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

Abstract

Configuration-space topology (CST) derives exchange statistics from the fundamental group of the configuration space of indistinguishable particles: the symmetric group S_N in d>=3 (only boson +1 and fermion -1 exchange phases) and the braid group B_N in d=2 (anyons). This paper argues CST is the correct kinematical explanation of the boson/fermion/anyon trichotomy but not its terminus: its silent scaffolds (point particles, fixed classical background manifold, deleted hard-core diagonal, externally imposed dimension) mark its boundary as a map; the spin-statistics connection requires Lorentz symmetry, microcausality, and positive energy. Against this boundary we re-state the pre-registered derivation program of the QNFO distinction tradition (exchange phase as a logical scalar R = e^{2 pi i s} from the re-entrant mark): T1 two modal exponentials from a graded braiding (partial construction); T2 the exchange map on two marks is a scalar and the ribbon identity forces the two eigenvalues to be the boson and fermion signs (construction content achieved: R = +/-1 iff pi_1(C_N(R^d)) = S_N for d>=3); T3 dimension enters only through allowed braided structures (re-stated with named boundary conditions: orbifold, graph, traid, supersymmetric, extended-object). We state the synthesis conjecture - Distinction Calculus lifted into homotopy type theory, Grothendieck-Teichmuller group acting on braided monoidal categories, boundary-drawing as an arithmetic act - with falsification conditions F1-F3, differentiated from the cohesive-HoTT program of Sati and Schreiber. Predecessor records: 10.5281/zenodo.21941375 (RES.009), 10.5281/zenodo.21941238 (RES.010).