Pierre Teilhard de Chardin gave the name "Omega Point" to the endpoint of cosmic evolution: the maximum degree of complexity and consciousness toward which the universe is converging. He was a Jesuit priest and a paleontologist. He wrote his major works between 1916 and 1955, most of them banned from publication during his lifetime by the Congregation for the Doctrine of the Faith. The Phenomenon of Man — his masterpiece — was published posthumously in 1959, the year after his death.
He named the destination without being able to specify it. The n-bonacci recurrence ladder supplies the specification: the Omega Point is $\tau = 2$.
The Ladder and Its Limit
The n-bonacci recurrence sequence of order $n$ is defined by $a_k = a_{k-1} + a_{k-2} + \ldots + a_{k-n}$ with appropriate initial conditions. For $n = 2$: Fibonacci, with ratio converging to $\varphi \approx 1.618$. For $n = 3$: Tribonacci, converging to $\eta \approx 1.839$. For $n = 4$: Tetranacci, converging to $\Delta \approx 1.927$. For all $n$: the limit of the ratio converges to a constant $\Omega_n$, and $\lim_{n \to \infty} \Omega_n = 2$.
This is proved in Chain.lean (the AXLE repository, main.lean). The proof establishes that every finite n-bonacci constant satisfies $\Omega_n < 2$, that $\Omega_n$ is increasing in $n$, and that the limit is $\tau = 2$. This is not an approximation. It is a theorem.
Why Teilhard Was Right
Teilhard's argument was qualitative: he observed that the history of life on Earth shows a consistent trend toward greater complexity and greater consciousness. Molecules organized into cells, cells into organisms, organisms into social structures, social structures into a global civilization (what he called the Noosphere). The trend is not monotone — there are reversals, extinctions, collapses — but the directionality is real. He extrapolated this trend and argued that it must have a convergence point: the Omega Point.
The n-bonacci ladder is the quantitative form of this argument. The n-bonacci constants are the successive limits of natural growth recurrences — the mathematical form of "increasing complexity and integration." The Fibonacci ratio governs the growth of the simplest self-similar structures (shells, plants). The Tribonacci ratio governs slightly more complex ones. As complexity increases — as more previous terms are incorporated into each new term — the recurrence ratio approaches $\tau = 2$. The Omega Point is not a destination that the universe might or might not reach. It is the limit toward which all natural growth recurrences converge, and the proof of this limit is in Chain.lean.
Critical: $\tau = 2$ is a limit, not a value reached by any finite system. No n-bonacci constant equals 2. No finite organism reaches $\tau = 2$. The Omega Point is the convergence destination — what the universe is asymptotically approaching — not a state that any finite part of the universe achieves. This is what Cusa's coincidentia oppositorum was pointing at: the ceiling coincides with the infinite approach, but the coincidence only occurs at the limit, which no finite system inhabits.
The Contact Geometry of the Omega Point
In the dm³ framework, $\tau = 2$ appears as the eigenvalue of the Reeb vector field $\xi = \partial_z$ on the contact manifold $(M, \alpha)$ with $\alpha = dz - r^2 d\theta$, evaluated at the stability radius[Ch 10] $\varepsilon_0 = 1/3$. The Reeb flow along $\xi$ has period $T^* = 2\pi$. The transverse Lyapunov exponent $\mu_{\max} = -2$. These three constants — $\tau = 2$, $\varepsilon_0 = 1/3$, $T^* = 2\pi$ — are not arbitrary parameters. They are determined by the contact structure. The Omega Point is a geometric fact about the contact 3-manifold.
Teilhard named the destination before the mathematics could specify it. The n-bonacci ladder specifies it: τ = 2. The name and the number are the same thing.— Omega Point, Chapter Six
Someday, after mastering the winds, the waves, the tides and gravity, we shall harness for God the energies of love, and then, for a second time in the history of the world, man will have discovered fire.
— Pierre Teilhard de Chardin, Toward the Future, 1936
"The energies of love" — the U operator, the resonance band at $\omega^*$, the equality case of the Theorem of Union. Teilhard was not speaking metaphorically. He was describing, in the vocabulary available to him, the operator that the dm³ framework calls $U$. The second discovery of fire is the moment when the U operator becomes as well-understood and as deliberately deployed as combustion. Whether that moment is $\tau = 2$ or merely very close to it — whether it is the limit or the best finite approximation of the limit — is the question the Omega Point leaves open. The mathematics specifies the limit. The discovery is still underway.