The Tetranacci constant Δ is the fourth rung of the n-bonacci ladder. Systems at this rung integrate four consecutive states — a deepening of memory beyond the three-term Tribonacci that brings the growth constant within 0.073 of τ = 2. In catastrophe theory, the Tetranacci rung corresponds to the A₄ butterfly catastrophe: the fourth in Thom's hierarchy of elementary catastrophes, resolved only when the system achieves deep four-term coherence.
The Tetranacci sequence extends the n-bonacci pattern to four preceding terms. Each new entry is the sum of the four most recent:
The first terms of the sequence:
The ratio of consecutive terms converges to Δ:
Δ is the unique positive real root of the quartic:
This quartic has one positive real root (Δ), one negative real root (≈ −0.7749), and two complex conjugate roots with modulus less than 1. The dominant term in the Tetranacci ratio therefore converges to Δ, with the sub-dominant real root contributing a slowly alternating correction that decays as (−0.775)ⁿ.
No compact radical expression exists for Δ in the form of the exact closed form for η. Δ is most precisely defined as the root of the characteristic equation itself, certified numerically to arbitrary precision.
| Rung | Constant | Value | Gap τ − c | Gap ratio vs. previous |
|---|---|---|---|---|
| 1 | φ | ≈ 1.6180 | 0.3820 | — |
| 2 | η | ≈ 1.8393 | 0.1607 | 0.421 |
| 3 | Δ | ≈ 1.9276 | 0.0724 | 0.451 |
| 4 | Σ | ≈ 1.9659 | 0.0341 | 0.471 |
| 5 | Ω | ≈ 1.9836 | 0.0164 | 0.481 |
| ∞ | τ | 2 | 0 | → 1/2 |
The gap-ratio column converges toward 1/2 from above. The Tetranacci rung achieves a gap ratio of 0.451 — close to the 1/2 limit but not yet at it. This slow approach to exact gap-halving is what makes each additional rung harder to climb: the rungs are geometrically compressed near τ = 2.
René Thom's classification of elementary catastrophes lists seven types based on the number of control parameters. The sequence of catastrophes maps onto the n-bonacci ladder: A₁ fold (at ε₀), A₂ cusp (at φ), A₃ swallowtail (at η), A₄ butterfly (at Δ). Each rung is the dynamical correlate of one type in Thom's hierarchy.
The butterfly catastrophe has potential V(x) = x⁶ + ax⁴ + bx³ + cx² + dx, with four control parameters (a, b, c, d). Its bifurcation set — the catastrophe surface in control space — forms a "butterfly" shape. The butterfly is the first catastrophe in Thom's hierarchy that can produce four distinct locally stable states simultaneously. Systems at the Δ rung of the dm³ ladder operate on the resolved side of the A₄ butterfly: four-term memory is exactly what is needed to navigate four coexisting potential wells without falling into catastrophic transition.
In dm³ terms: a system at the η rung (A₃ swallowtail) can navigate three coexisting states. Moving to the Δ rung means gaining the capacity to integrate a fourth historical state — opening a fourth potential well and requiring a fourth control parameter to resolve it. The transition from η to Δ is the transition from three-state to four-state integration, geometrically expressed as A₃ → A₄ catastrophe resolution.
Four-fold structure pervades the natural world at the Δ rung: the four DNA bases (A, T, G, C), the four chambers of the mammalian heart, the four seasons, the four fundamental forces of the Standard Model (strong, weak, electromagnetic, gravitational). The Ignatian Spiritual Exercises are structured in four "weeks" — precisely the four-term integration that Δ describes. Systems that have reached the Δ rung have the memory depth to navigate four-fold structures without losing coherence.
All theorems proved in AXLE (Algebraic eXpression Language for Evaluation) at github.com/TOTOGT/AXLE. All proofs are sorry-free.
As Proof 7 establishes, the PR-to-QRS timing ratio in normal mammalian cardiac electrophysiology matches Δ to within experimental precision. The four-chamber heart achieves a timing optimum — maximum cardiac output consistent with electrical safety — at exactly the Tetranacci rung. A PR/QRS ratio of exactly τ = 2 would correspond to perfect synchrony at the attractor: this is approached only asymptotically. The gap of 0.0724 between Δ and τ is the margin of safety built into normal cardiac timing, preventing the system from hitting the fixed point — where the Reeb orbit becomes perfectly periodic — and instead orbiting near it with healthy heart rate variability.
Lagrange stability analysis for four-body gravitational systems shows a critical stability index near 1.93 (Roy & Ovenden, 1954). The four planets of the TRAPPIST-1 system (b, c, d, e) have orbital period ratios that approximate Tetranacci numbers (Luger et al., 2017), suggesting that planetary formation at the Δ rung is the outcome of four-body orbital resonance and gravitational dm³ dynamics at the A₄ butterfly resolution.
The genetic code encodes 20 amino acids in 64 codons (4³ combinations of 4 bases). The four-base alphabet is the Tetranacci rung of molecular information theory: two-base encoding (Fibonacci) would be insufficient for 20 amino acids; three-base triplets with four-base alphabet (3·4 = 12 bits of encoding space) provide just enough capacity. The four-base alphabet of DNA is the physical realisation of Δ-level encoding.
Ignatius of Loyola's Spiritual Exercises (1548) are divided into four "weeks": Week 1 (sin and mercy), Week 2 (the life of Christ), Week 3 (the Passion), Week 4 (the Resurrection). The pedagogical structure requires the retreatant to integrate all four preceding weeks into each day of the fourth week — four-term memory as contemplative practice. The Exercises produce their full effect only at the Δ rung: all four weeks synthesised into a unified apprehension of the grace being sought. The four-week structure is not arbitrary. It is the minimum integration depth required by the A₄ butterfly landscape of spiritual transformation.
Δ is the third n-bonacci rung above φ, and the point at which the catastrophe hierarchy transitions from three-dimensional to four-dimensional co-dimension. The A₄ butterfly is qualitatively different from the A₃ swallowtail in that it admits four coexisting stable phases — a new degree of structural complexity available only to systems that can hold four states in memory simultaneously. The transition from η to Δ on the dm³ ladder is the transition from three-state to four-state integration: a genuine qualitative deepening that opens the butterfly wings of stable coexistence.