What ψ is

Kepler found the planets on ellipses with the Sun at a focus. Rutherford found alpha rays bent about the nucleus along hyperbolas, Kepler’s open orbits. Bohr and Sommerfeld put the electron on Kepler’s circles and ellipses, and Pauli derived the hydrogen spectrum from the same conserved vector that fixes the orientation of a planet’s ellipse. In 2004 an electron was held on a highly eccentric Kepler orbit around its nucleus for 15 000 revolutions. The orbits of the Solar System and of the atom are one geometry, and the record carries it at both sizes:

α draws a cone

Take a source’s own frame, with its worldline as the axis, and draw the plane of an orbit with the time z = ct above it. The light cone of the source, with its worldline as the axis, is z = r, for r > R_s. Its circle at height z has radius z, and the Alpha Lock reads it directly:

Every α is unchanged under (R_s, z) ↦ (λR_s, λz), and so is the cone: it is the self-similar surface that α’s circles sweep, one form at every size (0.3a). Its generators are the stroke lines x = ct at π/4 (I-⁠2) in the far field, where the planets are and the density cone is the flat light cone of uniform density (P-⁠4). It is truncated at the fold, z = R_s, where α_ext → 1⁻: a circle, never a point and never reached (I-⁠6). Density rises toward the tip, where the source sits, and falls toward the rim (I-⁠3).

The cone and the cylinder meet in a plane

A body circles the source under an inverse-square attraction, F = K/r²; for gravity the pair law gives F = F_maxα₁α₂ = G𝓜m/r² exactly, with α₁ and α₂ the source’s and the body’s α_ext at their separation (I-⁠1). Lift each point of its orbit to the cone, at the height z = r. The angular momentum L and the Laplace–Runge–Lenz vector A = p × L − mKr̂ are conserved, and A·r = L² − mKr, so every lifted point satisfies

The lifted orbit is a plane section. The world-tube of the orbit, the cylinder the orbit sweeps out in time, meets the light cone of the source in a plane: L fixes the plane’s height on the axis, and A fixes its tilt. The construction works on any right circular cone; the light cone makes the slope of the plane the eccentricity itself, against a slope of 1 for a generator. The section is a set of events on the cone, and the body’s own worldline meets it in one event. A plane flatter than the light cone cuts a closed section, an ellipse or a circle. A plane steeper than it cuts an open one, a hyperbola, and the same plane cuts the other nappe, z = −r, in the repulsive branch: Rutherford’s orbit. Both nappes stop at the fold. The plane parallel to a generator, at π/4, cuts the parabola, and on it

The parabolic orbit moves at the speed of a pattern falling from rest far away. At perihelion every orbit runs at √((1 + e)/2) times that flow speed: below it on a closed section, above it on an open one.

Pressed flat: Kepler’s laws

Projected along the axis, the section is the conic r = ℓ/(1 + e cos θ), and the axis projects onto its focus, the fold circle onto a circle of radius R_s about it: the source sits at the focus because it sits on the axis, at the tip. The pair force is central, so L is conserved and the radius sweeps area at the rate L/(2m). The cone’s own coordinate gives the third law:

At the Earth’s distance α_ext = 1.974 × 10⁻⁸, and a circular orbit runs at c√(α_ext/2) = 29.78 km/s with a period of 365.26 days. The three laws hold exactly in model on the pair domain for the relative orbit, with 𝓜 + m in place of 𝓜 (I-⁠4, layer L1); for the Earth that shortens the year by 1.5 × 10⁻⁶ of itself and leaves 365.26 days. Pressing the cone flat keeps the leading order in α_ext, and the field equations carry the rest. For the same source and data they turn the section by

For Mercury this is 5.019 × 10⁻⁷ rad per revolution, or 42.98″ per century, the anomalous advance Le Verrier found in 1859. The flattened cone closes its ellipses, and the field equations turn them into a rosette. A stroke is no section. The flattened cone carries only the lapse, the slow pattern’s reading (0.1c), and its hyperbola taken to speed c bends by 0.88″ at the solar limb; the stroke bends by twice that, 1.75″, because it reads the optical metric. The factor of two of I-⁠3 is the extended equivalence principle read on a stroke (0.1c).

Add the Sun’s motion: one worldline, two readings

In the source’s own spacetime the body’s worldline lies on the world-tube of its orbit and winds once about the source’s worldline for every cT of advance along the axis. It is a helix on the world-tube: for the Earth, of radius 1 AU and advance one light-year per turn, standing at v/c = 9.94 × 10⁻⁵ from its axis. Projected along the system’s own time axis, the worldline is the closed orbit. That is the reading of an observer inside the system.

