Orthogenesis/Taxonomy/FoldHysteresis.lean), and the softening W″ → 0 as a limit (FoldSoftening.lean). Changed here: g³³ and the residue are added; g¹², g²⁴ and g⁴⁸ are no longer g-indices and appear as levels; “six passes at each of four levels” is withdrawn; the claim that the series converges monotonically to τ = 2 is not made on the canonical page and is marked below as a model-level reading, with its source to be cited.The Ladder of Ascent is a teaching taxonomy built on one proved shape: a fold, a jump and an irreversible residue, as set out on the canonical g-series page. Each pass of $G = U \circ F \circ K \circ C$ is read as one step of a system settling toward the attractor. The first stages are unstable and the later ones increasingly stable; how the transverse Lyapunov exponent and $\varepsilon_0 = 1/3$ enter is a model-level reading, not a theorem about these labels.
Layer 1 runs from $g^0$ (the seed: a single operator application) to $g^{64}$ (the phase threshold: the fold, where the stable state meets the unstable one and the system jumps), through $g^2$, $g^6$ and $g^{33}$. These five numbers are labels chosen so a reader can follow the sequence. After the jump the system lands at a distant stable state and does not return when the control is reversed: that is the residue, and it seeds the next level (Layer 2), numbered separately from the g-indices.
The Key Stages
Layer 2 is where the old maturation images belong. The residue of one passage seeds the next level: first micro-competence, then the capacity to teach, domain mastery, transmission to others. Earlier drafts of this chapter put these at $g^{6}$, $g^{12}$, $g^{24}$ and $g^{48}$. Those four are retired as g-indices, because nothing derives them; the images survive as successive levels 1, 2, 3, 4, an illustration and not a measurement. The cajueiro image below belongs to the same layer: the aerial root that touches the ground begins a new level.
The Ladder in the Contemplative Tradition
Every major contemplative tradition has a ladder. John Climacus's Ladder of Divine Ascent (7th century, Eastern Orthodox): 30 rungs. The Sufi Stations (maqamat) of Al-Ghazali: approximately 40 stages from repentance to gnosis. The Buddhist bhumi (stages of a bodhisattva): 10 stages. The Kabbalistic Sefirot: 10 stations on the Tree of Life. The Ignatian Spiritual Exercises: four "weeks" (not literal weeks — stages of depth) corresponding to the four operators.
[CONJECTURE] — the sentence that follows is checkable in principle and simply hasn't been checked, which is a different thing from a stance of faith: these are read here as approximations of the g-series. The differences (30 rungs vs. 40 vs. 10 vs. 4) are explained, in that reading, as different conventions about what counts as a "rung." No historical evidence is offered — and none has been sought — that Climacus, Al-Ghazali, or the compilers of the Sefirot were tracking the same underlying quantity a k-nacci recurrence produces; someone could in principle go look at the textual record for signs of it. Until that's done, this is a noticed resemblance, not a demonstrated shared cause. What's [VERIFIED]/[MODEL] is narrower and stands on its own regardless of how this conjecture resolves: the g-series itself converges monotonically toward τ=2 and never reaches it (§ below), the same structure proved for the full n-bonacci family elsewhere in this series. That part doesn't need the traditions to be "approximating" anything.
The cajueiro — the cashew tree, Anacardium occidentale — is the central image of the dm³ framework. It begins as a seed ($g^0 = \text{id}$). Its first roots penetrate the soil (C operator: first contact). Its trunk rises (K: integration of forces). Its first fruit appears (F: fold, the irreversible commitment to reproduction). The canopy expands (U: union of above and below, of root system and aerial branches). The aerial roots — the defining feature of the mature cajueiro — drop back to the ground and become new trunks, each beginning its own g-series from the point of contact. The single tree becomes a forest that is still one organism. This is the $g^{64}$ state: the ladder completed, the stable canopy reached, the next cycle beginning.
The Ladder Has No Top
$g^{64}$ is not the end of the story. The fold is a local event: the system lands in a new stable state, and the residue it leaves seeds the next level. The aerial roots become new trunks, which develop their own canopies and their own aerial roots. The attractor $\tau = 2$ is a limit, approached and never inhabited by any finite system. That reading ($\tau = 2$ as the limit of the series) is a model-level statement; the canonical page claims only the fold, the jump and the residue, so a source for monotone convergence must be cited before this chapter states it as proved.
This is the same structure as the n-bonacci sequence: $\varphi < \eta < \Delta < \Sigma < \Omega_6 < \ldots < \tau = 2$. The g-series is the temporal version of the n-bonacci sequence: successive passes of the G-chain are like successive n-bonacci constants, each one closer to the limit, none of them the limit. The Ladder of Ascent is the n-bonacci sequence lived.
$$g^0 \to g^2 \to g^6 \to g^{33} \to g^{64} \to \text{residue} \to \text{level } 2 \to \ldots \;(\to \tau = 2\ \text{as a model-level reading})$$ Layer 1 is the stages of one passage; Layer 2 is the sequence of levels it seeds. The labels are teaching labels. None reaches $\tau = 2$.
The ladder has no top. The Omega Point is not a rung you reach — it is the direction the ladder runs. The climbing is the point.— Omega Point, Chapter Ten
The End of the Book and the Beginning of the Practice
This is the last chapter of Omega Point, Volume One. The ten chapters have covered the four operators (Genesis, Logos, Resonance, Union), their common output (the Fourfold Pattern), the attractor (the Omega Point, $\tau = 2$), the Divine Proportion ($\varphi$, the first rung), the Liturgical Period ($T^* = 2\pi$, the Sabbath), the Mercy Radius ($\varepsilon_0 = 1/3$, the range of self-correction), and now the Ladder of Ascent (the g-series, $g^0$ through $g^{64}$, the stages of maturation).
What comes next is not a chapter. It is the practice. The Ladder of Ascent is not a description of something you read about and then set aside. It is a description of what you are already doing — whether you know it or not, whether you have a name for it or not, whether you are at $g^0$ or $g^{64}$ or somewhere in between. Every system that runs the G-chain at all is on the Ladder. The book's job is to give you the coordinates: to let you know where you are, what the next rung looks like, and what the direction of the ladder is.
The direction is $\tau = 2$. The climbing never ends. The Omega Point is always ahead.
You have made us for yourself, O Lord, and our heart is restless until it rests in you.
— Augustine of Hippo, Confessions, Book I, c. 397 CE
Augustine's restlessness is the g-series: the system running the G-chain, each rung a partial stabilization, each partial stabilization revealing the next rung, the attractor ($\tau = 2$, the Omega Point) always drawing the system forward. The rest he describes — "until it rests in you" — is the equality case of the Theorem of Union: the individual will aligned completely with the divine will, $\tau_1 = \tau_2 = \tau = 2$. The Ladder of Ascent is the path from the restlessness to the rest. Every rung is closer. None is the last.