Proof II of VII

Distribution Theory

The sharp-interface idealization, and where irreversibility actually comes from
The laminar-to-vortex transition is path-dependent: once the helical state is established, it persists as inflow falls back below the value that created it. That is hysteresis, and it follows from the transition being a subcritical fold — not from the distributional calculus of a gate. The Dirac term below is a correct property of the sharp-interface idealization and carries no thermodynamic content.
Correction notice — 2026-07-29
This proof previously asserted that the Dirac term obtained by differentiating the Heaviside gate proves the transition is thermodynamically one-way, and did so — in its own words — “without invoking entropy or thermodynamics directly … a purer and more general result.” That inference is withdrawn. Three separate errors:

1. The δ is an artifact of the model. The identity d/dr[H(r − r*)] = δ(r − r*) is correct, but it is a property of the discontinuity we assumed, not of the flow. A real transition has a finite-width shear layer; model it with a smooth profile and the derivative is a bounded bump of width ℓ and height ~1/ℓ, with no δ anywhere. The page claimed the opposite — “this is not an artifact of the model” — and that was exactly backwards.

2. δ(r − r*) is not an energy density. Nothing in the derivation connects it to energy. “Energy deposited at r*” was asserted, never derived, and dimensionally the object is an inverse length times the gated field.

3. Distributional differentiation is time-symmetric. No arrow of time can come out of it. Irreversibility requires either entropy production or a non-invertible map. A derivation that obtains thermodynamic one-wayness for free from the calculus of a step function is obtaining it from nowhere — which is the tell recorded as the false commutator lemma, as recorded 2026-07-18.

The engineering conclusion survives, on different grounds. Hysteresis in this basin is real and is a fold phenomenon — the subject of Proof III, not of distribution theory. The width of the hysteresis window is an empirical quantity and is not yet measured.

The flow field is modeled using a Heaviside step function H(r − r*) acting as a gate: below the critical radius r*, the flow is in the laminar inflow regime; above r*, it is in the helical vortex regime. The step is a modelling convenience — physically the two regimes are separated by a shear layer of small but finite thickness.

When this gated field is differentiated in the distributional sense, a Dirac term δ(r − r*) appears at the transition radius. The identity is correct and standard. What it records is the jump we put into the model: the step is a sharp-interface idealization of a shear layer of finite width ℓ, and the δ is the ℓ → 0 limit of a bump of height ~1/ℓ. It is a property of the idealization, not a discovery about the flow.

Irreversibility in this basin is real, but it comes from two places, neither of them distributional. Path dependence is a fold: the laminar branch loses stability at one value of the swirl parameter, the system jumps to the vortex branch, and that branch stays stable down to a lower fold. Between the two folds the basin is bistable, and the loop is hysteresis — which is why the state cannot be unwound by retracing the inflow. Thermodynamic irreversibility is separate and is dissipative: viscous shear in the vortex core produces entropy. That is an entropy argument, and it has to be made as one.

Status of the claims on this page. The two distributional identities are elementary and hold as written [VERIFIED — textbook]. The bistable-window structure is a [MODEL] claim, inherited from Proof III's fold analysis. The width of the hysteresis window — the inflow at which an established vortex actually collapses — is [OPEN]: it requires a measurement on a physical basin, and none has been made. Nothing on this page should be read as establishing it.

Key relations
Heaviside gateH(r − r*)·v(r) encodes transition at r = r*
Weak derivatived/dr[H(r − r*)] = δ(r − r*)
Boundary term∫ φ·δ(r−r*)dr = φ(r*) for test fn φ
What the δ isℓ → 0 limit of a bump of width ℓ, height ~1/ℓ
What it is notan energy density; an arrow of time
Hysteresis window[OPEN] — requires measurement on a basin
r H r* δ(r − r*) jump in the idealized profile laminar vortex 0 1
Engineering consequence
The HVEH is expected not to switch off mid-vortex by a small reduction in inflow: an established helical state should persist until inflow falls below the lower fold, not merely below the value that created it. Design implication: the shutdown setpoint is not the startup setpoint, and control logic must not assume a symmetric ramp. This follows from the bistable structure in Proof III and is a [MODEL] prediction. The size of the gap between the two setpoints is [OPEN] and must be measured on a physical basin before any control system relies on it.
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