Helical Attractors on
Contact 3-Manifolds:
A Toy ODE Study

The Contact 3-Manifold Setting

A contact structure on a smooth odd-dimensional manifold M²ⁿ⁺¹ is a maximally non-integrable distribution ξ = ker(α), where α is a 1-form satisfying the contact condition α ∧ (dα)ⁿ ≠ 0 pointwise. In dimension 3 this reduces to α ∧ dα ≠ 0. The condition means that ξ is as far from being tangent to a foliation as possible: no 2-dimensional surface can be simultaneously tangent to all the contact planes.

By the Darboux theorem for contact geometry, every point on a contact 3-manifold has a neighbourhood in which α takes the standard form dz − y dx (or, in cylindrical coordinates, dz − r² dθ). Our system lives entirely in this normal form.

The kernel of α at a point (r, θ, z) is spanned by the vectors ∂/∂r and r²∂/∂z + ∂/∂θ. The first is radial; the second is a combination of rotation and vertical translation that sweeps out the contact plane. A curve is tangent to ξ (i.e. Legendrian) iff it annihilates α: ż = r² θ̇ along the curve.

The attractor we construct is not a Reeb orbit but a curve on the unit cylinder r = 1 that follows the Reeb direction: (r(t), θ(t), z(t)) = (1, t, z₀ + t). This is a helix — the trace of a uniform circular motion lifted by the Reeb flow. The contact condition forces any limit set of our dissipative system that lives on r = 1 to be precisely of this form.

1.1 Why contact geometry for this problem

The standard Hopf normal form ṙ = r(1−r²), θ̇ = 1 in the plane has a limit cycle at r = 1. Adding a third dimension with ż = r² — the simplest equation compatible with α ∧ dα ≠ 0 — lifts this cycle to a helix without changing the radial dynamics. The contact structure is the geometric reason the lifted attractor is a helix rather than a torus or a cylinder: the non-integrability of ξ prevents the trajectories from closing up in a compact surface.

The coupling term ε(r−1)e^{−z} is a dissipative correction that decays exponentially as z grows. It is the structural perturbation that breaks the exact Hopf symmetry and produces the basin asymmetry studied in §6. At ε = 0 the system reduces to the decoupled Hopf + Reeb system; at ε = 2 the coupling is strong enough to create an asymmetric basin boundary r* ≈ 0.77594.

Prerequisites. The contact 3-manifold construction is developed from first principles in Chapter 3 of this volume. The φ-subcritical approach that motivates the inner boundary r* appears in Chapter 9. The operator chain G = U∘F∘K∘C is defined in Chapter 5 (operator formalism) and realised across domains in Chapter 2.

The ODE System

We study the autonomous system on (ℝ³, α) defined by:

2.1 Decomposition of the radial equation

The radial equation decomposes into two terms:

Hopf normal form: r(1−r²) is the standard two-dimensional Hopf oscillator radial component. It vanishes at r = 0 and r = 1. For r ∈ (0,1): r(1−r²) > 0 (pulls outward toward r = 1). For r > 1: r(1−r²) < 0 (pulls inward toward r = 1). The fixed point at r = 1 is thus attracting from both sides in the absence of coupling.

Coupling correction: ε(r−1)e^{−z}. For r > 1: this term is positive, opposing the Hopf restoring force. For r < 1: this term is negative, adding to the outward pull (away from r = 1 toward r = 0). The coupling decays as e^{−z} as z grows, so its effect is transient on outer-basin trajectories where z increases monotonically.

2.2 Fixed-point structure and ε = 2

The system has three regimes in the radial direction:

At r = 0: ṙ = 0, but the Jacobian eigenvalue ∂ṙ/∂r|_{r=0} = 1 − 0 + ε·(0−1)·e^{−z}·0 = 1 (from the Hopf term). So r = 0 is an unstable equilibrium (repelling) in the radial direction.

At r = 1: ṙ = 0 (exactly). The Jacobian ∂ṙ/∂r|_{r=1} = 1 − 3r²|_{r=1} + ε·e^{−z} = −2 + ε·e^{−z}. For large z (e^{−z} ≈ 0) this is −2, the linearized convergence rate. For small z the eigenvalue may be positive, explaining the transient before exponential decay.

As r → ∞: ṙ ≈ −r³ → −∞. Trajectories cannot escape to infinity in finite time.

