The dm³ operator G = U ∘ F ∘ K ∘ C has three canonical invariants: the period T* = 2π, the transverse exponent μmax = −2, and the stability radius ε₀ = 1/3. Each chapter below shows one of these invariants — or one step of the recurrence ladder from Fibonacci (subcritical) to Hexabonacci (the fixed point of the sequence of sequences) — as a dm³ generative transition in its own domain. The threshold c* = 3 divides the ladder into three regimes; see the Criticality Bridge section between φ and μ.
Every dm³ system has a canonical period T*. In the toy model — and in every domain where the framework has been instantiated — T* = 2π. This chapter explains why 2π is forced by the contact geometry, not chosen by convention.
The dm³ contact manifold is M = S¹ × ℝ with contact form α = dz − p dq. The Reeb vector field R is defined by ι(R)dα = 0 and ι(R)α = 1. On M, the Reeb flow is:
The dm³ limit cycle Γ = {ρ = 1} is a Reeb orbit. Its period in the contact-normal-form coordinates is the period of the Reeb flow — which is 2π exactly. No other value is compatible with the contact structure being non-degenerate.
The canvas shows the dm³ limit cycle Γ as a closed orbit in (ρ, θ, z) space, projected to the (θ, ρ) plane. The coloured spiral is the transient — an orbit starting away from Γ converging exponentially. One full revolution = T* = 2π radians.
The Fibonacci sequence F(n) = F(n−1) + F(n−2) has characteristic polynomial x² − x − 1, whose dominant root is φ = (1 + √5)/2 ≈ 1.6180. This is the subcritical case in the dm³ criticality ladder: the curvature coefficient c = φ < c* = 3, and the fold operator F never activates.
The golden ratio appears everywhere in natural systems — phyllotaxis, mollusk shells, the Penrose tiling — but in the dm³ framework these are all pre-fold configurations. They have reached the curvature constraint K but have not activated the fold F. The spiral is visible; the transition has not happened.
The Fibonacci spiral in nature (nautilus, sunflower, galaxy arms) represents the compression C and curvature K phases of the dm³ cycle — the system has been compressed and curved to κ* — but the fold F has not fired. These are systems living permanently in the approach phase. Beautiful, stable, but incomplete. The full cycle needs c = 3.
The dm³ curvature coefficient c controls whether the fold operator F activates. It is the coefficient in the potential V(q) = q³ − cq. The criticality threshold c* = 3 is the unique value at which V has a double root at q = 1 — the Whitney A₁ fold. Every n-bonacci constant is associated with a curvature regime relative to c* = 3.
The seven Greek-letter constants in this file are not chosen for aesthetic reasons. They are the spectral signatures of the seven positions on the n-bonacci recurrence ladder, each sitting in one of the three regimes:
Drag c across the threshold c* = 3. Watch the fold point q* cross q = 1 (gold line). Below 3: fold left of the integer. At 3: fold lands exactly at 1. Above 3: fold overshoots.
The dm³ canonical triple is (T*, μ_max, τ) = (2π, −2, 2). The value μ_max = −2 is the transverse Lyapunov exponent — the rate at which deviations from the limit cycle Γ decay. This chapter shows why −2 is the only value compatible with the Whitney A₁ fold at q = 1.
The Tribonacci sequence T(n) = T(n−1) + T(n−2) + T(n−3) has characteristic polynomial x³ − x² − x − 1 = 0 with dominant root η ≈ 1.8393. In the dm³ framework, η is the critical constant: the Collatz map has c = 3, and the amplitude envelope is η⁻ᵏ — each level of the 12-phase orbit is weighted by the inverse Tribonacci power. The strict antitonicity of this sequence is machine-verified in Lean 4 (TribonacciMeasure.lean, sorry_count: 0).
The Tribonacci constant η sits exactly at c* = 3 on the criticality ladder — not below (like φ), not above (like the Tetranacci constant ≈ 1.927). The characteristic polynomial x³ − x² − x − 1 factors the Collatz curvature operator K at the critical threshold. This is why the Collatz map with c = 3 is the unique convergent: η is the only n-bonacci constant whose associated polynomial produces a double root at q = 1 in the dm³ potential.
The Tetranacci sequence T(n) = T(n−1) + T(n−2) + T(n−3) + T(n−4) has characteristic polynomial x⁴ − x³ − x² − x − 1 = 0, dominant root Δ ≈ 1.9275. This is the first step into the supercritical regime of the recurrence ladder — meaning depth n = 4 > 3, one step beyond Tribonacci criticality. Note carefully: the curvature coefficient c = Δ ≈ 1.928 is still less than c* = 3 in the potential V(q) = q³ − cq sense; the supercriticality here is in the recurrence depth (n > 3 summands), not in the fold-potential sense. The two uses of "supercritical" are distinct and both appear in the literature.
Pentanacci: T(n) = sum of previous 5. Characteristic polynomial x⁵ − x⁴ − x³ − x² − x − 1 = 0, dominant root Σ ≈ 1.9659. Four decaying complex roots. The entropic cost of enforcing integer coherence is near-maximum — the system is close to the limit where the integer lattice can no longer serve as the attractor boundary.
The Hexabonacci sequence sums the previous 6 terms. Its dominant root Ω ≈ 1.9837 → 2 as depth increases. This is the fixed point of the recurrence ladder: the limit of all n-bonacci dominant roots as n → ∞ is exactly 2. And 2 is τ — the dm³ embodiment threshold, the noise tolerance denominator, and the canonical invariant τ = 2 in the triple (T*, μ_max, τ) = (2π, −2, 2).