Source
Spivak (2026), LIT-1649, §§6.3, 6.6 and 7.3–7.4 (pp. 54–61, 67–71, 76–87 of arXiv 2606.28984v2); read in NOTE-1225.
What was actually shown
The arrangement. For a finite directed graph G = (V, E, src, tgt), each vertex v is a particle with parameter ℝ, constant sharp ξ ↦ ξ/m_v, output the diagonal broadcasting its position to its out-edges, and potential Σ_{e∈v⁻} (κ_e/2)(q − q′e)² over its in-edges. Wiring them by the graph’s prism lens gives one closed arrangement wire_G with parameter ℝ^V and composite potential U(q) = Σ{e∈E} (κ_e/2)(q_tgt(e) − q_src(e))², (89), p. 85. Its differential is dU = L_G q, the weighted graph Laplacian, and the orientation drops out because each edge term is symmetric in its endpoints (p. 85).
Two integrators. The configuration integrator stores q and updates q ↦ q − ♯_q(ξ) (41), p. 56. The phase integrator stores (q, ξ), presents q̃ = q + ♯_q(ξ), and updates ((q, ξ), ξ′) ↦ (q + ♯_q(ξ), ξ − ξ′) with ξ′ the gradient at q̃ (50), p. 61. Both share the factor Φ′sm, the polynomial interpretation of the arrangement, and differ only in the stored state and the update (p. 68).
The wave equation. 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)”, with q̈_v(t) = q_v(t+1) − 2q_v(t) + q_v(t−1). The arrangement is harmonic, so by Proposition 6.6.1 the step is a linear isomorphism preserving the canonical symplectic pairing (p. 86). §7.3 does the chain with pinned ends twice, composing in the syntax and composing the parts’ semantics, and the two agree (pp. 79–83).
The heat equation. Remark 7.4.2 (pp. 86–87): “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 q̇ = −L_G q with q̇_v(t) = q_v(t+1) − q_v(t) and M = diag(m_v). The author’s summary: “the second-order, symplectic regime and the first-order, descent regime are two semantics of one syntactic object” (p. 87).
Both computations were checked numerically on a four-vertex path graph (NOTE-1225): the wave residual was at machine precision, and U fell monotonically under the heat step.
What this does not say
- Anything about time scales. The paper does not speak of relaxation rates, separation or hierarchy. That the heat step damps each eigenmode of M⁻¹L_G at a rate set by its eigenvalue while the symplectic step damps none is a reading through CLAIM-481, not in the text.
- That the coupling is dissipative or conservative. The regime is fixed by the integrator, which the paper calls the semantics; physically it is the choice of whether the particles carry momentum. So the result shows that wiring, potential and response underdetermine the dynamics, not that one physical system shows both.
- That either step is stable for every step size. Neither the heat step nor the phase step comes with a stability bound in the paper; both are explicit, one tick per step.
- That the discrete step conserves energy. It preserves the symplectic pairing for harmonic systems; “The discrete step (65) need not conserve H exactly” (p. 70).
- Continuum equations. These are difference equations on a graph; the continuum limit of the chain is pointed to a reference, not derived (footnote 31, p. 76).
- That the regime is in the semantics for every arrangement. The author expects that for each quadratic arrangement f there is g on T*Q with Φphase(f) = Φconf(g) (p. 88). If so, the same split can also be expressed in the syntax.