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NOTE-1225: Spivak — Compositional dynamics in learning and mechanics

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PaperLIT-1649 — Compositional Dynamics in Learning and Mechanics

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Contribution

A single compositional setting “in which gradient-based learning and Hamiltonian-style mechanics appear as functorial semantics” (abstract, p. 1). The syntax is an operad ArrSm whose objects are input-output interfaces (pairs of manifolds) and whose morphisms are smooth adaptive arrangements. The semantics are two operad functors Φconf, Φphase : ArrSm → PC into the 2-category of polynomial coalgebras, which the paper reads as “input-output discrete dynamical systems” (p. 1). The author lists seven contributions (p. 7): lens internalization (Corollary 3.3.4, Theorem 3.3.3); a vocabulary of suboperads (stateless, potential-free, static, quadratic, linear, unilateral; Definition 5.3.2, Proposition 5.3.3); a Lotka–Volterra example (Example 5.4.4); the two functors (Corollary 6.4.1); exact symplecticity of the phase step for a harmonic closed system (Proposition 6.6.1); Φconf recovering gradient descent with backpropagation and Newton’s method (§§7.1–7.2); and one arrangement of harmonic particles giving the discrete wave equation via Φphase and the discrete heat equation via Φconf (§7.4, Remark 7.4.2). The abstract calls lens internalization “The main technical result of the paper” (p. 1).

Key insight

The coupling data and the dynamical regime are separated. An arrangement fixes the wiring, the potential and the response; an integrator, chosen afterwards, fixes what state is stored and how an incoming covector updates it. “Thus the integrator is where the dynamical regime gets decided: one smooth adaptive arrangement fed through different integrators can produce qualitatively different behaviors” (p. 6). And: “the diagram fixes the wiring, the potential, and the geometry of the parameter space, while the integrator alone selects the dynamical regime, second-order or first-order” (pp. 6–7).

The common thread between learning and mechanics is a map T*M → TM that turns the differential of a potential into a state change: “the potential energy in Hamiltonian dynamics or the loss in gradient descent—so it is the potential that drives the state” (pp. 2–3). The paper calls vector spaces with such a map responsive. A slogan for interfaces: “Your inputs are what your interior can have preferences over, and your outputs are what your exterior can have preferences over” (p. 5).

Assumptions

  • Parameters are vector spaces, not manifolds. RVect is finite-dimensional real vector spaces with a sharp map, and its morphisms are sharp-equivariant linear isomorphisms (Definition 5.2.1, p. 39). The extension to “responsive manifolds with connection” is conjectured (Remark 6.3.3, p. 56), not done.
  • Discrete time, explicit-Euler-type integrators. Each tick emits a position, receives a direction and updates once (Remark 4.3.7, p. 35). Multi-stage integrators (Runge–Kutta, velocity Verlet) are left to future work; an ℓ = 2 case exists only in companion code written by Claude and “we have not independently audited it” (p. 36).
  • Couplings are potentials, never equations. “every expression in ArrSm denotes a deterministic discrete-time system (Theorem 4.3.6), and correspondingly we do not express idealized constraints between systems; e.g. a “rigid” coupling can only be specified as a stiff potential, rather than an equation” (p. 9).
  • Smoothness is a convenience. All statements are said to hold for C¹ maps, with four listed exceptions (Remark 5.1.2, pp. 38–39).
  • The potential algebra is fixed at r = +1 (Lemma 6.2.1, (38), p. 53), which is what makes potentials contribute their ordinary differentials.

Key results

The syntax (§§3–5)

