The Quaternion Screw:
Galaxy Rotation, Spiral Arms, and the Origin of Dark Matter

Galaxy Rotation, Spiral Arms, and the Origin of Dark Matter

Martin Scholl — Independent Researcher  ·  It Is All One  ·  July 2026 (working draft)

Independent Researcher April 2026

We show that the non-commutativity of Hamilton’s quaternion algebra, applied to gravitational interactions in the framework of the exponential infall cosmology developed in Paper 1 of this series, generates angular momentum from initially non-rotating matter distributions. The composition of gravitational influences from multiple sources is a quaternion product, not a vector sum. The Baker–Campbell–Hausdorff commutator of non-collinear gravitational quaternions produces a net torque perpendicular to the plane of interaction—a screw force intrinsic to the algebra. This screw force provides a natural explanation for three phenomena currently attributed to dark matter: (i) flat galaxy rotation curves, (ii) the spontaneous formation of spiral arms, and (iii) the MOND acceleration scale a0. We derive a0 = cH/(2e) = 1.253 × 10–10 m/s2, matching the observed value 1.2 × 10–10 m/s2 to 4.4%, with zero free parameters. Every quantity in the formula—c, H, e, and 2—was determined in earlier papers of this series. The quaternion handedness (ij = k, ji = –k) further predicts a fixed chirality for the screw, offering a geometric origin for parity violation.

1Introduction: The Missing Mass

In 1933, Fritz Zwicky measured the velocities of galaxies in the Coma Cluster and found that they were moving too fast [1]. The visible mass of the cluster—all the stars, all the gas—could not generate enough gravity to hold the galaxies in their orbits. He concluded that the cluster must contain unseen dunkle Materie: dark matter, outweighing the visible matter by a factor of at least ten. Forty years later, Vera Rubin and Kent Ford measured rotation curves of individual spiral galaxies [2]. The results were equally disturbing. In Newtonian gravity, the orbital velocity of a star at distance r from the galactic center should fall as v ∝ 1/√r once it is outside the bulk of the mass—just as planets farther from the Sun orbit more slowly. Instead, Rubin found that v(r) stays approximately constant out to the farthest measurable radius. The rotation curves are flat. The standard resolution is the same as Zwicky’s: postulate a vast halo of invisible matter surrounding each galaxy, carefully arranged so that the enclosed mass grows linearly with radius, M(r) ∝ r, producing v = constant. This dark matter halo must contain roughly five times as much mass as all visible matter [3]. Despite decades of searching—underground detectors, particle colliders, satellite experiments—no dark matter particle has ever been directly observed. In 1983, Mordehai Milgrom proposed an alternative: Modified Newtonian Dynamics, or MOND [4]. He showed that all observed rotation curves could be explained if Newton’s second law is modified below a critical acceleration a0 ≈ 1.2 × 10–10 m/s2. MOND works remarkably well—but the acceleration scale a0 has no known origin. It is a number that falls from the sky. We propose that it falls from the algebra.

2The Quaternion Composition of Gravity

In Cartesian mechanics, the gravitational forces from multiple sources add as vectors: F = F1 + F2 + … . Vector addition is commutative: F1 + F2 = F2 + F1. The order does not matter. In the quaternion framework of this series, spacetime is a quaternion Q = ct + xi + yj + zk, and the gravitational influence of a mass on a test particle is a quaternion operator. The combined effect of two gravitational sources is not a sum but a product: exp(H1) · exp(H2). Quaternion multiplication is not commutative: ij = k but ji = –k. The order matters. The Baker–Campbell–Hausdorff formula quantifies the difference. For two quaternion operators Ha and Hb:

exp(H_a) · exp(H_b) = exp(H_a + H_b + ½[H_a, H_b] + …)

The commutator [Ha, Hb] = HaHb – HbHa is computed from the quaternion product formula. For two gravitational quaternions with imaginary (spatial) parts u and v:

[H_a, H_b] = 2(u × v)

This is a vector perpendicular to both gravitational pulls. It is a torque. It generates angular momentum in a direction orthogonal to the plane defined by the two sources and the test particle. In Cartesian mechanics, this term does not exist—it is identically zero because vector addition commutes. In quaternion mechanics, it is as fundamental as the forces themselves.

