A Physical Quantification Framework for Crowd Compression Risk: The P-index System and Its Contact-Mechanical Foundation

Criteria for crowd safety are conventionally expressed as densities. What injures a person is a force, and the relation between the two is not a function of density alone: it depends on how many bodies lie along a connected path, on whether exertion is oriented so as to accumulate, and on whether the configuration permits accumulation at all. The difficulty is sharper than a matter of precision. Sustained muscular exertion decays, and a body that ceases to hold itself upright does not stop delivering force but leans — transferring weight rather than effort, which is a larger contribution and one that does not fatigue. At a fixed density the borne load therefore rises with time. A criterion whose quantity is stationary while the quantity it stands for is not is not an imprecise relation but no relation. This paper separates what follows from geometry and conservation from what must be supplied from outside. A crowd is treated as having two phases, distinguished by whether floor area for a recovery step remains available; the framework addresses the second, in which contact is continuous and a disturbance is transmitted rather than absorbed. There the borne load accumulates along a connected path in closed form, and accumulates at all only where a growth condition holds — a comparison between the exertion a contributor delivers and the amount a receiving body absorbs irrespective of load. Where that condition fails, each individual bears one neighbour's exertion at any depth and for any duration. What the relations bound is the length of a connected run rather than a number of people. The deliverable before an event is accordingly a partition spacing together with the load it guarantees, and a demonstration that no accumulation can occur: a statement of geometric impossibility rather than of statistical rarity. After an event it is a trajectory of borne load, to which a reader applies whatever dose–response function they accept. Two quantities are declared — a reference body mass and an exertion scale — and a third, the packing limit, is needed only where a result is expressed as a length rather than a force. No physiological value enters any derivation, and the chapter cataloguing such values may be deleted without altering an equation or depriving a proposition of a premise. The account makes a prediction that criteria expressed in density do not: a dense crowd with no common target does not accumulate load at all. Everyday commuter crowding is that configuration, at an exposure large enough that a non-zero per-exposure risk would long since have been recorded as a statistical fact.