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 CONJECTURE — see note below
The Tetranacci constant $\Delta \approx 1.927$ is the first rung above the Widening Gate. [VERIFIED/MODEL] It corresponds to systems that integrate four previous states — a proved property of the k-nacci recurrence itself. [CONJECTURE] 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 — and the traditions that structure time in groups of four (the four elements, the four humors, the four-fold Ignatian Exercises) read here as operating at the Tetranacci rung — is this book's interpretive wager, not a demonstrated mechanism. No test is offered that would distinguish "the world groups things in fours because of this recurrence" from "four is a common small number that many independent systems land on for their own unrelated reasons."
Σ: The Pentanacci Rung CONJECTURE — see note below
The Pentanacci constant $\Sigma \approx 1.966$ integrates five previous states. [CONJECTURE] Five appears throughout the living world — five-fold symmetry in echinoderms, five fingers, five senses, the pentatonic scale, five books of the Torah, five pillars of Islam, five aggregates in Buddhist psychology — and this book reads that recurrence of "five" as the Pentanacci rung showing itself, "because that is what the recurrence produces at this scale." Stated plainly: this is a resonance the book finds and offers, not a causal chain it has traced. Echinoderm pentaradial symmetry has its own developmental-biology explanation (Hox gene expression in a bilaterally-symmetric ancestor); the pentatonic scale's near-universality has its own acoustic explanation (small-integer frequency ratios); the five pillars, five books, and five aggregates each have their own separate theological histories. None of those independent explanations is displaced by the resemblance in count. The pattern is real and worth noticing; "because" is the word doing the wagering.
Ω₆: The Hexabonacci Rung CONJECTURE — see note below
The Hexabonacci constant $\Omega_6 \approx 1.984$ is the last named rung before $\tau = 2$. [VERIFIED/MODEL] It integrates six previous states — proved, and the ratio of its sequential steps is within $0.016$ of $\tau=2$, closer to the limit than any lower rung. [CONJECTURE] The reading that follows is this book's, not a derivation: six as the days of creation before the Sabbath, the sides of the snowflake and the honeycomb cell, the dimensionality of the Calabi-Yau manifold, the number of quarks and of leptons — offered as consonance across very different domains, held out as meaningful by the author, not established as a shared cause. A snowflake's hexagonal symmetry has its own explanation in the geometry of hydrogen bonding in ice; the Standard Model's generation count is an open question in particle physics with no settled derivation from anything, let alone from this recurrence. The hexabonacci rung being "closer to the Omega Point than any lower rung" is the [VERIFIED/MODEL] part. Six showing up elsewhere in the world is the [CONJECTURE] part — checkable, not yet checked.
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.
Named honestly: this is the shape of a claim that cannot be wrong — any absence of correspondence in the traditions is read as confirming the theory ("too rare to describe") rather than counting against it. That doesn't make the intuition false. It does mean it cannot do the work of evidence, and this book would rather say so than let the shape of the argument do quiet work a reader might not notice.
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.