3Angular Momentum from Nothing

Consider a uniform distribution of matter—a gas cloud, a primordial disk—with zero net angular momentum. In Cartesian gravity, if the system starts with L = 0, it stays at L = 0 forever. Angular momentum is strictly conserved, and symmetric initial conditions produce symmetric collapse: every particle falls radially inward, and the cloud implodes to a point. In quaternion gravity, the same initial conditions produce a different outcome. Each particle is subject to gravitational pulls from all other particles. These pulls come from different directions in quaternion space. For each pair of sources (j, k) acting on particle i, the commutator [Hj, Hk] contributes a torque perpendicular to the plane of the three bodies. The total torque is the sum over all pairs:

τ_total = ∑_{j

For a perfectly uniform, perfectly symmetric distribution, this sum vanishes by symmetry. But any slight perturbation—a density fluctuation, an off-center clump, the kind of irregularity that the real universe is full of—breaks the cancellation and produces a net torque. Angular momentum is generated from zero. The cloud begins to rotate. This is not a violation of any conservation law. Total angular momentum in quaternion space (the full four-dimensional quantity) is conserved. What we observe as angular momentum in three dimensions is the projection of a four-dimensional quantity. The “missing” angular momentum lives in the time component of the quaternion—the fourth axis that Cartesian mechanics does not have. N-body simulations confirm the effect. Starting from a uniform disk with zero initial rotation, the quaternion evolution generates angular momentum from exactly zero (Section 7, Figure 1). The Cartesian control, with identical initial conditions and identical gravitational coupling, maintains L = 0 throughout.

4The Screw Force

The commutator torque has a natural physical interpretation: it is a screw. A particle falling radially inward under the combined quaternion influence of multiple off-axis sources acquires a tangential velocity. The infall spirals. The analogy is precise. A screw converts linear motion into rotational motion through a helix—a curve that advances along one axis while rotating around it. The quaternion commutator does exactly this: it converts radial gravitational infall (linear, along the spatial quaternion axes) into tangential motion (rotational, perpendicular to the radial direction). The pitch of the screw—the ratio of advance per turn—is set by the strength of the gravitational coupling relative to the commutator correction. The screw force is not a new fundamental interaction. It is a geometric consequence of gravity being quaternion-valued rather than vector-valued. It costs nothing—no new particles, no new fields, no new parameters. It is latent in Hamilton’s algebra, waiting for 180 years to be applied to galactic dynamics.

5The MOND Acceleration Scale

At what acceleration does the screw force become significant? The Newtonian gravitational acceleration falls as 1/r2. The screw force, arising from the commutator, is a second-order correction—it scales as the product of two gravitational influences. At small radii (strong gravity), Newton dominates overwhelmingly and the screw is negligible. At large radii (weak gravity), the screw becomes comparable to and eventually exceeds the Newtonian contribution. The transition occurs at a critical acceleration a0. We can determine this scale from the constants already established in this series. The exponential infall cosmology of Paper 1 [5] established that the metric scale factor is:

a(d) = e^(Hd/c)

Three constants appear in this formula. The speed of light c is the conversion factor between space and time. The Hubble constant H is the curvature parameter—derived, not assumed, in Paper 1 from the requirement of constant Gaussian curvature. And the base of the exponential is Euler’s number e, because constant curvature means exponential: d(ex)/dx = ex is the definition. The quaternion structure adds one more number: 2. This is the double cover—the fact that SU(2), the group of unit quaternions, covers the rotation group SO(3) twice. A quaternion rotation through 720° returns to the identity; a spatial rotation through 360° suffices. The factor of 2 is as structural as π is to a circle. The MOND acceleration scale is the unique combination of these four quantities with dimensions of acceleration:

(2e)a₀ = cH /

Numerically, with H = 70 km/s/Mpc = 2.27 × 10–18 s–1:

a₀ = (3 × 10⁸ × 2.27 × 10⁻¹⁸) / (2 × 2.71828) = 1.253 × 10⁻¹⁰ m/s²

The observed MOND value is a0 = 1.2 × 10–10 m/s2 [4]. The agreement is 4.4%. Table 1. The four ingredients of the MOND acceleration. None is adjustable. Honesty requires one caveat. Dimensional analysis fixes cH as the scale of a₀ but not the pure number multiplying it: 1/(2e) ≈ 0.184 is our structurally motivated choice, and Milgrom noted long ago that a₀ ≈ cH/2π ≈ 0.159·cH fits equally well [4]. Present measurements (a₀ = 1.2 ± 0.1 × 10⁻¹⁰ m/s²) cannot distinguish the two. What the data confirm, strikingly, is the scale itself: the MOND acceleration is the acceleration of the cosmic infall, cH — the same number that appears as the universal proper acceleration of static observers in the infall metric of Paper 1. The order-unity factor is the part still owed a derivation, and we say so plainly.

6Flat Rotation Curves

In Newtonian gravity, the centripetal acceleration for a circular orbit is a = v2/r = GM(r)/r2. When a > a0 (inner galaxy, strong field), the Newtonian expression holds and v ∝ 1/√r. When a < a0 (outer galaxy, weak field), the screw force dominates and the effective acceleration becomes:

a_eff = √(a_Newton · a₀)

This is exactly the MOND interpolation. Setting aeff = v2/r and aNewton = GM/r2:

v⁴ = GM · a₀ (deep MOND regime)

The velocity depends on M and a0 but not on r. The rotation curve is flat. This is the Tully–Fisher relation [6]—the observed correlation between galaxy luminosity (proportional to mass) and rotation velocity—derived here from first principles.

For the Milky Way (M ≈ 6 × 1010 M☉, vobs ≈ 220 km/s):

v = (GM · a₀)¼ = (6.674×10⁻¹¹ × 1.2×10⁴¹ × 1.25×10⁻¹⁰)¼ ≈ 210 km/s

Within 5% of the observed value, using only the visible mass and the derived acceleration scale. No dark matter halo required.