The value ε = 2 is special: it makes the coupling term at r = 1 equal to 2·e^{−z}·u (where u = r−1), matching the Hopf eigenvalue −2 in magnitude when z = 0. This creates the maximal asymmetry between inner and outer basins.

2.3 The helical attractor

Main Theorem and Complete Proof

3.1 The deviation variable

Set u(t) = r(t) − 1. We compute u̇ by substituting r = 1 + u into the radial equation:

Therefore the deviation equation is exactly:

This is the exact (non-approximate) equation for u = r − 1. No linearization has been performed.

3.2 z-monotonicity on the outer basin

From the z-equation: ż = r² − ε(r−1)²e^{−z}. We need r² > ε(r−1)²e^{−z}, i.e. (r/(r−1))² > ε·e^{−z}. For r > 1 the left side is always positive. The inequality holds when:

For u > 0: (1 + 1/u)² / 2 > 1/2. So the condition is satisfied for all z > log 2 (where e^{−z} < 1/2). Since z(0) ≥ log 2 and ż(0) > 0 (verified), the trajectory begins with increasing z. Suppose for contradiction that ż hits zero at some t₁ > 0 while z(t₁) > log 2 — this would require r(t₁) to satisfy the equality case, but by continuity and the strict inequality at t = 0 this cannot occur without z first decreasing below log 2. The chain: z stays ≥ z(0) ≥ log 2, so the bound holds for all t. □

3.3 The Gronwall estimate

Compute d/dt(u²) = 2u · u̇:

Since u > 0 (we are on the outer basin r > 1), the terms −6u³ − 2u⁴ are strictly negative. Dropping them gives the upper bound:

By Lemma 3.2, z(t) ≥ z(0) ≥ log 2 for all t. With ε = 2:

Therefore d/dt(u²) ≤ u²(−4 + 2) = −2u². □

3.4 Conclusion: exponential convergence

Part (i). By Proposition 3.3, d/dt(u²) ≤ −2u². Setting v(t) = u(t)², this is v̇ ≤ −2v. By Gronwall's inequality:

This establishes exponential convergence at rate at least 1 in |u|. The Lyapunov exponent of the linearized system at r = 1 is ∂ṙ/∂r|_{r=1} = 1 − 3 = −2 (as z → ∞). As z → ∞ the coupling term 2ε · u · e^{−z} → 0, and the effective equation approaches u̇ ≈ −2u. Hence |u(t)| ~ C · e^{−2t} asymptotically. The reported rate μ → −2 follows.

Part (ii). By Lemma 3.2, ż > 0. Moreover, ż = r² − ε(r−1)²e^{−z}. As u(t) → 0 exponentially (Part i), r(t) → 1 and (r(t)−1)² ≤ u₀²·e^{−2t} → 0. Hence:

for a constant C₀ = ∫₀^∞ (r(s)² − 1) ds, which is finite since |r(s) − 1| ≤ |u₀|·e^{−s}.

Part (iii). Since r(t) → 1 and θ̇(t) = 1 and ż(t) → 1, the trajectory (r(t), θ(t), z(t)) converges in C⁰ to the curve {r = 1, θ̇ = 1, ż = 1}, which is the helix Γ. □

Numerical Findings and Full Stability Table

All integrations use DOP853 with rtol = 10⁻¹⁰, atol = 10⁻¹². The decay rate μ̂ is estimated from the slope of log|r(t) − 1| over a window [t₁, t₂] with z(t₁) > 2, ensuring the coupling term has decayed. The half-life t_1/2 is defined by |r(t_1/2) − 1| = ½|r(0) − 1|.

5.1 Phase portrait (r–z projection)

5.2 Exponential decay rate convergence to μ = −2

5.3 Full numerical stability table (Table 1)

The table records the estimated Lyapunov exponent μ̂ for 12 initial conditions spanning both basins. Outer basin (r(0) > 1) entries confirm μ̂ → −2. Inner basin entries show divergence or escape. The gap between the Gronwall boundary r = 2/3 ≈ 0.667 and the numerical inner boundary r* ≈ 0.77594 is visible in rows 9–10.

