g`, and never changes `c`.
+
+The search state, its initialization `init` and its weighting and cap are those of the existing
+search: the state records the number of variables occurring in the input, and the leaf counter's
+contract is the occurring-support count `ψ.modelCount`. An externally declared number of variables
+`N` lives in the context; accounting for the `N - φ.numVars` unused declared variables, and
+normalizing any declared-`N` leaf count before it reaches the weighting, are left to the outer
+wrapper.
+
+For the thresholds `countingThreshold (MOf c)`, the context step is the existing step at base
+`MOf c` with leaf counter `g (c, ·)`, at every iterate
+(`Cslib.Complexity.CountingSparsify.iterate_stepWith_countingThreshold`), so the existing
+correctness theorems apply pointwise. One step is computable within a polynomial in the encoded
+context and state, plus the time and space of the leaf counter on a sparse head formula
+(`Cslib.Complexity.CountingSparsify.exists_computableInO_stepWith`); the context is charged in
+full, and the leaf counter's time and space are kept separate.
+
+This module certifies one step only. The resource bound for a whole uniform search, the
+outer wrapper for an externally declared count, and the assembly of Proposition 6.4 are separate.
+
+The context step `Cslib.Complexity.CountingSparsify.stepWith` and its agreement lemmas live with
+the search in `Complexity/Computability/Complexity/FineGrained/CountingSparsification.lean`; this
+module certifies its resources.
+
+## Main results
+
+* `Cslib.Complexity.CountingSparsify.exists_computableInO_stepWith`: the time and space of one
+  step.
+
+## References
+
+* [OpenAI, *The Exponential Time Hypothesis is False*][OpenAI2026ETH], Section 6.4,
+  Proposition 6.4 (pp. 43–44) and Appendix A (pp. 44–47).
+-/
+
+@[expose] public section
+
+namespace Cslib.Complexity
+
+open Cslib.Logic.PL Cslib.Logic.PL.CNF.Sparsification Turing.MultiTapeTM
+
+namespace CountingSparsify
+
+variable {C : Type}
+
+section PolyTime
+
+open ComputableTidilyInPolyTime
+
+/-- Search states with a context, encoded as framed tuples: the context, the weighting count in
+unary, the stack of formulas, and the accumulator in binary. -/
+local notation "encX" encC =>
+  encPair encC (encPair encUnary (encPair (encList encCNF) encNat))
+
+variable {encC : C ↪ List Bool} {θ : C → ℕ → ℕ}
+
+/-- Popping a sparse leaf and adding its weighted count is polynomial-time, keeping the context. -/
+private theorem computableTidilyInPolyTime_leafUpdateWith :
+    ComputableTidilyInPolyTime (fun p : (C × State) × ℕ => (p.1.1, leafUpdate p.1.2 p.2))
+      (encPair (encX encC) encNat) (encX encC) := by
+  have hvars := SAT.computableInPolyTime_distinctVars.tidy
+  have heq : (fun p : (C × State) × ℕ => (p.1.1, leafUpdate p.1.2 p.2)) = fun p =>
+      (p.1.1, (p.1.2.1, (p.1.2.2.1.tail,
+        if decide (2 ^ p.1.2.1 ≤ p.1.2.2.2 +
+            2 ^ (p.1.2.1 - (p.1.2.2.1.headD []).distinctVars.length) * p.2) then
+          2 ^ p.1.2.1
+        else
+          p.1.2.2.2 + 2 ^ (p.1.2.1 - (p.1.2.2.1.headD []).distinctVars.length) * p.2))) := by
+    funext p
+    simp [leafUpdate, leafWeight, min_def]
+  rw [heq]
+  unfold encCNF
+  fun_prop
+
+/-- Branching on the head formula is polynomial-time for runtime thresholds, keeping the context.
