Toward a Mathematical Theory of Integrated Positioning, Sensing, and Communication: Foundations for 7G

The sixth generation of cellular wireless has reached the normative standardization phase without an accompanying foundational theorem of the kind that preceded every prior generation: Shannon's coding theorem for 1G–3G, Telatar's multi-antenna capacity for 4G, Arıkan's channel polarization for 5G. Seventh-generation wireless is currently being defined at the level of use cases and marketing vocabulary; the mathematical problems that will actually constrain what 7G can become have not been posed. This survey argues that the foundational work required is substantial, that its natural language is the estimation-theoretic tradition developed over six decades of geodetic engineering, and that the open problems are numerous, well-defined, and presently unassigned. We organize the landscape around five research pillars; ISAC information theory, delay-Doppler capacity theory, Space-Air-Ground Integrated Network capacity with positioning constraints, AI-native physical-layer theory with provable guarantees, and electromagnetic information theory for near-field and holographic systems, and review the state of each pillar against its open problems. We argue that the Global Navigation Satellite System is the operational precedent for every system that Pillars I through V attempt to construct, and that the Cramér–Rao bound, Kalman filter, ambiguity function, dilution-of-precision formalism, multi-constellation estimation machinery, and antenna calibration theory of geodesy provide the mathematical toolkit that the current 6G and beyond-6G literature has not yet assimilated. We formalize a Fisher–Shannon unification object that appears across all five pillars, and we map the cross-pillar convergence zones—ISAC with OTFS, SAGIN with electromagnetic information theory, AI-native with semantic compression, quantum-enhanced positioning, and reconfigurable intelligent surfaces in integrated sensing—that will produce the highest-impact foundational results of the coming decade. We publish a numbered catalog of over one hundred open problems, each classified by difficulty, prerequisite knowledge, expected research effort, and the number of follow-on papers the problem is expected to generate. We close with a decade research roadmap tied to the ITU-R IMT-2030 evaluation phase, World Radiocommunication Conference 2027 spectrum decisions, and the 3GPP Release 20 through Release 22 standardization cycle, and we identify the 2026–2030 window as the critical period in which foundational theoretical work can still shape which systems get built.