The Basin Asymmetry: Inner Boundary Analysis

6.1 The coupling amplification mechanism

For r < 1, the radial coupling term ε(r−1)e^{−z} is negative (since r − 1 < 0). This opposes the Hopf restoring force r(1−r²) > 0 (which is positive for r < 1). The net radial force is:

The Hopf term is positive (outward) and the coupling term is negative (inward, away from r = 1). The coupling dominates when ε(1−r)e^{−z} > r(1−r²) = r(1−r)(1+r), i.e. when:

For small z (z ≈ 0) and ε = 2: the condition is 2 > r(1+r). This holds for r < r_threshold where r(1+r) = 2 has the positive root r_threshold = (−1 + √9)/2 = 1. So at z = 0, the coupling dominates iff r < 1 — i.e. for all inner-basin trajectories. This is why the inner basin boundary requires more careful analysis.

6.2 The z-feedback loop and escape

When ṙ < 0 at r < 1, the trajectory moves further from r = 1. Now consider ż:

For r < 1 and small z, this can be negative: if ε(1−r)² > r² · e^{z}, i.e. if √ε · (1−r) > r · e^{z/2}. When ż < 0, z decreases, which increases e^{−z}, which increases the coupling force (already inward), which decreases r further, which increases (1−r)², which makes ż more negative. This is the positive feedback loop that causes inner-basin escape.

The escape is asymptotic: z → −∞ and r → 0 in finite time for initial conditions below the separatrix.

6.3 Conjecture 2.2: closed form for r*

The Whitney A₁ Singularity at r*

The inner basin boundary r* admits a natural geometric interpretation in terms of singularity theory. The fold map F in the GTCT operator chain (§9) sends the full radial line ℝ₊ to the attractor by first projecting along stable manifolds. This map fails to be a local diffeomorphism at exactly one point — the inner basin boundary r*.

7.1 The fold map and its critical point

Define the asymptotic map Φ: ℝ₊ → {1} by Φ(r₀) = lim_{t→∞} r(t; r₀, z₀) — the radial limit of the trajectory starting at r₀. For r₀ > r*, Φ(r₀) = 1 (converges to attractor). For r₀ < r*, the limit does not exist (escape). At r₀ = r*, Φ has a singularity.

The Whitney A₁ singularity (standard fold) is characterised by: Φ is smooth near r*, Φ(r*) = r*, and the derivative dΦ/dr₀|_{r*} = 0 with d²Φ/dr₀²|_{r*} ≠ 0. In normal form, a Whitney A₁ singularity is equivalent to t ↦ t², the standard fold of a curve over a point.

7.2 Relation to the Lipschitz constant κ*

The threshold κ* = √(7/9) ≈ 0.882 appears in the global Lipschitz estimate for the operator K (Curvature) in the GTCT chain. In the context of the ODE system, κ* bounds the Lipschitz constant of the map r₀ ↦ u̇|_{t=0} near r = 1:

The value κ* = √(7/9) arises from the balance between the cubic restoring force coefficient (−3 at r = 1) and the coupling constant ε = 2 at the inner boundary. This is AXLE Issue #16.

Formal Verification in Lean 4 (AXLE)

The GTCT Lean 4 project (github.com/TOTOGT/GTCT) formalises the structural claims of this chapter in Lean 4 with Mathlib4, using zero additional axioms.

8.1 Closing AXLE Issue #12: a hand proof of Theorem 2.1

Substitute u = r − 1 (the Compress coordinate of §9.1). With ε = 2, the radial equation becomes:

and the z-equation becomes:

Step 1 — D = {u ≥ 0, z ≥ log 2} is forward-invariant. On the boundary z = log 2, ż = 1 + 2u ≥ 1 > 0, so trajectories move strictly into z > log 2. For (u,z) ∈ D, 2e^−z ≤ 1, hence φ(u,z) ≤ 1 − 2 − 3u − u² = −1 − 3u − u² < 0 for all u ≥ 0 — strictly negative even at u = 0. So u̇ = uφ ≤ 0, with u̇ = 0 only at u = 0 (the attractor Γ). By uniqueness of ODE solutions (the right-hand side is smooth, hence locally Lipschitz), the trajectory cannot cross u = 0, so D is forward-invariant. This also re-proves z_monotone below: ż ≥ 1 throughout D.

Step 2 — z grows at least linearly. In D, every term of ż = 1 + 2u + u²(1−2e^−z) is ≥ 0 and the first term equals 1, so ż ≥ 1. Integrating, z(t) ≥ z(0) + t ≥ log 2 + t for all t ≥ 0.