+-/
+private theorem computableTidilyInPolyTime_innerWith
+    (hθ : ComputableInPolyTime (fun x : C × ℕ × ℕ => decide (θ x.1 x.2.1 ≤ x.2.2))
+      (encPair encC (encPair encUnary encUnary)) encBool) (K : ℕ) :
+    ComputableTidilyInPolyTime (fun x : C × State =>
+      (x.1, (x.2.1, (countingNext (θ x.1) K (x.2.2.1.headD []) false ::
+        countingNext (θ x.1) K (x.2.2.1.headD []) true :: x.2.2.1.tail, x.2.2.2))))
+      (encX encC) (encX encC) := by
+  have h := (computableInPolyTime_countingNextWith encNat hθ K).tidy
+  have hH : ComputableTidilyInPolyTime (fun x : C × State => x.2.2.1.headD [])
+      (encX encC) encCNF := by
+    unfold encCNF
+    fun_prop
+  have hb (b : Bool) : ComputableTidilyInPolyTime
+      (fun x : C × State => countingNext (θ x.1) K (x.2.2.1.headD []) b) (encX encC)
+      (encList (encList (encPair encNat encBool))) := by
+    simpa only [Function.comp_def] using
+      h.comp ((fst encC _).prod (hH.prod (const (encMid := encBool) b)))
+  have h₀ := hb false
+  have h₁ := hb true
+  unfold encCNF
+  fun_prop
+
+private theorem exists_computableTidilyInO_stepWith
+    (hθ : ComputableInPolyTime (fun x : C × ℕ × ℕ => decide (θ x.1 x.2.1 ≤ x.2.2))
+      (encPair encC (encPair encUnary encUnary)) encBool) (K : ℕ)
+    {g : C × CNF ℕ → ℕ} {t s : C × CNF ℕ → ℕ}
+    (hg : ComputableTidilyInO g (encPair encC encCNF) encNat t s) :
+    ∃ d, ComputableTidilyInO (stepWith θ K g) (encX encC) (encX encC)
+      (fun x => (((encX encC) x).length + 1) ^ d +
+        if x.2.2.1.isEmpty = false ∧ isSparse (θ x.1) K (x.2.2.1.headD []) = true then
+          t (x.1, x.2.2.1.headD []) +
+            (((encX encC) x).length + (encNat (g (x.1, x.2.2.1.headD []))).length + 1) ^ d
+        else 0)
+      (fun x => (((encX encC) x).length + 1) ^ d +
+        if x.2.2.1.isEmpty = false ∧ isSparse (θ x.1) K (x.2.2.1.headD []) = true then
+          s (x.1, x.2.2.1.headD []) +
+            (((encX encC) x).length + (encNat (g (x.1, x.2.2.1.headD []))).length + 1) ^ d
+        else 0) := by
+  have hH0 : ComputableTidilyInPolyTime (fun x : C × State => x.2.2.1.headD [])
+      (encX encC) encCNF := by
+    unfold encCNF
+    fun_prop
+  obtain ⟨dE, eE, hE⟩ := (show ComputableTidilyInPolyTime (fun x : C × State => x.2.2.1.isEmpty)
+    (encX encC) encBool by unfold encCNF; fun_prop).exists_degree
+  obtain ⟨dI, eI, hI⟩ := (id' (encIn := encX encC)).exists_degree
+  have hS0 : ComputableTidilyInPolyTime
+      (fun x : C × State => isSparse (θ x.1) K (x.2.2.1.headD [])) (encX encC) encBool := by
+    simpa only [Function.comp_def] using
+      (computableInPolyTime_isSparseWith encNat hθ K).tidy.comp ((fst encC _).prod hH0)
+  obtain ⟨dS, eS, hS⟩ := hS0.exists_degree
+  obtain ⟨dH, eH, hH⟩ := (show ComputableTidilyInPolyTime
+      (fun x : C × State => (x.1, x.2.2.1.headD [])) (encX encC) (encPair encC encCNF) from
+    (fst encC _).prod hH0).exists_degree
+  obtain ⟨dN, eN, hN⟩ := (computableTidilyInPolyTime_innerWith hθ K).exists_degree
+  obtain ⟨dU, eU, hU⟩ := (computableTidilyInPolyTime_leafUpdateWith (encC := encC)).exists_degree
+  set D := dE + dI + dS + dH + dN + dU + 1
+  have hE' := (hE.mono (d' := D) (e' := eE) (by omega) le_rfl).inO
+  have hI' := (hI.mono (d' := D) (e' := eI) (by omega) le_rfl).inO
+  have hS' := (hS.mono (d' := D) (e' := eS) (by omega) le_rfl).inO
+  have hH' := (hH.mono (d' := D) (e' := eH) (by omega) le_rfl).inO
+  have hN' := (hN.mono (d' := D) (e' := eN) (by omega) le_rfl).inO
