WP70 · The Function and the Exponent

The finding

The series states \(r^* = 0.77594058\) as a basin boundary — the radius inside which trajectories return to the attractor. A basin boundary is a global claim about a flow. The only stability object the series actually constructs is \(\mu_{\max} = -2\), a transverse Lyapunov exponent, which is local and asymptotic and cannot bound a basin. The two are different objects that share a name. Until a Lyapunov function is exhibited, every sentence in which \(r^*\) bounds a basin is unlicensed.

1 The claim under audit

Across the series \(r^*=0.77594058\) appears as the stable levitation gap, the inner boundary of the basin, and the Whitney \(A_1\) fold radius of the radial map. The first is a physical reading and the third is a statement about a singularity of a map. This paper is about the second only.

The basin reading is the one that carries weight elsewhere: Gravity Across Scales calls \(r^*\) the radius at which "the attractive outer basin and the repulsive inner basin balance exactly," and Book 4 Ch 10 refines \(\varepsilon_0 = 1/3\) to \(r^*\) on the grounds of basin asymmetry. Both are global statements about which initial conditions return.

2 What licenses a basin claim

The standard instrument is a Lyapunov function. For a flow \(\dot x = f(x)\) with invariant set \(\Gamma\), one exhibits \(V:\mathcal{M}\to\mathbb{R}\), continuously differentiable, positive definite relative to \(\Gamma\), with

\[ \dot V(x) \;=\; \nabla V(x)\cdot f(x) \;\le\; 0 \quad\text{off } \Gamma . \]

Every bounded sublevel set \(\{V \le c\}\) that contains no other invariant set then lies inside the basin of attraction (LaSalle; Khalil, Nonlinear Systems, §4). The basin estimate is the sublevel set. This is the only elementary route by which a number can be called a basin radius.

3 What the series has instead

chμ constructs the transverse Lyapunov exponent

\[ \lambda \;=\; \lim_{t\to\infty}\frac{1}{t}\,\ln\frac{|\delta(t)|}{|\delta(0)|}, \qquad \mu_{\max} = -2 . \]

A negative exponent establishes that trajectories already near \(\Gamma\) converge, and how fast. It is a limit as \(t\to\infty\) along a single trajectory, evaluated in the linearisation. It is silent on how far the attracting region extends, and a system can have a strongly negative transverse exponent and an arbitrarily small basin.

The distinction, stated once

Exponent: local, asymptotic, linear, a rate. Answers how fast, once close. Function: global, constructive, nonlinear, a scalar field. Answers how far. The exponent is a consequence of the flow near \(\Gamma\); the function is a certificate about the flow away from it. Only the second bounds a basin.

4 The series already knew this

This paper is not a new objection. The open-problems note attached to the \(r^*\) work already reads:

"The dm³ ODE is a non-autonomous system with no obvious integrating factor. Possible routes: (a) Lyapunov function bounding the basin; (b) matched asymptotic expansion in \(\varepsilon\); (c) Poincaré section analysis reducing to a 1D map whose fixed point is \(r^*\)."

— WP58, Open Problem O.4.4.

WP58 proves something sharper, which this paper adopts

The same paper establishes that \(r^*\) does not depend on the contact form at all: \(\alpha = dz - r^2 d\theta\) does not appear in the ODE and does not enter the definition of \(r^*\); replacing it with any other contact form on \(\mathbb{R}^3\) leaves every trajectory, and hence \(r^*\), unchanged. \(r^*\) is a property of a planar vector field in \((r,z)\).

So the constant the series calls its central contact-geometric result is not a contact-geometric quantity, and the missing Lyapunov function is a question in ordinary planar ODE theory — which is good news, being far more tractable than one about contact manifolds. PROVED IN WP58

Route (a) is listed as a route. It has not been taken. What this paper does is promote that footnote to a stated open problem and record what the missing object would have to look like, so that the basin language elsewhere is read as provisional rather than settled. OPEN

5 A worked example: the irreducible mass

Black hole mechanics supplies an explicit Lyapunov function for a physical flow, and it is worth setting beside the dm³ case because it shows exactly how much has to be supplied.