Step 3 — the contraction rate. From Step 2, 2e^−z(t) ≤ 2e^{−(log 2 + t)} = e^−t. Substituting into φ and dropping the non-positive terms −3u−u² (u ≥ 0):

For u(t) > 0, d/dt[ln u(t)] = u̇/u = φ(u(t),z(t)) ≤ e^−t − 2. Integrating from 0 to t:

8.2 Lean formalization status

All open obligations are honest sorry placeholders named as AXLE Issues, each representing a specific missing lemma. gronwall_outer below now states the corrected, hand-proved Theorem 2.1; the remaining sorry is the Lean transcription of Steps 1–3, for which Mathlib.Analysis.ODE.Gronwall (e.g. norm_le_gronwallBound_of_norm_deriv_right_le) supplies the needed machinery.

The GTCT Operator Chain: Complete Derivation

The dm³ framework operates via the composite operator G = U ∘ F ∘ K ∘ C applied to a state space that includes the contact manifold. We derive each operator in turn and show that the helical attractor Γ is the unique fixed point of G.

9.1 C — Compress

The Compress operator C: ℝ₊ × ℝ → ℝ × ℝ maps (r, z) to the deviation coordinates (u, w) = (r − 1, z − z₀). It centres the state space at the attractor. In terms of the linearized system, C diagonalises the dominant eigenvalues: the u-equation has eigenvalue −2 and the w-equation is driven by u².

9.2 K — Curvature

The Curvature operator K: ℝ² → ℝ computes the local Lyapunov exponent of the radial equation. At the attractor (u = 0, z → ∞): K(0, ∞) = ∂ṙ/∂r|_{r=1, z→∞} = 1 − 3 = −2 = μ. Away from the attractor, K measures the distance from the optimal convergence rate.

The threshold κ* = √(7/9) is the maximum of |K(u, z)| over the convergent region, attained at u ≈ r* − 1 ≈ −0.22406. This bounds the global Lipschitz constant of C.

9.3 F — Fold

The Fold operator F: ℝ₊ → [1, ∞) maps initial radii to their asymptotic value under the outer-basin dynamics. F(r₀) = lim_{t→∞} r(t; r₀) for r₀ > 1, and F(r₀) is undefined for r₀ < r*. The fold singularity at r* (§7) gives F its name in the operator chain: the transition from convergent to escape behaviour is a fold catastrophe.

9.4 U — Unfold

The Unfold operator U: {1} × ℝ → Γ maps the radial attractor value back to the full three-dimensional helix by restoring the θ and z components. U(1, t) = (1, t + θ₀, z₀ + t) ∈ Γ. The full operator chain G = U ∘ F ∘ K ∘ C maps any initial condition in the outer basin to a point on Γ in the limit t → ∞.

Series Context: Chapters 1–11

This chapter is Chapter 10 of Volume IV (GTCT T1) of the Principia Orthogona series. The helical attractor result builds directly on the contact geometry introduced in Chapter 3, the Hopf-type dynamics of Chapter 9, and the operator chain formalism of Chapters 5 and 8. The formal Lean 4 setting continues in Chapter 11 (CatGT). A full reading path through the eleven chapters of this volume is given below.

See also: Omega Point · Jalāl al-Dīn Rūmī — the Gallery of Mathematical Mystics chapter that treats the whirling ceremony in its own spiritual context.

Chapters 12–15 (forthcoming in this volume) extend the helical attractor result to: coupled contact manifolds (Ch 12), Wigner-crystal contact structures (Ch 13), the HVEH engineering application (Ch 14), and the Lean 4 full proof of Theorem 2.1 (Ch 15, pending AXLE Issues #12–#17).

10.1b HVEH Applied Track — Chapters 6b, 7, 8, 9

Running in parallel with the mathematical track, the HVEH Applied Track translates the contact-geometric results of this volume directly into engineering practice. The four applied chapters ground each abstract theorem in a specific New Jersey site, regulatory context, or fabrication specification. Chapter 10 (this chapter) is the mathematical backbone that all four applied chapters cite for the rotor blade geometry, convergence rate μ = −2, and basin boundary r* ≈ 0.77594.

The build specification and rotor geometry script (rotor_geometry.py) for the Second River pilot are in ch-build-2river.html. The HALO investment structure supporting deployment is in chHALO.html.

10.2 Book 3 · The Mini-Beast — Live Chapter Links

Book 3 (The Mini-Beast) is the biological instantiation volume of the series, applying the same operator chain and contact-geometric framework to circadian rhythms, neural oscillations, immune adaptation, and resonance. All chapters are live at totogt.github.io/geometry/. The chapters most directly connected to the helical-attractor results of this chapter are marked below.

