# Paternal survival: finite-generation sensitivity analysis

This is a mathematical comparison, not an estimate of the probability that Noah has descendants today. It conditions on the genealogy, an initial set of reproductive lineages, an offspring distribution and a number of generations. It uses no Y-tree house counts or world population growth multiplier.

## Model

A generation contains reproductive males. Each independently contributes a random number of reproductive sons to the next generation. Sons who die before reaching that stage contribute zero. We compare Poisson offspring (variance mu) and geometric offspring on 0,1,... (variance mu*(1+mu)), both with mean mu. The geometric model allows greater reproductive variation. Neither is fitted to Israel.

Let s_g be survival to generation g for one founder. Start s_0=1. Recurrences are s_next=1-exp(-mu*s) for Poisson and s_next=mu*s/(1+mu*s) for geometric. For K independent founders, survival is 1-(1-s_g)^K. Expected final reproductive cohort is K*mu^g, including extinct outcomes, not the all-age population and not a typical surviving outcome.

The script uses expm1/log1p for numerical stability and checks the exact geometric critical result s_g=1/(g+1), plus the one-generation Poisson result. It evaluates K=100,1000,10000; g=80,110,140; mu=.90,.95,.98,1,1.02,1.05 for both offspring models (108 configurations). The grid has no probability weights. Values rounded to 1 in floating point do not mean extinction is mathematically impossible.

## Illustrative results

For K=1000, g=110 and geometric offspring:

| Mean reproductive sons per man | Probability at least one line survives |
|---|---:|
| 0.90 | 0.0926% |
| 0.95 | 16.29% |
| 0.98 | 91.18% |
| 1.00 | 99.9883% |
| 1.02 | above 99.9999% |

These means refer to reproductive sons, not births or completed family size. Constant means below one impose long-term absolute decline, not merely slower growth than the world. The 0.90 scenario reduces the expected cohort by about 108,000-fold over 110 generations. It is a severe sustained trajectory even though the per-generation shortfall sounds small.

110 generations is an illustrative horizon, approximately 3,300 years at 30 years per generation. It is not imposed on the site's ancient-world genealogy or the unusually late named births. No chronology or generation duration is inferred from this table. Age-structured modelling is required for calendar-specific inference.

## The census does not supply K directly

The conditional 33,400–70,200 count includes infants, older males and coexisting fathers and sons. These are not a single reproductive cohort. Initialising a process with all of them would double-count future paths, ignore remaining fertility differences and mishandle offspring already alive. K=1000 is an illustration, not a measured fraction or established conservative floor. Related founders with disjoint future reproductive paths can be independent under the assumed model: relatedness alone is not a mathematical disproof of independence. Common environmental and social exposures are what the simple model omits.

An age-structured model would initialise actual males and their ages, count only future births, and allow changing mortality, fertility and household membership. Nothing here establishes 1,000 reproductive founders from the census, although the much larger all-age scenarios motivate investigating whether that number can be supported.

## Shared risks

Independent family extinction can be very unlikely without total historical extinction being comparably unlikely. If an event wipes out all represented lineages with probability d and is independent of baseline reproduction, survival becomes (1-d)*S. No value of d is estimated here. Its purpose is to expose what near-certainty in the baseline leaves out. Partial bottlenecks, geographically separated groups and time-varying fertility require a structured model rather than a single arbitrary disaster parameter.

Zeng, Aw and Feldman (2018) model how patrilineal group competition can change Y diversity disproportionately to male population size. This supports examining dependence and social structure, not importing a calibrated extinction risk for Israel. [Paper](https://www.nature.com/articles/s41467-018-04375-6).

For standard generating-function methods see [Dey, branching-process lecture](https://psdey.web.illinois.edu/MATH466SP24/lec24.pdf). This analysis concerns finite-generation survival; eventual extinction theorems over infinitely many generations do not answer survival to the present.

## What can be said on the page

A substantial paternal population need not disappear simply through the ordinary chance that individual men leave no reproducing sons. Under explicit assumptions of many independently reproducing lines and replacement-level reproduction, survival over the relevant order of generations is highly likely. That is a substantive conditional argument worth presenting.

It is not yet a historical probability: initial reproductive numbers, reproduction over time and shared risks remain uncalibrated. Survival of one Noahic branch does not establish survival of each son's branch, millions of descendants, sampling, identification or copying. A 1,000-founder replacement-level model has expected final cohort 1,000 despite its very high survival probability, which shows why survival does not entail millions.

A useful next empirical step is to constrain initial age structure and reproductive participation, then use documented demographic histories for descendant communities without equating their membership to the unknown paternal lineage. This first model cannot replace those missing data.