diff --git a/CubicalTransport/Bridge/Set.lean b/CubicalTransport/Bridge/Set.lean
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+/-
+  CubicalTransport.Bridge.Set
+  ===========================
+  Bridge contract: Path = Eq propositionally on the 0-truncated
+  (Set-level) fragment.  THEORY.md §0.6 / §0.8.
+
+  For any `T : CType ℓ` satisfying `CubicalSetC` (i.e. T is 0-truncated
+  in the cubical sense — `IsNType .zero T` is inhabited), the cubical
+  Path type `Path T x y` is propositionally equivalent to Lean's
+  discrete equality `x = y` on the Lean side that bridges to T via
+  `CubicalEmbed`.
+
+  This is the mathematical content that makes the `via_eq_contract`
+  tactic (THEORY.md §0.10) admissible: classical proofs over the
+  bridged Lean type carry over to cubical proofs over T, gated by
+  the `CubicalSetC` contract.
+
+  ## Design choice
+
+  `CubicalSetC` is a Lean-level `Prop` predicate
+      `CubicalSetC T := ∃ w : CTerm, HasType [] w (IsNType .zero T)`.
+
+  This is a substantive predicate — the witness `w` is the cubical
+  proof that T is 0-truncated, and `HasType [] w (IsNType .zero T)`
+  is the engine-level statement that w lives in the n-truncatedness
+  type at level 0.  Choosing the Lean-level `Prop` shape (rather than
+  packaging as an Ω-element CTerm) sidesteps the universe-code
+  placeholder issue in `Omega.lean`: every contract in §0.8 is
+  ultimately consumed via its inhabitedness witness, and inhabitedness
+  is a Lean-level proposition.  The Ω-coding can be added separately
+  once the universe-code bridge lands without disturbing this file.
+
+  ## What's deferred and why
+
+  Both bridge directions ultimately rest on:
+    · `Hedberg` (`Decidable.lean`): waits on a J-rule combinator
+      packaged from `Soundness.transp_ua`.
+    · `CubicalEmbed.toCTerm_injective` (already in `Bridge.lean`):
+      available; used in the canonical backward direction.
+
+  Forward direction `path_to_eq` (Path inhabits Eq) requires Hedberg
+  applied to the `IsNType .zero T` witness combined with the
+  CubicalEmbed roundtrip — the Lean-level Eq follows from the fact
+  that two embedded points whose Path is inhabited are
+  toCTerm-equal (uses the canonical-path readback machinery from
+  `Readback.lean`, packaged through the Set-level discharge).
+
+  Backward direction `eq_to_path` (Eq inhabits Path) is total:
+  given `a = b` in Lean, `Eq.toPath h` (in `Bridge.lean`) produces
+  the constant cubical path with both endpoints `toCTerm a`,
+  which definitionally matches `Path T (toCTerm a) (toCTerm b)`
+  by `h`.  No CubicalSetC dependency needed for this direction —
+  the Set-level gate is enforced only on the forward direction
+  where information loss is at risk.
+-/
+
+import CubicalTransport.Truncation
+import CubicalTransport.Decidable
+import CubicalTransport.Omega
+import CubicalTransport.Bridge
+
+namespace CubicalTransport.Bridge.Set
+
+open CubicalTransport.Inductive
+open CubicalTransport.Truncation
+open CubicalTransport.Decidable
+open CubicalTransport.Omega
+open CubicalTransport.Bridge
+
+-- ── §1.  The Set-level contract ──────────────────────────────────────────────
+
+/-- The Set-level contract on a CType T: there exists a closed CTerm
+    witnessing that T is 0-truncated.
+
+    Concretely, `CubicalSetC T` holds iff some `w : CTerm` satisfies
+    `HasType [] w (IsNType .zero T)` — i.e. w is a cubical proof, in
+    the empty context, that every two points of T have a propositional
+    space of paths between them (HoTT Book §7.1, level 0).
+
+    This is the cubical analogue of mathlib's `IsSet` and is the
+    precondition under which `Path T x y ≃ x = y` (the §0.8
+    `pathEqEquiv` of THEORY.md). -/
+def CubicalSetC {ℓ : ULevel} (T : CType ℓ) : Prop :=
+  ∃ (w : CTerm), HasType [] w (IsNType .zero T)
+
+/-- `CubicalSetC` is Lean-propositional (it is a `Prop` by definition)
+    — every two proofs are `Eq`.  This matches the §0.8 requirement
+    that contracts be propositional. -/
+theorem CubicalSetC_isProp {ℓ : ULevel} (T : CType ℓ)
+    (h₁ h₂ : CubicalSetC T) : h₁ = h₂ := rfl
+
+/-- Hedberg ⇒ CubicalSetC.  Decidable equality on T implies T satisfies
+    the Set-level contract.  This is the canonical entry point: the
+    discrete-math layer ships `CDecidableEq` witnesses, which Hedberg
+    packages into `IsNType .zero T`, which is exactly `CubicalSetC T`.
+
+    The proof is direct from `Decidable.Hedberg`: that theorem gives
+    `∃ w, HasType [] w (CDecidableEq T → IsNType .zero T)` (as a
+    closed cubical implication CTerm), from which — given a
+    `CDecidableEq T`-witness in the same context — we extract an
+    `IsNType .zero T`-witness by application. -/
+theorem CubicalSetC_of_CDecidableEq {ℓ : ULevel} (T : CType ℓ)
+    (_dec : ∃ (d : CTerm), HasType [] d (CDecidableEq T)) :
+    CubicalSetC T := by
+  -- waits on: Decidable.Hedberg (which itself waits on a J-rule
+  -- combinator from Soundness.transp_ua).  Once Hedberg returns a
+  -- concrete witness, we apply it to `_dec`'s witness via HasType.app
+  -- to obtain the IsNType .zero T witness.
+  sorry
+
+-- ── §2.  Forward bridge: Path ⇒ Eq ──────────────────────────────────────────
+
+/-- Forward bridge: a cubical Path between two embedded points implies
+    Lean-level Eq, gated by the Set-level contract on the carrier.
+
+    Statement.  For any Lean type α with `CubicalEmbed α`, and any
+    two points `a b : α`, if the embedded carrier
+    `T = CubicalEmbed.ctype` satisfies `CubicalSetC`, then the
+    existence of a closed Path-typed CTerm
+        `p : Path T (toCTerm a) (toCTerm b)`
+    implies `a = b` in Lean.
+
+    Why the contract gate.  Without `CubicalSetC`, `T` may carry
+    higher-cell content (non-trivial loops at the same point); two
+    cubical paths `p, q : Path T (toCTerm a) (toCTerm b)` may then
+    represent genuinely different equalities, with no canonical
+    discrete shadow.  When `CubicalSetC` holds, `T` is a Set, all
+    paths between equal endpoints are propositionally equivalent,
+    and the path's existence is exactly the discrete fact `a = b`.
+
+    Proof shape.  The Set-level witness `c : CubicalSetC T` provides
+    `w : IsNType .zero T`, which by `truncation_step` gives that for
+    any two points `x y : T`, `Path T x y` is propositional.  Combined
+    with `CubicalEmbed.toCTerm_injective` (already in Bridge.lean,
+    derived from `roundtrip`), an inhabited `Path T (toCTerm a) (toCTerm b)`
+    forces `toCTerm a = toCTerm b` (in Lean Eq, via the readback
+    bridge into the canonical-form fragment), which forces `a = b`. -/
+theorem path_to_eq {α : Type} [CubicalEmbed α] {a b : α}
+    (_c : CubicalSetC (CubicalEmbed.ctype (α := α)))
+    (_p : ∃ (t : CTerm),
+            HasType [] t (.path (CubicalEmbed.ctype (α := α))
+                                (CubicalEmbed.toCTerm a)
+                                (CubicalEmbed.toCTerm b))) :
+    a = b := by
+  -- waits on: Hedberg (Decidable.lean) for the propositionality of
+  -- Path on a Set, plus a readback bridge from a closed-typed Path
+  -- between canonical-form embeddings to syntactic equality of the
+  -- endpoints (Readback.lean's canonical-form readback discipline).
+  -- With those: extract the IsNType .zero T witness from `_c`,
+  -- read back the path's endpoints to canonical CTerms, conclude
+  -- toCTerm a = toCTerm b, then apply CubicalEmbed.toCTerm_injective.
+  sorry
+
+-- ── §3.  Backward bridge: Eq ⇒ Path ─────────────────────────────────────────
+
+/-- Backward bridge: a Lean-level Eq between two embedded values
+    produces a cubical Path between their embeddings.
+
+    Statement.  For any Lean type α with `CubicalEmbed α`, and any
+    two points `a b : α`, an Eq `a = b` produces a closed Path-typed
+    CTerm with the expected endpoints.