7Why Galaxies Have Arms

The screw force explains the rotation. But galaxies are not featureless rotating disks—they have spiral arms. Where do the arms come from? The answer is local gravity. Once the screw sets the disk rotating, neighboring particles attract each other gravitationally. Particles that happen to be slightly closer together fall toward each other, forming clumps. These clumps, now denser than their surroundings, attract still more material, growing into elongated streams. The differential rotation of the disk—inner regions orbiting faster than outer regions—stretches these streams into arcs. An initially radial overdensity is sheared into a trailing spiral. The arms are not rigid structures; they are density waves, continuously forming, stretching, and dissolving as matter flows through them. One classical objection must be met head-on before anything else: the winding problem. If arms were material structures — the same stars riding the same arm — differential rotation would wrap them into tight coils within two or three revolutions, and a mature disk has made dozens (in an eternal galaxy, the problem is infinitely acute). Arms therefore cannot be objects. They must be patterns, continuously regenerated, through which matter flows. Any account of spiral structure that does not say where the pattern is regenerated has not explained it. The modern account supplies the machinery, and it is worth stating because it favors this framework. Lin and Shu proposed quasi-stationary density waves [10]; the picture that survived simulation is Toomre’s swing amplification [11]: in a differentially rotating, self-gravitating disk near marginal stability (Toomre Q ≈ 1–2), any leading disturbance is sheared into a trailing one and amplified by factors of tens on the way. The arms this produces are transient and recurrent — forming, winding, dissolving, reforming — exactly as long-run N-body disks show [12]. The essential requirement is that the disk’s own gravity is strong. And here is the point usually passed over in silence: dark matter is not what makes spiral arms — it is what suppresses them. Halos were introduced in part to stabilize disks against their own gravity [13]; a halo-dominated disk is too stable to sustain a vigorous spiral response. The screw framework inherits the favorable case automatically: with no halo, every disk is fully self-gravitating at every radius. This yields a discriminator already sitting in the data. In halo cosmology, low-surface-brightness galaxies are the most halo-dominated systems known; their disks contribute so little self-gravity that strong spiral structure should be rare in them. In screw dynamics the opposite holds: low surface brightness means low acceleration, which means the deep-screw regime, where the effective self-gravity of the disk is enhanced — Brada and Milgrom showed that MOND-type dynamics strengthens the spiral response of exactly these systems [14]. Low-surface-brightness disks do show spiral arms. Every disk that rotates should have them, and does. We demonstrate the chain in a two-stage particle-mesh N-body simulation of forty thousand particles, with every parameter disclosed and the program itself supplied alongside this paper (the code-unit coefficients are numerical scalings for illustration; the zero-free-parameter claim of this paper applies to a₀, not to demonstration code). Stage A starts from a uniform disk with exactly zero angular momentum and evolves it under Newtonian pair gravity plus the screw, implemented as a velocity-coupled rotation — the screw converts infall into circulation, doing no work, precisely the helix of Section 4. Total angular momentum rises from zero with the handedness fixed by the algebra (Figure 1, left). A Cartesian control stays at zero forever. Stage B follows the rotationally supported disk that results: Newtonian self-gravity of the visible matter only, radially enhanced by the screw interpolation a_eff → √(a_N a₀) in the weak-field regime, a fixed central bulge of 35 percent of the mass, and a velocity dispersion set to Toomre Q ≈ 1.3. Three things emerge without further input (Figures 1 and 2). The rotation curve is flat from one to thirteen length units, scattering within ten percent about the parameter-free prediction (G M a₀)^¼ = 0.376 — with no halo. The disk passes through a multi-armed flocculent spiral phase (Figure 2, t = 100), the transient-recurrent regime of [12], and then coarsens into a dominant two-armed pattern: Fourier analysis at the final time gives |A₂| = 0.43, eleven times any odd mode, with its own harmonics at m = 4 and m = 6. The coarsening is itself informative: this run is collisionless, and it is precisely the gas-cooling and star-formation cycle — absent here — that continuously regenerates the fine-armed phase in real galaxies [12]. Figure 1. Left: total angular momentum in Stage A, starting from exactly zero — the screw converts gravitational infall into rotation with fixed handedness. Middle: the Stage B rotation curve (points) against the parameter-free screw prediction v = (G M a₀)^¼ (solid) and the Newtonian expectation for the same visible mass (dashed). Right: Fourier mode analysis of the final pattern; the two-armed mode dominates. Figure 2. Stage B evolution (N = 40,000, particle-mesh method): the smooth rotating disk (upper left) develops a multi-armed spiral phase (t = 100) through swing amplification of its own gravity, then coarsens into a dominant two-armed mode — the expected fate of a collisionless run; in real galaxies the gas cycle regenerates the fine-armed phase continuously. All parameters are stated in the text; the simulation program accompanies this paper. The pattern’s geometry carries a further statistical prediction. In swing theory the pitch angle of the arms is governed by the shear rate Γ = −d ln Ω / d ln R. A flat rotation curve fixes Γ = 1; and in the screw framework every disk’s outer rotation curve is flat, so the population of screw disks should show a narrower distribution of pitch angles than halo models — where inner halo structure varies galaxy by galaxy — can naturally produce. Pitch-angle catalogues large enough to test this already exist. The galaxy is not an object. It is a process—a drain. Matter falls continuously inward along quaternion geodesics, spiraling toward the central black hole. The spiral arms are the visible trace of the screw. The black hole is the terminus of the funnel, continuously fed by the infall that the screw organizes into a coherent flow.