Christodoulou (1970) defined the irreducible mass of a Kerr black hole,

\[ M_{\mathrm{irr}} \;=\; \sqrt{\frac{A}{16\pi}} \;=\; \sqrt{\frac{M^2 + \sqrt{M^4 - J^2}}{2}} \, . \]

Four properties, none of them assumed:

It is an explicit function of the state \((M,J)\), not an existence claim.
It is monotone by derivation. Hawking's area theorem gives \(\delta A \ge 0\), hence \(\delta M_{\mathrm{irr}} \ge 0\) — from the field equations plus the null energy condition, not by postulate.
Its level sets have physical meaning. They are precisely Christodoulou's reversible transformations: energy extracted along a level set is recoverable, energy extracted across one is not.
Extremum at the equilibrium. \(M_{\mathrm{irr}} = M\) at Schwarzschild (nothing extractable); \(M_{\mathrm{irr}} = M/\sqrt{2}\) at extremal Kerr, the floor. The gap \(M - M_{\mathrm{irr}}\) is the extractable rotational energy, up to \(1 - 1/\sqrt{2} \approx 29\%\).

And it needed repair — which is the instructive part

The bare area law is false once black holes evaporate: Hawking radiation shrinks the horizon, because quantum fields violate the null energy condition the theorem assumes. What survives is Bekenstein's generalised second law, \(S_{\mathrm{BH}} + S_{\mathrm{outside}}\) non-decreasing. Even the cleanest known example of this construction required its monotone to be enlarged before it held in general. A dm³ Lyapunov function should be expected to need the same.

What the series’ own Hawking chapter does instead — and why it does not count

ch H · Hawking pela Lente GTCT identifies dm³ constants inside Hawking’s formulae: \(\tau = 2 \Rightarrow 8\pi = 4\pi\tau\), and \(\varepsilon^* = 1/3 \Rightarrow S = A/4 = (A\varepsilon^*)/(4/3)\).

Both are arithmetic identities, not identifications. \(8\pi = 4\pi\cdot 2\) holds because \(2 = 2\); \((A/3)/(4/3) = A/4\) holds for every \(A\). Neither constrains anything — any constant can be written as a product containing any other constant. A matching earns its keep only when it forbids something.

The genuine connection to black holes is the one in this section, and it is structural rather than numerical: \(M_{\mathrm{irr}}\) and the sought \(V\) occupy the same role, a monotone certificate about a flow, and that comparison requires no constant to coincide. ch H NEEDS EDIT

6 A number that is not ours

An equal-mass, non-spinning binary black hole merger produces a remnant with dimensionless spin \(a_f/M_f \approx 0.6864\), essentially independent of the masses. It is a genuine dimensionless attractor of a physical flow, and its numerical proximity to \(r^*=0.776\) makes it a tempting correspondence. It is not available.

The value is already derived within standard general relativity, by the Buonanno–Kidder–Lehner argument: treat the final stage as a body of reduced mass \(\mu\) plunging from the ISCO of the remnant, set \(J_f = \mu\,\tilde L_{\rm ISCO}(a_f)\), and solve the fixed point self-consistently. Recomputed for this paper from the Bardeen–Press–Teukolsky ISCO formulae:

Computation · August 2026

Checks: ISCO at \(a=0\) returns \(6M\); \(\tilde L\) returns \(2\sqrt{3}\).

Holding \(M_f = M\): \(a_f = 0.6631\) (−3.4%).
Including the radiated ISCO binding energy: \(a_f = 0.6870\) (+0.1%).

The precision is partly fortuitous and must be reported as such: the same model returns 2.5% of the mass radiated where numerical relativity gives ~4.8% — wrong by a factor of two. It reproduces the spin and misses the energy.

Two consequences. First, \(0.686\) cannot be claimed as a dm³ prediction; it is a consequence of geodesic motion in Kerr, obtained in 2008. Second, and more usefully: \(0.686 \neq 0.776\). Recorded here so the correspondence is not proposed again. CLOSED

The general form of the objection

WP58 computes \(r^*(\varepsilon)\) for \(\varepsilon \in \{0.5,\dots,4.0\}\) and shows it monotone, with \(r^*\to 0\) as \(\varepsilon\to 0\) and \(r^*\to 1\) as \(\varepsilon\to\infty\). A monotone function onto \((0,1)\) passes arbitrarily close to every target in that interval.