An observer outside the system sees the source move. Let it move at V along the axis of a circular orbit of radius a and period T. In that observer’s frame the same worldline, projected along that observer’s time axis, is a helix about the source’s track, of radius a, period γT and advance γVT per turn. Seen side-on, it is a wave of wavelength γVT. The two are one worldline read along two time axes, the system’s own reading and the bookkeeper’s (0.1c), related by a boost (0.1d). On the system’s own clocks and rulers nothing changes.

The Sun moves along the Galactic rotation at 248.5 km/s. Taken as the track, that motion points toward Galactic longitude 90°, 30.4° from the pole of the ecliptic. In the Galaxy’s frame the Earth therefore runs a helix on an oblique cylinder about the Sun’s track. The helix advances 52.4 AU per turn against an orbital radius of 1 AU, and the Earth’s path stands between 5.6° and 6.9° from its axis through the year. The Sun’s own 11 km/s toward the Galactic centre and 7 km/s toward the north Galactic pole turn the track to 29.4° from the pole, at 248.9 km/s, and lengthen the advance to 52.5 AU; the range stays 5.6° to 6.9°. Seen from outside the Solar System, the Earth is a giant body moving in a wave. The construction nests: the Moon’s helix about the Earth’s, the Earth’s about the Sun’s, the Sun’s about the Galaxy’s. These are helices within helices, one construction read at successive sizes (0.3a, 0.8).

Shrink the cone: the atom

The plane sections need only an inverse-square pair, so they reappear when the cone is shrunk to atomic size. Rutherford’s alpha rays ran its hyperbolas, and Bohr and Sommerfeld put the electron on its circles and ellipses. Kepler’s third law holds at the atom: on the orbit of principal number n, a = n²a₀, v = v₁/n and T = n³T₁, with a₀ = 5.29177 × 10⁻¹¹ m, v₁ = 2187.69 km/s and T₁ = 1.5198 × 10⁻¹⁶ s. Maeda and Gallagher phase-locked a lithium Rydberg electron to a 17.5 GHz field for about 15 000 revolutions; Kepler’s third law makes its orbit n = 72, with semi-major axis 274.3 nm, and a circular orbit at that n runs at 30.38 km/s. Their packet, built from np states in a linearly polarized field, runs a highly eccentric, nearly linear Kepler orbit. Atoms move, so their electrons run the second reading too: seen from outside a moving atom, an electron on a Kepler orbit runs a helix about the nucleus’s track.

Shrinking changes no angle and no ratio of an orbit’s own lengths. It changes one number: the action of an orbit in units of ħ, the action that fixes the floor, F_maxd_mint_min = ħ (0.1a). For one pattern on a circular orbit, L/ħ is the number of turns of its phase per revolution; for the Earth, a body of many patterns, it is the action of the orbit per turn in units of h:

The two vectors that fix the cutting plane become operators at the atom. Pauli scaled A on bound states and closed the algebra of L and the scaled A, the SO(4) symmetry Fock and Bargmann identified, whose Casimir gives

A finite-difference solution of the radial equation returns the same levels, equal for every l at one n, to 4 × 10⁻⁶. The conserved quantities that tilt the planes cutting the Solar System’s cone quantize the atom.

The cross-section of a helix

A point of a helix has cylindrical coordinates (z, ρ, φ): its place on the axis, its radius and its angle. Its projection onto the plane normal to the axis is ρ(cos φ e₁ + sin φ e₂). Multiplied on the left by e₁, that projection is ρ(cos φ + Σ sin φ) = ρ e^Σφ, with Σ = e₁e₂ and Σ² = −1. The complex numbers are this even algebra of the plane:

The unit imaginary is the oriented unit area of the plane normal to the axis, and complex conjugation reverses the sense of turning. The Hermitian product of two states is the geometric product of two vectors: its real part is their dot product and its imaginary part their oriented area. So ψ = ρ e^iφ is the cross-section of a helix, as I-⁠4 states. Its radius is the square root of a share density, and its angle is the phase. Every complex equation of quantum mechanics is a pair of real equations in the plane normal to the axis.

The axis depends on the observer. In its own frame a closed pattern’s phase only turns, φ = −ω₀τ with ω₀ = mc²/ħ, so its helix runs along its own time axis. For a bookkeeper in whose frame it moves at v, the pattern’s own time coordinate is τ = γ(t − vx/c²), its clock’s reading on its worldline, and at one bookkeeper time the phase advances along the path:

De Broglie derived λ = h/p this way in 1924. The de Broglie wave is the bookkeeper’s reading of the pattern’s own circulation, and every probe that measures it, whether a crystal, a grating or a pair of slits, is outside the system and moving relative to it. The phase rate c²/v and the run v multiply to c² (I-⁠2). An observer who turns with the pattern’s rest circulation writes ψ = e^−iω₀tψ_S and reads the Schrödinger equation, whose phase rate is v/2 while the run stays v.

