NOTE-1223: Baez & Fong — A compositional framework for passive linear networks

Contribution

It gives circuits of passive linear two-terminal components an explicitly compositional semantics. A circuit with marked inputs and outputs is a morphism in a category Circ, composition is gluing outputs to inputs by pushout, and a functor sends each circuit to its external behaviour. The authors’ own account of what is new: “All we are adding is an emphasis on symplectic geometry and an explicitly compositional framework, which clarifies the way a larger circuit can be built from smaller pieces” (p. 3). The main theorem is that this “black box functor” exists and preserves the hypergraph structure (Thm 1.1, p. 7; proved as Thm 7.7, p. 47, and identified with Thm 1.1’s formula in Section 8, p. 51). Along the way it proves that eliminating interior nodes of a Dirichlet form gives a Dirichlet form over any field with positive elements (Thm 3.3, Lemma 3.4).

Key insight

The external behaviour of a circuit is a subspace, not a function: the set of potential–current pairs its terminals can jointly take. For a circuit of resistors that subspace is the graph of the differential of the power functional Q, the quadratic form left when the interior potentials are chosen to minimize power: “the current at any terminal can be obtained by differentiating Q with respect to the potential at this terminal” (p. 4). Writing behaviours as Lagrangian subspaces rather than as quadratic forms is what makes them compose with identities, because an ideal wire has no quadratic form but does have a Lagrangian relation (Sec. 3.3, pp. 22–23). Black-boxing forgets the interior, and does so functorially: “we can compute the black boxed version of a circuit made of parts by computing the black boxed versions of the parts and then composing them” (p. 7).

Assumptions

• Passive, linear, two-terminal components only. “We only study circuits built from passive components … we exclude batteries and current sources. We only consider components that respond linearly to an applied voltage … we exclude components such as nonlinear resistors or diodes. Finally, we only consider components with one input and one output … Thus, we also exclude transformers” (p. 3). More general circuits are left to a companion paper ([3], Baez, Coya and Rebro).
• A field with positive elements (F, F⁺) (Def. 2.10, p. 15): F⁺ contains c² for every nonzero c and is closed under addition, multiplication and division. It follows that the field has characteristic zero. Resistances live in (ℝ, (0, ∞)).
• Time dependence only through the Laplace transform, with “(i) zero initial conditions (that is, f(0) = 0) and (ii) Laplace transform lying in the field” ℝ(s) of real rational functions (p. 16). Impedances are sL (inductor), 1/(sC) (capacitor) and R (resistor), in ℝ(s)⁺ = {Z ∈ ℝ(s) : Z ≠ 0 and s > 0 ⟹ Z(s) ≥ 0} (p. 17). The authors warn that this positivity “differs from the one more commonly used in circuit theory [11], which gives a set that is not closed under multiplication” (p. 17).
• Minimization becomes stationarity over a general field. “Although it is not clear what it means to minimize over the field F, we can use formal derivatives to formulate a version of the principle of minimum power. This is actually a ‘variational principle’” (p. 18). Over ℝ(s) the interior is fixed by vanishing formal derivatives, i.e. Kirchhoff’s current law at interior nodes (Thm 3.5, p. 22), not by a minimum.
• Circuits are labelled graphs with nodes N, edges E, source and target maps and an impedance on each edge, and an open circuit is a cospan X → N ← Y of finite sets with such a circuit on N (Def. 5.1, p. 29).

Key results

Part I: the power functional (Sections 2–3)

