The P-index Framework
A single dimensionless quantity that resolves a long-standing failure of crowd safety assessment: density-based indicators cannot distinguish safe high-density scenarios from lethal ones.
Tokyo's Yamanote Line operates routinely at 6–7 persons/m2 without recorded compressive-asphyxia incidents. The 2022 Itaewon disaster occurred at 8–10 persons/m2 and produced 159 fatalities. The two scenarios are not separated by density. They are separated by physical quantities that density does not measure.
The P-index measures those quantities directly — cumulative seconds, per person-hour, during which individuals are subject to above-threshold contact pressure. Grounded in Hertzian contact theory, granular force-chain physics, and the biomechanical lethality envelope of Kroll, Still and Griffin (2017).
Documents
A Physical Quantification Framework for Crowd Compression Risk
The P-index System and Its Contact-Mechanical Foundation
arXiv ID pending — link will appear here upon release. Email for early access.
The P-index Framework — Commercial White Paper v1
Engagement architecture and applied framing for insurance, regulatory, and venue-design audiences.
Three application domains
Insurance
Three-tier output aligned with Solvency II catastrophe-model disclosure: Pbaseline as underwriting threshold, Pmax as base-rate driver, full EP curve for reinsurance pricing. Compatible with existing RMS, Verisk/AIR, and KatRisk pipelines.
Regulation
Nmax as a physics-based legal capacity ceiling, derived from contact mechanics rather than fire-evacuation time. Supplements or replaces per-person area rules under NFPA 101, ISO 16730, and Eurocode performance-based standards.
Venue design
Iterative design evaluation, real-time risk monitoring via sub-region PΩ(x,y,t), quantified retrofit prioritization, and cumulative-risk evaluation in evacuation planning. Full operational lifecycle.
Engagement
The P-index framework, the C-value formulation, and the non-repulsive computational architecture are held by PneumaTheorem, Inc. (Delaware C-Corporation, 2026). Engagement structures range from standard commercial licensing through to full acquisition.
Counterparty profile: insurance and reinsurance carriers with crowd-density underwriting exposure; national and supranational regulatory bodies responsible for venue safety certification; venue operators, design consultancies, and event organizers with dense-crowd portfolios.
Initial inquiry: written technical or commercial communication through the email channel above. Responded to within five business days. No application form, no scoring, no required pre-qualification — the white paper and the companion arXiv manuscript together provide a self-contained basis for initial counterparty evaluation.