# Labeled-token transmission and finite samples

Run `python reproduce.py` with numpy and matplotlib. Standalone exact mathematical study: nine pair combinations,1,143 binomial probability rows, two original figures in SVG/PNG/PDF, runtime/generator hash and validation. No measured organisms or human traits are modeled.

## Contract

`cross(left,right)` accepts canonical labels AA,Aa,aa. Each pair has two labeled token positions. Select one position uniformly from each pair, independently, and combine the two labels without regard to order. Four equally likely positional outcomes give exact AA,Aa,aa probabilities. For Aa×Aa these are1/4,1/2,1/4. A and a are just token labels; no dominance, phenotype, health, ability or learned knowledge is encoded.

This is an abstract representation of simple one-locus segregation assumptions, not all biological inheritance. Real inheritance can violate this simplified independent equal-transmission model. It contains no linkage, mutation, recombination map, selection, development, environmental effect or population dynamics. It is not a model of whole evolution or transmission of learned knowledge. Passing a token is a different kind of process from teaching a skill.

`binomial(n,k,p)` returns choose(n,k) p^k (1-p)^(n-k), the probability of k AA outcomes in n independent repetitions with fixed probability p and replacement. Export each of the nine pair combinations at n=4,20,100, every k=0..n. Degenerate p=0 or1 gives an exact endpoint mass. The plotted probability mass is assigned by the model, not a measured histogram or a confidence interval. The three panels show Aa×Aa with p=0.25. A finite sample need not contain exactly the expected fraction; this does not change the underlying probability.

Independent checks derive pair probabilities from each side's A-token proportion, verify reversal symmetry and normalization, and check binomial mean np and variance np(1-p) for all27 distributions. A separate enumeration of256 equally likely four-draw label sequences verifies the n=4,p=1/4 distribution without using the binomial formula. Six invalid guards pass. This arithmetic demonstrates transmission assumptions and sampling variability; it does not forecast a biological outcome or prescribe an experiment.