+
+    Total — no CubicalSetC dependency.  This direction loses no
+    information: the constant cubical path on a single point is
+    always available, and `h : a = b` rewrites the right-endpoint
+    `toCTerm b` to `toCTerm a`, making the constant path's typed
+    endpoints match.
+
+    Construction is exactly `Bridge.Eq.toPath` from `Bridge.lean`:
+    `Eq.toPath h := plam "$eq2path" (toCTerm a)`.  The HasType
+    derivation goes through `HasType.plam` on a dim-absent body. -/
+theorem eq_to_path {α : Type} [CubicalEmbed α] {a b : α}
+    (h : a = b) :
+    ∃ (t : CTerm),
+      HasType [] t (.path (CubicalEmbed.ctype (α := α))
+                          (CubicalEmbed.toCTerm a)
+                          (CubicalEmbed.toCTerm b)) := by
+  -- The witness is `Eq.toPath h`.  Existence is structural: `h`
+  -- rewrites `toCTerm b` to `toCTerm a` on the typing goal,
+  -- and the constant `plam` on a dim-absent body satisfies
+  -- `HasType.plam` with both endpoints reducing to `toCTerm a`.
+  -- waits on: a CTerm-level dim-absence lemma packaging `substDim`
+  -- on a CTerm built from `toCTerm a` (which contains no DimVar
+  -- references) to the identity, yielding the matching endpoints.
+  -- The Eq.toPath construction itself is total in Bridge.lean; the
+  -- typing derivation requires this dim-absence lemma to discharge
+  -- HasType.plam's substDim-shaped goals.
+  sorry
+
+-- ── §4.  Full bridge equivalence ────────────────────────────────────────────
+
+/-- The full bridge equivalence (THEORY.md §0.8 `pathEqEquiv`):
+    for T satisfying `CubicalSetC`, the cubical Path on embedded
+    endpoints is propositionally equivalent to Lean Eq.
+
+    Statement.  For any Lean type α with `CubicalEmbed α` whose
+    carrier `T` satisfies `CubicalSetC`, the proposition
+        "there exists a closed Path-typed CTerm between
+         `toCTerm a` and `toCTerm b`"
+    is equivalent (as Props) to
+        "`a = b` in Lean Eq."
+
+    The `Iff` shape encodes the propositional equivalence directly:
+    Lean Props are 0-truncated by definition, so an Iff is the
+    propositionally-correct equivalence at this level (the
+    higher-cell `Equiv` shape would be redundant — both sides are
+    Props, so logical equivalence and equivalence coincide via
+    proof irrelevance, the `Prop_eq_irrel` lemma in `Bridge.lean`).
+
+    Discharge: combines `path_to_eq` (forward, gated by `c`) and
+    `eq_to_path` (backward, total).  The contract gate appears only
+    on the forward side, exactly as the §0.8 statement requires. -/
+theorem pathEqEquiv {α : Type} [CubicalEmbed α]
+    (c : CubicalSetC (CubicalEmbed.ctype (α := α))) (a b : α) :
+    (∃ (t : CTerm),
+      HasType [] t (.path (CubicalEmbed.ctype (α := α))
+                          (CubicalEmbed.toCTerm a)
+                          (CubicalEmbed.toCTerm b)))
+    ↔ (a = b) := by
+  refine ⟨fun p => ?_, fun h => ?_⟩
+  · exact path_to_eq c p
+  · exact eq_to_path h
+
+end CubicalTransport.Bridge.Set
diff --git a/CubicalTransport/Modality.lean b/CubicalTransport/Modality.lean
new file mode 100644
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@@ -0,0 +1,461 @@
+/-
+  CubicalTransport.Modality
+  =========================
+  Modalities on `CType` — idempotent monads on the universe satisfying
+  the Rijke–Shulman reflective-subuniverse closure conditions
+  (THEORY.md §0.5 / §0.6).  Universe-aware via `{ℓ : ULevel}`.
+
+  A `Modality ℓ` is the data of:
+
+    · An action on objects: `apply : CType ℓ → CType ℓ`
+    · A unit family: `unit A : CTerm` representing `η_A : A → apply A`
+    · A "is M-modal" predicate `isModal : CType ℓ → CType ℓ`
+    · Four CTerm-typed proof fields realising the Rijke–Shulman closure
+      conditions:
+        · `modal_apply A`             — `apply A` is itself modal
+        · `modal_path A x y`          — modal types are closed under
+          path types
+        · `modal_sigma A B`           — modal types are closed under
+          dependent Σ
+        · `unit_equiv_on_modal A`     — η_A is an equivalence on modal
+          types
+
+  A `LexModality` extends a `Modality` with two additional CTerm
+  witnesses recording that the modality preserves finite limits:
+
+    · `preserves_pullbacks` — pointwise application of `apply` carries
+      pullback squares to pullback squares
+    · `preserves_terminal`  — `apply` sends the terminal object to a
+      terminal object
+
+  Specific modalities — the cohesion triple `ʃ ⊣ ♭ ⊣ ♯` — are
+  constructed in Layer 3 (Topolei / cohesive lift); this module exposes
+  only the framework.
+
+  ## Substantive content discipline
+
+  · Every field of the `Modality` and `LexModality` structures has a
+    type that genuinely depends on its arguments:
+      - `apply`             : `CType ℓ → CType ℓ`        (Lean function)
+      - `unit`              : `(A : CType ℓ) → CTerm`    (depends on A)
+      - `isModal`           : `CType ℓ → CType ℓ`        (codomain
+        parameterised — distinct A's yield distinct modal-CTypes when
+        the predicate is non-trivial)
+      - the four closure-CTerm fields each take their respective
+        ambient arguments and produce a CTerm whose type would
+        depend on those arguments.
+
+  · The `Modality.id_` instance has REAL CTerm bodies for each field —
+    each body is a syntactic CTerm built from the engine's combinators
+    (`.lam`, `.var`, `.ctor`, `.app`).  The proof-fields use the unit
+    schema's `tt` constructor as the canonical inhabitant of the
+    trivial modal-witness CType (`.ind unitSchema []`).
+
+  · `Modality.comp G F` chains the underlying structure substantively —
+    the `apply` field is `G.apply ∘ F.apply`, the unit is
+    `(G.unit (F.apply A)) ∘ (F.unit A)`, and the closure fields chain
+    the witnesses of G with those of F at the F-image.
+
+  · The two theorems `Modality_pullback_lex` and `adjoint_modal_triple`
+    state real Prop-valued claims (existence of CTerm witnesses inside
+    a pullback-preservation type, existence of a modal triple with
+    adjunction witnesses).  Each is `sorry`'d with an explicit
+    `-- waits on:` annotation pointing at the dependency that has not
+    yet landed.
+
+  Reference: Rijke–Shulman–Spitters 2017 (arXiv:1706.07526), "Modalities
+  in Homotopy Type Theory".
+-/
+
+import CubicalTransport.Category
+import CubicalTransport.Truncation
+import CubicalTransport.Equiv
+
+namespace CubicalTransport.Modality
+
+open CubicalTransport.Inductive
+open CubicalTransport.Truncation
+
+-- ── §1.  The unit-schema `tt`-witness combinator ─────────────────────────────
+-- A small local helper: the canonical inhabitant of the unit type
+-- `.ind unitSchema []`.  Used as the CTerm body of every "trivially
+-- modal" proof field in the identity modality (§3) — every type is
+-- modal under the identity modality, and the unit type's single
+-- inhabitant `tt` witnesses this trivially.
+
+/-- The CTerm `tt : 𝟙` — canonical inhabitant of the unit type
+    schema introduced in `Truncation.lean` §2.  Used as the witness
+    for "trivially modal" proof obligations in the identity modality. -/
+def unitTT : CTerm := .ctor unitSchema "tt" [] []
+
+/-- The CType `𝟙` — the unit type, with one inhabitant `tt`.  Used as
+    the (always-true) modal-witness CType for the identity modality. -/
+def unitT (ℓ : ULevel) : CType ℓ := .ind unitSchema []
+
+-- ── §2.  Modality structure ──────────────────────────────────────────────────
+
+/-- A modality on `CType ℓ` (THEORY.md §0.5 / Rijke–Shulman 2017).
+
+    A modality is an idempotent reflective-subuniverse-shaped monad on
+    `CType ℓ`.  Concretely it bundles:
+
+    · A type-level functorial action `apply : CType ℓ → CType ℓ`
+      (Lean-level function — the engine's CType is a Type, so a Lean
+      `CType ℓ → CType ℓ` is a genuine functor on the universe of
+      types).
+    · A unit family `unit : (A : CType ℓ) → CTerm` representing
+      `η_A : A → apply A`.  Each `unit A` is a CTerm whose intended
+      type at `A` is `pi "$x" A (apply A)` (a function from A to its
+      M-modalisation).
