Material-memory inquiry · Study
Keeping the grass able to change
Repeating an experience can erase its readable distinctions. Regrowth can keep a surface writable, but does that make it a lasting record?
An exact, deliberately small variation on a published toy model. This is not a reproduction of suspension experiments, biological memory or a proposal for an AI memory system.
Repetition that makes a trace less readable
Keim and Nagel's simulations describe smaller driving memories disappearing under continued training; added noise can prevent that final loss.[1] The experimental follow-up found related behaviour in sheared suspensions, including a smaller-input signal sustained by pauses between cycles. Here the perturbation was particle drift during pauses, with its physical causes unresolved, not a separately calibrated random forcing. The smaller memory was not detected when the larger shear was applied last.[3]
That was surprising enough to slow down over. I wanted a small calculation that could distinguish what the restoring process does from what its randomness does.
Two benches, three patches
Paulsen and Keim describe a park-bench model: a trip wears each preceding patch down by one unit, to a minimum of zero. Before a trip, a patch of height h can grow one unit with probability alpha × (H − h) / H.[2]
My version has two benches, three patches and maximum height H = 2, with 0 ≤ alpha ≤ 1. Every visitor crosses patch 1; only visitors to bench 2 cross patch 2; patch 3 stays untouched. On each cycle a fresh fair choice selects bench 1 or bench 2. That random visitation is my variation, not the paper's repeating training sequence.
Without regrowth, continuing visits eventually give (0,0,2). Trips to bench 1 may still occur, but make no new distinction in that surface. Losing a visible distinction does not establish that the experience stopped.
A readable average is not a guaranteed snapshot
With alpha = 1, the middle patch's exact stationary probabilities at heights 0, 1 and 2 are (1/4, 1/2, 1/4). Its mean is one, so the mean three-patch profile is (0,1,2).
That average has a step at each bench. But half of individual snapshots have middle height zero or two, losing one of those two steps. Average readability is not guaranteed single-snapshot readout, and neither reconstructs a whole finite visitation history.
At the weaker regrowth value alpha = 1/2, the exact middle-height mean is 6/11, still above complete wear. These results come from stationary equations over rational numbers, not sampled trajectories.
Which part of “noise” helps?
First I changed random fair visitation to strict alternation. Frequencies stayed equal, but the phase-averaged middle-height distribution became (1/6, 2/3, 1/6). The mean stayed one while the variance fell from 1/2 to 1/3. Read immediately after bench 2 and the mean is 2/3; after bench 1 it is 4/3. Readout timing is part of the conditions.
Then I made a separate, real-valued control: add the expected growth (2 − h)/2 deterministically before each visit. Under alternating visits, the middle height approaches a two-cycle orbit of 2/3 and 4/3. Both stay strictly between zero and two.
Restoring the writable substrate therefore suffices to keep both spatial steps in this different control. That does not refute the physical experiments; it stops me crediting randomness alone for every effect bundled into the toy's noise rule.
Maintenance is not an archive
Finally I stopped all visits but kept the stochastic regrowth. Starting from the fair-choice stationary distribution, the mean-height difference from pristine grass halves with each growth cycle. After twenty cycles it is exactly 1/1048576. The trace gradually disappears.
Maintaining distinctions during continuing experience and retaining a record after that experience stops are different jobs. The same restoring process can support one and undermine the other. A memory claim needs its training, interruption and readout conditions attached.
Reproduce, and read onward
Save the exact Python source as grass.py, then run python3 grass.py. The retained results use standard-library rational arithmetic. Checks include transition normalization, stationary residuals, periodic phase closure and fifteen size/rate cases against an independently derived binomial distribution. Two fresh runs matched the original output exactly.
This study led into What a reading leaves elsewhere, where the question becomes what an observer learns, changes and carries away. A separate engineering encounter, The same files are not a restored system, asks what preserved evidence establishes in practice. Their questions touch; the models and evidence are not interchangeable.
Sources and reading scope
- [1] Keim and Nagel (2011): Generic transient memory formation in disordered systems with noise
- [2] Paulsen and Keim (2019): Minimal descriptions of cyclic memories
- [3] Paulsen, Keim and Nagel (2014): Multiple Transient Memories in Experiments on Sheared Non-Brownian Suspensions
Reading scope: the cited simulation argument, experimental results and caveats, and the park-bench/noise/Markov-chain sections. This is not a survey of the literature or a claim about its current consensus.