+  have hU' := (hU.mono (d' := D) (e' := eU) (by omega) le_rfl).inO
+  have hleaf := hU'.comp (hI'.prod (hg.comp hH'))
+  have h := hE'.ite hI' (hS'.ite hleaf hN')
+  have heq : stepWith θ K g = fun x =>
+      if x.2.2.1.isEmpty then x
+      else if isSparse (θ x.1) K (x.2.2.1.headD []) then
+        (x.1, leafUpdate x.2 (g (x.1, x.2.2.1.headD [])))
+      else
+        (x.1, (x.2.1, (countingNext (θ x.1) K (x.2.2.1.headD []) false ::
+          countingNext (θ x.1) K (x.2.2.1.headD []) true :: x.2.2.1.tail, x.2.2.2))) := by
+    funext x
+    unfold stepWith
+    split_ifs <;> rfl
+  have hpair_le (x : C × State) (c : ℕ) :
+      ((encPair (encX encC) encNat (x, c)).length + 1) ^ D ≤
+        4 ^ D * (((encX encC) x).length + (encNat c).length + 1) ^ D := by
+    rw [← mul_pow]
+    refine Nat.pow_le_pow_left ?_ D
+    simp only [length_encPair]
+    omega
+  rw [heq]
+  refine ⟨D, h.mono (4 ^ D + 5) (fun x => ?_) (fun x => ?_)⟩ <;>
+  · simp only [Function.comp_apply]
+    have hp := hpair_le x (g (x.1, x.2.2.1.headD []))
+    have h4D : 1 ≤ 4 ^ D := Nat.one_le_pow _ _ (by norm_num)
+    generalize 4 ^ D = F at hp h4D ⊢
+    generalize (((encX encC) x).length + 1) ^ D = P
+    generalize t (x.1, x.2.2.1.headD []) = T
+    generalize s (x.1, x.2.2.1.headD []) = S
+    generalize (((encX encC) x).length + (encNat (g (x.1, x.2.2.1.headD []))).length + 1) ^ D = Q
+      at hp ⊢
+    generalize ((encPair (encX encC) encNat (x, g (x.1, x.2.2.1.headD []))).length + 1) ^ D = R
+      at hp ⊢
+    cases x.2.2.1.isEmpty <;>
+      cases isSparse (θ x.1) K (x.2.2.1.headD []) <;>
+      simp only [Bool.false_eq_true, ite_false, ite_true, and_true, and_false,
+        Bool.true_eq_false] <;> nlinarith
+
+/-- **The time and space of a context step**: a polynomial in the length of the encoded context
+and state, plus the time (respectively space) of the leaf counter and a polynomial in the state and
+counter output lengths when the head formula is sparse. -/
+theorem exists_computableInO_stepWith
+    (hθ : ComputableInPolyTime (fun x : C × ℕ × ℕ => decide (θ x.1 x.2.1 ≤ x.2.2))
+      (encPair encC (encPair encUnary encUnary)) encBool) (K : ℕ)
+    {g : C × CNF ℕ → ℕ} {t s : C × CNF ℕ → ℕ}
+    (hg : ComputableInO g (encPair encC encCNF) encNat t s) :
+    ∃ d, ComputableInO (stepWith θ K g) (encX encC) (encX encC)
+      (fun x => (((encX encC) x).length + 1) ^ d +
+        if x.2.2.1.isEmpty = false ∧ isSparse (θ x.1) K (x.2.2.1.headD []) = true then
+          t (x.1, x.2.2.1.headD []) +
+            (((encX encC) x).length + (encNat (g (x.1, x.2.2.1.headD []))).length + 1) ^ d
+        else 0)
+      (fun x => (((encX encC) x).length + 1) ^ d +
+        if x.2.2.1.isEmpty = false ∧ isSparse (θ x.1) K (x.2.2.1.headD []) = true then
+          s (x.1, x.2.2.1.headD []) +
+            (((encX encC) x).length + (encNat (g (x.1, x.2.2.1.headD []))).length + 1) ^ d
+        else 0) :=
+  let ⟨d, h⟩ := exists_computableTidilyInO_stepWith hθ K (ComputableInO.tidy hg)
+  ⟨d, ComputableInO.of_tidy h⟩
+
+end PolyTime
+
+end CountingSparsify
+
+end Cslib.Complexity
diff --git a/ComplexityTests/CountingSparsifyContext.lean b/ComplexityTests/CountingSparsifyContext.lean
new file mode 100644
index 0000000000..c763c02729
--- /dev/null
+++ b/ComplexityTests/CountingSparsifyContext.lean
@@ -0,0 +1,75 @@
+/-
+Copyright (c) 2026 Samuel Schlesinger. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Samuel Schlesinger
+-/
+
+import Complexity.Computability.Complexity.FineGrained.CountingSparsifyContext
+
+/-!