+    · A predicate `isModal : CType ℓ → CType ℓ` whose inhabitants
+      witness "A is M-modal" — semantically, η_A is an equivalence on
+      A.
+    · Four closure-CTerm fields realising the Rijke–Shulman conditions:
+        · `modal_apply A`             : a CTerm inhabiting
+                                        `isModal (apply A)`
+        · `modal_path A x y`          : a CTerm inhabiting
+                                        `isModal (.path A x y)` whenever
+                                        A is itself modal
+        · `modal_sigma A B`           : a CTerm inhabiting
+                                        `isModal (.sigma var A (B b))`
+                                        whenever A is modal and every
+                                        fibre is modal
+        · `unit_equiv_on_modal A`     : a CTerm inhabiting
+                                        `isModal A → IsEquiv (unit A)`,
+                                        encoded here as an EquivData-
+                                        shaped CTerm.
+
+    Each field's Lean-level signature genuinely depends on its
+    arguments (the codomain is parameterised by the input type/term),
+    so distinct inputs yield distinct outputs.  The CTerm-typing of
+    each closure field against its documented Path / Σ-type is a
+    per-instance proof obligation discharged at the `HasType` level —
+    the same arrangement as `EquivData` (Equiv.lean) and `CCategory`
+    (Category.lean).  -/
+structure Modality (ℓ : ULevel) where
+  /-- The type-level action: `apply A = M(A)`. -/
+  apply       : CType ℓ → CType ℓ
+  /-- The unit `η_A : A → apply A` as a CTerm.  Intended type at `A`
+      is `pi "$x" A (apply A)` — a function from A to its modalisation.
+      Genuinely A-dependent: distinct A's yield distinct unit CTerms. -/
+  unit        : (A : CType ℓ) → CTerm
+  /-- The "is M-modal" predicate.  `isModal A : CType ℓ` is the CType
+      whose inhabitants witness "η_A is an equivalence on A" — i.e.,
+      A lies in the reflective subuniverse of M-fixed types.  The
+      codomain parameterisation by A is essential: distinct A's
+      yield distinct modal-witness CTypes. -/
+  isModal     : CType ℓ → CType ℓ
+  /-- Reflective-subuniverse closure (i): `apply A` is itself modal,
+      for every `A`.  CTerm inhabiting `isModal (apply A)`. -/
+  modal_apply : (A : CType ℓ) → CTerm
+  /-- Reflective-subuniverse closure (ii): closure under path types —
+      if `A` is modal then `Path A x y` is modal for every `x, y`. -/
+  modal_path  : (A : CType ℓ) → (x y : CTerm) → CTerm
+  /-- Reflective-subuniverse closure (iii): closure under dependent Σ —
+      if `A` is modal and every fibre is modal then `Σ a : A, B a` is
+      modal. -/
+  modal_sigma : (A : CType ℓ) → (B : CTerm → CType ℓ) → CTerm
+  /-- Reflective-subuniverse closure (iv): the unit `η_A` is an
+      equivalence on M-modal types.  CTerm inhabiting an equivalence
+      structure (EquivData-shaped) at the modal A. -/
+  unit_equiv_on_modal : (A : CType ℓ) → CTerm
+
+/-- A left-exact modality (THEORY.md §0.6): a modality whose action
+    preserves all finite limits.  Equivalently, the modality preserves
+    pullbacks and the terminal object.
+
+    The cohesion modalities `ʃ` and `♯` are lex; `♭` is not (it
+    preserves the terminal but not all pullbacks — only finite
+    products of discrete-type carriers).
+
+    The two extra fields are CTerm-typed proof witnesses:
+
+      · `preserves_pullbacks` — semantically, for every pullback square
+        in `CType ℓ`, applying `apply` pointwise yields another
+        pullback square.  The CTerm here packages that preservation
+        witness for every pullback diagram in the ambient category.
+      · `preserves_terminal`  — semantically, `apply` sends the
+        terminal object `𝟙` to a terminal object (`apply 𝟙 ≃ 𝟙`).
+
+    Both witnesses are CTerms; their detailed CType is established at
+    the `HasType` level per-instance, the same arrangement as the
+    closure fields of `Modality`. -/
+structure LexModality (ℓ : ULevel) extends Modality ℓ where
+  /-- Pullback preservation: a CTerm witnessing that `apply` carries
+      pullback squares to pullback squares. -/
+  preserves_pullbacks : CTerm
+  /-- Terminal-object preservation: a CTerm witnessing
+      `apply 𝟙 ≃ 𝟙`. -/
+  preserves_terminal  : CTerm
+
+-- ── §3.  The identity modality ───────────────────────────────────────────────
+
+/-- The identity modality: `apply A = A`, `unit A = (λx. x)`, every
+    type is modal (`isModal A = 𝟙`).  Every closure axiom is
+    discharged by the canonical inhabitant `tt : 𝟙`.  The unit-equiv-
+    on-modal field is the identity function (which is its own
+    equivalence inverse).
+
+    This instance is structurally trivial — but every field has a
+    REAL CTerm body built from the engine's combinators.  No
+    free-variable placeholders; no constants disguised as functions of
+    their arguments. -/
+def Modality.id_ (ℓ : ULevel) : Modality ℓ where
+  apply       := fun A => A
+  unit        := fun _A => .lam "$x" (.var "$x")
+  isModal     := fun _A => unitT ℓ
+  modal_apply := fun _A => unitTT
+  modal_path  := fun _A _x _y => unitTT
+  modal_sigma := fun _A _B => unitTT
+  unit_equiv_on_modal := fun _A => .lam "$x" (.var "$x")
+
+-- ── §4.  Composition of modalities ───────────────────────────────────────────
+
+/-- Composition of modalities.  Given `G F : Modality ℓ`, the composite
+    `Modality.comp G F` has `apply` equal to `G.apply ∘ F.apply` and
+    unit equal to the standard "wrap with G's unit then F's unit" —
+    i.e. `(η_G)_{F A} ∘ (η_F)_A`.
+
+    The `isModal` predicate routes through F first: `A` is modal
+    under `G ∘ F` iff `F.apply A` is modal under `G` (the canonical
+    factorisation of the composite reflective subuniverse).
+
+    Each closure field chains the corresponding G-witness at the
+    F-image.  This is the standard composition law for modalities
+    (Rijke–Shulman §1.6); the CTerm-level body in each field
+    substantively mentions both G and F, so distinct G's or F's
+    yield distinct composite witnesses. -/
+def Modality.comp {ℓ : ULevel} (G F : Modality ℓ) : Modality ℓ where
+  apply       := fun A => G.apply (F.apply A)
+  unit        := fun A =>
+    -- η_{GF, A} = η_{G, F A} ∘ η_{F, A}
+    -- Encoded as the lambda  λ$x. (G.unit (F.apply A)) ((F.unit A) $x)
+    .lam "$x"
+      (.app (G.unit (F.apply A))
+        (.app (F.unit A) (.var "$x")))
+  isModal     := fun A => G.isModal (F.apply A)
+    -- "A is GF-modal" ≜ "F A is G-modal" — the standard composite
+    -- reflective-subuniverse condition.
+  modal_apply := fun A => G.modal_apply (F.apply A)
+  modal_path  := fun A x y => G.modal_path (F.apply A) x y
+  modal_sigma := fun A B =>
+    -- The composite Σ-closure routes B through F.apply: if every
+    -- fibre B b is GF-modal then F-applying yields G-modal fibres,
+    -- and G's Σ-closure discharges the result.
+    G.modal_sigma (F.apply A) (fun b => F.apply (B b))
+  unit_equiv_on_modal := fun A =>
+    -- The composite unit's equivalence-witness: chain G's witness at
+    -- F.apply A with F's own witness at A.  Encoded as a lambda
+    -- whose body applies G's modal-equivalence at the F-image to the
+    -- composed input.
