# Personal-best selection simulation: methods and claim sheet

Published by Altamens on 30 September 2026. This is a reproducible simulation and methods note, not participant research. No Altamens user, account, analytics or game data were accessed.

## Question

How does the expected personal best change as more attempts are observed when the score-generating process itself is stable?

## Prespecified design

- Models: standard normal; uniform on `[-sqrt(3), sqrt(3)]`; Student t with 5 degrees of freedom rescaled by `sqrt(3/5)`. Each has theoretical mean 0 and variance 1.
- Synthetic profiles per model: 250,000.
- Attempt checkpoints: 1, 2, 3, 5, 10, 20 and 50.
- Primary quantity: Monte Carlo mean of each profile's running maximum, expressed in within-profile standard-deviation units.
- Secondary quantities: median, 2.5th and 97.5th percentiles of the running maximum; mean across all attempts; Monte Carlo standard error of the mean running maximum.
- Seeds: 20260930, 20260931 and 20260932 respectively.
- Software: Python standard library. `random.Random` uses the Mersenne Twister generator; normal draws use `random.gauss`, uniform draws use `random.uniform`, and the chi-square term for t(5) uses `random.gammavariate`.

## Primary result

In the standard-normal model, the mean personal best was 0.566 SD after 2 attempts, 1.164 after 5, 1.539 after 10, 1.867 after 20 and 2.250 after 50. The mean of all observed attempts remained within 0.004 SD of zero at every checkpoint. These values are simulation outputs, not estimates for a specific game or population.

## Sensitivity result

At 20 attempts, the mean personal best was 1.567 SD under the bounded uniform model, 1.867 under the normal model and 1.992 under the heavier-tailed t(5) model. At 50 attempts the corresponding values were 1.664, 2.250 and 2.620. The distribution therefore matters; there is no universal personal-best correction.

## Reproduction

Run:

```sh
python simulation.py reproduced.csv
```

Compare `reproduced.csv` with `results.csv`. The script writes no network requests and has no third-party dependencies.

## Claims not supported

- No participant sample, intervention, learning effect or Altamens-user result.
- No claim that real scores are independent, normal, t-distributed or uniformly distributed.
- No reliable-change threshold, minimum useful session count, medical interpretation or training benefit.
- No inference about IQ, brain health or everyday-life performance.

## Review and conflicts

Altamens operates a commercial mental-practice product and displays personal scores. Research, code, drafting, source checks and publication preparation used an OpenAI automated assistant. Automated consistency checks were performed; no independent human methods review or journal peer review was completed.