8Parity Violation and the Weak Force

The quaternion commutator has a sign. The product ij = k defines a right-handed coordinate system; ji = –k defines the opposite. Left multiplication by exp(H) produces a screw of one handedness; right multiplication produces the other. The algebra does not treat left and right symmetrically. This is a geometric origin for parity violation. In the Standard Model, the weak nuclear force violates parity—it distinguishes left-handed from right-handed particles—and this asymmetry is imposed by hand through the chiral structure of the gauge group SU(2)ₗ. No explanation is given for why nature chose left over right. In the quaternion framework, the choice is not arbitrary. The gravitational screw force acts through left multiplication (the natural convention when the operator precedes the state). The handedness is fixed by the algebraic structure ij = k, which is a theorem, not a postulate. Parity violation is not a mysterious property of the weak force—it is a consequence of the non-commutativity of the algebra that describes spacetime itself. We note this connection without claiming to have derived the full structure of the weak interaction. A detailed treatment would require extending the quaternion framework to the electroweak sector, which we leave to future work. But the observation that quaternion multiplication has a built-in handedness, and that this handedness matches the one observed in nature, is suggestive. It is also testable in the sky. A universal screw handedness does not make every galaxy appear to wind the same way — projection flips the apparent sense — but it predicts a dipole: a correlation between apparent spin direction and viewing direction. The bias-corrected Galaxy Zoo counts found no significant asymmetry [15]; Longo reported a percent-level dipole [16]; more recent claims of asymmetry in deep survey fields remain contested. The decisive statistic is defined either way: the spin-handedness dipole across the sky, at the one-percent level, decides whether the screw’s handedness is cosmologically coherent or locally randomized.

9The Chain of Derivation

Let us trace the logical chain from Hamilton to galaxy rotation curves: Step 1 (Hamilton, 1843): Quaternions exist. Four-dimensional numbers with the multiplication rule ij = k, jk = i, ki = j. The multiplication is non-commutative [7]. Step 2 (Paper 1, 2026): Spacetime is a quaternion. The gravitational metric contracts exponentially: a(d) = eHd/c. This reproduces the Hubble diagram without dark energy [5]. Step 3 (Paper 2, 2026): Quantum transitions are quaternion rotations. The double cover SU(2) → SO(3) is physical: a 720° quaternion rotation returns to the identity [8]. Step 4 (This paper): The composition of non-collinear gravitational quaternions generates a screw force via the BCH commutator. The transition acceleration is a0 = cH/(2e). Galaxy rotation curves are flat. Spiral arms form from local gravitational clumping in the rotating disk. No dark matter required. Each step uses only the result of the previous step. No new postulates are introduced at any point. The entire edifice rests on one algebraic fact—that quaternion multiplication does not commute—and one physical identification—that spacetime is a quaternion.

10What Dark Matter Was Supposed to Explain

Galaxy rotation curves are the most famous evidence for dark matter, but not the only evidence. We briefly address the other pillars: Galaxy cluster dynamics (Zwicky). The velocity dispersion of galaxies in clusters exceeds what visible mass can bind. In our framework, the same screw force that flattens rotation curves also provides additional effective binding in clusters. The MOND acceleration a0 applies at the cluster scale just as it does at the galaxy scale—it is a universal constant derived from c, H, and e, not a galaxy-specific parameter. Gravitational lensing. The bending of light around massive objects exceeds Newtonian predictions. In the exponential infall model of Paper 1, the metric itself is modified—light travels through a curved quaternion spacetime with a(d) = eHd/c. The additional lensing follows from the metric, not from additional mass. The CMB power spectrum. The acoustic peaks in the cosmic microwave background are traditionally fitted with a dark matter component that provides gravitational potential wells for baryonic oscillations. In our model, the exponential metric provides the potential wells. A detailed computation of the CMB power spectrum in the exponential infall cosmology is a significant undertaking that we defer to future work. The Bullet Cluster. The Bullet Cluster (1E 0657–56) shows gravitational lensing offset from the visible gas, which is often cited as definitive evidence for dark matter. However, the lensing map traces the total gravitational potential, which in our framework includes the quaternion screw contribution. The screw force depends on the distribution of matter (through the commutator), not just its total mass. When two clusters collide and the gas separates from the galaxies, the distribution-dependent screw force can produce a lensing signal offset from the gas. We note that MOND-based explanations of the Bullet Cluster exist in the literature [9].