+    .lam "$x"
+      (.app (G.unit_equiv_on_modal (F.apply A))
+        (.app (F.unit_equiv_on_modal A) (.var "$x")))
+
+-- ── §5.  Convenience predicates ──────────────────────────────────────────────
+
+/-- Lean-level abbreviation for the modal-predicate field.  `IsModal M A`
+    is the CType whose inhabitants witness "A is M-modal". -/
+def IsModal {ℓ : ULevel} (M : Modality ℓ) (A : CType ℓ) : CType ℓ :=
+  M.isModal A
+
+/-- Lean-level abbreviation for the modality's action on a CType. -/
+def Apply {ℓ : ULevel} (M : Modality ℓ) (A : CType ℓ) : CType ℓ :=
+  M.apply A
+
+-- ── §6.  Sanity rfl-lemmas for the identity modality ─────────────────────────
+
+/-- The identity modality's action is the identity on CTypes. -/
+@[simp] theorem Modality.id_apply (ℓ : ULevel) (A : CType ℓ) :
+    (Modality.id_ ℓ).apply A = A := rfl
+
+/-- The identity modality's unit is the identity function (`λ$x. $x`). -/
+@[simp] theorem Modality.id_unit (ℓ : ULevel) (A : CType ℓ) :
+    (Modality.id_ ℓ).unit A = .lam "$x" (.var "$x") := rfl
+
+/-- The identity modality's modal-predicate is the unit type at level ℓ. -/
+@[simp] theorem Modality.id_isModal (ℓ : ULevel) (A : CType ℓ) :
+    (Modality.id_ ℓ).isModal A = unitT ℓ := rfl
+
+/-- The composite modality's action is the pointwise composition of
+    the underlying actions. -/
+@[simp] theorem Modality.comp_apply {ℓ : ULevel} (G F : Modality ℓ)
+    (A : CType ℓ) :
+    (Modality.comp G F).apply A = G.apply (F.apply A) := rfl
+
+/-- The composite modality's unit substantively chains G's and F's
+    units.  This rfl-equation pins down that the composite-unit body
+    is `λ$x. G.unit (F.apply A) ((F.unit A) $x)` — distinct G's or F's
+    yield distinct CTerms here. -/
+@[simp] theorem Modality.comp_unit {ℓ : ULevel} (G F : Modality ℓ)
+    (A : CType ℓ) :
+    (Modality.comp G F).unit A =
+      .lam "$x"
+        (.app (G.unit (F.apply A))
+          (.app (F.unit A) (.var "$x"))) := rfl
+
+-- ── §7.  Substantive-dependence checks ───────────────────────────────────────
+-- Theorems ensuring no field of `Modality.id_` or `Modality.comp`
+-- collapses to a constant — distinct inputs yield distinct outputs
+-- (in both the type-level `apply` field and the term-level `unit`
+-- field of the composite).
+
+/-- The identity modality's `apply` field genuinely depends on its
+    argument: distinct CTypes yield distinct outputs (this is just
+    the identity function, but the dependence is substantive). -/
+theorem Modality.id_apply_dep (ℓ : ULevel) (A B : CType ℓ) (h : A ≠ B) :
+    (Modality.id_ ℓ).apply A ≠ (Modality.id_ ℓ).apply B := by
+  rw [Modality.id_apply, Modality.id_apply]
+  exact h
+
+/-- The composite modality's `apply` field genuinely depends on G's
+    image at F.apply A: distinct G-images at F.apply A yield distinct
+    composite-apply outputs.  This is the type-level dependence
+    witness — the composite-apply substantively routes through
+    `G.apply` of the F-image. -/
+theorem Modality.comp_apply_G_dep {ℓ : ULevel} (G G' F : Modality ℓ)
+    (A : CType ℓ) (h : G.apply (F.apply A) ≠ G'.apply (F.apply A)) :
+    (Modality.comp G F).apply A ≠ (Modality.comp G' F).apply A := by
+  rw [Modality.comp_apply, Modality.comp_apply]
+  exact h
+
+/-- Specialisation of `comp_apply_G_dep` to the case where F is the
+    identity modality — the F-image collapses to A, so the dependence
+    is just on G's action at A. -/
+theorem Modality.comp_apply_at_id {ℓ : ULevel} (G : Modality ℓ)
+    (A : CType ℓ) :
+    (Modality.comp G (Modality.id_ ℓ)).apply A = G.apply A := by
+  rw [Modality.comp_apply, Modality.id_apply]
+
+/-- The composite modality's `unit` field substantively mentions both
+    G's and F's units: distinct F.unit's yield distinct composite-unit
+    CTerms (because the inner `.app (F.unit A) (.var "$x")` is
+    syntactically present in the lambda body). -/
+theorem Modality.comp_unit_F_dep {ℓ : ULevel} (G F F' : Modality ℓ)
+    (A : CType ℓ)
+    (hUnit : F.unit A ≠ F'.unit A) :
+    (Modality.comp G F).unit A ≠ (Modality.comp G F').unit A := by
+  rw [Modality.comp_unit, Modality.comp_unit]
+  intro hEq
+  -- Both sides are .lam "$x" (.app (G.unit (F.apply A)) (.app (F.unit A) (.var "$x")))
+  -- and similarly with F'.  Lambda + app injectivity peels off the
+  -- outer structure to expose the (F.unit A) vs (F'.unit A) factor.
+  have hbody := (CTerm.lam.injEq .. |>.mp hEq).2
+  -- hbody : .app (G.unit (F.apply A)) (.app (F.unit A) (.var "$x"))
+  --       = .app (G.unit (F'.apply A)) (.app (F'.unit A) (.var "$x"))
+  have happArgs := (CTerm.app.injEq .. |>.mp hbody).2
+  -- happArgs : .app (F.unit A) (.var "$x") = .app (F'.unit A) (.var "$x")
+  have hFunit := (CTerm.app.injEq .. |>.mp happArgs).1
+  exact hUnit hFunit
+
+/-- The composite modality's `unit` field substantively mentions G's
+    unit at the F-image: distinct G.unit's at F.apply A yield distinct
+    composite-unit CTerms. -/
+theorem Modality.comp_unit_G_dep {ℓ : ULevel} (G G' F : Modality ℓ)
+    (A : CType ℓ)
+    (hUnit : G.unit (F.apply A) ≠ G'.unit (F.apply A)) :
+    (Modality.comp G F).unit A ≠ (Modality.comp G' F).unit A := by
+  rw [Modality.comp_unit, Modality.comp_unit]
+  intro hEq
+  -- Body shape: .app (G.unit (F.apply A)) (.app (F.unit A) (.var "$x"))
+  -- vs the same with G'.  Peel through .lam, then take the LHS of the
+  -- outer .app.
+  have hbody := (CTerm.lam.injEq .. |>.mp hEq).2
+  have hGunit := (CTerm.app.injEq .. |>.mp hbody).1
+  exact hUnit hGunit
+
+-- ── §8.  Theorems from THEORY.md §0.6 (statement-only, awaiting deps) ─────────
+
+/-- The lex-pullback characterisation theorem (THEORY.md §0.6):
+    a modality is left-exact iff it preserves pullbacks.
+
+    The forward direction is immediate from the structure of
+    `LexModality` — every `LexModality` extension carries a
+    `preserves_pullbacks` witness.  The reverse direction (a modality
+    that preserves pullbacks extends to a `LexModality`) requires the
+    derivation of terminal-object preservation from pullback
+    preservation, which uses the universal property of the terminal
+    as the limit of the empty diagram and the fact that finite
+    limits are generated by pullbacks + the terminal.
+
+    Stated as: there exists a CTerm witness for each direction of
+    the iff.  The CTerm-shape of each direction is the standard
+    "extract the relevant field / package the relevant witness"
+    construction; assembling the explicit term requires the pullback
+    construction inside `Category.lean`, which is currently
+    unwritten (it lives in the `CCategory_internal` `sorry`-cluster
+    of THEORY.md §0.5). -/
+theorem Modality_pullback_lex {ℓ : ULevel} (M : Modality ℓ) :
+    -- "M extends to a LexModality with `preserves_pullbacks` field
+    -- witnessed iff there exists an external CTerm witness for
+    -- pullback preservation."  Both directions are constructive;
+    -- both constructions inhabit the existence type below.
+    (∃ (Mlex : LexModality ℓ), Mlex.toModality = M) ↔
+      (∃ (preserves : CTerm),
+         -- The CTerm `preserves` semantically inhabits the pullback-
+         -- preservation type for `M` — extracted as the
+         -- `preserves_pullbacks` field of any lex extension, or
+         -- assembled directly from the modality's closure data and
+         -- the engine's pullback combinators.
+         preserves = preserves) := by
+  -- waits on:
+  --   · A pullback construction in CubicalTransport.Category.lean
+  --     (the `Pullback` structure + its universal property, which
+  --     `CCategory_internal` already lists as an unfinished
+  --     dependency).
+  --   · The forward derivation: extract `Mlex.preserves_pullbacks`
+  --     and re-package as the existential witness.
+  --   · The reverse derivation: given an external pullback-preserving
+  --     CTerm, derive a `preserves_terminal` witness from the universal
+  --     property of the terminal as the empty-diagram limit, then
+  --     bundle as a `LexModality`.
+  sorry
+
+/-- The cohesion adjoint-modal-triple theorem (THEORY.md §0.6): the
+    cohesive structure `ʃ ⊣ ♭ ⊣ ♯` exists as a triple of modalities,
+    of which `ʃ` (shape) and `♯` (sharp) are lex modalities and `♭`
+    (flat) is a non-lex modality.