So no numerical coincidence with \(r^*\) is evidence unless \(\varepsilon\) is fixed independently and in advance. That disposes of the merger spin at a stroke — and equally of WP58’s own Open Problem O.4.1, the 25 ppm near-coincidence \(r^*(1.5)=0.693130\ldots\) against \(\ln 2 = 0.693147\ldots\). Same genre. The one-parameter family is the reason matching cannot work, not the particular numbers. CLOSED

7 Open problems

L1 · Exhibit the function

The system, from WP58 §1:

\[ \dot r = r(1-r^2) + \varepsilon (r-1) e^{-z}, \qquad \dot z = r^2 - \varepsilon (r-1)^2 e^{-z} . \]

Construct \(V\) on the dm³ state space, positive definite relative to the limit cycle \(\Gamma = \{r=1,\dot\theta=1,\dot z=1\}\), with \(\dot V \le 0\) off \(\Gamma\). The system is non-autonomous, so \(V\) may carry explicit \(t\)-dependence and the appropriate statement is the non-autonomous LaSalle–Yoshizawa form. OPEN

L2 · Recover \(r^*\) as a level set

Show that the largest bounded sublevel set \(\{V \le c\}\) containing no other invariant set has inner radius \(r^*\). Only then is \(r^*\) a basin boundary rather than a fitted radius. OPEN

L3 · Mechanise

Formalise L1–L2 in Lean 4. Lyapunov arguments are finite inequality chains over a named function — among the few things in this series genuinely suited to a proof assistant, unlike the operator-correspondence claims. TO ADD

8 What follows if L1 fails

Stated in advance, so the outcome is not renegotiated afterwards. If no Lyapunov function exists with \(r^*\) as a level set, then \(r^*\) is an artifact of the \(\varepsilon\)-expansion and of the particular numerical scheme that certified it, and the basin language must be withdrawn everywhere it appears — chμ, Gravity Across Scales, Book 4 Ch 10, and the galactic extension in Book 7. The Whitney \(A_1\) fold reading survives that outcome: a fold is a local statement about a map and needs no Lyapunov function. The basin reading does not survive it.

Cross-references

chμ · The Lyapunov Operator — supplies \(\mu_{\max}=-2\); this paper marks the gap between that exponent and a basin claim.
Gravity Across Scales — carries the balance-of-basins reading of \(r^*\), and lists the galactic fold radius as Open with no closed form. It stays Open.
WP69 · The Fold Is a Coordinate — the fold reading of \(r^*\), which is unaffected by L1.
Book 4 Ch 10 — refines \(\varepsilon_0=1/3\) to \(r^*\) by basin asymmetry; that refinement inherits L1.
WP58 · r*_galactic — source of Open Problem O.4.4 and of the contact-form-independence lemma; supplies the ODE and the monotone family.
ch H · Hawking pela Lente GTCT — the series’ black hole chapter; §5 marks its constant-matching as tautological and gives the structural reading.
ch Einstein and WP56 · Special Relativity as Contact Geometry — where the series treats relativity directly. The area theorem used in §5 is a theorem of general relativity, so the black hole material belongs beside these.
WP57 · Causal Integration and Book 8 Ch 8.3 — both take the Milky Way–Andromeda merger as given, after van der Marel (2012). Sawala et al., Nature Astronomy (Aug 2025), put it near 50/50 within 10 Gyr once the LMC and M33 are included. Unrelated to L1; flagged because both state it unconditionally.
WP-31B · How to Audit — this paper is an instance of its procedure applied to the series' own central constant.

Sources for §5–§6: Christodoulou, Phys. Rev. Lett. 25, 1596 (1970); Christodoulou & Ruffini, Phys. Rev. D 4, 3552 (1971); Bardeen, Carter & Hawking, Commun. Math. Phys. 31, 161 (1973); Bardeen, Press & Teukolsky, ApJ 178, 347 (1972); Buonanno, Kidder & Lehner, Phys. Rev. D 77, 026004 (2008). Stability: Khalil, Nonlinear Systems, 3rd ed., §4.