Source
Baez and Fong, LIT-1644 (arXiv 1504.05625v6), Sections 2–3, pp. 9–23; read in NOTE-1223.
What was actually shown
The setting. A Dirichlet form on a finite set S is Q(ψ) = Σ_{i,j∈S} c_ij(ψ_i − ψ_j)² with nonnegative c_ij (Def. 2.11, p. 15), over a field with positive elements (Def. 2.10). A circuit’s extended power functional P(φ) = ½ Σ_e (1/Z(e))(φ(t(e)) − φ(s(e)))² is one. The principle of minimum power for a boundary value ψ ∈ F^R asks that ∂P/∂φ(n) = 0 at every n ∉ R (p. 18). Over ℝ this is a minimum. Over ℝ(s) it is a formal variational principle: “it is not clear what it means to minimize over the field F” (p. 18). Either way it is Kirchhoff’s current law at the interior nodes (Prop. 2.5, Thm 3.5).
Thm 3.3 (p. 18). A solution exists; all solutions give the same P; they differ by constants on connected components; there is a unique one f(ψ) vanishing on components that miss R; it is linear in ψ; and “the power functional Q : F^R → F given by Q(ψ) = P(f(ψ)) is a Dirichlet form on R.” Existence is proved from nonnegativity: if im Bᵀ ⊄ im A, P could be made −1 (p. 19).
Lemma 3.4 (pp. 21–22). Eliminating one node s sets φ_s = Σ_k c_ks φ_k / Σ_k c_ks and gives min_{s} P(ψ) = Σ_{i,j} (c_ij + c_is c_js / Σ_k c_ks)(ψ_i − ψ_j)² (reconstructed from the extraction). The interior value is a weighted average of its neighbours, and the new coupling c_is c_js / Σ_k c_ks between i and j exists whether or not i and j were joined.
Example 2.9 (pp. 14–15). For A —r_AB— B —r_BC— C, the minimizing interior potential is 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 one resistor r_AB + r_BC. With conductances k = 1/r this is ½ · k₁k₂/(k₁ + k₂) · (ψ(A) − ψ(C))², CLAIM-480’s computation in the paper’s variables.
Composition and its gap (Sec. 3.3, p. 23). Dirichlet forms on S + T and T + U compose by (P ∘ Q)(α, γ) = min_T Q(α, β) + P(β, γ). By Thm 3.3 the result is a Dirichlet form, and composition is associative. For a one-element S, an identity k(β − γ)² would need c = kc/(k + c) for every c, which “only holds when c = 0”. Large k gives “‘approximate identities’”, but the ideal wire’s power functional “involves division by zero” (p. 23). Prop. 3.6: Dirichlet forms form a semicategory. That gap is why the paper’s black box lands in Lagrangian relations (THEORY-1063).
What this does not say
- Not a statement about energy at mechanical equilibrium. The paper’s form is half the power a resistor network dissipates, minimized at fixed terminal potentials. The algebra of Thm 3.3 and Lemma 3.4 holds for any Dirichlet form, so it covers a chain of springs at static equilibrium too, but the paper does not discuss springs.
- Not that the elimination is a minimization in the dynamic case. Over ℝ(s) it is stationarity, and the reduced form is a frequency-dependent impedance, exact at every frequency with no time-scale separation assumed. Zero initial conditions are assumed (p. 16).
- Not that the reduced form is new information. It is fixed by the parts’ coefficients. Prop. 2.12 adds that every real Dirichlet form on ∂N is some resistor circuit’s power functional, so the map from circuits to forms is many-to-one and onto; its proof is a citation to the star-mesh transform.
- Not that no Dirichlet form is an identity for every S. The computation is for one element; for larger sets the paper says “typically” (p. 23).
- Not named Kron reduction or Schur complement in the paper. The identification is standard but the record’s, not the authors’.