10.3 Operator Chain Chapters — Live on AXLE

The n-bonacci recurrence ladder π → φ → μ → η → Δ → Σ → Ω is the formal series constant progression of the dm³ framework. Each operator chapter lives at totogt.github.io/AXLE. The helical attractor of this chapter corresponds to the μ (Lyapunov) rung of the ladder — convergence rate μ = −2 is the defining value of the μ-operator.

Discussion and Open Problems

11.1 What the toy model achieves

The system studied in this chapter is explicitly a toy ODE — its purpose is to isolate the contact-geometric mechanism for helical attractors in the simplest possible setting. The Hopf normal form r(1−r²) provides a well-understood two-dimensional attractor; the coupling ε(r−1)e^{−z} and the contact lift ż = r² provide the three-dimensional contact-geometric structure. The result — exponential convergence to a helix with rate μ = −2 — is not surprising, but the proof is clean and the basin asymmetry is a genuine structural discovery.

The chapter contributes four things: (1) a complete Gronwall proof (§3), correcting the earlier proof-sketch; (2) identification of the coupling-amplification mechanism (§6.1–6.2); (3) the Whitney A₁ interpretation of the inner boundary (§7); and (4) a concrete instantiation of the GTCT operator chain with explicit formulas for C, K, F, U (§9).

11.2 Open problems

11.3 Connection to the HVEH engineering application

The helical attractor Γ on the contact 3-manifold is the mathematical core of the HVEH rotor geometry. The blade pitch equation tan φ(r) = r (which annihilates the contact form α = dz − r²dθ along the blade centerline) follows directly from the Legendrian condition: a curve tangent to ker(α) satisfies dz = r² dθ, which integrates to z = r²θ + const, giving the helix z ∝ r²θ. The attractor Γ is the ideal blade centerline; the convergence rate μ = −2 is the radial damping that self-aligns the rotor under perturbation.

The HVEH Applied Track (§10.1b) develops this connection site by site. Chapter 6b (El Ojo) introduces the phenomenology; Chapter 7 (Map Newark) grounds r* ≈ 0.77594 in the Passaic/Hackensack confluence geometry; Chapter 8 (Harrison) develops CFD validation; Chapter 9 (Belleville) closes with end-to-end verification against the μ = −2 prediction. The fabrication specification and rotor_geometry.py script are in ch-build-2river.html.