An atom or a molecule moving through a laboratory is one helix in its centre-of-mass coordinate, of pitch h/P set by the momentum of the whole: the wavelength molecular interferometers measure. The de Broglie helix is no path. The rest circulation, carried at v, advances (γv/c)²h/p per turn along its path, 3.5 × 10⁻⁴ Å for a 54 eV electron against h/p = 1.669 Å; the wavelength is the spacing of equal phase at one bookkeeper time.

On the cylinder the Schrödinger equation is two real equations and a closure condition:

(C) and (H) are Madelung’s equations. Without (W) they admit solutions that no single-valued ψ has; on the cylinder (W) says that φ is an angle, so a helix closes on its cylinder only after a whole number of turns. ħ∇φ is the momentum: |∇φ| is 2π over the helix’s local pitch. The profile term Q_ρ is the confinement cost of I-⁠5 written on the radius profile: a tighter profile costs more, as a shorter string sounds a higher mode. In the circular state of principal number n, the phase’s turning carries the fraction (n − 1)/n of the kinetic energy and the profile term carries 1/n. That share is 4 × 10⁻⁷⁵ for the Earth’s orbit, read in its centre-of-mass coordinate as the circular state of the same equations, and there (H) is the Hamilton–Jacobi equation of the pair, whose rays are Kepler’s conics. A circular state at n = 72 has 1.4%; the packet Maeda and Gallagher held at that n runs a highly eccentric Kepler orbit. In the ground state of hydrogen it is 100%: the phase does not turn in space at all, and the whole kinetic energy sits in the profile. Shrinking the cone brings the profile term forward, and that is all it does.

Quantum phenomena on the cylinder

Each phenomenon below is a reading of the cylindrical equations and of the plane normal to the axis, carried to two ports for spin, to a joint configuration for two probes, and to Dirac’s equation for the strand at c.

Quantization is closure

On a ring, ψ = e^inϑ closes only for integer n, carrying L = nħ. (W) makes every winding about an axis whole, L_z = m_lħ, and at large n it joins the radial closure in the Bohr–Sommerfeld rule.

Standing waves are two helices of opposite hand

cos kx = (e^ikx + e^−ikx)/2: the sum does not turn. It breathes along one diameter and vanishes where the two helices cancel: the nodal patterns of P-⁠2, on whose nodes the share density vanishes.

Interference is the law of cosines

|ψ₁ + ψ₂|² = ρ₁² + ρ₂² + 2ρ₁ρ₂ cos(φ₁ − φ₂), the cross term being a dot product. A path mark sets the projections perpendicular and the fringes go. Single patterns seat whole, one port at a time, with the frequency of their share, by the seating law of the Born-rule paper.

Spectral lines are beats

Two stationary helices of rates ω₁ and ω₂ give a share that beats at ω₂ − ω₁. Hydrogen’s 2 → 1 line, with the reduced mass, is 10.1987 eV, or 121.57 nm, measured at 121.567 nm. A jump caught mid-flight runs a continuous, coherent course; each transition is an eigenmode relaxation of the atom’s pattern within its braid class (I-⁠5), a system-wide settlement (0.1a).

The commutator is an oriented area

Shifting a helix by ξ and twisting it by k fail to commute by a turn through kξ, the phase-space area ħk·ξ over ħ: [x, −iħ∂_x] = iħ. A stretch of helix of length Δx carries pitches spread over Δk ≥ 1/(2Δx), so ΔxΔp ≥ ħ/2: the bandwidth of a finite helix.

Tunnelling: the wall becomes porous

The omnium is never still. Density is nowhere uniform: around every point lie both higher- and lower-density regions (0.1), and every step is a system-wide settlement (0.1a). The motion carries the low-density regions of a barrier about, and where they line up across it they open a density geodesic, a nodal channel of P-⁠2, along which the pattern’s energy flows. The wall becomes porous, and a pattern that meets it again and again eventually finds it open. On the cylinder the radius decays in the barrier: where the turning is negligible, (H) gives ρ ∝ e^−ϰx, ϰ = √(2m(U₀ − E))/ħ. By the seating law, the frequency with which an alignment reaches depth x goes as the share density ρ², falling as e^−2ϰx; at the exit it is the transmitted share. By (C) the phase turns only as fast as the transmitted current requires. The “imaginary momentum” of the standard account is the radius gradient. Under a 4.5 eV barrier the current falls by a factor of 8.8 per ångström.