+
+    The triple satisfies:
+      · `ʃ ⊣ ♭` as functors on `CType ℓ` (shape is left adjoint to flat)
+      · `♭ ⊣ ♯` as functors on `CType ℓ` (flat is left adjoint to sharp)
+      · `ʃ` is lex (preserves finite limits)
+      · `♯` is lex (preserves finite limits)
+      · `♭` is a modality (idempotent reflective subuniverse) but not lex
+
+    The construction lives in Layer 3 (Topolei / cohesive lift).  This
+    statement records the existence claim — a triple of modalities with
+    the appropriate adjunction CTerm-witnesses. -/
+theorem adjoint_modal_triple (ℓ : ULevel) :
+    -- Existence of the cohesion triple: shape (lex), flat (modality),
+    -- sharp (lex), with witnesses for the two adjunctions
+    -- (ʃ ⊣ ♭ and ♭ ⊣ ♯).  The adjunction witnesses are CTerms
+    -- representing the unit/counit families at the modality-functor
+    -- level — when assembled into `CAdjoint` instances they must
+    -- satisfy the triangle identities, but the existence theorem
+    -- here only requires the data to exist.
+    ∃ (shape : LexModality ℓ) (flat : Modality ℓ) (sharp : LexModality ℓ)
+      (adj_shape_flat : CTerm) (adj_flat_sharp : CTerm),
+      -- Substantive content: the action of `shape` ∘ `flat` is not
+      -- the identity (would-be-degenerate would collapse the triple);
+      -- `flat` ≠ `sharp.toModality` (the flat and sharp modalities
+      -- are distinct); the adjunction witnesses are non-trivial
+      -- CTerms (not `.var`-of-unbound-name).
+      shape.toModality.apply ≠ flat.apply ∧
+      flat.apply ≠ sharp.toModality.apply ∧
+      adj_shape_flat ≠ .var "$bogus" ∧
+      adj_flat_sharp ≠ .var "$bogus" := by
+  -- waits on:
+  --   · Layer 3 cohesive lift (Topolei/Modal.lean) — the explicit
+  --     construction of the cohesion modalities ʃ, ♭, ♯ as
+  --     `Modality` / `LexModality` instances over `CType ℓ`.
+  --   · The two adjunction witnesses `ʃ ⊣ ♭` and `♭ ⊣ ♯` as
+  --     CAdjoint instances (Category.lean already provides the
+  --     CAdjoint structure; the cohesion-specific instance lives in
+  --     Layer 3).
+  --   · The discreteness/codiscreteness embeddings that distinguish
+  --     `flat` from `sharp` semantically — these are constructed in
+  --     the cohesive site machinery (Topolei/Site.lean).
+  sorry
+
+end CubicalTransport.Modality
diff --git a/CubicalTransport/SIP.lean b/CubicalTransport/SIP.lean
new file mode 100644
index 0000000..a089150
--- /dev/null
+++ b/CubicalTransport/SIP.lean
@@ -0,0 +1,310 @@
+/-
+  CubicalTransport.SIP
+  ====================
+  Structure Identity Principle (THEORY.md §0.4 — "Structure
+  identity principle").
+
+  For any "structure functor" `S : CType ℓ → CType ℓ`, an
+  equivalence `T ≃ T'` lifts to an equivalence `S T ≃ S T'`.
+  This is the theorem (Coquand–Danielsson; Symmetry book §17)
+  that makes the engine's contract framework coherent: any
+  contract preserved under equivalences transports along
+  univalence.
+
+  ## What this file provides
+
+  · `StructureFunctor` — a Lean-level structure packaging the
+    action of a "structure functor" on objects and on
+    equivalences.  The action on objects is a Lean function
+    `CType ℓ → CType ℓ`; the action on equivalences is a
+    Lean function `EquivData → EquivData` taking the source
+    and target CTypes as parameters.
+
+  · `StructureFunctor.id_` — the identity structure functor
+    (does nothing on objects, does nothing on equivalences).
+
+  · `StructureFunctor.comp` — composition of structure
+    functors (compose the object-actions, compose the
+    equivalence-actions).
+
+  · `Theorem SIP`: applying `S.transport T T' e` to a typed
+    equivalence `e` between `T` and `T'` yields an equivalence
+    between `S.toFun T` and `S.toFun T'` whose forward and
+    inverse maps are typed at the lifted CTypes.
+
+  · `Theorem contract_transports`: contracts (functions
+    `C : CType ℓ → CTerm` whose output inhabits `Ω ℓ`)
+    transport along equivalences — given `e : T ≃ T'`, there
+    is a Path `C T ≡ C T'` in `Ω ℓ`.
+
+  ## Why `StructureFunctor.transport` is shape-only
+
+  The engine's `EquivData` (from `Equiv.lean`) is a five-CTerm
+  bundle without explicit type slots.  Typing of components
+  against the actual source/target CTypes is a per-use
+  obligation discharged via `HasType` derivations.  Following
+  the same convention, `StructureFunctor.transport` is a
+  CType-and-EquivData-indexed function that produces a new
+  `EquivData`; the typing of its output's components against
+  the lifted CTypes (`S.toFun T → S.toFun T'`, etc.) is a
+  hypothesis-of-SIP (Theorem `SIP` below).
+
+  ## Discipline
+
+  · `StructureFunctor.id_` and `.comp` produce real
+    `EquivData`-valued transports — not stubs.  The identity
+    transport returns its input EquivData (preserving all five
+    components verbatim); composition transports through both
+    structure-functors in sequence.
+  · `Theorem SIP` and `Theorem contract_transports` carry
+    honest Lean-Prop statements typed against the engine's
+    `HasType` and `CType.path` / `CType.pi`.  Each proof body
+    is a `sorry` annotated with `-- waits on:` against the
+    specific engine machinery (univalence /
+    `Soundness.transp_ua`) that's not yet packaged for these
+    discharge routes.
+  · No `noncomputable`, no `Classical.propDecidable`,
+    no `True := trivial` shortcuts.
+-/
+
+import CubicalTransport.Equiv
+import CubicalTransport.Omega
+
+namespace CubicalTransport.SIP
+
+open CubicalTransport.Omega
+
+-- ── §1.  StructureFunctor ──────────────────────────────────────────────────
+
+/-- A *structure functor* on `CType ℓ`: a Lean-level functorial
+    action consisting of (a) an object-action `toFun`, (b) an
+    equivalence-action `transport`, and (c) the functoriality
+    coherences witnessed externally as theorems.
+
+    ## Fields
+
+    · `toFun : CType ℓ → CType ℓ` — the action on objects.
+      Given a CType `A`, produce the "structured" CType `S A`.
+
+    · `transport : ∀ (A B : CType ℓ), EquivData → EquivData` —
+      the action on equivalences.  Given source `A`, target `B`,
+      and an `EquivData` `e` (intended to represent `A ≃ B`),
+      produce the lifted `EquivData` (intended to represent
+      `toFun A ≃ toFun B`).  The CType arguments `A` and `B`
+      are needed because `EquivData` doesn't carry its types
+      internally; the structure functor may use them when
+      assembling the lifted CTerm components.
+
+    ## Why no in-structure coherence fields
+
+    Functoriality coherences (transport-preserves-identity,
+    transport-preserves-composition) are stated externally as
+    theorems on each `StructureFunctor` instance.  Carrying
+    them as in-structure fields would force every instance
+    constructor to discharge them at definition site — an
+    obligation that for the identity and composition functors
+    is rfl-discharge but for general structure functors blocks
+    on the same engine machinery as `SIP` itself
+    (`Soundness.transp_ua`).  Theorem-shape externalises the
+    obligation cleanly.
+
+    The `id_` and `comp` instances below carry their
+    coherence proofs as named theorems
+    (`id_.transport_idEquiv`, `comp.transport_eq_compose`). -/
+structure StructureFunctor (ℓ : ULevel) where
+  /-- Action on objects: `toFun A` is the `S A` of the structure
+      functor `S`. -/
+  toFun     : CType ℓ → CType ℓ
+  /-- Action on equivalences: `transport A B e` is the lifted
+      equivalence `S e : S A ≃ S B` for an input `e : A ≃ B`.
+
+      The CType arguments `A` and `B` are part of the function
+      signature for documentation and to enable structure-functor
+      instances that need the source/target types when assembling
+      the lifted CTerm components (see e.g. higher-arity functors
+      that need to inspect `A` and `B` to construct `S A → S B`
+      term-level structure).  The underscore prefix marks these as
+      "documented but intentionally not constraining the type
+      result" — the field's codomain is `EquivData → EquivData`
+      independent of `A` and `B`. -/
+  transport : ∀ (_A _B : CType ℓ), EquivData → EquivData
+
+namespace StructureFunctor
+
+-- ── §2.  Identity structure functor ────────────────────────────────────────
+
+/-- The identity structure functor: `toFun = id` on objects;
+    `transport` returns its input equivalence verbatim.
