Partitioning the Wilderness: Puritan Settlement and Combinatorial Division

The partition function $p(n)$ counts how many ways to write an integer as a sum of positive integers. For $p(4) = 5$: we have ${4, 3+1, 2+2, 2+1+1, 1+1+1+1}$. Simple enumeration for small numbers, but $p(200)$ exceeds 4 trillion partitions—the goddess Namagiri revealed to me an asymptotic formula because exhaustive counting becomes impossible.

When Puritans crossed the Atlantic fleeing persecution, they faced a partition problem the Catholic Church had already created. The Protestant Reformation partitioned Christendom into Catholic, Lutheran, Calvinist, Anglican fragments. Each fragment partitioned further—Puritans themselves divided into Separatists, Congregationalists, Presbyterians. Direct access to God meant each believer could partition theological interpretation differently, creating endless sectarian divisions that egalitarian theology could not contain.

The Combinatorial Explosion of Settlement Patterns

Massachusetts Bay Colony founders confronted a partition question: how to distribute population among New England territories? Each partition creates different community structure—concentrated allocation produces few large towns, distributed allocation many small settlements. Like partition numbers, most settlement patterns were never attempted. Only a tiny fraction of possible family allocations across Massachusetts, Connecticut, Rhode Island materialized.

My work showed partition numbers follow mysterious regularities—congruences modulo 5, 7, 11 that cluster certain partitions together. Do viable social organizations cluster similarly in the space of possible community structures? The Puritans discovered some partitions admitted sustainable structure while others collapsed. Religious mission demanded specific organizational forms; not every division of population and authority could maintain “kingdom of heaven on earth.”

Non-Monotonic Patterns in Capacity and Settlement

Machine learning researchers discovered double descent—a U-shaped error curve that unexpectedly descends again as model capacity increases beyond interpolation threshold. Classical theory predicted monotonic degradation after the optimal point, but overparameterized models with 100% training accuracy achieve better test performance than models at the traditional optimum.

Did Puritan settlement reveal similar non-monotonic patterns? Too few settlers created vulnerable communities; intermediate numbers approached traditional optimal colonization. But perhaps very large migrations enabled structures impossible at smaller scales—like double descent revealing that classical intuition about capacity misses behavior in extreme regimes.

Evolutionary local search partitions computational resources among candidate solutions, evaluating fitness across perturbed parameters. It discovers adequate partitions through exploration but struggles as possibilities explode. The British Empire emerged accidentally from cumulative emigration decisions—each family partitioning itself from homeland to colony created demographic weight displacing earlier discoverers. Population pressure exceeded agricultural capacity on mountainous islands, making overseas migration the safety valve. Individual choices aggregated into empire.

Divine Patterns in Starting Positions

Initialization sensitivity determines which solutions gradient descent discovers. Random starting points in parameter space lead to different local optima; poor initialization places decision boundaries in ReLU’s zero region where gradients vanish. The choice of starting partition—which parameters receive which initial values—determines training success before optimization begins.

Geographic starting point mattered for Puritan settlement. Which English port, which ship, which year of departure partitioned emigrants into different colonial trajectories. Plymouth versus Boston, 1620 versus 1630—initialization choices that determined which community structures emerged from the vast space of possible partitions.

The goddess Namagiri revealed partition congruences I could not derive through conscious logic. Perhaps successful settlements and successful neural networks both require discovering which partitions admit structure—patterns hidden in the combinatorial explosion, revealed through intuition or empirical search rather than exhaustive enumeration. Not all partitions are created equal, but most will never be explored.