TOTOGT/io → ZeoliteCommutation.lean (gate_commutes). Nothing about a gate “at the fold” can generate non-commutativity.
coupling_not_commute in the same file). Order-dependence is carried by whichever operator moves amplitude between locations — never by a gate acting alone. This page contains a reasoned argument, not a machine-checked proof of the continuum statement.
The HVEH basin contains two operators acting on the incoming flow field: K, the curvature gate (basin sills and vanes that enforce safe flow geometry), and F, the fold operator (the nonlinear self-amplification that drives vortex tightening).
In standard linear systems, operator order is irrelevant. Here it is not. The commutator [K, F] = KF − FK measures the failure of the two operators to commute, and it is nonzero — but for a specific reason worth stating precisely, because the obvious reason is wrong.
Why a gate alone is not enough. Write K as multiplication by a radial mask χ(r) and suppose F acted pointwise, F[v](r) = f(v(r)) with f(0) = 0. Then K(F(v)) = χ·f(v) and F(K(v)) = f(χv); where χ = 1 both equal f(v), where χ = 0 both equal 0. They agree everywhere, so [K,F] = 0. A static gate composed with a sitewise map always commutes. No amount of steepness at r* changes this.
What actually breaks the symmetry. F is not pointwise. Vortex tightening is advective: it transports vorticity between radii, so F[v](r) depends on v at radii other than r. Gate-then-transport and transport-then-gate now differ, because the second moves amplitude across the mask boundary that the first had already zeroed. The non-commutativity is a property of transport, not of the gate and not of any concentration at the fold.
Applying F before K amplifies the vortex before geometry is locked, driving the system into a chaotic or cavitating state. Applying K before F locks the geometry first, then lets nonlinear amplification build on a stable foundation. The stable energy-producing attractor is only reachable via the correct sequence.
The ordering law is therefore a consequence of the transport term, not a tuning parameter. Status: the finite-dimensional statement — a static gate commutes with a sitewise map, and fails to commute with an inter-site coupling — is machine-checked (gate_commutes, coupling_not_commute). The continuum claim for this basin is [MODEL]: argued here, not formalized. The commissioning sequence below follows from the model and from the hysteresis in Proof III, and the quantitative thresholds remain [OPEN] pending measurement.
| Operator sequence | G = U ∘ F ∘ K ∘ C |
| Commutator | [K, F] = KF − FK ≠ 0 |
| Source of ≠ 0 | advective transport inside F (not the gate) |
| If F were pointwise | [K, F] = 0 exactly — kernel-checked |
| Wrong order | F ∘ K → chaotic attractor |
| Correct order | K ∘ F → stable helical attractor Γ |