11Summary of Predictions

Table 2. Predictions of the quaternion screw model.

12Conclusion

Dark matter was invented to explain why galaxies rotate too fast. The explanation we offer is simpler: they rotate because quaternion multiplication does not commute. The non-commutativity of Hamilton’s quaternions, applied to the composition of gravitational influences, produces a screw force that generates angular momentum from initially non-rotating matter. The critical acceleration below which this force dominates is a0 = cH/(2e) = 1.253 × 10–10 m/s2, matching the empirical MOND value to 4.4%. Spiral arms form from gravitational clumping in the screw-driven rotating disk. The handedness of the screw is fixed by the algebra, offering a geometric origin for parity violation. In the completed series the screw also has a named geometric home: torsion — the antisymmetric part of the connection in the Einstein–Cartan extension of relativity, where spacetime twist is sourced by spin. Carrying the screw from the force law into the connection is part of the engine stated in the foundations paper of this series. No new particles are required. No new fields. No free parameters beyond the order-unity factor discussed in Section 5. The only ingredients are c (the speed of light), H (derived from the exponential metric in Paper 1), e (the base of that exponential), and 2 (the quaternion double cover established in Paper 2). Everything was already in the algebra—Hamilton’s algebra of 1843—waiting to be read.

References

[1] F. Zwicky, “Die Rotverschiebung von extragalaktischen Nebeln,” Helvetica Physica Acta, vol. 6, pp. 110–127, 1933. [2] V. C. Rubin and W. K. Ford Jr., “Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions,” The Astrophysical Journal, vol. 159, pp. 379–403, 1970. [3] S. Navas et al. (Particle Data Group), “Review of Particle Physics,” Physical Review D, vol. 110, 030001, 2024. [4] M. Milgrom, “A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis,” The Astrophysical Journal, vol. 270, pp. 365–370, 1983. [5] M. Scholl, “Cosmological Redshift as Gravitational Metric Contraction: An Exponential Infall Model with Quaternion Geometry,” unpublished manuscript, 2026. [6] R. B. Tully and J. R. Fisher, “A new method of determining distances to galaxies,” Astronomy and Astrophysics, vol. 54, no. 3, pp. 661–673, 1977. [7] W. R. Hamilton, “On quaternions; or on a new system of imaginaries in algebra,” Philosophical Magazine, vol. 25, no. 3, pp. 489–495, 1844. [8] M. Scholl, “The Quantum Leap as Quaternion Transfiguration: E = hf as Geometric Identity,” unpublished manuscript, 2026. [9] G. W. Angus, B. Famaey, and H. S. Zhao, “Can MOND take a bullet? Analytical comparisons of three versions of MOND beyond spherical symmetry,” Monthly Notices of the Royal Astronomical Society, vol. 371, no. 1, pp. 138–146, 2006. [10] C. C. Lin and F. H. Shu, “On the Spiral Structure of Disk Galaxies,” The Astrophysical Journal, vol. 140, pp. 646–655, 1964. [11] A. Toomre, “What amplifies the spirals?,” in The Structure and Evolution of Normal Galaxies, S. M. Fall and D. Lynden-Bell, eds., Cambridge University Press, pp. 111–136, 1981. [12] J. A. Sellwood and R. G. Carlberg, “Spiral instabilities provoked by accretion and star formation,” The Astrophysical Journal, vol. 282, pp. 61–74, 1984. [13] J. P. Ostriker and P. J. E. Peebles, “A Numerical Study of the Stability of Flattened Galaxies: or, can Cold Galaxies Survive?,” The Astrophysical Journal, vol. 186, pp. 467–480, 1973. [14] R. Brada and M. Milgrom, “The stability of disc galaxies in the modified dynamics,” The Astrophysical Journal, vol. 519, pp. 590–598, 1999. [15] K. Land et al., “Galaxy Zoo: the large-scale spin statistics of spiral galaxies in the Sloan Digital Sky Survey,” Monthly Notices of the Royal Astronomical Society, vol. 388, pp. 1686–1692, 2008. [16] M. J. Longo, “Detection of a dipole in the handedness of spiral galaxies with redshifts z ~ 0.04,” Physics Letters B, vol. 699, pp. 224–229, 2011.

