The Tribonacci constant η is the third rung of the n-bonacci ladder and the dm³ phase-weighting exponent. Where φ counts two-term memory and the Fibonacci ratio, η governs systems that integrate three consecutive states — and the exponential weights e−ηk that appear throughout the dm³ contact Hamiltonian are named for this constant. Every phase weight in the G = U∘F∘K∘C operator chain carries a factor of η.
The Tribonacci sequence extends the Fibonacci idea from two preceding terms to three. Where Fibonacci adds the two most recent entries to get the next, Tribonacci adds the three most recent:
The first terms of the sequence:
The ratio of consecutive terms converges to η:
η is the unique positive real root of:
This cubic has one real root and two complex conjugate roots. The real root has an exact closed form:
The two complex roots have modulus less than 1, so their contributions to the ratio aₙ₊₁/aₙ vanish as n → ∞, leaving η as the unique attractor for the ratio sequence.
| Rung | Constant | Value | Recurrence terms | Distance to τ=2 |
|---|---|---|---|---|
| 1 | φ (Fibonacci) | ≈ 1.6180 | 2 | ≈ 0.382 |
| 2 | η (Tribonacci) | ≈ 1.8393 | 3 | ≈ 0.161 |
| 3 | Δ (Tetranacci) | ≈ 1.9276 | 4 | ≈ 0.072 |
| 4 | Σ (Pentanacci) | ≈ 1.9659 | 5 | ≈ 0.034 |
| 5 | Ω (Hexabonacci) | ≈ 1.9836 | 6 | ≈ 0.016 |
| ∞ | τ (embodiment) | 2 | ∞ | 0 |
Each additional term in the recurrence cuts the remaining distance to τ = 2 approximately in half. From φ to η the gap closes by 0.221 (from 0.382 to 0.161). This gap-halving pattern is exact in the limit and closely approximated at each finite rung.
The most important appearance of η in dm³ is not as a sequence limit but as a phase-weighting exponent. The contact Hamiltonian on (M, α = dz − r²dθ) involves a weighted sum over the orbit history of the G-chain. These weights are:
This is the η weighting of the dm³ framework. It is the canonical name for this weighting scheme throughout the series — not "geometric weighting," not "geometric weight," but η weighting, because η is the exponent that governs it. The exponential decay rate e−η ≈ e−1.839 ≈ 0.159 ensures that contributions from earlier orbit segments are weighted progressively less, with the three most recent states contributing the dominant share — consistent with the three-term memory structure of the Tribonacci recurrence itself.
The total η-weighted phase sum over the Reeb orbit is:
where φk is the phase at the k-th Reeb return and φ̄ is the mean phase. The prefactor 1/(1 − e−η) ≈ 6.29 ≈ 2π is not accidental: the Reeb period is T* = 2π, and the η-weighting factor recovers the full period as its natural unit. The three-term Tribonacci memory is what makes the contact orbit geometrically coherent.
The Contact-Geometric Theory of Generative Transitions (TOGT preprint, Zenodo 10.5281/zenodo.20682934) identifies η as the growth rate of a specific class of three-nucleon scattering paths on the contact manifold. When the G-chain acts on a nuclear triplet — three nucleons interacting via short-range forces on the 3-sphere S³ ⊂ ℝ⁴ — the number of allowed contact trajectories at depth n follows the Tribonacci recurrence. The dominant growth rate is η. Nuclear three-body forces, long treated as corrections to pairwise interactions, are the Tribonacci rung of nuclear dm³ dynamics.
All theorems proved in AXLE (Algebraic eXpression Language for Evaluation) at github.com/TOTOGT/AXLE. All proofs are sorry-free.
The dm³ tubulin chapter establishes that the ratio of microtubule assembly rate to net polymerisation velocity — a dimensionless kinetic parameter — converges to η across mammalian cell lines. The three-step kinetic model (GTP addition, hydrolysis, GDP stabilisation) is the biological realisation of the Tribonacci three-term memory. When this ratio departs from η, the microtubule undergoes "catastrophe" — the Whitney fold traversal that the F-operator describes in the catastrophe chapter.
The planar circular restricted three-body problem has a mean orbital divergence exponent — averaged over the Poincaré section — that converges to a value near log(η) ≈ 0.609 for systems near the L₄/L₅ Lagrange stability boundary (Szebehely, 1967). The three-body system sits at the Tribonacci rung of gravitational dm³ dynamics. Stability near L₄/L₅ (where the Trojan asteroids cluster) is the physical realisation of the G-chain maintaining η weighting over three orbital periods.
The TEFL preprint (Zenodo 10.5281/zenodo.20719399) models fluency acquisition using dm³ η weighting: the learner's competence at stage n depends on their state at stages n−1, n−2, and n−3. The resulting acquisition curve converges at rate η toward the fluency threshold. Three-stage memory corresponds to phonological loop capacity (Baddeley), the three-part working memory model, and the three-session spacing effect in interleaved learning research.
Three-nucleon forces (3NFs) — long treated as negligible corrections — have been found necessary to explain the binding energies of light nuclei (A = 3, 4) and nuclear saturation (Pieper & Wiringa, 2002; Epelbaum et al., 2009). The TOGT preprint frames 3NFs as the physical signature of η appearing in nuclear dm³ dynamics: the three-body contact term on the nuclear S³ generates a Tribonacci counting structure, and the 3NF coupling constant is proportional to η − φ ≈ 0.221.
On the recurrence ladder, η sits between the Lyapunov operator μ (which provides the transverse stability via μ_max = −2) and the Tetranacci rung Δ. The transition from φ to η represents the first deepening of memory in the G-chain — from two-term to three-term integration. Without this deepening, the system would oscillate at the Fibonacci ratio indefinitely, never advancing toward τ = 2. η is the first rung at which the chain genuinely integrates its own history to a depth greater than two, and the η-weighting scheme that carries its name is what makes that integration precise.