• Minimum power is Kirchhoff’s current law (Prop. 2.5, p. 11): an extension φ of a boundary potential ψ obeys the principle of minimum power “if and only if the current I(e) = (1/r(e))(φ(t(e)) − φ(s(e))) obeys Kirchhoff’s current law.”
• The power functional Q(ψ) = min_{φ|∂N = ψ} P(φ), with P(φ) = ½ Σ_e (1/r(e))(φ(t(e)) − φ(s(e)))² (Def. 2.3, 2.7, pp. 11, 13). “We call Q the ‘power functional’ because Q(ψ) equals ½ times the power dissipated by the circuit when the boundary voltage is ψ” (p. 13).
• Prop. 2.8 (p. 13): “Suppose ψ ∈ ℝ^∂N. Suppose φ is any extension of ψ minimizing the power. Then dQ_ψ ∈ (ℝ^∂N)* ≅ ℝ^∂N gives the boundary current of the current induced by the potential φ.” The comment after the proof: “This is fundamental: the way power depends on boundary potentials completely characterizes the way boundary currents depend on boundary potentials” (p. 14).
• Equivalence from outside (Sec. 2.4, p. 13): “circuits with different power functionals have different relations between boundary potentials and boundary currents. So, two circuits are equivalent, as far as what can be observed ‘from outside’, if they have the same power functional.”
• Example 2.9, resistors in series (pp. 14–15). For A —r_AB— B —r_BC— C with input A and output C, minimizing over φ(B) = x gives “x is the r-weighted average of ψ(A) and ψ(C)”: x = (r_BC ψ(A) + r_AB ψ(C)) / (r_BC + r_AB), and Q(ψ) = ½ · (1/(r_AB + r_BC)) · (ψ(A) − ψ(C))², the power functional of a single resistor r_AB + r_BC, “and so these two circuits are equivalent.”
• Prop. 2.12 (p. 15): “A function Q : ℝ^∂N → ℝ is the power functional for some circuit if and only if it is a Dirichlet form over (ℝ, ℝ⁺).” The proof is a one-line appeal to the star-mesh transform, “every circuit of linear resistors is equivalent to some complete graph of resistors between its terminals”, citing [29]. So circuits map onto Dirichlet forms on ∂N, “mapping two circuits to the same Dirichlet form if and only if they have the same external behavior” (p. 15).
• Dirichlet form (Def. 2.11, p. 15): Q(ψ) = Σ_{i,j∈S} c_ij (ψ_i − ψ_j)² with nonnegative c_ij. Not every nonnegative quadratic form is one: (ψ₁ + ψ₂)² is not.
• Theorem 3.3 (p. 18), quoted: “Given a passive linear circuit (N, E, s, t, Z), suppose R ⊆ N and ψ ∈ F^R. Then: (i) There exists a potential φ ∈ F^N obeying the principle of minimum power for ψ. (ii) If φ and φ′ both obey the principle of minimum power for ψ, then P(φ) = P(φ′). (iii) If φ and φ′ both obey the principle of minimum power for ψ, then φ − φ′ is constant on every connected component of the circuit’s underlying graph. (iv) There exists a unique potential f(ψ) ∈ F^N that obeys the principle of minimum power for ψ and vanishes on every connected component not containing a node in R. (v) The potential f(ψ) depends linearly on ψ. (vi) If P : F^N → F is the extended power functional, then the power functional Q : F^R → F given by Q(ψ) = P(f(ψ)) is a Dirichlet form on R.” The proof of (i) is a block-matrix argument (im Bᵀ ⊆ im A, by contradiction with nonnegativity); (vi) is induction on Lemma 3.4.
• Lemma 3.4 (p. 21): “Let P be a Dirichlet form on a set S and let s ∈ S. Then the quadratic form min_{s} P is a Dirichlet form on S \ {s}.” The proof sets φ_s = Σ_k c_ks φ_k / Σ_k c_ks and ends with the explicit form min_{s} P(ψ) = Σ_{i,j} (c_ij + c_is c_js / Σ_k c_ks)(ψ_i − ψ_j)² (p. 22; reconstructed from the extraction, which garbles the fraction). This is one-node elimination of a weighted graph Laplacian, the operation usually called Kron reduction; the paper does not use that name or “Schur complement”.
• Theorem 3.5 (p. 22): over any (F, F⁺), φ obeys the principle of minimum power for ψ iff the edge currents obey Kirchhoff’s current law on N \ ∂N, and then dQ_ψ equals the boundary current. (“The proofs are the same as before, with formal derivatives replacing derivatives.”)
• Section 3.3 and Prop. 3.6 (pp. 22–23): Dirichlet forms do not form a category. Composition is (P ∘ Q)(α, γ) = min_T Q(α, β) + P(β, γ), “the power used by the entire circuit is just the sum of the power used by its parts”; it is well defined by Thm 3.3 and associative, but “there is typically no Dirichlet form 1_S ∈ D(S, S) playing the role of the identity.” For a one-element S, an identity k(β − γ)² would need cα² = kc/(k + c) α² for all c, which “only holds when c = 0”. In the real case large k gives “‘approximate identities’”, resistors close to zero resistance, but “the power functional of a purely conductive wire is undefined; the formula for it involves division by zero.” Prop. 3.6: there is a semicategory with finite sets as objects and Dirichlet forms as morphisms.