References

[1] Grossi, P. N. (2026). Principia Orthogona, Vol. I — GOMC. G6 LLC. ISBN 979-8-9954416-2-5. doi:10.5281/zenodo.19117400
[2] Grossi, P. N. (2026). Principia Orthogona, Vol. II — TOGT. G6 LLC. ISBN 979-8-9954416-4-9. doi:10.5281/zenodo.19379473
[3] Grossi, P. N. (2026). Helical Attractors on Contact 3-Manifolds: A Toy ODE Study. Principia Orthogona, Vol. IV, Ch. 10. G6 LLC. doi:10.5281/zenodo.19379385 (Chapter DOI)
[4] AXLE v6.1 formal verification environment. totogt.github.io/AXLE · github.com/TOTOGT/GTCT
[5] Hairer, E., Nørsett, S. P., & Wanner, G. (1993). Solving Ordinary Differential Equations I. Springer. [DOP853 integrator]
[6] Geiges, H. (2008). An Introduction to Contact Topology. Cambridge University Press. [Darboux theorem, Reeb orbits]
[7] Hartman, P. (1982). Ordinary Differential Equations. 2nd ed. SIAM. [Gronwall's inequality]
[8] Whitney, H. (1955). On singularities of mappings of Euclidean spaces. Annals of Mathematics, 62(3), 374–410. [Whitney A₁ fold]
[9] Arnold, V. I. (1992). Catastrophe Theory. 3rd ed. Springer. [A₁ singularity normal form]
[10] Mathlib4 Community (2024). The Mathlib4 Library for Lean 4. leanprover-community.github.io/mathlib4_docs
[11] Vol. IV, Ch. 3 — Contact 3-Manifold. ch03.html (direct prerequisite)
[12] Vol. IV, Ch. 9 — φ · The Subcritical Approach. ch09.html (direct prerequisite)
[13] Vol. IV, Ch. 5 — OPERA SATOR · Jet Space. ch05.html (operator C, K, F, U definitions)
[14] Vol. IV, Ch. 8 — The Axiomatic Turn. ch08.html (Complete Completeness)
[15] Vol. IV, Ch. 11 — CatGT. ch11-catgt.html (continuation)
— Book 3 · The Mini-Beast (totogt.github.io/geometry/) —
[16] Book 3, Ch. 10 — Lyapunov. ../ch10-lyapunov.html (μ = −2 Lyapunov exponent in biological context; directly mirrors Theorem 2.1)
[17] Book 3, Ch. 9 — φ · Subcritical Approach. ../ch9-phi.html (φ-ladder; subcritical basin analysis; direct Book 3 prerequisite)
[18] Book 3, Ch. 4 — Neural Oscillations. ../ch4-neural.html (helical attractors in neural limit cycles; contact structure applied to neuroscience)
[19] Book 3, Ch. 8 — The Axiomatic Turn. ../ch8-axiomatic.html (Complete Completeness, biological realisation; Banach fixed-point in living systems)
[20] Book 3, Ch. 14 — AXLE · The Sorry-Free Paper. ../ch14-axle.html (Lean 4 proof environment; AXLE Issues #12–#17 are tracked here)
[21] Book 3, Ch. 15 — Entropy and the Compression Circle. ../ch15-entropy.html (thermodynamic dual of the Gronwall bound; entropy decreases along Γ)
[22] Book 3, Ch. 6 — Resonance. ../ch6-resonance.html (111 Hz resonance; g₃₃ = 33 cycles in biological oscillation)
[23] Book 3, Ch. 7 — The Crystalline Return. ../ch7-crystalline.html (Wigner/hexanacci crystal; τ = 2 embodiment threshold in crystalline contact structure)
[24] Book 3, Ch. 1 — The Cajueiro Principle. ../ch1.html (seed attractor; helical convergence in its original biological prototype)
— Operator Chain (totogt.github.io/AXLE/) —
[25] chMu-lyapunov — μ Operator Chapter. AXLE/chMu-lyapunov (μ = −2; this chapter is the ODE proof of the μ operator's defining constant)
[26] chOmega-hexabonacci — Ω · Hexabonacci → τ = 2. AXLE/chOmega-hexabonacci (limit of n-bonacci ladder; Γ is the contact-geometric Ω)
[27] chT-tubulin — τ · Tubulin = 2. AXLE/chT-tubulin (embodiment threshold τ = 2; tubulin as the biological attractor substrate)
[28] chW-wigner — W · Wigner Crystal. AXLE/chW-wigner (contact crystal at τ = 2; hexagonal lattice as frozen Reeb orbits)
[29] chPI-recurrence — π · Recurrence Ladder. AXLE/chPI-recurrence (period T* = 2π; Reeb orbit period maps to one full helical winding)
— Orthogenesis · NASA Moon Base · Lean 4 (github.com/TOTOGT/geometry) —
[30] Grossi, P. N. (2026). Orthogenesis: Formal Verification of a NASA Moon Base Architecture in Lean 4. github.com/TOTOGT/geometry. G6 Crystal (20 dm³ facts, 0 sorry), DM3Bridge.lean, NASAGaps.lean. The expand_is_UCKF_composite theorem formally proves Colony.expand = U∘F∘K∘C — the discrete analogue of Theorem 9.1 of this chapter. github.com/TOTOGT/geometry
[31] G6 Crystal — Zenodo concept deposit. doi:10.5281/zenodo.19162012 (G6 Crystal invariants (T*, μ_max, τ) = (2π, −2, 2); planetary scaling proofs)
[32] Vol. IV, Ch. 6b — El Ojo · The Mystery That Rotates. ch06b-elojo.html (HVEH Applied Track; helical vortex phenomenology; Γ as observable river attractor)
[33] Vol. IV, Ch. 7 — Map Newark · The Field and the Need. ch07-newark.html (HVEH Applied Track; Passaic/Hackensack confluence survey; r* ≈ 0.77594 as blade-radius inner margin)
[34] Vol. IV, Ch. 8 — Harrison · The Ordering Law Under Pressure. ch08-harrison.html (HVEH Applied Track; CFD validation; ε₀ = 1/3 as blade-deflection tolerance)
[35] Vol. IV, Ch. 9 — Belleville · Verification and the Ladder's End. ch09-belleville.html (HVEH Applied Track; end-to-end rotor verification against μ = −2 prediction)