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: The ring's three-state generator embeds in positive-degree section spaces under the same Killing action, but this does not identify a degree-zero electron sector.

What can be unified algebraically

The demo uses a separate ring coefficient generator ν diag(0,−1,0), with initial state (A,B/√2,B/√2), and a constant-radius readout whose angle is the relative clock phase. Its projective orbit is not generated by the one-dimensional H⁰(O(0)) space.

There is nevertheless a simple embedding in the same section spaces used for the petalled cases. Denote the section degree by k here, keeping it distinct from the demo's ring label L = 0. For every k ≥ 2 choose normalized monomials proportional to

s0=Z1Z2Z4k−2,s−=Z3Z4k−1,sr=Z4k. s_0=Z_1Z_2Z_4^{k-2},\qquad s_-=Z_3Z_4^{k-1},\qquad s_r=Z_4^k.

They are distinct sections of O(k). Under the same generator Ĝ_k = N₁−N₂−N₃ their weights are exactly (0,−1,0). Hence

Φ(t)=As0+B2eiνts−+B2sr \Phi(t)=A s_0+\frac B{\sqrt2}e^{i\nu t}s_- +\frac B{\sqrt2}s_r

has precisely the demo's three-state coefficient evolution and returns after 2π/ν. The prescribed ring readout Γ_x+iΓ_y = R times the normalized clock coherence is Re^{iνt}. This is an actual time-dependent coefficient orbit in a selected section subspace, not a stationary field substituted for the ring.

The selected spinor-line construction lifts this field into Σ⊗O(k+2), just as it lifts the degree-k petalled orbit. At k = 2, the 10-dimensional section space therefore contains both the ring's three-state generator and the four-state generator used for four petals. No new generator weights or electric charge have been fitted to obtain this fact.

What this does not unify

This is not a degree-zero construction. The earlier dimension obstruction for H⁰(O(0)) remains intact, and the k ≥ 2 ring cannot be declared the physical L = 0 atomic state without a separate identification. It shows that the generator algebra need not require an unrelated carrier space; it does not derive which carrier the physical circular state occupies.

The ring and petalled prescriptions select different mode subspaces and readout rules. An algebraic embedding alone does not derive a single apparatus response that selects between those rules, a nuclear preparation mechanism, energies, or transition strengths. In particular, the petal-count rule 2k applies to the prepared extremal exchange construction, not to every trajectory in O(k). Degree alone does not force a petalled outline.

The smooth geometric clock dipole from the probe calculation also exists here, but it already gives circular dipole signals for some petalled preparations. It cannot by itself distinguish those detector images.

Research consequence

A single positive-degree field sector may support multiple coefficient-orbit families under the same Killing action. This supplies a concrete candidate bridge for the ring, with no change to the displayed petals or their code. The remaining problem is physical family selection and the degree/flux interpretation, not merely finding three suitable abstract frequencies.

The scalar effective-drive route also applies algebraically in this space, but inherits the omitted-mode leakage and locking corrections audited in the companion note. The embedding is not a stability proof.

Reproduction

orbit_drive_origin_checks.py checks the three weights, normalized evolution, clock readout, and full recurrence for k = 2, 3, 4, while retaining the one-dimensional degree-zero check.