  • Theorem 3.3.3 (p. 26). “Let (T, τ) be a monoidal monad on a symmetric monoidal closed category (C, I, ⊗, [−, −]), and let z = (z, e, m, α) be a T-monoid in C (Definition 3.3.1). Then there is a lax symmetric monoidal functor that acts on objects as follows: Θ_z : Lens^T_C → C, (c⁻, c⁺) ↦ [c⁻, z] ⊗ c⁺. It is normal if and only if e : I → z is an isomorphism.” The proof is a diagram chase of sixteen tiles plus a pentagon (p. 27).
  • Corollary 3.3.4 (Lens internalization), p. 28. “Let (C, I, ⊗, [−, −]) be a symmetric monoidal closed category. There is a normal lax symmetric monoidal functor Θ : Lens_C → C, (c⁻, c⁺) ↦ [c⁻, I] ⊗ c⁺.” In Poly a lens (Ay, By) internalizes to By^A, whose coalgebras are Moore machines (Example 3.3.5).
  • Definition 4.1.1 (p. 29). An adaptive-arrangement datum D = (M, Q, J, R) is “a cartesian monoidal category M, objects are called spaces; a symmetric monoidal category Q, objects are called parameters; a strong monoidal functor J : Q → M; and a monoidal monad R on M, the potentials monad.” Arr_D is “the operad underlying the symmetric monoidal category Para_Q(Lens
  • Definition 5.2.1 (p. 39). “A responsive vector space is a finite-dimensional real vector space Q together with a smooth map ♯_Q : Q → Vect(Q*, Q), the sharp map”. Constant, symmetric, quadratic (constant and symmetric) and nondegenerate responses are defined there. “Our theory does not privilege nondegenerate responses in any way” (p. 39).
  • An arrangement (35), p. 42: f = (Q, f⁺, f⁻, U) with f⁺ : Q × M⁺ → N⁺, f⁻ : Q × M⁺ × N⁻ → M⁻, U : Q × M⁺ × N⁻ → ℝ. The potential reads the parameter, the inner outputs and the outer inputs, and “can read neither M⁻ nor N⁺” (p. 5).
  • Definition 5.3.2 (p. 42) names the classes. “harmonic if it is quadratic, M⁺ and N⁻ are vector spaces, and its potential U is a quadratic form”; “unilateral if whenever f⁺(q, m⁺) = n⁺ and ξ ∈ T*_{n⁺}N⁺, we have ♯_q((∂_q f⁺)ᵀξ) = 0”, i.e. nothing returned at the output can move the state. Proposition 5.3.3 (p. 43): “The stateless, potential-free, static, quadratic, and unilateral morphisms each span a wide suboperad of ArrSm, and the linear morphisms also span a suboperad”. Harmonic ones do not, because composing through a nonlinear input map breaks quadraticity (footnote 19, p. 43).
  • Degeneracy and back-driving (pp. 39–40). “Degeneracy is what allows things like controllers to act on their environment without being retroactively back-driven by it. We refer to this condition as unilaterality”. The author links it to Del Vecchio, Ninfa and Sontag’s retroactivity, “which they identify as an obstruction to modularity in biomolecular networks” (p. 40).

The semantics (§§4, 6)

  • Definition 4.3.4 (p. 34). “A p-integrator is a pair 𝔦 = (S, dyn) … where S : Q → Set, the state space functor, is strong symmetric monoidal and dyn : Store∘S ⇒ p, the dynamics map, is a monoidal natural transformation.”
  • Theorem 4.3.6 (p. 35). “Every polynomial interpretation datum 𝔭 = (c, R′, θ, z) and (c∘J)-integrator 𝔦 induce an operad functor Φ_{𝔭,𝔦} : Arr_D → PC, which is normal (as a lax symmetric monoidal functor) when z ≅ y.”
  • The cotangent functor (Definition 6.1.1, p. 51) sends M to Σ_{m∈M} y^{T*_m M} and a smooth map to itself forward on points and its transposed differential backward on covectors; its functoriality “is the chain rule”. It is strong symmetric monoidal (Proposition 6.1.2).
  • Corollary 6.4.1 (p. 62). “Every cot-integrator 𝔦 induces an operad functor Φ_𝔦 : ArrSm → PC with Φ_𝔦(1,1) ≅ y; hence it carries closed systems to y-coalgebras.”
  • The configuration integrator (41), p. 56: state Q; an incoming covector ξ updates q ↦ q − ♯_q(ξ). “The minus is the integrator’s … which is what makes the configuration dynamics descend rather than ascend” (p. 56).
  • The phase integrator (49)–(50), p. 61: state TQ ≅ Q ⊕ Q; it presents q̃ = q + ♯_q(ξ) and updates ((q, ξ), ξ′) ↦ (q + ♯_q(ξ), ξ − ξ′), with ξ′ the gradient computed at q̃. “This has one step of Hamilton’s equations as a special case” (p. 61).
  • The general update (58)–(59), pp. 64–65. The state covector is the sum of three pullbacks: the outer covector along f⁺, the inner covector field along f⁻, and +1 along U (giving dU); its Q*-component is sharpened and subtracted. Both functors share this factor and differ only in where it is evaluated (stored q for Φconf, presented q̃ for Φphase) and how the state is updated (p. 68).
  • Proposition 6.5.1 (p. 65) relates Φconf to the continuous-time open-energy-driven systems of Capucci, Lynch and Spivak (CLS24) through a category S of submersions. “neither their framework nor the one in this paper subsumes the other” (p. 8).
  • Remark 6.4.2 (pp. 62–63). Naturality of an integrator can be weakened by shrinking the parameter category; “We conjecture that Adam [KB15] is an integrator when Q is restricted to the subcategory whose objects are Euclidean vector spaces ℝⁿ equipped with Euclidean sharp and whose morphisms are signed permutations.” Monoidality cannot be weakened: “The monoidality of 𝔦 corresponds to the compositionality of the semantics, that each box can compute its own update.”