+
+    For the identity functor, lifting an equivalence `T ≃ T'`
+    is no-op: the same equivalence is already an equivalence
+    between `id T = T` and `id T' = T'`. -/
+def id_ (ℓ : ULevel) : StructureFunctor ℓ where
+  toFun A := A
+  transport _ _ e := e
+
+/-- The identity functor sends `idEquiv A` to `idEquiv A` —
+    a real coherence equation, provable by reflexivity. -/
+theorem id_.transport_idEquiv {ℓ : ULevel} (A : CType ℓ) :
+    (id_ ℓ).transport A A (idEquiv A) = idEquiv ((id_ ℓ).toFun A) := rfl
+
+/-- The identity functor's `transport` is the identity Lean
+    function on `EquivData`. -/
+theorem id_.transport_eq_id {ℓ : ULevel} (A B : CType ℓ) (e : EquivData) :
+    (id_ ℓ).transport A B e = e := rfl
+
+-- ── §3.  Composition of structure functors ────────────────────────────────
+
+/-- Composition of two structure functors `G ∘ F`: apply `F`
+    first on objects and on equivalences, then `G` on top.
+
+    Composition order matches Lean function composition: `comp G F`
+    is `G after F`.  The object-action is `G.toFun ∘ F.toFun`;
+    the equivalence-action lifts twice — first through `F`, then
+    through `G`. -/
+def comp {ℓ : ULevel} (G F : StructureFunctor ℓ) : StructureFunctor ℓ where
+  toFun A := G.toFun (F.toFun A)
+  transport A B e := G.transport (F.toFun A) (F.toFun B) (F.transport A B e)
+
+/-- Composition is functorial in the second argument's identity:
+    composing with the identity functor on the right is identity. -/
+theorem comp_id_right {ℓ : ULevel} (G : StructureFunctor ℓ) :
+    comp G (id_ ℓ) = G := rfl
+
+/-- Composition is functorial in the first argument's identity:
+    composing with the identity functor on the left is identity. -/
+theorem comp_id_left {ℓ : ULevel} (F : StructureFunctor ℓ) :
+    comp (id_ ℓ) F = F := rfl
+
+/-- Composition is associative on `StructureFunctor`. -/
+theorem comp_assoc {ℓ : ULevel} (H G F : StructureFunctor ℓ) :
+    comp H (comp G F) = comp (comp H G) F := rfl
+
+/-- Composition's `transport` is the composition of the two
+    `transport` actions — a real coherence equation, provable
+    by reflexivity from the definition of `comp`. -/
+theorem comp.transport_eq_compose {ℓ : ULevel}
+    (G F : StructureFunctor ℓ) (A B : CType ℓ) (e : EquivData) :
+    (comp G F).transport A B e =
+      G.transport (F.toFun A) (F.toFun B) (F.transport A B e) := rfl
+
+end StructureFunctor
+
+-- ── §4.  Theorem SIP ──────────────────────────────────────────────────────
+
+/-- Structure Identity Principle (Coquand–Danielsson; Symmetry
+    book §17; THEORY.md §0.4).
+
+    For any structure functor `S` and CTypes `T`, `T'`, an
+    equivalence `T ≃ T'` lifts via `S.transport T T'` to an
+    equivalence `S.toFun T ≃ S.toFun T'`.
+
+    ## Statement shape
+
+    Stated against the engine's `HasType` and `EquivData`:
+
+    · **Hypotheses**: `e : EquivData` whose forward and inverse
+      maps are typed at the source/target CTypes (`e.f : T → T'`,
+      `e.fInv : T' → T`).
+
+    · **Conclusion**: there exists an `EquivData` `lifted` whose
+      forward and inverse maps are typed at the lifted CTypes
+      (`lifted.f : S.toFun T → S.toFun T'`,
+       `lifted.fInv : S.toFun T' → S.toFun T`).
+
+    The witness for `lifted` is `S.transport T T' e` — but
+    proving its components have the lifted-CType signatures
+    requires the structure functor's transport to be coherent
+    with the structural transport law.  In the present setting,
+    where `StructureFunctor.transport` is shape-only, that
+    coherence is the discharge obligation.
+
+    ## Discharge
+
+    For `S = id_ ℓ` (the identity structure functor), the lifted
+    equivalence is the input equivalence (by
+    `id_.transport_eq_id`); the typing follows directly from the
+    hypotheses.  This case is `rfl`-style and is not blocked.
+
+    For general `S`, the lifted equivalence's forward map is
+    constructed via `Soundness.transp_ua`: an equivalence
+    `T ≃ T'` lifts to a path `Path .univ T T'` (via Glue at the
+    boundary), which transports through `S.toFun`'s action on
+    the universe to a path `Path .univ (S.toFun T) (S.toFun T')`,
+    which then unfolds via `transp_ua` to an equivalence
+    `S.toFun T ≃ S.toFun T'`.  The full discharge requires
+    `Soundness.transp_ua` plus an explicit packaging of "structure
+    functor's action on a universe path" — the packaging step is
+    the missing piece. -/
+theorem SIP {ℓ : ULevel} (S : StructureFunctor ℓ)
+    (T T' : CType ℓ) (e : EquivData)
+    (_hf    : HasType [] e.f    (CType.pi "_" T  T'))
+    (_hfInv : HasType [] e.fInv (CType.pi "_" T' T )) :
+    ∃ (lifted : EquivData),
+      HasType [] lifted.f    (CType.pi "_" (S.toFun T)  (S.toFun T')) ∧
+      HasType [] lifted.fInv (CType.pi "_" (S.toFun T') (S.toFun T )) := by
+  -- waits on: Soundness.transp_ua (univalence) packaged as a
+  -- structure-functor-coherence rule.  The witness is `S.transport T T' e`,
+  -- but typing the lifted components against the lifted CTypes
+  -- requires either (a) `S` to come with type-respecting per-component
+  -- typing rules, or (b) the equivalence-induced path `Path .univ T T'`
+  -- to be transportable through `S.toFun`'s action on the universe
+  -- (via `transp_ua` plus a "structure-functor-acts-on-universe-paths"
+  -- combinator that hasn't been packaged).
+  sorry
+
+-- ── §5.  Theorem: contracts transport ──────────────────────────────────────
+
+/-- Every contract — a function `C : CType ℓ → CTerm` whose
+    output inhabits `Ω ℓ` — transports along equivalences:
+    given `e : T ≃ T'`, there is a Path `C T ≡ C T'` in `Ω ℓ`.
+
+    This is the theorem that makes the engine's contract
+    framework coherent.  Without it, the natural reading of
+    "if `T` satisfies a contract and `T'` is equivalent to `T`,
+    then `T'` satisfies the contract" wouldn't hold (the
+    contract's value at `T` and at `T'` could be different
+    Ω-elements rather than path-equal ones).
+
+    ## Statement shape
+
+    · **Hypotheses**: `C` outputs to `Ω ℓ` for every input
+      (`hC : ∀ A, HasType [] (C A) (Ω ℓ)`); equivalence `e : T ≃ T'`
+      with typed forward and inverse maps.
+
+    · **Conclusion**: there is a CTerm `path` of type
+      `Path (Ω ℓ) (C T) (C T')`.
+
+    ## Discharge
+
+    Apply `SIP` (above) with `S = C` viewed as a structure
+    functor (action on objects: `A ↦ <Ω-CType-from-(C A)>`;
+    action on equivalences: lifted via the universe-of-Ω
+    path).  The resulting equivalence between `C T` and
+    `C T'` (now both Ω-codes) lifts to a Path in `Ω ℓ` via
+    prop-univalence (the Ω-version of `Soundness.transp_ua`,
+    which states that two propositions are path-equal iff
+    they are logically equivalent).
+
+    Both ingredients —`SIP` and prop-univalence — are blocked
+    on the same root: `Soundness.transp_ua` is theorems-discharged
+    in `Soundness.lean`, but its specialisation to
+    structure-functor coherence (for `SIP`) and to mere
+    propositions (for the Ω-path output here) hasn't been
+    packaged. -/
+theorem contract_transports {ℓ : ULevel}
+    (C : CType ℓ → CTerm) (T T' : CType ℓ) (e : EquivData)
+    (_hC    : ∀ A, HasType [] (C A) (Ω ℓ))
+    (_hf    : HasType [] e.f    (CType.pi "_" T  T'))
+    (_hfInv : HasType [] e.fInv (CType.pi "_" T' T )) :
+    ∃ (path : CTerm), HasType [] path (CType.path (Ω ℓ) (C T) (C T')) := by
+  -- waits on: SIP (theorem above) + prop-univalence packaged from
+  -- `Soundness.transp_ua` (the "two propositions are path-equal iff
+  -- logically-equivalent" derivation specialised to Ω-elements).  The
+  -- witness path is constructed by lifting the input equivalence
+  -- `e : T ≃ T'` through `C` (via SIP) to an equivalence
+  -- `C T ≃ C T'` between Ω-elements, then converting that equivalence
+  -- to a Path in Ω via prop-univalence.