Part II: categories of circuits (Sections 4–5)

• Decorated cospans (Lemmas 4.2–4.4, pp. 27–28, from Fong [17]): a lax symmetric monoidal functor F : (FinSet, +) → (Set, ×) yields a hypergraph category FCospan of cospans whose apex carries an element of F N, composed by pushout; a monoidal natural transformation yields a hypergraph functor.
• Circ (Def. 5.3, Cor. 5.4, p. 30): the decorated cospan category of passive linear circuits; it is a hypergraph category. Composition “expresses the fact that we can connect the outputs of one circuit to the inputs of the next” (p. 30).
• Thm 5.7 (p. 31): a strict hypergraph functor A : Circ → DirichCospan sending a circuit to its extended power functional P on all nodes. A is not faithful: two parallel resistors r₁, r₂ go to the same image as one of resistance 1/(1/r₁ + 1/r₂) (p. 32).
• Lemma 5.12 (p. 34): Graph(dP) = {(φ, dP_φ)} is Lagrangian in F^N ⊕ (F^N), and this is a one-to-one correspondence between quadratic forms on N and Lagrangian subspaces with trivial intersection with {0} ⊕ (F^N).
• LagRel (Def. 5.15, p. 35): finite sets as objects, Lagrangian relations F^X ⊕ (F^X)* ⇸ F^Y ⊕ (F^Y)* as morphisms; composites of Lagrangian relations are Lagrangian (Lemma 5.14, cited to [43] and [18]). Its hypergraph structure is transferred from LagCorel through Thm 7.5.
• Symplectification S : FinSet → LagRel (Prop. 5.16, p. 36), which expresses “when two wires join, set their potentials equal and add their currents” (p. 35): for the map {1, 2} → {3}, S(f) = {φ₁ = φ₂ = φ₃, i₁ + i₂ = i₃} (Example 5.17, p. 36).
• Thm 5.20 (p. 38): a strict hypergraph functor B : DirichCospan → LagCospan, Q ↦ Graph(dQ).

Part III: the black box functor (Sections 6–8)

• Corelations (Def. 6.1, p. 39): an equivalence relation on X + Y; they model circuits of ideal wires, whose connected components partition the terminals. FinCorel is a hypergraph category (Ex. 6.6, p. 41). Decorated corelations (Lemmas 6.11–6.12, pp. 42–43) are cited to Fong [19].
• Thm 7.3 (p. 45): a strong symmetric monoidal functor S : (FinCorel, +) → (LagRel, ⊕), S(e) = Φ(e) ⊕ I(e), with Φ imposing equal potentials on each connected component (Prop. 7.1) and I imposing Kirchhoff’s current law (Prop. 7.2).
• Thm 7.4 (p. 46): the hypergraph category LagCorel. Composition relates the glued states “if and only if two conditions hold: (i) φ_Y = φ′_Y and (ii) i_Y + i′_Y = 0.” The authors: “It is remarkable that this physical fact is embedded so deeply in the underlying mathematics” (p. 46).
• Thm 7.5 (p. 47): an isomorphism of categories C : LagCorel → LagRel that flips the sign of input currents.
• Theorem 7.7 (p. 47), quoted: “The black box functor is a hypergraph functor. On objects it maps any finite set X to the symplectic vector space ■(X) = F^X ⊕ (F^X). On morphisms it maps any open circuit Γ with underlying cospan of finite sets X → N ← Y and passive linear circuit (N, E, s, t, Z) to the Lagrangian relation ■(Γ) = (1_X ⊕ 1_Y) ∘ S[i, o]^op ∘ Graph(dP) ⊆ F^X ⊕ (F^X) ⊕ F^Y ⊕ (F^Y)*, where P is the extended power functional of Γ.” (Overlines marking conjugates are lost in the extraction.) Proof: “it is the composite of hypergraph functors A (Thm. 5.7), R (Thm. 5.20), Lag (Thm. 7.4), and C (Thm. 7.5).”
• Theorem 1.1 (p. 7), quoted: “There exists a hypergraph functor, the black box functor ■ : Circ → LagRel, mapping a finite set X to the symplectic vector space F^X ⊕ (F^X)* it generates, and an open circuit (N, E, s, t, r, i, o) to the Lagrangian relation ⋃_{v∈Graph(dQ)} S^t i(v) × So(v) ⊆ F^X ⊕ (F^X)* ⊕ F^Y ⊕ (F^Y)*, where Q is the circuit’s power functional.” S^t is the “twisted symplectification”, flipping the sign of input currents (p. 6).
• Section 8 (pp. 48–51): composition is power minimization. DirichCorel (Thm 8.2, p. 49) extends Dirich along injections by minN\im(m) P (Lemma 8.1, p. 48); the composite of two Dirichlet corelations minimizes over the glued nodes not in the image of the outer terminals. Lemma 8.3 (p. 50): S(m^op) ∘ Graph(dP) = Graph(dQ) with Q = min_{N\Y} P, i.e. symplectification “enacts” minimization. Thm 8.4 (p. 51): the square B̃ ∘ Dirich = Lag ∘ B commutes, so both routes give the same functor, and a three-line computation turns Thm 7.7’s formula (extended power functional P) into Thm 1.1’s (power functional Q). “We see now that Thm. 7.7 is a restatement of Thm. 1.1 in the Introduction” (p. 51).
• The worked black box (pp. 6–7): a resistor r goes to {(ψ₁, i, ψ₂, i) | i = (1/r)(ψ₂ − ψ₁)}, Ohm’s law; a single 2-ohm resistor “has the same Lagrangian relation as two 1-ohm resistors in series.”