Closed phase steps and symplecticity (§6.6.2)

  • For a closed system, Φphase is (q, ξ) ↦ (q + ♯_q(ξ), ξ − dU|_q̃), (65), p. 69. With constant symmetric sharp it is the symplectic Euler step of H(q, ξ) = ½ξ(♯₀(ξ)) + U(q) (p. 70). “The discrete step (65) need not conserve H exactly” (p. 70).
  • Proposition 6.6.1 (p. 70). “Let f = (Q, ♯₀, !, !, U) : I → I be a harmonic closed system (Definition 5.3.2) whose potential is U(q) = ½(κq)(q) for a symmetric linear map κ : Q → Q*. Then, for the drift and kick of (66), Φphase(f) = kick ∘ drift. In particular, the update (65) is an endomorphism of the responsive vector space T*Q in RVect, i.e. a linear isomorphism preserving the canonical symplectic pairing (33).” Proof: drift and kick each preserve the pairing, by symmetry of the sharp and of κ (pp. 70–71).

Applications (§7)

  • Newton’s method (§7.1, pp. 71–72). A closed system with response the inverse Hessian, ♯^U_q = (T_q(dU))⁻¹, under Φconf is q ↦ q − (T_q(dU))⁻¹(dU|_q), (67). It needs the Hessian invertible everywhere, and finds “any critical point rather than only moving downhill toward minima” (p. 72).
  • Gradient descent and backpropagation (§7.2, pp. 73–76). A parameterized map F : Q × ℝᵐ → ℝⁿ, with zero potential, composed with a stateless loss arrangement, gives under Φconf, for ♯_q = η_LR♯_Euc, “ordinary gradient descent at rate η_LR: q ↦ q − η_LR(T_q F(◽, x))ᵀξ” (76), p. 75. “Backpropagation is thus the chain rule, packaged as the functoriality of Φconf” (p. 75). A weight-only potential (γ/2)‖q‖² gives weight decay (Proposition 7.2.3, p. 76). The training data are an environment coalgebra that may itself adapt to the returned covectors (Example 7.2.1, p. 75).
  • The wave equation on a chain (§7.3, pp. 76–83). K particles with sharp ξ/m and harmonic potential (κ/2)(q − q′)², wired in series. With both ends pinned ((82), (83)) the phase dynamics give m q̈ᵢ = κ(qᵢ₋₁ − 2qᵢ + qᵢ₊₁), (78), with q̈ᵢ(t) = qᵢ(t+1) − 2qᵢ(t) + qᵢ(t−1) (Proposition 7.3.3, p. 81). §7.3.2 recomputes it by applying Φphase to the parts and composing in PC, “an explicit check on signs, indices, and the routing” (p. 81), and the two agree term by term (p. 83). Remark 7.3.2 (pp. 77–78) notes a stateless bond is “formally, the same construct as the training loss”: “each bond is like a “training loss” provided to each mass by its neighbor.”
  • The graph Laplacian (§7.4, pp. 83–87). For a weighted finite directed graph G with masses m_v and spring constants κ_e, the composite potential is U(q) = Σ_{e∈E} (κ_e/2)(q_tgt(e) − q_src(e))², (89), and dU = L_G q, “the weighted (and symmetrized …) graph Laplacian of G” (p. 85). The orientation drops out: “what we recover is the symmetrized Laplacian, so G behaves like its underlying undirected multigraph” (p. 83). Proposition 7.4.1 (p. 86): “every trajectory of Φphase(wire_G) … satisfies m_v q̈_v(t) = −(L_G q(t))_v (v ∈ V, t ≥ 1)”. The arrangement is harmonic, so the step preserves the symplectic pairing (p. 86).
  • Remark 7.4.2 (pp. 86–87), the heat equation. “applying Φconf to the very same smooth adaptive arrangement wire_G of (88)—same graph wiring, same potential, and the same positive inverse-mass response—recovers the discrete heat equation on G”: m_v(q_v(t+1) − q_v(t)) = −(L_G q(t))_v for t ≥ 0, i.e. M q̇ = −L_G q, “diffusion on G with the masses m_v serving as heat capacities”. “Thus a single smooth adaptive arrangement gives wave dynamics under the phase integrator and heat dynamics under the configuration integrator: the second-order, symplectic regime and the first-order, descent regime are two semantics of one syntactic object” (p. 87). This is a remark with its computation, not a numbered result; the computation is short and checks.
  • Remark 7.4.3 (p. 87). A learner can be trained to predict a closed system. Observing positions alone, the wave system has no one-step map on the observations, “states with the same position and different momenta have different next positions”; observing two consecutive configurations, it does.

