The Exchange Phase as a Logical Scalar: R = e^(2 pi i s) from the Re-Entrant Calculus

The boson/fermion dichotomy is conventionally presented as a primitive classification of nature. Recent work established that the underlying invariant is the relation between the exchange phase of identical particles and their topological spin, R = e^(2 pi i s), with the dichotomy as its shadow in three spatial dimensions. This paper examines whether a calculus whose only primitive is the act of drawing a distinction can generate this invariant rather than importing it as an axiom. The re-entrant mark of the calculus of indications, disciplined by linearity, already generates the constants e and pi as logical scalars: e as the fixed point of D f = f, and pi as the trace of the identity on the circle type. The central observation: the exchange phase is the (2s)-fold half-turn of the re-entrant mark, R = (e^(i pi))^(2s) = e^(2 pi i s) = (-1)^(2s), so the boson/fermion dichotomy is the parity of 2s. The arithmetic is established; the identification is a model; the claim that the calculus derives the invariant as a logical scalar within a single formal system is a conjecture with explicit falsifiability conditions. Source repository (provenance): https://github.com/QNFO/qnfo-research/tree/res/paper/exchange-phase-logical-scalar **So what (v1.1).** This revision adds the P3-verified citation corpus (including the closest prior art: Kauffman's mark-to-fermion-algebra construction, arXiv:1301.6214, and the 2013 World Scientific predecessor) and the P4 formal derivation of the exchange phase as the (2s)-fold half-turn monodromy power in the treatise's Part VIII system. Reader value: a minimal-premise engine for exchange statistics (parity of 2s), a sharpened handle for the ZBW-Majorana Z2 diagnostic (spin noise / EELS-RIXS / Gromov delta protocols), and executable toy models (T1-T7 suite, 10.5281/zenodo.21943007). **v1.3 (2026-08-16):** adds the FQ2 consilience result — the p-adic anyon phases (10.5281/zenodo.21208491/21208368) are, under the archimedean embedding, exactly the (eiπ)²ˢ family at rational spins s = m/(2p^k); the monodromy-power reading is CONSISTENT with the p-adic braid-group construction (disconfirmation not met). Bridge artifact + ZBW-Majorana P4 cross-reference deposited.