The transverse system, as ch-strogatz establishes, is Strogatz's Example 7.1.1: $\dot r = r(1-r^2)$, $\dot\theta = 1$. Because $\dot\theta = 1$, a trajectory crosses any ray $\theta = \text{const}$ at intervals of exactly $2\pi$, so that ray is a surface of section and the first-return map is available in closed form. Example 8.7.1, page 282:
Block [1] of the verify script integrates $\dot r = r(1-r^2)$ over $t \in [0, 2\pi]$ from eight starting radii between $0.02$ and $40$ and compares. Agreement is at the $10^{-15}$ level — the map is not an approximation to the flow, it is the flow, sampled once per turn.
Strogatz's Liapunov exponent for a $p$-cycle (p. 374, Example 10.5.1) is $\lambda = \frac1p\ln|(f^p)'(x_0)|$; per unit time for a flow, $\frac1{T^*}\ln|\text{multiplier}|$. Block [2] runs it: $\frac{1}{2\pi}\ln e^{-4\pi} = -2$ exactly. The series' use of $\mu_{\max} = -2$ alongside $e^{-4\pi}$ is correct — they are one fact in two units. That is stated here before §4, which is about the one place they are conflated.
chEps-gronwall Proof IV writes $\rho = r-1$, $f(\rho) = -\rho(1+\rho)(2+\rho)$ — which block [3] confirms is $r(1-r^2)$ rewritten — bounds $|P'|$ by Grönwall as $\exp\!\big((\mu_{\max} + 6\varepsilon)T^*\big)$ on $|\rho| \le \varepsilon$, and reads off $\varepsilon_0 = 1/3$ as the radius where that bound reaches 1. The derivation is correct. The bound is very loose.
| quantity on |ρ| ≤ 1/3 | value |
|---|---|
| Grönwall bound on |P′| at ε = 1/3 | 1.000000 |
| exact sup |P′|, attained at ρ = −1/3 | 1.176970 × 10−5 |
| the bound exceeds the truth by | 8.496 × 104 |
| radius at which |P′| actually reaches 1 | r = 0.015049224 |
So $P$ is a contraction from $r \approx 0.0150$ outward — a ball about three times wider in $\rho$ than $|\rho| \le 1/3$, and the contraction inside it is five orders of magnitude stronger than the bound admits.
| instrument | where | the corpus question it answers |
|---|---|---|
| Trapping region + Poincaré–Bendixson | §7.3, pp. 205–206 | Example 7.3.1 perturbs this very system to $\dot r = r(1-r^2) + \mu r\cos\theta$ and proves a closed orbit survives for $\mu < 1$, inside $0.999\sqrt{1-\mu} < r < 1.001\sqrt{1+\mu}$. The proof is two inequalities that hold for every $\theta$ because $|\cos| \le 1$. WP-62 asks what it takes for a basin result to be “publishable as a certificate rather than an illustration”; this is the analytic answer, and block [7] checks the annulus at five values of $\mu$. |
| Dulac's criterion | §7.2, pp. 201–204 | Rules out closed orbits in a simply connected region by exhibiting one function $g$ with $\nabla\cdot(g\mathbf{f}) < 0$. The series asserts $\Gamma$ is the cycle and has nowhere proved there is no other. One mention in the corpus, in WP-92. |
| Liapunov functions | §7.2, p. 203 | The same $V$-decreases argument the contact chapters use informally, with the two conditions stated as conditions. Strogatz is candid that there is no systematic way to construct one, which is itself worth quoting in a chapter that produces one. |
chRho-spectral, Argument V of seven, reads: “the linearised Poincaré map at $\Gamma$ has eigenvalues $\{\mu_{\max}, \pm i\} = \{-2, \pm i\}$… the first zero at $t_1 \approx 14.135$ gives $t_1/(2\pi) \approx 2.25 \approx \tau\cdot\varepsilon_0^{-1}\cdot(1/3)$ — within the dm³ scaling of the first spectral gap.” Block [5] runs it.
| as written | computed | |
|---|---|---|
| $t_1/(2\pi)$ | 2.249611375552 | |
| $\varepsilon_0^{-1}\cdot(1/3)$ | 1.000000000000 | the two factors cancel |
| $\tau\cdot\varepsilon_0^{-1}\cdot(1/3)$ | 2.000000000000 | so the right-hand side is $\tau$ |
| relative gap | 12.4806 % |
The units. $-2$ is an exponent and $e^{-4\pi}$ is the multiplier; a return map's eigenvalues are multipliers. The two numbers differ by a factor of $5.7\times10^{5}$. Strogatz p. 374 gives the conversion, and §1 above shows the corpus elsewhere applies it correctly.
The count. Strogatz p. 281: a surface of section for an $n$-dimensional flow is $(n-1)$-dimensional, so the linearisation of $P$ is $(n-1)\times(n-1)$ — one multiplier for the planar transverse system, two on a contact 3-manifold. A spectrum listed as three numbers fits neither, and a modulus-one entry would assert a neutral direction this system does not have.
The scaling. $\varepsilon_0^{-1}\cdot(1/3) = 1$ identically, so $\varepsilon_0$ does not appear in the expression at all; the right-hand side is $\tau = 2$ wearing two cancelling factors. And $2.2496$ against $2$ is a 12.5 % gap presented as agreement. This is vocabulary-correspondence in its exact form — two numbers near each other, joined by an expression that looks structural and is not.
Scope. The other six arguments on that page were not examined and nothing here is a verdict on them, on RH, or on whether a contact reading of the critical strip is worth pursuing. The page also carries no MODEL / OPEN tag on any of the seven, which the house rules require.
Every result above is about the uncoupled transverse system. The full model is $\dot r = f(r)(1-e^{-z})$, $\dot\theta = 1$, $\dot z = 1$, and it has no Poincaré map at all: $\dot z = 1$ everywhere, so no trajectory ever returns to a section $\{z = z_0\}$. What the series computes over one period is a time-$2\pi$ flow map, not a first-return map. The two agree on the transverse coordinate and nowhere else, and the drift of the per-period exponent with base point — $-4\pi + 2e^{-z_0}(1-e^{-2\pi})$, reaching $-4\pi$ only in the limit — is exactly that gap, measured.