[problem] derived_solidification_free_CW_homology is false without a Hausdorff/T2 hypothesis

[hongzhoulin89]

Problem id

derived_solidification_free_CW_homology

What is wrong

At Lean-Eval commit 0e093293afbe96be91504ae6e44947876d3fc84e, Hole 10 asks for

def derivedSolidification_free_CW_derivedNatIso :
    derivedSolidificationFreeCWFunctor ≅
      singularChainsLightCondAbCWDerivedFunctor := sorry

The fixed category CWTopCat and the pointwise hypotheses in Holes 10–12 only require

[Topology.CWComplex (Set.univ : Set X)]

with no Hausdorff/T2 hypothesis. This admits a non-Hausdorff zero-cell thickening of an
infinite light profinite space. On that object, the requested comparison would make the
non-discrete measure object a retract of a discrete singular-homology object, which is
impossible.

The counterexample has been formalized independently of the unfinished Hole 10:

def counterexample : CWTopCat :=
  cwThickening (LightProfinite.toTopCat.obj (ℕ∪{∞}))

theorem no_natural_comparison :
    IsEmpty
      (CWTopCat.toTopCat ⋙ freeDerivedFunctor ≅
        singularChainsLightCondAbCWDerivedFunctor)

theorem no_natural_comparison_of_adjunction
    {L : DLightCondAb ⥤ DSolid} (a : L ⊣ derivedInclusion)
    {F : CWTopCat ⥤ DLightCondAb}
    (spec : F ≅ CWTopCat.toTopCat ⋙ freeLightCondAbOfTopFunctor ⋙
      singleAmbient ⋙ L ⋙ derivedInclusion) :
    IsEmpty (F ≅ singularChainsLightCondAbCWDerivedFunctor)

There is also a degree-zero obstruction showing that changing the left-adjoint choices
cannot make the pointwise conclusion true:

theorem not_exists_CW_zero_homology_adjunction :
    ¬ ∃ L : DLightCondAb ⥤ DSolid,
      Nonempty (L ⊣ derivedInclusion) ∧
        ∀ (X : TopCat) [Topology.CWComplex (Set.univ : Set X)],
          Nonempty
            (isSolid.ι.obj
              ((DerivedCategory.homologyFunctor Solid 0).obj
                (L.obj (singleAmbient.obj (freeLightCondAbOfTop X)))) ≅
              singularHomologyLightCondAb X 0)

Verification:

• The formal counterexample was independently rebuilt in a fresh, isolated Lean 4.34.0
  workspace with no project/dependency build artifacts.
• lake build Submission and an explicit build of all 71 editable helper modules,
  including Submission.CWComparisonObstruction, passed.
• The exported axiom audit checked 2,655 declarations and rejected none; the two
  negative certificates use only propext, Quot.sound, and Classical.choice.
• The required benchmark API was 9/12: the only unavailable declarations were Holes
  10–12, and lake build Solution failed exactly on those three missing identifiers.
• The current Lean-Eval commit
  0e093293afbe96be91504ae6e44947876d3fc84e retains the same unrestricted signatures
  under Lean v4.35.0-rc3. A direct overlay build of the old proof sources reaches
  unrelated 4.34→4.35 compatibility failures in helper modules before compiling the
  obstruction; no 4.35 success is claimed here.

Suggested repair: restrict the CW domain used by Hole 10 to the appropriate
Hausdorff/T2 full subcategory (matching the intended mathematical theorem), and add the
corresponding separation hypothesis to Holes 11–12 before regenerating the workspace.

Relevant link (optional)

lean-eval/generated/derived_solidification_free_CW_homology/Challenge.lean

Lines 229 to 267 in 0e09329

	/-- **Hole 10.** The derived comparison theorem, naturally in a CW complex `X`: after applying the
	exact derived inclusion from solid light condensed abelian groups to light condensed abelian groups,
	the derived solidification of the free light condensed abelian group on `X` is the integral singular
	chain complex of `X` as a derived object of light condensed abelian groups. -/
	def derivedSolidification_free_CW_derivedNatIso :
	derivedSolidificationFreeCWFunctor ≅ singularChainsLightCondAbCWDerivedFunctor := sorry
	/-- The pointwise component of `derivedSolidification_free_CW_derivedNatIso`. -/
	def derivedSolidification_free_CW_derivedIso
	(X : TopCat) [Topology.CWComplex (Set.univ : Set X)] :
	derivedInclusion.obj
	(derivedSolidification.obj
	((DerivedCategory.singleFunctor LightCondAb 0).obj (freeLightCondAbOfTop X))) ≅
	singularChainsLightCondAbDerived X :=
	(derivedSolidificationFreeCWFunctorSpec.app ⟨X, ⟨inferInstance⟩⟩).symm.trans
	(derivedSolidification_free_CW_derivedNatIso.app ⟨X, ⟨inferInstance⟩⟩)
	/-- **Hole 11.** For a CW complex `X`, the homology of the derived solidification of the free
	light condensed abelian group on `X` is integral singular homology. Since the derived category
	is cohomologically indexed, the `n`-th singular homology group occurs in degree `-n`. -/
	def derivedSolidification_free_CW_homologyIso
	(X : TopCat) [Topology.CWComplex (Set.univ : Set X)] (n : ℕ) :
	isSolid.ι.obj
	((DerivedCategory.homologyFunctor Solid (-(n : ℤ))).obj
	(derivedSolidification.obj
	((DerivedCategory.singleFunctor LightCondAb 0).obj (freeLightCondAbOfTop X)))) ≅
	singularHomologyLightCondAb X n := sorry
	/-- **Hole 12.** The theorem form of `derivedSolidification_free_CW_homologyIso`: the derived
	solidification of the free light condensed abelian group on a CW complex computes integral
	singular homology. -/
	theorem derivedSolidification_free_CW_homology
	(X : TopCat) [Topology.CWComplex (Set.univ : Set X)] (n : ℕ) :
	Nonempty
	(isSolid.ι.obj
	((DerivedCategory.homologyFunctor Solid (-(n : ℤ))).obj
	(derivedSolidification.obj
	((DerivedCategory.singleFunctor LightCondAb 0).obj (freeLightCondAbOfTop X)))) ≅
	singularHomologyLightCondAb X n) := sorry