The Poincare disk is alive. It breathes (conformal scaling) and rotates (Mobius isometry). Points move but distances are preserved.
Challenge: Watch the colored dots. During breathing, do they change distance from each other? (Hint: look at the hyperbolic distance readout — it stays constant even though the dots visually move.)
L6 — Breathing Transform: The disk periodically contracts and expands (conformal scaling).
This is NOT random — it's a controlled temporal modulation that makes static attacks fail. An adversary
who calibrates their input to a specific disk state finds the state has changed by the time the input arrives.
L7 — Mobius Phase: An isometric rotation of the Poincare disk. Points orbit but their
hyperbolic distances to each other remain exactly the same. This prevents attackers from predicting
which region of the disk they need to target — the target moves, but the geometry is preserved.
Combined: Breathing changes the scale. Mobius changes the orientation. Together they create
a moving target that preserves all security invariants. The math guarantees that ALLOW/DENY decisions
are the same regardless of the current breathing/rotation state.