Hemodynamic Homeostasis · §7 · spinodal(κ*) = |load|

Chronic heart failure as a saddle-node basin collapse

Chronic heart failure is a basin collapse, not a reset. As contractility κ falls at fixed load, the cardiac high-output fixed point exists only while spinodal(κ) > |load| and annihilates in a saddle-node fold where spinodal(κ*) = |load|; the closed-form κ* = 1.629 matches the swept collapse at κ ≈ 1.6 (load 0.8). Collapse shape [V], markers cited [L], rate open [O].

Where hypertension moves an attractor, decompensated heart failure destroys one. The cardiac high-output basin exists iff spinodal(κ) > |load| and annihilates in a saddle-node fold at spinodal(κ*) = |load|; the closed-form κ* = 1.629 matches the sweep collapse at κ ≈ 1.6 [V]. Progression markers are cited [L]; absolute event rates open [O].

The high-output fixed point annihilates in a fold

Heart failure is modelled as the loss of an attractor on the same R19 substrate, which gives a sharper picture than “the pump gets weak.” The cardiac high-output fixed point exists only while the spinodal exceeds the load, spinodal(κ) > |load|; as contractility κ falls at fixed load it reaches a saddle-node fold and the basin is annihilated (research target RP5). Below that point there is no high-output steady state to fall back to — the system is not merely depressed, it has lost the attractor it used to live in.

The fold has a closed form: the collapse occurs at κ* solving spinodal(κ*) = |load|. With load 0.8 the closed-form κ* = 1.629 matches the swept collapse at κ ≈ 1.6, so the mechanism is confirmed by two independent calculations — an analytic fold condition and a numerical sweep — that agree. That agreement is the discriminant for “saddle-node” rather than a gradual decline.

A collapse is not a reset

This failure mode is qualitatively different from hypertension, and the difference dictates therapy. A reset moves a defended attractor to a new value (the integrator still holds a setpoint); a collapse removes the attractor entirely, so there is no setpoint left to defend. Treating the two the same way is a category error.

The distance to collapse is the barrier margin M = spinodal(κ) − |load|, the quantity that makes therapy falsifiable. Therapy must grow this margin — by reducing load and breaking the maladaptive neurohormonal cycle — rather than flogging the effector, which shrinks it. That asymmetry, and its match to the clinical mortality evidence in both directions, is the subject of the therapy chapter. The collapse dynamics are reproduced [V]; progression markers are cited [L]; absolute event rates are open [O].