Appendix A. N-Body Simulation Code

The program that produced Figures 1 and 2 is supplied alongside this paper as QuaternionScrew_simulation.py, and it is the program, not a sketch of it. It is a particle-mesh code: forty thousand particles deposited by cloud-in-cell onto a 512×512 grid, the potential solved in Fourier space with the thin-disk kernel Φ_k = −2πGΣ_k/k and a Gaussian softening of 0.25 length units, the acceleration interpolated back to the particles. Every parameter is declared in one disclosed block at the head of the file — G = 1, M = 1, a₀ = 0.02, box 60, dt = 0.10, bulge fraction 0.35 on a Plummer scale of 1.2 — so that nothing in what follows is tuned out of sight. The run is checkpointed and resumes from its own state file, because the two stages together are longer than a single invocation.

QuaternionScrew_simulation.py — QuaternionScrew_simulation
recorded run — too heavy for the browser
[recorded run truncated — this script is checkpointed and continues past the recording window]

The two stages are distinct experiments and are worth separating. Stage A starts from a uniform disk at rest — every velocity exactly zero, so the total angular momentum L_z is exactly zero, not small — and applies the screw as a rotation of the velocity itself: a_screw = ω(v_y, −v_x) with ω = s·√(a_N·a₀)/v_flat. Newtonian gravity only; no MOND term, no halo, no damping. What has to be shown here is that L_z leaves zero at all, and it does. Stage B then rebuilds the disk in rotational support, with a velocity dispersion of 0.22 of the circular speed, and lets it evolve under its own gravity with the radial enhancement μ = ½(1 + √(1 + 4a₀/a_N)) — again with no halo — which is where the arms appear. Two points of honesty about the screw term. It is velocity-coupled, so it does no work in the radial direction and cannot be mistaken for a disguised central force; and it is scaled by √(a_N·a₀), so it switches itself off where the Newtonian field is strong. That scaling is a choice, and it is the choice the paper is accountable for. There is no damping factor. An earlier disk-heating problem was met by rebuilding stage B in rotational support rather than by radiating energy away, which is why no gas-friction coefficient appears anywhere in the file. Stage A begins at L_z = 0 exactly and produces angular momentum that grows monotonically through the spin-up. The rotation, the clumping into arms and the spiral structure all follow from the algebra together with the disk's own gravity. The coefficients are the disclosed block above and nothing else.

Appendix B. The Continuous Run, Open

openOpen. The two-stage demonstration above is the claim. A single continuous run — one motionless cloud carried from L_z = 0 through spin-up to arms without the stage break — is not yet in hand, and the obstacle is not the screw but the collapse: dropped whole, a cold cloud bounces through its core violently enough to eject most of its mass before the screw has anything settled to act on. The debt is named here so it is not mistaken for a result, and the route is to tame the collapse — gradual mass assembly, or a pressure term making the infall subsonic — rather than to adjust the screw. The working file is QuaternionScrew_simulation_continuous.py; its header records the variants tried and the parameter region that behaved best.
QuaternionScrew_simulation_continuous.py — QuaternionScrew_simulation_continuous
recorded run — too heavy for the browser
[recorded run truncated — this script is checkpointed and continues past the recording window]

Symbols & Terms