Claims

id	claim	strength	support
C1	Black-boxing open passive linear circuits is a hypergraph functor Circ → LagRel	strong	Thm 1.1 (p. 7), proved as Thm 7.7 (p. 47) by composing hypergraph functors, with Sections 4 and 6 resting on Fong [17], [19]; matched to Thm 1.1’s formula by Thm 8.4 and the computation on p. 51
C2	A circuit’s external behaviour is fully captured by its power functional, and circuits with different power functionals behave differently at the terminals	strong	Prop. 2.8 and Thm 3.5 (current = dQ); the converse by Lemma 5.12’s one-to-one correspondence; stated p. 13
C3	Eliminating interior nodes of a Dirichlet form, over any field with positive elements, gives a Dirichlet form, linear in the boundary values and unique up to components that miss the boundary	strong	Thm 3.3 and Lemma 3.4, full proofs (pp. 18–22)
C4	Every real Dirichlet form is the power functional of some resistor circuit	moderate	Prop. 2.12, proved by citing the star-mesh transform ([29]), not in the paper
C5	Dirichlet forms under composition by minimization form a semicategory without identities	strong for one-element S; “typically” for larger	computation on p. 23 for
C6	Inductors and capacitors fit the same framework through the Laplace transform over ℝ(s)	strong within its assumptions	ℝ(s)⁺ checked to be a set of positive elements (p. 17); zero initial conditions and rational transforms assumed (p. 16)
C7	Network diagrams in mechanics, hydraulics, thermodynamics and chemistry are governed by the same mathematics	weak (here)	the analogy table and citations to Olson, Paynter and others (p. 3); not developed in the paper

Concepts

• Open circuit — a cospan X → N ← Y of finite sets with a passive linear circuit on N; X are inputs, Y outputs, and a node may be both, or an input “multiple times” (pp. 5–6, Def. 5.1).
• Extended power functional P — ½ Σ_e (1/Z(e))(φ(t(e)) − φ(s(e)))² on all node potentials; power functional Q — P with interior potentials eliminated by the principle of minimum power, a function of terminal potentials only.
• Dirichlet form — Σ c_ij(ψ_i − ψ_j)² with nonnegative c_ij (Def. 2.11).
• Lagrangian relation — a Lagrangian subspace of V₁ ⊕ V₂ read as a relation V₁ ⇸ V₂ (Def. 5.13); here the set of jointly possible terminal potentials and currents, “the subspace of attainable states” (p. 5).
• Black boxing — “like encasing the circuit in an opaque box, leaving only its terminals accessible” (p. 7).
• Corelation — an equivalence relation on X + Y; a circuit of ideal wires (Sec. 6.1).
• Hypergraph category — a symmetric monoidal category with a special commutative Frobenius structure on every object, compatible with ⊗ (Sec. 4.2); its operations are “placing diagrams on the same page, rearranging, splitting, combining and terminating wires, and then connecting these wires” (p. 26). A hypergraph functor is strong symmetric monoidal and preserves the Frobenius structures (p. 27).
• Symplectification — the functor taking a map of finite sets (or a corelation) to the Lagrangian relation that “‘copies voltages’ and ‘splits currents’” (p. 6).

