This is a transcript of the sheet above, re-keyed from the object. Section numbering follows the original. Marginal hands are reproduced where legible: AR in red, E.C. in pencil-blue. Neither hand was placed by Authority personnel.
The Echo thinks they are free.
The primary difficulty in containing "Rogue Echoes" is their non-linear movement. They do not follow Brownian Motion (random, small steps). They follow Levy Flight Foraging patterns.
I want to note, before the mathematics: I did not begin this research to help the Authority catch anyone. I began it because the pattern is beautiful. The mathematics of how a person searches for meaning in a fragmented existence turns out to be one of the most elegant equations in temporal physics.
A Levy Flight consists of a series of short, localized movements punctuated by rare, massive "Jumps." This is an optimal search strategy for finding resources in a "Sparse Environment" — in this case, the resource is Meaning or Anchor Points.
The Levy Flight is "Scale-Invariant," but it is also Energy-Inefficient in a closed system. Each "Long Jump" generates a massive Ripple Vector (R). Eventually, the Echo will jump into a "Causal Vacuum" — a place where their Momentum (M) is higher than the era's Elasticity (C).
M > C // CAUSAL VACUUM IMPACT — TOTAL SYSTEMIC COLLAPSE PREDICTEDThe Levy Flight predicts its own crash.
We should stop chasing the Echo during their "Short Steps." Instead, we must identify the "Long Jump" trajectories and place Mnemonic Traps at the most likely "Distant Nodes." The Echo thinks they are free because they jump far. They don't realize that the longer the jump, the easier the math becomes to solve.
[CHEN-MATH-101 // NOTE: The author wishes to add, outside the formal findings, that knowing the mathematics of how Echoes search for meaning does not make it acceptable to weaponize that knowledge. The Levy Flight is how a person looks for home. Using it as a trap is using homesickness as a weapon. The author notes this and submits the paper anyway because the mathematics will be discovered with or without her and she would prefer the discoverer to have moral qualms about it. — E.C.]
[CHEN-MATH-101 // NOTE: The author wishes to add, outside the formal findings, that knowing the mathematics of how Echoes search for meaning does not make it acceptable to weaponize that knowledge. The Levy Flight is how a person looks for home. Using it as a trap is using homesickness as a weapon. The author notes this and submits the paper anyway because the mathematics will be discovered with or without her and she would prefer the discoverer to have moral qualms about it. — E.C.]