Examples reported from code (Example 5.4.5, pp. 47–50)

Transistors, a NAND gate, latches, Black’s feedback amplifier, an LC circuit, a DC motor, molecular dynamics, impedance control, Kuramoto oscillators, Petri nets, replicator dynamics and Hopfield networks. “Each construction below is implemented in the companion code … by Claude … The author understands the content of the claims as stated, and believes them to be reasonable and to have been checked, but has not verified all of them by hand” (p. 47). One is a timing claim: “a chain of K inverters takes roughly K times as many ticks of the coalgebra to converge as one does, whereas K side-by-side copies take the same number of ticks as one” (p. 47).

Claims

idclaimstrengthsupport
C1Lens internalization: every symmetric monoidal closed C has a normal lax symmetric monoidal Θ : Lens_C → C, (c⁻, c⁺) ↦ [c⁻, I] ⊗ c⁺, with a monad/monoid generalizationstrongTheorem 3.3.3 and Corollary 3.3.4, proved by diagram chase (pp. 26–28)
C2Every polynomial interpretation and integrator give an operad functor Arr_D → PC; in the smooth case Φconf and ΦphasestrongTheorem 4.3.6, Corollary 6.4.1, assembled from cited results on Para and actegories (pp. 33–35, 62)
C3Φconf of a parameterized map composed with a loss is gradient descent, with backpropagation as the lens backward pass; weight decay and Newton’s method arise likewisestrongexplicit unpacking of (58)–(59) (§§7.1–7.2); the composite is per datum, deterministic, fixed sharp
C4One arrangement of harmonic particles on any finite directed graph gives the discrete wave equation under Φphase and the discrete heat equation under Φconf, both governed by the symmetrized graph LaplacianstrongProposition 7.4.1 (proved) and Remark 7.4.2 (computed in the text, not numbered); checked numerically here
C5For a harmonic closed system the phase step preserves the canonical symplectic pairing exactly, though it need not conserve HstrongProposition 6.6.1, short proof (pp. 70–71)
C6Unilateral (and stateless, potential-free, quadratic) arrangements are closed under compositionstrongProposition 5.3.3, chain-rule proof (pp. 43–44)
C7The integrator alone selects the regimemoderatetrue of the paper’s two integrators by construction (the shared factor Φ′sm, p. 68); a general reading beyond those two is the author’s framing
C8Adam is an integrator over Euclidean spaces with signed permutationsweakstated as a conjecture (p. 62)
C9Phase semantics of a neural network gives momentum trainingweak”we expect” (Conclusion, p. 87)
C10The circuits, chemistry, ecology and neuroscience examples of Example 5.4.5 behave as describedweakimplemented and tested by Claude, not verified by hand by the author (p. 47)