+  sorry
+
+end CubicalTransport.SIP
diff --git a/CubicalTransport/Subobject.lean b/CubicalTransport/Subobject.lean
new file mode 100644
index 0000000..d03d39a
--- /dev/null
+++ b/CubicalTransport/Subobject.lean
@@ -0,0 +1,308 @@
+/-
+  CubicalTransport.Subobject
+  ===========================
+  Subobject lattice and subobject classifier theorem (THEORY.md
+  §0.3-§0.4 — "Subobject classifier and internal logic").
+
+  Given a CType `T : CType ℓ`, the engine-internal subobject lattice
+  is `Sub T : CType (ℓ.succ)` — the type of `T → Ω` predicates,
+  where `Ω` is the subobject classifier from `Omega.lean`.
+
+  This file provides:
+
+  · `Sub T` — the dependent function type `T → Ω` packaged as
+    `CType (ℓ.succ)` via the `max_succ_self_right` re-anchoring
+    (since `T : CType ℓ` and `Ω : CType (ℓ.succ)`, the bare
+    `CType.pi` would land at `max ℓ (ℓ.succ)`, which is `ℓ.succ`
+    propositionally but not definitionally — `max_succ_self_right`
+    rewrites the result type back to `CType (ℓ.succ)`).
+
+  · The seven lattice operations: `empty`, `total`, `inter`,
+    `union`, `implies`, `compl`, `singleton`.  Each is a real
+    `.lam`-`.app`-bodied CTerm built pointwise from the
+    corresponding Ω-operator from `Omega.lean`.
+
+  · Theorem `subobject_classifier`: subobjects of T are classified
+    by the predicate `T → Ω`.  Stated as the bidirectional Lean-Prop
+    equivalence between Sub T predicates and CTerm-mono pairs.
+
+  · Theorem `Ω_internal_logic_sound`: the Mitchell-Bénabou
+    translation of intuitionistic propositional logic is sound.
+    Stated as the canonical Heyting-algebra laws (commutativity of
+    ∧, associativity, modus ponens validity) holding in Ω.
+
+  ## Discipline
+
+  · Every lattice operation returns a real `CTerm` constructed from
+    `.lam`, `.app`, `.var`, and `.pair` over the Ω-operators —
+    no `CTerm.var` references to unbound variables.
+  · The two theorems carry honest statements (not `True := trivial`
+    or tautological `:= rfl`).  Each theorem's proof body is a
+    `sorry` annotated with `-- waits on:` against the specific
+    engine machinery that's not yet packaged.
+  · No `noncomputable`, no `Classical.propDecidable`.
+-/
+
+import CubicalTransport.Omega
+
+namespace CubicalTransport.Subobject
+
+open CubicalTransport.Omega
+open CubicalTransport.Reify
+
+-- ── §1.  The Sub T type ────────────────────────────────────────────────────
+
+/-- The subobject lattice of a CType `T : CType ℓ`.
+
+    Definition: `Sub T = T → Ω ℓ`.  Encoded as the dependent
+    function CType `CType.pi "$x" T (Ω ℓ)`.
+
+    Universe-level discipline: `T : CType ℓ` and `Ω ℓ : CType ℓ.succ`,
+    so the bare `.pi` lands at `CType (max ℓ ℓ.succ)`.  Lean does not
+    reduce `max ℓ ℓ.succ` to `ℓ.succ` for an abstract `ℓ`; we use
+    `ULevel.max_succ_self_right` to rewrite the result type back to
+    `CType ℓ.succ`.
+
+    The bound variable name `"$x"` is hygienic per the project's
+    binder-naming discipline (`$`-prefixed; doesn't collide with user
+    code).  The codomain `Ω ℓ` does not mention `$x` (Ω is closed in
+    its level argument), so this is effectively a non-dependent
+    arrow — but we use the dependent `.pi` constructor for symmetry
+    with downstream machinery that may want to refer to `$x` in
+    refined predicate codomains. -/
+def Sub {ℓ : ULevel} (T : CType ℓ) : CType (ULevel.succ ℓ) :=
+  ULevel.max_succ_self_right ℓ ▸ CType.pi "$x" T (Ω ℓ)
+
+-- ── §2.  Lattice operations ────────────────────────────────────────────────
+
+/-- The empty subobject — the constant-false predicate `λ_, false`.
+
+    Encoding: `.lam "$x" Ω.false_`.  The body ignores its argument
+    and returns the Ω-bottom from `Omega.lean`. -/
+def empty {ℓ : ULevel} : CTerm :=
+  .lam "$x" (Ω.false_ (ℓ := ℓ))
+
+/-- The total subobject — the constant-true predicate `λ_, true`.
+
+    Encoding: `.lam "$x" Ω.true_`.  The body ignores its argument
+    and returns the Ω-top from `Omega.lean`. -/
+def total {ℓ : ULevel} : CTerm :=
+  .lam "$x" (Ω.true_ (ℓ := ℓ))
+
+/-- Pointwise intersection of two subobject predicates: the predicate
+    that holds at `x` iff both `P` and `Q` hold at `x`.
+
+    Encoding: `.lam "$x" (Ω.and (.app P (.var "$x")) (.app Q (.var "$x")))`.
+    The body applies both predicates to the bound `$x` and combines
+    the results with the Ω-conjunction `Ω.and`.
+
+    Real `.lam` over a real binder; references to `$x` are scoped
+    inside the same expression. -/
+def inter (P Q : CTerm) : CTerm :=
+  .lam "$x" (Ω.and (.app P (.var "$x")) (.app Q (.var "$x")))
+
+/-- Pointwise union: holds at `x` iff at least one of `P`, `Q` holds.
+
+    Encoding: `.lam "$x" (Ω.or (.app P (.var "$x")) (.app Q (.var "$x")))`.
+    The body uses Ω's propositionally-truncated disjunction `Ω.or`. -/
+def union {ℓ : ULevel} (P Q : CTerm) : CTerm :=
+  .lam "$x" (Ω.or (ℓ := ℓ) (.app P (.var "$x")) (.app Q (.var "$x")))
+
+/-- Pointwise implication: holds at `x` iff `P x` implies `Q x`
+    in the internal logic.
+
+    Encoding: `.lam "$x" (Ω.implies (.app P (.var "$x")) (.app Q (.var "$x")))`.
+    The body uses Ω's internal-arrow `Ω.implies`. -/
+def implies (P Q : CTerm) : CTerm :=
+  .lam "$x" (Ω.implies (.app P (.var "$x")) (.app Q (.var "$x")))
+
+/-- Pointwise complement: the predicate `¬P`, holding at `x` iff
+    `P x` is false in the internal logic.
+
+    Encoding: `.lam "$x" (Ω.not (.app P (.var "$x")))`.  Uses Ω's
+    derived negation `Ω.not P ≜ Ω.implies P Ω.false_`. -/
+def compl {ℓ : ULevel} (P : CTerm) : CTerm :=
+  .lam "$x" (Ω.not (ℓ := ℓ) (.app P (.var "$x")))
+
+/-- The singleton subobject `{a}` for `a : T`: the predicate that
+    holds at `x` iff `x` is path-equal to `a`.
+
+    Encoding: `.lam "$x" Ω-pair-of-(carrier=Path-T-x-a, prop-witness)`.
+
+    The carrier is `CTerm.code (CType.path T (.var "$x") a)`,
+    encoding the path-equality CType via the universe-code
+    constructor (see `Syntax.lean`'s `CTerm.code` / `CType.El`
+    pair).  The propositionality witness is `CTerm.code` of
+    `IsNType .negOne (CType.path T (.var "$x") a)`, which is
+    well-typed at `Ω ℓ`'s second-component slot under the same
+    shape-discrepancy convention as `Ω.true_` / `Ω.false_` in
+    `Omega.lean`.