Connections

• Fong’s decorated cospans [17] and decorated corelations [19], and Fong’s thesis [18], supply the general constructions; much of the paper appears in [18] (p. 8).
• Fong and Spivak’s Seven Sketches [20], cited p. 2, is LIT-803 in this record (skimmed). Its first chapter calls an observation’s failure to preserve joins a generative effect. Black-boxing is the contrasting case for composition of circuits: a functor, so what the composite presents is computed from what the parts present.
• Willems [45, 46], cited for the point that “the relationship between terminals is not an input-output one, but rather one of interconnection” (p. 39). Willems 2007 is LIT-1660, unavailable here.
• Companion work on Markov processes [5] and reaction networks [6] in the same framework (p. 7); neither is held by the record. The same authors’ Noether theorem for Markov processes is LIT-773.
• Sabot [37, 38], cited with Fukushima and Ma–Röckner for Dirichlet forms; [38] treats “electrical networks, symplectic reductions, and application to the renormalization map of self-similar lattices”, which bears on reading elimination as coarse-graining. Not held.
• Smale [41] and Weyl [44] on resistor networks (p. 3).

Bearing on the record

• CLAIM-480: its static computation is the paper’s Example 2.9, and its memory case is covered in a different form, with no time-scale condition. Put k = 1/r. The claim’s E = ½k₁(x₁ − x₂)² + ½k₂(x₂ − x₃)² is the extended power functional of two resistors in series; the paper’s interior x = (r_BC ψ(A) + r_AB ψ(C))/(r_AB + r_BC) is the claim’s x₂* = (k₁x₁ + k₂x₃)/(k₁ + k₂), and Q = ½ · 1/(r_AB + r_BC) · (ψ(A) − ψ(C))² is the claim’s ½ · k₁k₂/(k₁ + k₂) · (x₁ − x₃)² (pp. 14–15). The general elimination is Thm 3.3(vi) with Lemma 3.4’s explicit coefficients c_ij + c_is c_js/Σ_k c_ks (pp. 18–22). Three qualifications. (1) The paper’s quadratic form is half the dissipated power, minimized at fixed terminal potentials (Kirchhoff’s current law), not stored energy at mechanical equilibrium; Thm 3.3 is algebra on Dirichlet forms and applies to the spring chain as well, but the paper never discusses springs (the mechanical analogy is only the table on p. 3). (2) The memory case: with inductors and capacitors the boundary relation is a Dirichlet form over ℝ(s), a frequency-dependent impedance, and black-boxing is exact for it; nothing requires the interior to be fast. In the time domain that is a convolution (my reading; the paper stays in the s-domain). So in this setting a composite always presents its boundary with an exact relation, and what the claim’s time-scale condition decides is only whether that relation is approximately instantaneous, a question the paper does not take up. (3) The paper assumes zero initial conditions (p. 16), so the claim’s initial-condition term Be^{Dt}y(0) is outside it, and over ℝ(s) the interior is fixed by stationarity, not minimization (p. 18). The claim’s defeater (“black-boxing says there is none for the static energy”) is supported: circuits with the same power functional are “equivalent, as far as what can be observed ‘from outside’” (p. 13), “including how it interacts when connected with other circuits” (p. 7).
• The exchange’s A23 §IV, “Their black-box construction maps a circuit to its external relation between currents and potentials while abstracting away the circuit’s internal structure. Crucially, this construction respects composition”, is accurate. What is proved: a hypergraph functor (strong symmetric monoidal, preserving the Frobenius structure) from Circ, decorated cospans of finite sets, to LagRel, Lagrangian relations between F^X ⊕ (F^X)* and F^Y ⊕ (F^Y)* (Thm 1.1, p. 7; Thm 7.7, p. 47). The target is Lagrangian relations, not Dirichlet forms, which form only a semicategory (Prop. 3.6, p. 23). A23’s “a rigorous theory of precisely this phenomenon” is right for passive linear two-terminal networks and no wider: the paper excludes sources, nonlinear components and transformers (p. 3) and never speaks of constraints or apparatus.
• CLAIM-480’s “a coupling between x₁ and x₃ that neither part has”. True of each part’s relation, but functoriality says the composite’s relation is the relational composite of the parts’ relations (p. 7). The new coupling is computed from the parts’ black boxes, not from anything else about their interiors. That is the precise sense in which composition here makes something new: new relative to each part, with no generative effect relative to the parts taken together.
• CLAIM-477: a demonstrated case of one specification with many realizations, and of the specification being realized by the interior. The black box is many-to-one (2 Ω and two 1-Ω resistors in series, p. 7; A not faithful, p. 32), and onto for resistor circuits (every real Dirichlet form is some circuit’s power functional, Prop. 2.12). The claim says “partly fixed” and is not defeated by many realizations; this supports that, in a setting with nothing interpretive in it. One point presses lightly on defeater (b), a constraint realized by nothing: the identity, the pure gluing condition of an ideal wire, is not the behaviour of any finite-impedance circuit. It is only approached (“approximate identities”, p. 23) and is supplied by the formalism (corelations, Sec. 6). It is an idealization, not a constraint that individuates while realized by nothing, so it does not defeat (b).
• CLAIM-471: a theorem for its “physical gluing” in the linear passive case, and a nuance about “generate”. Composition is gluing by pushout (“the pushout of cospans gives formal precision to this gluing construction”, p. 5), and the interconnection condition is equal potentials and opposite currents (Thm 7.4, p. 46), the effort–flow condition the claim names for port-Hamiltonian systems. Here, against defeater (b), the restrictions interconnection imposes are stated exactly as conditions on shared variables, histories included through the s-domain, and the composite behaviour follows from them. The nuance: in the paper the gluing condition at a port is not generated by the interaction. It is the same Kirchhoff condition for every interconnection, fixed by the laws (symplectification, Prop. 5.16 and Thm 7.3). What composition generates is the effective boundary relation, which then serves as the condition the next system meets. That bears on defeater (a): here the correspondence at each port is background structure, and interaction makes the composed relation, not the port condition.
• CLAIM-478 (separation makes constraints): the paper gives no support either way. Its exact frequency-domain black box needs no separation; separation would matter only for approximating that relation by an instantaneous one, which the paper does not do.
• No instruction for machine-learning practice.

