9 November 2026. Exploration at the stated request: which properties that need convoluted explanations in three dimensions (VSEPR, hybridisation, the octet rule, Hund's rule) look simpler when electrons are orbits in four directions. Status of each point is stated. No molecular calculation has been done.

Setting: an electron orbits a nucleus in four directions (the observed three and the fourth), on an orbit that is a closed curve in a two-plane through the nucleus, with a smaller fourth-direction oscillation. Shells are sets of equal orbits.

1. Two exact facts about equal orbits in four directions

(a) Two circular orbits of equal size about one centre lie on a three-sphere. In four directions two such circles generically do not meet (two planes through the origin in R^4 generically share only the origin). In three directions they always meet.

(b) Two equal circles keep a constant distance from each other all the way round if and only if their planes are isoclinic (Clifford parallels). The set of all circles Clifford-parallel to a given one, with the same sense, is one Hopf fibration of the three-sphere: a family of circles filling the sphere, no two meeting, every pair equidistant, all with one handedness relative to one pairing of the four directions. Circles of the opposite handedness are not equidistant from them.

Reading for shells (candidate): equal orbits that are merely non-crossing still approach and recede once per revolution, a periodic kick. Equidistant orbits never do. A stable shell of several electrons on equal orbits is therefore one Hopf family, and every orbit in it circulates with the same handedness. That is a geometric version of Hund's rule (electrons sharing a shell take parallel orientations) that needs no exchange energy.

2. The space of orbits in a shell is a sphere

The orbits of one Hopf family are labelled by the points of an ordinary two-sphere (the Hopf base). So the question "how do k electron pairs arrange themselves in a shell" becomes "how do k points arrange themselves on a sphere under mutual repulsion". That is the Thomson problem, whose answers are: 2 antipodal, 3 equilateral triangle, 4 tetrahedron, 5 trigonal bipyramid, 6 octahedron. These are exactly the VSEPR shapes, with the awkward five-fold case coming out the way VSEPR has to assume it.

What changes relative to VSEPR: the points are orbit planes, not electron clouds; there is no need for hybrid orbitals (an orbit is an orbit; nothing has to be mixed to point somewhere); and there is no octet to be exceeded, because the capacity of a shell is set by the orbit dynamics (open, note.md section 3), not by a count of orbitals. Six pairs around sulphur in SF6 are six points on the base sphere, an octahedron, with no d-orbital story.

Open: the map from a point on the base sphere to a bond direction in ordinary space. A plane in four directions projects to ordinary space as a squashed ellipse; which direction counts as "where the bond points" has to come from where the orbit passes nearest the second nucleus. Until that map is written, "the shapes agree" is a statement about arrangements of orbit planes, not yet about measured bond angles.

3. Lone pairs and bent molecules

In VSEPR a lone pair "takes more room" to explain water's 104.5 degrees against 109.5. In the orbit picture a bonding orbit is anchored (it must pass near the other nucleus) and a lone orbit is free. Two anchored and two free points on the base sphere repelling will not sit at tetrahedral positions; the free pair can move apart and the anchored pair closes. Direction of the effect is right; the size needs the base-sphere repulsion law, which is not known. Open.

4. Where the tetrahedral number already appears

The case document (section 9.10) has four orientation patterns of the three hidden pairs, (+++), (+--), (-+-), (--+), with pairwise alignment -1/3, the tetrahedral cosine. One pattern is the lepton-type value and three are the quark-type values. That is colour geometry, not chemistry; no link to bond angles is claimed. Recorded so the two occurrences of -1/3 are not later confused.

5. What this picture would make simple, if it holds

  • Shell shapes without hybridisation or orbitals: Thomson arrangements of orbit planes.
  • Hund's rule as a geometric fact about equidistant orbits.
  • No octet rule to break: hypervalent molecules are just more points on the base sphere.
  • A natural atomic axis: the pairing that defines a Hopf family joins one observed direction with the fourth, so a many-electron shell singles out one direction in ordinary space.

6. What it does not yet do

  • Give capacities (2, 8, 18, 32) or the Madelung doubling: needs the locked-orbit dynamics.
  • Give bond angles in degrees: needs the base-sphere-to-direction map and the repulsion law.
  • Treat the actual orbit shape: the facts in section 1 are for circles; the model's orbits have a fourth-direction oscillation and are not circles. Whether equidistant families exist for the real orbit shape is a computation that has not been done. It is the first thing to do.