+# Tests for the counting-sparsification step with a captured runtime context
+-/
+
+namespace ComplexityTests.CountingSparsifyContext
+
+open Cslib.Logic.PL Cslib.Logic.PL.CNF.Sparsification Cslib.Complexity
+open Cslib.Complexity.CountingSparsify Turing.MultiTapeTM
+
+/-- With the threshold base as the context, the context search is the existing search at that base,
+so the existing correctness theorems apply to each input. -/
+example (K k B M : ℕ) (g : ℕ × CNF ℕ → ℕ) (φ : CNF ℕ) :
+    ((stepWith (fun c => countingThreshold c) K g)^[rounds B φ] (M, init k φ)).2.2.2 =
+      search M K k B (fun ψ => g (M, ψ)) φ := by
+  rw [iterate_stepWith_countingThreshold (MOf := fun c => c)]
+  rfl
+
+/-- Two clauses sharing the heart `x₁`, on three occurring variables, with five models. -/
+abbrev ψ : CNF ℕ := [[(1, true), (2, true)], [(1, true), (3, true)]]
+
+/-- With context `(1, 7)` the base is `1`, the heart `x₁` is heavy, and one step replaces the
+formula on the stack by its two counting children, keeping the context and the accumulator. -/
+example : (stepWith (fun c : ℕ × ℕ => countingThreshold c.1) 2 (fun x => x.2.modelCount)
+    ((1, 7), init 2 ψ)).1 = (1, 7) ∧
+    (stepWith (fun c : ℕ × ℕ => countingThreshold c.1) 2 (fun x => x.2.modelCount)
+      ((1, 7), init 2 ψ)).2.2.1.length = 2 ∧
+    (stepWith (fun c : ℕ × ℕ => countingThreshold c.1) 2 (fun x => x.2.modelCount)
+      ((1, 7), init 2 ψ)).2.2.2 = 0 := by
+  decide
+
+/-- With context `(100, 7)` the formula is sparse, so one step pops it and adds its five models. -/
+example : (stepWith (fun c : ℕ × ℕ => countingThreshold c.1) 2 (fun x => x.2.modelCount)
+    ((100, 7), init 2 ψ)).2.2.1 = [] ∧
+    (stepWith (fun c : ℕ × ℕ => countingThreshold c.1) 2 (fun x => x.2.modelCount)
+      ((100, 7), init 2 ψ)).2.2.2 = 5 := by
+  decide
+
+/-- On the empty formula the existing search counts one assignment, with any context. -/
+example (N : ℕ) : ((stepWith (fun _ : ℕ => countingThreshold 2) 2 (fun x => x.2.modelCount))
+    (N, init 2 [])).2.2.2 = 1 := by
+  rw [stepWith_countingThreshold (MOf := fun _ => 2)]
+  change (step 2 2 (fun ψ => ψ.modelCount) (init 2 [])).2.2 = 1
+  decide
+
+/-- One step for a runtime base `M` given in binary, with any leaf counter certificate. -/
+example {g : ℕ × CNF ℕ → ℕ} {t s : ℕ × CNF ℕ → ℕ}
+    (hg : ComputableInO g (encPair encNat encCNF) encNat t s) (K : ℕ) :
+    ∃ d, ComputableInO (stepWith (fun c => countingThreshold c) K g)
+      (encPair encNat (encPair encUnary (encPair (encList encCNF) encNat)))
+      (encPair encNat (encPair encUnary (encPair (encList encCNF) encNat)))
+      (fun x => ((encPair encNat (encPair encUnary (encPair (encList encCNF) encNat)) x).length +
+          1) ^ d +
+        if x.2.2.1.isEmpty = false ∧ isSparse (countingThreshold x.1) K (x.2.2.1.headD []) = true
+        then t (x.1, x.2.2.1.headD []) +
+          ((encPair encNat (encPair encUnary (encPair (encList encCNF) encNat)) x).length +
+            (encNat (g (x.1, x.2.2.1.headD []))).length + 1) ^ d
+        else 0)
+      (fun x => ((encPair encNat (encPair encUnary (encPair (encList encCNF) encNat)) x).length +
+          1) ^ d +
+        if x.2.2.1.isEmpty = false ∧ isSparse (countingThreshold x.1) K (x.2.2.1.headD []) = true
+        then s (x.1, x.2.2.1.headD []) +
+          ((encPair encNat (encPair encUnary (encPair (encList encCNF) encNat)) x).length +
+            (encNat (g (x.1, x.2.2.1.headD []))).length + 1) ^ d
+        else 0) :=
+  exists_computableInO_stepWith computableInPolyTime_countingThreshold_le_binary K hg
+
+end ComplexityTests.CountingSparsifyContext