Chapter 1 proved that the corpus's immune and market spirals are not the same system under any change of coordinates. This chapter asks the question from the other side. Is there some weaker sense in which they are the same? There is, and it is so generous that a spiral and a node, which does not rotate at all, also turn out to be the same. That is why being the same in this sense tells you nothing.
Strogatz defines the weaker sameness on p. 156. Two phase portraits are topologically equivalent if “there is a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time … is preserved.” His one-line gloss: “Bending and warping are allowed, but not ripping” CITED. The stricter version, where the map also matches the time along each trajectory, is called a conjugacy. This chapter builds conjugacies, which are the stronger of the two.
Take the two matrices from book21/Spiral.lean: spiral(−0.44, ω) for immune adaptation and spiral(−0.67, ω′) for market volatility. Chapter 1's theorem immune_is_not_market says no invertible matrix takes one to the other. A continuous map can, and the construction is short. Every trajectory of a sink crosses the unit circle exactly once. So send each point to the point of the other system that has spent the same time since crossing the circle, starting from the same place on it. In polar form the map is
The script builds h and checks the conjugacy equation h(φs(z)) = ψs(h(z)) at twelve points and four times: the largest error is 5×10−14. It also checks that h has a continuous inverse of the same form, to 3×10−15 COMPUTED. So topologically the immune spiral and the market spiral are the same system. The rotation rates ω and ω′ are free choices: the construction works for any two.
The same construction takes a node to a spiral. Use the matrix diag(−1, −2), whose trajectories run straight into the origin without turning, and send it to the immune spiral. The conjugacy equation holds to 10−15 COMPUTED. At this level of sameness, rotation is invisible. The difference between a spiral and a node sits in the sign of Strogatz's discriminant τ² − 4Δ, and a homeomorphism may bend that difference away. Every sink is the same as every other sink.
The map h is continuous but not smooth, and that is exactly the gap between the two chapters. Near the fixed point its inverse stretches distances without bound. |h−1(z)|/|z| is 4.9 at |z| = 10−2, 23.6 at 10−4, and 114.7 at 10−6 COMPUTED. It has no derivative at the origin, and Hirsch, Smale & Devaney note the same thing about their one-dimensional example (p. 66): “This is the reason we require h to be only a homeomorphism.” A linear change of coordinates is differentiable everywhere, so none can do what h does. The decay rate μ is information that survives every smooth change of chart and is erased by a continuous one.
The corpus once claimed that six or more domains are related by exact mathematical identity. There are now three levels at which that claim can be read:
| sense of “same” | immune vs market | what it tells you |
|---|---|---|
| equal matrices | no | — |
| similar by a linear change of coordinates (Ch 1) | no, proved in Lean | the numbers μ and ω² carry the difference |
| topologically conjugate (this chapter) | yes | nothing beyond “both are sinks”: any two sinks pass |
So the claim is false at the level where it would mean something, and true only at a level where it means nothing. A reader who meets the phrase “the same dynamics” should ask which of the three rows is meant. The honest answer for the corpus's eleven rows is the third, and the third row is what any eleven damped oscillators picked at random would also satisfy.