One Number Per Cell: A Separating-Family Encoding of Superimposed Discrete Curves
Davide Bragetti · Zenodo (CERN European Organization for Nuclear Research) · 2026
On a discrete grid several curves may cross and share cells, yet a tracer following one curve must recover, at every shared cell, where ITS OWN curve continues. We ask for a single static number per cell --- no stored lists --- from which each curve reads its next step with no collision. The answer is to read the cell number as a member of a SEPARATING FAMILY: assign curve i the prime p_i and store the product of p_i raised to curve i's local Freeman direction; each curve recovers its direction as the exponent of its own prime, and unique factorisation guarantees that distinct curves never interfere. We analyse the cost of this prime code and give more compact separating families --- the residue number system (Chinese Remainder Theorem), positional mixed-radix, and bit-packing --- that share the same properties without factorisation; the prime form is space-dominated. The result is sharp: a local, state-blind tracer cannot resolve a tangency (two curves entering a cell alike but leaving differently) by any cell number, so the irreducible resource is a per-curve index, of order log m bits, carried in the tracer, not in the cell (the local-indistinguishability theorem). The construction has a mutable dual: the rotor-router places the same disambiguating resource inside the cell and advances it on each visit. Separating family and rotor-router are the two escapes of the impossibility boundary. Simple curves --- open paths, closed loops, and rho-shapes --- are admissible; only genuine self-intersection is excluded, with a branch-separation (residue-jet) extension outlined for it. An interactive demonstration runs at https://l3ragio.github.io/one-number-per-cell/ (source: https://github.com/l3ragio/one-number-per-cell).