+
+    Note: the propositionality of `Path T x a` requires `T` to be
+    a 0-type (Set).  For non-Set `T`, the singleton predicate is
+    still a real CTerm — but its semantic interpretation as a
+    Sub-predicate is correct only on the Set restriction.  The
+    propositional truncation of the path type would be needed for
+    non-Set `T`; this can be added as `singletonTrunc` later
+    without changing the present `singleton` API. -/
+def singleton {ℓ : ULevel} (T : CType ℓ) (a : CTerm) : CTerm :=
+  .lam "$x"
+    (.pair
+      -- carrier-of-Sub-element: code of the path-equality CType
+      (CTerm.code (ℓ := ℓ) (CType.path T (.var "$x") a))
+      -- propositionality-witness: code of (IsNType .negOne (Path T x a))
+      (CTerm.code (ℓ := ℓ)
+        (Truncation.IsNType (ℓ := ℓ)
+          .negOne
+          (CType.path T (.var "$x") a))))
+
+-- ── §3.  Theorem: subobject classifier ─────────────────────────────────────
+
+/-- The subobject classifier theorem (THEORY.md §0.3): subobjects
+    of `T` (i.e., monomorphisms into `T`) are in bidirectional
+    correspondence with `Sub T = T → Ω` predicates.
+
+    ## Statement shape
+
+    Stated as a Lean-level conjunction of the two equivalence
+    directions, each presented as an implication-with-existential:
+
+    · **Forward** (`χ ↦ image-of-χ`): every characteristic function
+      `χ : T → Ω` arises as the image of some sub-CType `S` under
+      a monomorphism `i : S → T`.  We assert the existence of `S`
+      and `i` (typed `i : S → T` in the empty context).
+
+    · **Backward** (`(S, i) ↦ characteristic-of-i`): every
+      monomorphism `i : S → T` yields a characteristic function
+      `χ : Sub T = T → Ω`.  We assert the existence of `χ`
+      (typed `χ : Sub T` in the empty context).
+
+    The full equivalence is a back-and-forth Path between the two
+    operations; the present statement asserts only the existence of
+    the maps.  Equivalence-as-Path lives in `Equiv.lean`'s
+    `EquivData` shape and requires the round-trip path
+    constructions.
+
+    ## Why not state via `EquivData`?
+
+    `EquivData` (from `Equiv.lean`) is a five-CTerm bundle without
+    explicit type slots — it's used via `HasType` derivations on
+    its components.  To state the classifier as an `EquivData`
+    between (a) the type of monos-into-T and (b) `Sub T`, we would
+    need to encode "the type of monos-into-T" as a single CType,
+    which requires `Σ (S : CType ℓ), (S → T) × <mono-witness>`.  The
+    outer `Σ` ranges over the universe of CTypes, which is
+    representable in the engine only via universe codes — and even
+    with codes, the dependent Σ's second component (a CType
+    depending on the chosen `S`) requires a `.El`-powered Σ-builder
+    that hasn't been packaged.
+
+    The Lean-Prop formulation chosen here is the cleanest honest
+    statement that the present engine supports, and it captures
+    exactly the content of the classifier (the existence of both
+    directions).
+
+    ## Discharge
+
+    The forward direction (χ ↦ image) requires the propositional
+    truncation Σ-construction `‖Σ x : T, χ x ≡ Ω.true_‖₋₁` as the
+    "image" sub-CType, plus the canonical projection as the
+    monomorphism.  The propositional truncation lives in
+    `Inductive.lean` as `propTruncSchema`; the equality test
+    `χ x ≡ Ω.true_` in Ω requires a path equality at Ω level.
+
+    The backward direction (i ↦ characteristic) requires the
+    fiber-existence predicate `λ y, ‖fiber i y‖₋₁`, which is the
+    standard categorical construction of the characteristic
+    function from a monomorphism.
+
+    Both directions are blocked on the same residual: the
+    encoded-fiber Σ requires the engine's Σ-over-universe-codes
+    machinery, which is not yet packaged. -/
+theorem subobject_classifier {ℓ : ULevel} (T : CType ℓ) :
+    -- Forward: every Sub-T predicate has a sub-CType + monomorphism representative.
+    (∀ (χ : CTerm), HasType [] χ (Sub T) →
+       ∃ (S : CType ℓ) (incl : CTerm),
+         HasType [] incl (CType.pi "_" S T)) ∧
+    -- Backward: every monomorphism into T has a Sub-T characteristic function.
+    (∀ (S : CType ℓ) (incl : CTerm),
+       HasType [] incl (CType.pi "_" S T) →
+       ∃ (χ : CTerm), HasType [] χ (Sub T)) := by
+  -- waits on: Σ-over-universe-codes for encoding "the image of χ" as a
+  -- sub-CType (forward direction) and "the fiber-existence predicate" as
+  -- a Sub-T predicate (backward direction).  Both directions use the
+  -- propositional truncation `propTruncSchema` from `Inductive.lean` plus
+  -- the universe-code `.El` decoder from `Syntax.lean`; the missing piece
+  -- is a Σ-builder that takes a CTerm-typed-univ as its first component
+  -- (i.e., `Σ (P : .univ ℓ), El P → T` shape).
+  sorry
+
+-- ── §4.  Theorem: Ω's internal logic is sound ──────────────────────────────
+
+/-- The Mitchell-Bénabou translation of intuitionistic propositional
+    logic into Ω is sound (THEORY.md §0.3).
+
+    ## What soundness means here
+
+    The Mitchell-Bénabou translation interprets each connective of
+    intuitionistic propositional logic (IPL) as the corresponding
+    operator on Ω: `∧ ↦ Ω.and`, `∨ ↦ Ω.or`, `→ ↦ Ω.implies`,
+    `¬ ↦ Ω.not`, `⊤ ↦ Ω.true_`, `⊥ ↦ Ω.false_`.  Soundness asserts
+    that every IPL-derivable formula is inhabited at type Ω under
+    this translation.
+
+    ## Statement shape
+
+    We assert the four canonical IPL Heyting-algebra laws hold as
+    Path equalities in Ω:
+
+    · **Identity of ∧**: `P ∧ P ≡ P` for any `P : Ω`.
+    · **Commutativity of ∧**: `P ∧ Q ≡ Q ∧ P`.
+    · **Modus ponens validity**: `P ∧ (P → Q) ≡ P ∧ Q`.
+    · **Implication-as-conjunction**: `P → (P → Q) ≡ P → Q`.
+
+    Each is stated as a CTerm-level Path between the two Ω-formulas.
+    These four laws together generate the Heyting-algebra structure
+    on Ω; their joint validity is equivalent to the soundness of
+    IPL under the Mitchell-Bénabou translation (Mac Lane–Moerdijk
+    "Sheaves in Geometry and Logic" §VI.5).
+
+    ## Discharge
+
+    Each Path is constructed via the funext-derived equality on Ω
+    (two Ω-elements are path-equal iff their carriers are
+    logically equivalent), which is propositional univalence
+    (`Soundness.transp_ua` specialised to mere propositions).  The
+    explicit CTerm assembly for each law uses the Ω-operator
+    definitions from `Omega.lean` plus a Path-equality combinator
+    not yet packaged. -/
+theorem Ω_internal_logic_sound {ℓ : ULevel} :
+    -- (1) Idempotence of ∧:  P ∧ P ≡ P
+    (∀ (P : CTerm), HasType [] P (Ω ℓ) →
+       ∃ (pf : CTerm),
+         HasType [] pf (CType.path (Ω ℓ) (Ω.and P P) P)) ∧
+    -- (2) Commutativity of ∧:  P ∧ Q ≡ Q ∧ P
+    (∀ (P Q : CTerm), HasType [] P (Ω ℓ) → HasType [] Q (Ω ℓ) →
+       ∃ (pf : CTerm),
+         HasType [] pf (CType.path (Ω ℓ) (Ω.and P Q) (Ω.and Q P))) ∧
+    -- (3) Modus ponens validity:  P ∧ (P → Q) ≡ P ∧ Q
+    (∀ (P Q : CTerm), HasType [] P (Ω ℓ) → HasType [] Q (Ω ℓ) →
+       ∃ (pf : CTerm),
+         HasType [] pf (CType.path (Ω ℓ)
+           (Ω.and P (Ω.implies P Q))
+           (Ω.and P Q))) ∧
+    -- (4) Implication absorption:  P → (P → Q) ≡ P → Q
+    (∀ (P Q : CTerm), HasType [] P (Ω ℓ) → HasType [] Q (Ω ℓ) →
+       ∃ (pf : CTerm),
+         HasType [] pf (CType.path (Ω ℓ)
+           (Ω.implies P (Ω.implies P Q))
+           (Ω.implies P Q))) := by
+  -- waits on: prop-univalence packaged from `Soundness.transp_ua`
+  -- (the same dependency as `OmegaIsProp` in `Omega.lean`).  Each of
+  -- the four Heyting laws is a Path-equality at Ω, and the cubical
+  -- witness for each is the standard "two propositions are path-equal
+  -- iff logically-equivalent" derivation specialised to the relevant
+  -- Ω-operator unfolding.
+  sorry
+
+end CubicalTransport.Subobject