Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Restores the actual-orbit target and Coulomb-plus-precession history, separating the existing mechanism from auxiliary models and its unresolved physical source.

Actual trajectories and auxiliary orbitals

The target in V2 is a literal deterministic electron trajectory in six ordinary physical spatial dimensions, represented by Hamiltonian evolution and its observer projection. Probability clouds and stationary orbital wavefunctions do not describe the electron's position in this proposal. Circulation and second moments alone do not specify this target. The existing 6D orbit demo already specifies the complete projected curve and its time-occupation measure.

The recent scalar Hartree and oscillator-state calculations remain correct within their stated auxiliary models. They do not establish the actual trajectory, the clock, the selected four-state orbit representation, or its observer map. Calling them progress toward the observed shapes without establishing this connection was unjustified. A stationary field carrying an angular expectation is not a derivation of a particle trajectory. The coupled scalar branch is an auxiliary field solution, not an established source for the demo's generator or proof of stability of its orbits.

Some preserved note IDs and filenames contain “orbital”; they name the auxiliary calculation that was actually performed. Replacing that word everywhere with “orbit” would falsely relabel the mathematics. Those notes now carry this correction explicitly. Standard terms such as spin-orbit coupling and orbital angular momentum do not imply a conventional-orbital ontology and are retained where mathematically appropriate.

Position, fields and registration

The physical trajectory is q(t)∈Q6q(t)\in Q_6. A classical field evaluated on physical space, a coefficient vector representing that field, and a homogeneous-coordinate lift of qq are different objects. Classical field oscillations and spectral eigenfunctions do not imply a probabilistic position law, and they do not by themselves derive q(t)q(t).

Squared homogeneous coordinates are geometric invariants. Their sum can be normalized to one without making them probabilities of electron positions. Likewise, a normalized time measure records the fraction of a deterministic orbit spent in each region. A broadened registration density describes the apparatus response accumulated along that orbit, not an electron cloud. The apparatus coupling and any sampling law still require physical equations.

The radial rule ∣W∣2/R2=q|W|^2/R^2=q in the observer module is a conditional coordinate readout. Calling it a Born-intensity premise was misleading: no Born probability rule is derived or adopted by that formula.

The tetrahedral simplex theorem and the 1⊕3\mathbf1\oplus\mathbf3 representation remain useful geometry. Conventional ss--pp hybrid orbitals, Slater-orbital energies and a Born--Oppenheimer calculation are external comparisons. They do not supply the deterministic electron--nucleus dynamics or explain molecular bonding in IDWT merely by sharing a four-component vector space.

Existing orbit construction

Provenance correction: Coulomb binding plus precession

The starting construction is a Coulomb-bound six-dimensional orbit with precession producing the petal pattern. The successive mathematical models below specify that motion and identify its remaining physical assumptions.

The history of visualizations/6d-orbit-slice.html records:

  • ce71f75: computeShell rotates the four-component direction in six coordinate planes, with rates abs(I[i]-I[j])*W1, where I[i]=exp[i]+1. It explicitly sets the precession lock κ = ω₁, takes component moduli, and combines them with commensurate phase motion to generate the path. The paths are computed from this motion, not spherical-harmonic surfaces. The assigned rates, phases, and small radial breathing term are not a numerical integration of an electron–nucleus force law. This is a mathematical replacement, not evidence that the legacy ordered rotations obeyed that same Hamiltonian.
  • c6d6bcf and 4701c3a: construct the exchange and clock as parts of one degree-L representation of a higher-dimensional symmetry action. Their frequency ratio follows within the selected construction; identifying its source with physical electromagnetism and fixing its rate remain conditional. Later corrections distinguish sector points from section coefficients and constrain the observer map.

The earlier Coulomb-plus-confinement Hamiltonian in the page's prose (for example c50ef57) does not by itself derive the later precession term. The historical intent and the implemented dynamics must both be recorded without conflating them.

The research baseline is therefore the existing orbit-plus-precession construction, not an arbitrary petal curve awaiting a mechanism. The next physical task is to derive or test that precession and its lock from the coupled electron–nucleus dynamics. Nuclear feedback is a candidate contribution or correction, not an already demonstrated prerequisite for petals. The recent nuclear-polarization diagnostic treats an electric dipole response; induced magnetic feedback remains uncalculated.

Scope of the geometric corrections

The completed Coulomb-petal calculation already integrates an autonomous Hamiltonian with literal six-coordinate positions and momenta. Its signed projection has four and eight petals at L=2,4L=2,4; the odd-LL images differ from the modulus-folded demo. Source preparation, dynamical integer selection and stable multi-electron binding remain open. These actual-orbit results must be retained separately from auxiliary stationary field calculations.

The characteristic-class obstructions concern full-rank global bundle identifications, not the existence of the coefficient trajectory in P(V_L). They do not invalidate the demo's orbit formula. The relation between the section-coefficient state, the sector coordinate, and the Dirac field on physical space must retain the distinctions already made in the demo.