Addendum 1: bonds as orbits around all the bonded nuclei (user guidance, 9 November 2026)

the stated picture: in a bond, the electron's orbit goes around all the nuclei involved; how many electrons do so depends on the kind of bond. Consequences for the notes above, as readings:

  • A bonding orbit is a two-centre orbit: its plane contains both nuclei and so contains the bond axis. A lone-pair orbit goes around the central nucleus only. The arrangement problem of section 2 is therefore an arrangement of two kinds of orbit about the central nucleus, anchored two-centre planes and free one-centre planes, which is what section 3 assumed.
  • Bond angle = angle between the axes of two two-centre orbits sharing the central nucleus. The two orbits both pass close to that nucleus; keeping them apart there is a constraint on the angle. That is the quantity to compute for water and ammonia.
  • Bond order = number of electron pairs on orbits around the same two nuclei. Equal orbits around the same two nuclei have their far points on the axis in common, so a double or triple bond needs either orbits of different size or phase offsets that keep the electrons from meeting at those points. Which of these the dynamics allows is not known.
  • Metallic bonding: orbits around many nuclei. Aromatic rings: electrons on orbits around all the ring nuclei (benzene's six). Ionic bonding: the orbit moves from one nucleus to the other. Whether the 4n + 2 rule for rings is a count of phase slots on such orbits is an open question; nothing is claimed.

Addendum 2: equal orbits, phase arrangement (user guidance, 9 November 2026)

Changing level is hard for an electron (established earlier in the orbit work), so electrons on a bond do not take orbits of different size. Multiple bonds are electrons on equal orbits around the same nuclei, kept from meeting by their phases; with many electrons present they rearrange until the phases fit. Sharing is literal.

Consequence for the open items above: bond order is a count of phase slots on one two-centre orbit, set by how many electrons can be spaced around it without meeting at the shared far points on the axis. That count depends on the electron's extent along the orbit relative to the orbit's length, which the orbit model can supply. It would also say why bond order stops at three for ordinary atoms. Not computed.

Independent verification (Opus agent, 10 November 2026)

Verified exactly (with numerical checks): (a) concentric equal circles meet in R^3 (planes share a line) and in R^4 iff the planes share a line (det[a1 a2 b1 b2] = 0), generically not; (b) the set distance is constant iff the two principal-angle cosines are equal (isoclinic), and the distance between two co-moving electrons at equal angular speed is constant for every relative phase iff the transfer matrix is a positive multiple of a rotation (isoclinic with matched senses); (c) the fibres span{x, ux} of a Hopf family are pairwise isoclinic, base S^2, base angle psi with isoclinic angle psi/2 and minimum chord 2 r sin(psi/4); Thomson minimisers for Coulomb: k = 2 antipodal, 3 triangle, 4 tetrahedron, 5 trigonal bipyramid, 6 octahedron.

Corrections accepted:

  1. Non-isoclinic equal orbits approach and recede twice per revolution (second harmonic), not once.
  2. Opposite-handed circles are not equidistant from the family as a whole, but every circle lies in exactly one left and one right Hopf family, and circles in its right family are isoclinic to it.
  3. k = 2, 3, 4, 6 hold for every completely monotone repulsion; the k = 5 trigonal bipyramid is a property of soft laws (a square pyramid wins for Riesz exponent above about 15).
  4. Reducing a shell to points on the base sphere discards each electron's phase along its orbit. Point electrons with free phases, carried by the common Hopf flow, have constant mutual distances for any configuration on S^3, so their repulsion problem is Thomson on S^3, with different minimisers (for k = 5 a regular simplex). The base-sphere picture holds if each orbit acts as a phase-averaged ring or if the phases are fixed by something not stated here.
  5. The map from a bond axis to a Hopf base point (the fibre containing the axis a is span{a, ua}, base point 2(a.u)a - u) is a doubling map from RP^2 to S^2. It sends a base tetrahedron to bond axes at 54.7 and 90 degrees and sends the six octahedral axes to two base points. The claim that the standard molecular shapes are the arrangements of k points on the Hopf base is therefore not supported under the map the premises fix; it stays an open reading. The Case no longer states it.
  6. Two-centre bonding orbits (addendum 1) are not circles about a common centre, so the section-1 results do not apply to them as written. Equal concentric orbits whose planes share the bond axis meet at the axis points, so a multiple bond is one orbit with several phase slots, not distinct isoclinic orbits.
  7. The shell's average fourth-direction fraction is exactly 1/4 for any Thomson configuration on the base (a fibre's fraction in w is (1 + b.u)/4, and the base points sum to zero); only one orbit of a family can lie nearly in the observed three directions. This conflicts with a setting of a small fourth-direction oscillation.
  8. The complex structure of a left family pairs the fourth direction w with u = Jw, which necessarily lies in observed three-space; u is arbitrary on S^2 and the statement presupposes that w is distinguished.
  9. Conventional physics: points-on-a-sphere repulsion is VSEPR's own model (Gillespie and Kepert); VSEPR itself uses no hybrid orbitals and d-orbital participation in SF6 is no longer taught, so neither is a gain over VSEPR. The tetrahedral cosine -1/3 is exact.

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