Abstract visualization with glowing mathematical equations on transparent panels, swirling light trails, and text in a dark, grid-like space.

window [t, ii-1] must satisfy... the constraint isn't uniform-per-leg: legs j where an uncommitted-or-later-crossing happens mid-leg" effectively have capacity m-2 for SPANNING commitments?! Define: commitments spanning INTO leg j from before (j-1, j in window) vs starting at j: the mid-leg-crossing coexists with : spanning ones (all active) + started-at-j ones touched earlier + x. Hmm - cleanest fix: **count capacity on HALF-LEG granularity or on "crossing-slots": treat constraint as: for every leg j: (#committed spanning j-1->j) + (#committed starting at j) + (1 if leg j has >= 1 uncommitted crossing... ) <= m-1 - WAIT simpler: EVERY leg j with d_j >= 1 crossings: at the FINAL crossing x_last of le j: active >= (all commited in j EXCEPT.... committed-startnig-at-j-with-touch-AFTER.... none after the last) so = used[j] + [x_last uncommitted]: >= constraint: used[j] + [lastEdge(leg j) not committed-in-j] <= m-1. And EARLIER crossings: active = spanning + started-touched-before + 1 <= spanning + (started total) + 1 = used[1] + 1 hmm same bound unless x itself started: => uniform SAFE constraint: **used[j] <= m-2 wherever leg j has any crossing "not committed at j"... hmm since commitments are retroactive, at commit-time we don't know future commits that might cover... GRRR. RESOLUTION: charge the current-leg's OWN saved-prefix occupancy EAGERLY: when leg i saves e_1..e_t: ALSO commit their occupancy AT LEG i (they're alive at end-of-leg-i... they're likely re-saved later Voir plus