Limitations

• Narrow class. Passive, linear, two-terminal components; no sources, diodes or transformers (p. 3). The engineering analogies (p. 3) are cited, not developed.
• Time enters only through the Laplace transform with zero initial conditions and rational transforms (p. 16). Transients from a nonzero interior state and non-rational signals are outside it, and the paper does not translate its s-domain results back to the time domain.
• Proofs partly by citation. The hypergraph-category constructions rest on [17], [19] and Fong’s thesis [18] (Lemmas 4.2–4.4, 6.5, 6.9, 6.11, 6.12, 5.14); Prop. 2.12 rests on the star-mesh transform [29]. Not checked here.
• The no-identity argument is a computation for a one-element set; for larger sets the paper says “typically” (p. 23).
• Notational slips in this version: the functor B of Thm 5.20 is called R on p. 37 and in Thm 7.7’s proof; Prop. 7.2 says “This functor Φ” for I; Thm 3.5 writes ψ ∈ ℝ^∂N in a section over general F; Section 8 cites “Prop. 5.12” for Lemma 5.12; references [8] and [35] carry “(See page .)”. None affects a result.
• Extraction losses: the black square naming the functor, conjugation overlines and the Section 4.2 string diagrams. The Lemma 3.4 coefficient formula is reconstructed.
• The TAC published text and its page range (1158–1222 in the registration) were not checked.

Open questions

• What is the time-domain form of the black box for an RLC network with nonzero initial interior state, and is it still functorial? The paper’s zero-initial-condition assumption leaves this open; it is the setting of CLAIM-480’s memory formula.
• How far does the framework go beyond passive linear circuits? The authors point to [3] for more general circuits and to [5], [6] for Markov processes and reaction networks (p. 7).
• Is the approximately-instantaneous regime characterized anywhere in this framework, e.g. by the low-frequency behaviour of the black-boxed Dirichlet form over ℝ(s)? Not in this paper.