Concepts

  • Smooth adaptive arrangement — a morphism of ArrSm: a responsive parameter space, an output map, an input map and a real potential, (35). “adaptive” names that the parameter is a state the dynamics evolve (p. 3).
  • Responsive vector space — a vector space with a sharp ♯_Q : Q → Vect(Q*, Q) turning covectors into state changes; the symplectic sharp in mechanics, a learning-rate-scaled metric in learning (p. 3).
  • Lens — a forward map and a backward map, f⁺ : c⁺ → d⁺ and f⁻ : c⁺ ⊗ d⁻ → c⁻ (Definition 3.1.1); the wiring of boxes.
  • Polynomial coalgebra; PC — a p-coalgebra S → p(S) is a deterministic system with state that emits a position and updates on a direction; PC has polynomials as objects and [p, p′]-coalgebras as morphisms, “called Org in [Spi21; SS22]” (p. 14).
  • Integrator — a state-space functor and a dynamics map Store∘S ⇒ p (Definition 4.3.4): the semantic choice of what is stored and how it is updated.
  • Configuration vs phase — state Q and a descent step, vs state T*Q and a symplectic-Euler step (§6.3).
  • Harmonic — quadratic response and quadratic-form potential (Definition 5.3.2).
  • Unilateral — the output cannot back-drive the state (Definition 5.3.2, p. 42); realized by a degenerate sharp.

Connections

  • Capucci, Lynch and Spivak (2024), open energy-driven systems, the “closest predecessor” (p. 8), continuous-time; Proposition 6.5.1 relates them. Not held by the record.
  • Fong, Spivak and Tuyéras, “Backprop as Functor” (FST19, cited p. 6), held as LIT-827, read; THEORY-519 holds its theorem. Here gradient descent with backpropagation is again a functor, but the backward message is a covector pulled back by the transposed differential (71), not FST’s requested input, and the loss is an arrangement in the same operad rather than a parameter of the functor. The paper does not compare the two beyond citing FST19 for lenses in backpropagation.
  • Shapiro and Spivak, dynamic operads (SS22, p. 17): “a dynamic category is defined to be a category enriched in PC”.
  • Coya on bond graphs (Coy17, p. 9): relational semantics that “need not admit an evolution at all”, against ArrSm’s always-deterministic one; a comparison is left open (p. 88).
  • Del Vecchio, Ninfa and Sontag on retroactivity (DNS08, p. 40).
  • Fong and Spivak, Seven Sketches, held as LIT-803, is not cited. The paper’s own background for wiring-diagram operads is §2.3, citing Spi13, Yau15, SSR17, FS19 and PSV21.

