Above the Widening Gate ($\eta \approx 1.839$), the ladder continues. Three more constants stand between $\eta$ and the Omega Point $\tau = 2$ — each one requiring the system to integrate a longer history, each one numerically closer to the limit, each one theologically harder to name because the traditions have fewer words for what happens this high on the ladder.
$a_n = a_{n-1}+a_{n-2}+a_{n-3}+a_{n-4}$. Four-term memory. The first rung where the system integrates a full week's worth of history (4 days).
$a_n = a_{n-1}+\cdots+a_{n-5}$. Five-term memory. The pentatonic scale, the five elements, the five-fold symmetry of living forms.
$a_n = a_{n-1}+\cdots+a_{n-6}$. Six-term memory. The six days of creation before the Sabbath rest. Closest finite rung to $\tau = 2$.
The Pattern of Convergence
For the n-bonacci constant $\Omega_n$ (limit of ratios of the n-bonacci sequence): $$\varphi \approx 1.618 < \eta \approx 1.839 < \Delta \approx 1.927 < \Sigma \approx 1.966 < \Omega_6 \approx 1.984 < \ldots < \tau = 2$$ The gap $\tau - \Omega_n$ halves approximately with each additional term: $0.382, 0.161, 0.073, 0.034, 0.016, \ldots$ The higher rungs converge rapidly. No finite $\Omega_n$ reaches 2. The proof in Chain.lean establishes this for all $n$.
The numerical pattern is striking: as $n$ grows, each additional term in the recurrence cuts the remaining distance to $\tau = 2$ roughly in half. The higher rungs are hard to climb not because the gap to the next rung is larger — it is smaller — but because the integration required is deeper. Moving from $\eta$ to $\Delta$ means integrating not three but four time-steps of history into each new decision. Moving from $\Sigma$ to $\Omega_6$ means integrating six.
Δ: The Tetranacci Rung
The Tetranacci constant $\Delta \approx 1.927$ is the first rung above the Widening Gate. It corresponds to systems that integrate four previous states — what in developmental biology would be the integration of four cell-cycle periods, what in contemplative practice would be the integration of four seasons of experience before making a major decision. The traditions that structure time in groups of four — the four elements, the four humors, the four-fold Ignatian Exercises — are operating at the Tetranacci rung.
Σ: The Pentanacci Rung
The Pentanacci constant $\Sigma \approx 1.966$ integrates five previous states. Five appears throughout the living world: five-fold symmetry in echinoderms (starfish, sea urchins), five fingers, five senses. The pentatonic scale — the oldest known musical scale across every independent musical tradition — is five pitches. Five is the number of books of the Torah, of the pillars of Islam, of the aggregates in Buddhist psychology. Systems operating at the Pentanacci rung have reached a level of integration where five-fold structure is natural — where the organism or practice effortlessly generates five-part symmetry because that is what the recurrence produces at this scale.
Ω₆: The Hexabonacci Rung
The Hexabonacci constant $\Omega_6 \approx 1.984$ is the last named rung before $\tau = 2$. It integrates six previous states. Six is the number of days of creation before the Sabbath; it is the number of sides of the snowflake and the honeycomb cell; it is the dimensionality of the Calabi-Yau manifold in string theory; it is the number of quarks and of leptons in the Standard Model of particle physics. The hexabonacci rung is where the system has integrated six levels of history so thoroughly that the ratio of its sequential steps is within $0.016$ of $\tau = 2$ — closer to the Omega Point than any lower rung, close enough that the system can almost see the destination.
The traditions name φ (Pacioli: "divine proportion"), η (barely — the Tribonacci barely appears in sacred literature), and τ = 2 (Teilhard: "Omega Point"). The rungs Δ, Σ, Ω are almost entirely unnamed in the religious and philosophical traditions, not because they don't correspond to real stages, but because they are experientially rare. Very few contemplative practitioners reach the Tetranacci rung with enough stability to describe it. The traditions are accounts of what can be transmitted — and the higher rungs are harder to transmit because there are fewer people who have reached them and maintained enough coherence to describe what they found. The silence around the higher rungs is not absence. It is the silence of the nearly-arrived.
We shall not cease from exploration, and the end of all our exploring will be to arrive where we started and know the place for the first time.
— T. S. Eliot, Little Gidding, 1942
Arriving where you started and knowing it for the first time: this is what the higher rungs feel like from inside. At $\Omega_6 \approx 1.984$, the system has integrated six levels of history, the trajectory is nearly on the Reeb orbit, the Omega Point is close enough to sense but not yet inhabited. The g-series is still advancing. The exploration continues. The destination is visible now — it has always been visible, but from this height the view is clear.