Configuration-Space Topology and the Distinction Calculus: The Exchange Scalar, Its ±1 Shadow, and a Pre-Registered Derivation Program

In the paper's words

Configuration-space topology (CST) derives exchange statistics from the topology of the space in which identical particles move: the fundamental group pi_1(C_N(M)) is the symmetric group S_N in d >= 3 spatial dimensions (exactly two exchange phases: boson +1, fermion -1) and the braid group B_N in d = 2 (anyons, including the non-abelian braiding used in topological quantum computation). This paper draws the boundary of that framework. It names the five silent scaffolds CST imports from classical geometry -- point-like particles, a fixed background manifold, the deleted hard-core diagonal, externally imposed dimension, adiabatic exchange -- and engages the active counter-literature testing each one (orbifolds, traid groups, graph configuration spaces, extended objects, supersymmetric anyons).

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. This dimension-dependent dichotomy is not an empirical accident; it is a theorem about the topology of the space in which particles move [ESTABLISHED — ].

   from 1. Introduction

2. Configuration-space topology shows that the dichotomy is not an arbitrary empirical fact but a theorem about the space in which particles move: in $d \ge 3$ the exchange phase can only be $\pm 1$, because the fundamental group of the configuration space of identical particles is the symmetric group.

   from 2. So What? Why Should a Reader Care About This Research?

3. The dimension theorem is then [ESTABLISHED —; ]: $$\pi_{1}\bigl(C_{N}(\mathbb{R}^{d})\bigr) = \begin{cases} B_{N} & d = 2 \\ S_{N} & d \ge 3, \end{cases}$$ where $B_{N}$ is the braid group (Artin presentation: generators $\sigma_{1}, \ldots, \sigma_{N-1}$ with relations $\sigma_{i} \sigma_{i+1} \sigma_{i} = \sigma_{i+1} \sigma_{i} \sigma_{i+1}$ and $\sigma_{i} \sigma_{j} = \sigma_{j} \sigma_{i}$ for $\lvert i - j \rvert \ge 2$) and $S_{N}$ is the symmetric group, obtained from $B_{N}$ by imposing the involutive relations $\sigma_{i}^{2} = 1$ (Artin's theorem: $S_{N} \cong B_{N} / \langle \sigma_{i}^{2} = 1 \rangle$).

   from 3.2 The fundamental group and the dimension theorem

4. The spin-statistics theorem is a separate result of relativistic quantum field theory: integer spin fields are bosonic, half-integer spin fields are fermionic, derived from Lorentz invariance, microcausality, and positivity of the energy [ESTABLISHED —;;; ].

   from 5.1 The spin-statistics connection (C3)

What would falsify it

1. Report the derivation program and the conjecture (Sections 6–8): the current construction status of the pre-registered QNFO program — exchange statistics from the mark calculus — and the synthesis conjecture lifting the Distinction Calculus into homotopy type theory under the Grothendieck–Teichmüller group, with explicit falsification conditions F1–F3.

   from 1. Introduction

2. The movement is from what is proven (Section 3), to what is assumed (Section 4), to what is conceded (Section 5), to what is under construction (Section 6), to what is conjectured and how it could be falsified (Sections 7–8).

   from 1. Introduction

3. What a mathematician gets: a falsifiable conjecture with mature ingredients.

   from 2. So What? Why Should a Reader Care About This Research?

Where the claim stops

1. Its scaffolds (point particles, fixed background, deleted diagonal, imposed dimension, adiabatic exchange) are under active external test; its spin-statistics boundary requires relativistic input the framework cannot supply; and its non-abelian and emergent sectors lie beyond the fundamental group, in braided fusion categories.

   from 10. Conclusion

2. Because the boson/fermion dichotomy is an input to every quantum theory — and this paper tells you exactly which parts of it are proven, which are assumed, which are contested, and which could one day be derived from a more primitive logical act.

   from 10. Conclusion

3. It rests on scaffolds — point-like particles, a fixed classical background manifold, a deleted hard-core diagonal, an externally imposed dimension — that the framework itself cannot derive [ACKNOWLEDGED — this paper, Section 4].

   from 1. Introduction

Where to read

1. 1. Introduction381 words · 2 min
2. 2. So What? Why Should a Reader Care About This Research?671 words · 3 min
3. 3. Configuration-Space Topology: The Framework412 words · 2 min
4. 4. The Scaffolds446 words · 2 min
5. 5. The Boundary392 words · 2 min
6. 6. The Distinction Calculus: Pre-Registered Derivation Program686 words · 3 min
7. 7. The Synthesis Conjecture (C2): Distinction Calculus into Homotopy Type Theory372 words · 2 min
8. 8. Falsification Conditions186 words · 1 min
9. 9. Applications197 words · 1 min
10. 10. Conclusion209 words · 1 min
11. Acknowledgements and Provenance74 words · 1 min