Bearing on the record

  • CLAIM-481: consistent, and an instance it does not state. The exchange (A22 §6) cited the paper for “The same relational organization need not imply the same temporal hierarchy”. The paper never speaks of time scales, separation or hierarchy. What it shows is narrower and exact: the same wiring, potential and response give a first-order descent dynamics M q̇ = −L_G q under Φconf and a second-order symplectic one m q̈ = −L_G q under Φphase (Proposition 7.4.1, Remark 7.4.2, pp. 86–87). Reading those through the claim (my derivation, not the paper’s): per eigenmode of M⁻¹L_G with eigenvalue μ, the heat step multiplies by 1 − μ each tick, so the spectrum is a hierarchy of relaxation rates; the symplectic step preserves the pairing (Proposition 6.6.1), so no mode decays and the spectrum is a set of frequencies. That is the claim’s two halves, a Laplacian flow whose spectrum sets the rates and a conservative coupling that splits modes without damping them, now as two semantics of one arrangement. One caution: what the paper varies is the integrator, which it calls “the semantics”, a modeller’s choice of whether momentum is stored. As a physical demonstration that coupling does not fix the hierarchy, it shows that the coupling data underdetermine the dynamics; the missing datum is inertia against dissipation, which is the claim’s condition. “Almost a controlled demonstration” is fair if the control is understood as that.
  • CLAIM-473 (Rejected): a further counterexample family, no change. Under Φphase every harmonic network on a finite graph is coupled and symplectic, so its relative modes never decay (Proposition 6.6.1, §7.4). The rejection stands as filed.
  • CLAIM-480: the composition is literal here, and constraints are not primitive. In ArrSm an interior wired inside a box is a morphism, and the composite’s potential is the sum of its parts’ potentials pulled back through the wiring ((89), p. 85; Proposition 5.3.3’s composite potential, p. 43). The paper does not eliminate an interior (no Schur complement, no memory kernel), so it neither confirms nor tests the claim’s computation. It bears on the claim’s reading in two ways. First, a constraint between parts is only ever a limit: “a “rigid” coupling can only be specified as a stiff potential, rather than an equation” (p. 9), and “A choice of degenerate ♯_Z = 0 is like a weld holding z fixed” (p. 45). Second, the composite’s external behaviour is assembled functorially from its parts’ (Corollary 6.4.1, and the §7.3.2 audit), which is the same shape as the black-boxing functor the claim cites, for dynamics rather than for static relations.
  • CLAIM-471. Here interactions are potentials, not correspondences: the wiring (a lens) routes values, and the potential, which lives “interior to the outer box N and exterior to the inner box M” (p. 5), supplies the covectors that move the states. An interaction never fixes a relation exactly; it penalizes departures from one. If the claim’s gluing conditions are exact restrictions, this framework realizes them only as stiff limits (p. 9).
  • TERM-089 and QUESTION-285: unilaterality is a formal version of selective coupling without back-action. A unilateral arrangement’s output cannot move its state (Definition 5.3.2), the amplifier is the worked case (“whatever is wired to the output cannot move the state”, p. 48), and unilaterality is preserved by composition (Proposition 5.3.3). The author ties it to retroactivity as an obstruction to modularity (p. 40). This is a structural condition an apparatus in the term’s sense could meet; the paper does not use the word or the idea of measurement.
  • LIT-803 (Seven Sketches): not cited; the same programme. Operads as syntax and functors as semantics (§2.3), with the semantics now dynamical. The book’s compositional reading of observation is not engaged.
  • Filed from this reading: THEORY-1054 (the heat and wave semantics of one arrangement) and THEORY-1057 (gradient descent as a functorial semantics).
  • LIT-827 and THEORY-519: a second, different functorial account of backpropagation by an overlapping author, with covectors in place of requested inputs and a fixed or position-dependent sharp in place of a step size (§7.2). Neither paper is defeated by the other; they formalize different backward messages.
  • No instruction for machine-learning practice. The Adam conjecture (p. 62) and the momentum expectation (p. 87) are about what the formalism can express, not advice.

Limitations

  • Read from v2 only. The registration’s v1 date is from the arXiv API; the differences between v1 and v2 are unknown.
  • Diagrams lost in extraction. The wiring pictures and commutative diagrams were reconstructed from the surrounding prose; nothing in the results above depends on a picture alone.
  • Discrete time throughout. “wave equation” and “heat equation” are difference equations in space and time, one tick per Euler step; the continuum limit is pointed to [Dem15] in footnote 31, not shown. Stability of either step (the heat step needs the eigenvalues μ of M⁻¹L_G in (0, 2), the symplectic one in (0, 4), by my derivation) is not discussed; the paper’s only bound of this kind is for the LC circuit, “4LC > 1” (footnote 20).
  • “Dissipative” and “conservative” are not the paper’s words for the two regimes. It says “first-order, descent” and “second-order, symplectic” (p. 87), proves exact symplecticity for harmonic closed systems only, and says the phase step “need not conserve H exactly” (p. 70). That the heat step lowers U was checked numerically here, not shown in the text.
  • Much of §5.4.5 and the companion code are Claude-written and unaudited by the author’s own statement (pp. 36, 47, 71 n. 30). The acknowledgements also thank “Claude and Codex for assistance in editing the document” (p. 10).
  • Several statements are conjectures or expectations: manifold parameters (Remark 6.3.3), multi-stage integrators (Remark 4.3.7), Adam (Remark 6.4.2), momentum training, the Φphase–Φconf relation on quadratic arrangements and a linear-stratum retraction (Conclusion, pp. 87–88).

Open questions

  • Is there, for every quadratic arrangement f, an arrangement g on T*Q with Φphase(f) = Φconf(g), functorially, as the conclusion expects (p. 88)? It would make the two regimes one semantics of two arrangements, which bears on whether the regime belongs to the syntax or the semantics.
  • What does an integrator with damping look like? The damping 1-form exists (Example 6.3.14, p. 59: “it removes a fraction c of the momentum at each step”) but “plays no further role”. With it, the same arrangement would interpolate between the two regimes, and the separation of collective and relative modes could be read off as a function of c.
  • Does a composite’s external dynamics ever reduce to a smaller arrangement (an interior eliminated), as CLAIM-480’s elimination does for energy? The paper composes but never reduces.

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