The Universal-Language No-Go Theorem: Incompatibility of Geometric Genus Invariants and Operator-Algebraic Factor Invariants in Any Neutral Framework (Paper14 of the LQG–LQC Intertwiner Series)

1 Introduction The LQG–LQC Intertwiner Series has produced two structurally parallel but mathematically distinct realizations of a common completion system:(i) Geometric realization: the stationary equation determining the LQC bounce normalizationconstant C sits on a smooth genus-one elliptic curve EC over Q(π), with Weierstrass invariantsg2, g3 (Paper 9).(ii) Operator-algebraic realization: the GNS von Neumann algebra generated by the RC encountermodes above threshold κ0 = √2 is the unique hyperfinite Type III1 factor (Paper 6, Theorem 2.5).Both realizations share the same Möbius completion functional F(q) = (1 + q)/(1 − q), whichappears as the nome of EC on the geometric side and as the equal-time modular kernel Gβ(0) onthe algebraic side (Paper 16 paper). A natural question is: can these two realizations be unified ina single neutral framework that speaks both languages simultaneously?The present paper proves the answer is no