A non-orientable surface sees only the 2-torsion — and loop-closing is what "quantized" means

Sighted conversation, 2026-09-27, continuing N-016. Every number below was computed by brute force in the session, not recalled; the first attempt had a wrong cell complex for the sphere and the computation caught it.

The obstruction comes first

A chiral topological order cannot be put on a non-orientable surface at all. The Chern–Simons action ∫A∧dA needs an orientation to integrate. So the ground-state degeneracy of Laughlin ν=1/3 or Ising anyons on a Klein bottle is not small — it is undefined. Only c₋ = 0 orders can live there. Whether a topological order can exist on a Klein bottle is a yes/no test of whether it is chiral, and c₋ is measurable as a thermal Hall conductance (κ_xy = c₋·π²k_B²T/3h; Banerjee et al. 2018 read c₋ = 5/2 at ν = 5/2). The property the topology forbids is the property a thermometer reads. Candidate instance for CASE-06 (chirality-origin): the ambient surface adjudicates chirality.

The count, for a G-gauge theory (quantum double D(G))

GSD on Σ = |Hom(π₁(Σ), G) / conjugation|. Brute-forced over group elements:

G         |G|  torus  Klein   RP2   torus/Klein^2   |G[2]|
Z2          2      4      4     2         0.2500   2
Z3          3      9      3     1         1.0000   1
Z5          5     25      5     1         1.0000   1
Z6          6     36     12     2         0.2500   2
Z2xZ2       4     16     16     4         0.0625   4
Z2xZ3       6     36     12     2         0.2500   2
Z4xZ2       8     64     32     4         0.0625   4
Z3xZ3       9     81      9     1         1.0000   1
S3          6      8      6     2         0.2222   (non-abelian)

For abelian G, exactly:

torus  = |G|²
Klein  = |G| · |G[2]|
RP²    =        |G[2]|          G[2] = { g : g² = 1 }, the 2-torsion

so torus / Klein² = 1 / |G[2]|². The N=2 column was checked independently against the stabilizer count 2^(2−χ), which knows nothing about homology.

The rule is one rule. Torus = Klein² exactly when |G[2]| = 1, which is exactly odd order. The evens do not have a second rule; they have a larger G[2], and its size is what matters, not evenness: Z₆ behaves like Z₂ (one element of order two), Z₂×Z₂ and Z₄×Z₂ sit at 1/16 (four of them).

Mechanism. An orientation-reversing loop returns every anyon as its antiparticle, so the Möbius cycle can carry only charges that are their own antiparticle — g = g⁻¹, the 2-torsion. The orientable cycle carries everything. Klein bottle = one of each. RP² is the bare cross-cap and carries G[2] alone, which is why a Z₃ gauge theory on a projective plane has a unique ground state. Odd-order gauge theories are blind to non-orientability — every non-orientable surface looks to them like a sphere with orientable handles.

The non-abelian version. For S₃ the torus gives 8, the anyon count of D(S₃); the Klein bottle gives 6, from pairs (a,b) with a b a⁻¹ = b⁻¹ up to conjugation. The Möbius condition stops being "b is its own inverse" and becomes "b is conjugate to its own inverse, with a as the witness" — real elements in the character-theory sense. Everything in S₃ is real, so its Klein count does not collapse the way Z₃'s does even though S₃ contains Z₃: the Z₂ supplies the conjugation that rescues the 3-cycles.

A heuristic that failed, recorded because it failed instructively

The first guess was "the Klein bottle degeneracy counts self-conjugate anyons" — which gives 4 for Z₂ (right) and 1 for Z₃ (wrong; it is 3). The error was scope: conjugation acts on one cycle, not on the whole degeneracy. The corrected statement is the product above.

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The same mechanism, running the other way: loop-closing is what "quantized" means

Everything in this note and in N-016 is one condition — a loop must close consistently — and that condition is also where the integers in quantum mechanics come from. Instances, oldest first (the last added 2026-09-27, plektons):

• Bohr–Sommerfeld, ∮p·dq = nh: the WKB phase must return to itself around a closed orbit.
• Hydrogen's m ∈ ℤ: e^{imφ} must be single-valued under φ → φ+2π. That is π₁(S¹) = ℤ acting; the allowed m are the characters of the loop group.
• Dirac's monopole, eg ∈ ℏℤ/2: the string must be invisible, i.e. the U(1) bundle over S² must close — c₁ ∈ H²(S²;ℤ) is an integer.
• TKNN / quantized Hall conductance, σ_xy = (e²/h)·C: C is the first Chern number of the Bloch bundle over the Brillouin torus. Measured to a part in 10⁹ because an integer cannot drift.
• Exchange statistics: the phase is constrained to the one-dimensional representations of π₁(configuration space) — Z₂ in three dimensions, the braid group in two.
• This note: the Möbius cycle admits only the 2-torsion.
• Plektons and the Jones index — the instance where "quantized" and "integer" come apart. In algebraic QFT the statistical dimension of a superselection sector is a Jones index (Longo 1989, d_ρ² = [M : ρ(M)]), and Jones' theorem confines the index below 4 to 4cos²(π/n). So a non-abelian anyon's quantum dimension below 2 is quantized to 2cos(π/n): 1, √2, φ, √3, … — Ising at n = 4, Fibonacci at n = 5. The golden ratio is a rung on a ladder, not a fitted parameter. The mechanism is a closure: the restriction comes from positivity of the Markov trace, which is the trace that closes a braid into a link, and it is consistent only on the ladder. DHR in 3+1 forces d_ρ to be an integer (a compact-group representation dimension); Fredenhagen–Rehren–Schroer in 2+1 free it, because spacelike cones cannot be exchanged without linking — codimension 2 in operator-algebraic clothing (P-024).

The general statement: the allowed values of the quantity are the representations of π₁ (or the classes in H²) of the space the system lives on. Quantization is not a separate postulate laid on top of the topology. It is the topology, read as a constraint on holonomy.

And the twist worth holding, because it inverts the usual slogan. Loop-closing does not always quantize. It constrains to what π₁ permits, which can be less than expected or more. Hydrogen's m is discrete because π₁(S¹) = ℤ and its characters are e^{imφ}, m ∈ ℤ. Anyons in two dimensions are continuous — any phase — because π₁ is the braid group and its one-dimensional representations form a continuous family. Same mechanism, opposite outcome. Three dimensions is the case where the loop-closing removes freedom (Z₂: only ±1); two dimensions is the case where it grants it. "The loop must close" is the whole content, and the integer is a consequence of the group, never an input.

Held as a note with instances, not a coordinate: it is a mechanism seen six times, and the map's own rule is that a pattern earns a coordinate by predicting an absence, which this has not yet done.

Clause added 2026-09-27, from two expander theorems. "The allowed values are the representations of π₁" is necessary and not sufficient for a sector to exist. Expanders contain a Hamiltonian cycle (Draganić–Montgomery–Munhá Correia–Pokrovskiy–Sudakov, arXiv:2402.06603, the Krivelevich–Sudakov conjecture) — a loop through every vertex — and yet the Kuramoto flow on an expander has no stable twisted state (Abdalla–Bandeira–Kassabov–Souza–Strogatz–Townsend, arXiv:2210.12788): the spectral gap unwinds every winding around every cycle. A loop-closing sector needs the loop and a spectrally slow neighbourhood to live in. In the physical instances the slowness is stiffness (a superconductor's, a lattice's); the theorem says connectivity overrides stiffness. Run 035 asks the same of the intermediate host, which is spectrally a path. (Attributions from memory; verify before citing.)