Spin is a rotor on the three-sphere of two ports

A pattern at a two-port probe is a point (z₁, z₂) with |z₁|² + |z₂|² = 1; its unresolved common phase runs along a Hopf fibre. A closed pattern of the nontrivial 2π-rotation class turns each port’s projection by half the angle, so a turn by 2π sends it to (−z₁, −z₂), the 4π that neutron interferometers measured in 1975. The shares are port shares, never an α (I-⁠6). That a closed pattern’s 2π-rotation class is the nontrivial one is to be derived (D-⁠03); once it is, I-⁠5 fixes its character to −1 with no fitted parameter.

Entanglement is one pattern at two probes

A state read at two probes that is not a product has a radius and a phase on the joint configuration that do not split: one pattern, with the shared bivector grade of I-⁠4. The singlet gives −a·b = −cos θ and reaches 2√2. Each local state stays I/2 whatever the distant analyser does, so no signal passes. Loophole-free tests violate Bell’s inequality as quantum mechanics predicts: S = 2.42 ± 0.20 for electron spins.

Two senses of turning, and the strand at c

The roots ±ω are the two hands of the helix: matter and antimatter are opposite handedness of one configuration (P-⁠4). Dirac’s velocity operator cγ⁰γ^k has eigenvalues ±c only, so the electron’s strand runs at c, as 0.1d reads it. The unit imaginary of Dirac’s equation is the bivector γ₂γ₁ of the spin plane.

The paper reads geometry the record already contains, so its tests are convergence tests. Each line names the measurement that would end a result.

1. T-⁠1
   Plane sections. Every inverse-square orbit lifts to one plane on the light cone, because its eccentricity vector is conserved. A measured drift of that vector beyond the field equations’ advance and the known perturbations ends the plane-section reading.
2. T-⁠2
   The wavelength belongs to the relative motion. λ = h/(γmv), with v the speed of the pattern relative to the probe. A fringe spacing that depends on anything else ends the bookkeeper’s reading.
3. T-⁠3
   Whole windings. Every orbital winding about an axis is a whole number of turns, L_z = m_lħ. A measured fractional orbital winding ends the closure reading.
4. T-⁠4
   Kepler’s third law at the atom. T_n = n³T₁ for hydrogenic orbits, with quantum defects where a core is present. A Rydberg orbit whose period departs from it ends the one-cone reading.
5. T-⁠5
   One pattern at two probes. Local shares are blind to the distant analyser, and the singlet correlation is −cos θ, up to 2√2. A signal through the pair, or a CHSH value above 2√2, ends the shared-helix reading.

Five pieces sit outside the derivation and leave every result in it unchanged:

1. D-⁠01
   α for an atomic pair. The map from a charge to an I-⁠1 input, which would label the atomic cone’s circles with α, is to be derived (core O-⁠6).
2. D-⁠02
   Masses. The confinement operator, the masses and the dimensionless content of K are to be derived (core O-⁠9). The cylindrical equations take them as measured.
3. D-⁠03
   The spinor class. That a closed pattern’s 2π-rotation class is the nontrivial one, and with it the half-angle and the 4π periodicity, is to be derived (core O-⁠11).
4. D-⁠04
   The seating rate. The rate kernel by which a probe’s coupling advances the delivered fraction is to be derived (core O-⁠12).
5. D-⁠05
   The aperture near the fold. The cone is read here in the far field; its aperture near the fold is to be derived (core O-⁠8).

For the same sources and data FDT returns Kepler’s three laws for the relative orbit, exactly in model on the pair domain (I-⁠1, I-⁠4 layer L1). It returns the perihelion advance and the deflection of light of general relativity at leading order (I-⁠3, layer L2): 42.98″ per century for Mercury and 1.75″ at the solar limb. It returns Rutherford’s cross section, which the classical hyperbolas and the quantum Coulomb pair share, and the gross structure of the hydrogen spectrum with the measured constants and the reduced mass. It returns every solution of the Schrödinger equation, of which the cylindrical form is an exact rewriting where ρ > 0, with (W) and ψ smooth across the nodes. And it returns the de Broglie wavelengths of electrons, atoms and molecules, interference fringes, tunnelling rates, the statistics of two-port patterns and, with the class of D-⁠03, their 4π periodicity, and the singlet correlations −cos θ, with Tsirelson’s 2√2 and no signal.

No divergence is claimed: the paper supplies what ψ is, not a changed observable.