# How large a footprint should Adam's male line have today?

STATUS: model record, rebuilt 2 September 2026, Adelaide. Supersedes the first
draft of the same date, which counted only Terah's line and so undercounted the
men carrying Adam's chromosome by between two and fourteen times. Mike found that
error; `flags.md` §1 records it.

This is not a preregistered test and it is not labelled as one. Nothing here was
frozen in advance, because nothing here is scored against data that could have
come out otherwise. It is an arithmetic estimate built from figures this site
already publishes, run so that the companion page can cite a number and a reader
can check it.

Sources of the inputs: `numbers/census.html` and `data/census.json` for the
ledger reading of the wilderness musters; `pages/nations-after-noah.html` for how
this site reads Genesis 10; `pages/can-ancient-dna-test-cains-arrival.html` for
Cain's crossing; `research/family-space/T9-RESULTS.md` for the frozen count of
ancestral houses. Outside figures are cited in `sources.md`. Every point where
this record makes a choice, or cannot support one, is logged in `flags.md`.

Scripture quotations are NIV.

---

## 0. The question, and the two that hide inside it

Genesis, read the way this site reads it, puts Adam at about 5159 BC and puts
Noah's household on the far side of a flood that ended the ancient world without
emptying ours. On that reading a definite population of men descends from Adam
in the male line, and some number of them were alive in Moses' day.

The male family tree of living men has been measured, and it is the one fixed
thing every reading of Genesis has to face. Two questions follow, and they are
not the same question.

1. **Should any of Adam's male line be alive today at all?** The answer is yes on
   every reading of the numbers I can construct, and it is not close.
2. **If some is alive, how large is it, and should the tree have caught it?** The
   answer here has changed since the first draft, and it has changed against the
   comfortable version. The reading now predicts a large branch, not a modest one.

A third question runs alongside, and it needs the same arithmetic without needing
the tree at all: **does the Bible's own account of Israel in Egypt hold together
as history?** Section 11 takes that separately, because the numbers that decide
it barely move the answer to the first two.

---

## 1. A1: how many Israelite males

Numbers 3:43 counts 22,273 firstborn males, "listed by name", from a month old
and upward. That is the one figure in the muster material that reads as a real
count of individuals, and this site's census page treats it as one.

To get from firstborn males to all males, multiply by **c**, the number of living
males for each living firstborn male.

What c is depends on what "firstborn" counts, and the two readings differ by
about a factor of two. Mike's ruling, 2 September 2026, is the household reading,
taken from Exodus 12:30, "there was not a house without someone dead": every
household had a firstborn son. On that reading firstborn males equal households
and **c is the number of living sons per household**. The alternative, the
mother's first child, is offered as a toggle rather than argued away.
`flags.md` §17 sets out both, and records that the choice was made on the text
before any footprint had been computed from it.

| | c | Israelite males |
|---|---|---|
| low | 2.5 | 55,700 |
| central | 3.0 | 66,800 |
| high | 3.5 | 78,000 |

The face-value reading, in which 603,550 counts men aged twenty and over, is
carried through every table below as a contrast. It is not this site's reading;
`numbers/census.html` rejects it on four independent legs, and section 11 adds
three more.

## 2. A2: the fraction of Israelite males on Adam's male line

The quantity the model needs is the fraction of Israelite males biologically on
**Adam's** male line, not Jacob's. A man absorbed into Israel who descends from
Noah by another route still carries the chromosome; a man descended from the
peoples already living here does not.

Three channels put men into Israel who were not born into Jacob's line: the mixed
multitude of Exodus 12:38; bought slaves circumcised into the fathers' houses
under Genesis 17:12-13, which covers men "bought with money from any foreigner";
and captives, as in Genesis 34:29 after the assault on Dinah.

Foreign wives are not one of them. Joseph married Asenath, an Egyptian, and
Ephraim and Manasseh still carry Joseph's chromosome. Only men entering the
household change the male line, which is why Israel's reputation for marrying
within itself does not raise this fraction.

The fraction is bracketed **0.60 / 0.80 / 0.90**, and applied uniformly to Israel
and to Terah's other houses. `flags.md` §2a records why it is not differentiated,
and that it is applied more harshly than the reading requires.

## 3. A3: Terah's other houses

Israel is not the only male line running back through Terah. Edom, Ishmael,
Midian and Keturah's other sons, Moab and Ammon through Lot, and Nahor's house
are all on the same line at the same date. Scripture gives no numbers for any of
them, so each is bracketed as a multiple of Israel's Adamic males:

| house | line | low | high |
|---|---|---|---|
| Edom | Esau | 0.3 | 1.0 |
| Ishmael | Ishmael | 0.3 | 1.0 |
| Midian and Keturah's sons | Abraham | 0.2 | 0.6 |
| Moab and Ammon | Lot, via Terah | 0.3 | 0.8 |
| Nahor's house | Nahor | 0.3 | 1.0 |
| **total** | | **1.4** | **4.4** |

The central value is **2.5**, below the 2.9 the midpoints give. Each of these
peoples took over lands that already had people in them: Edom lists the Horite
chiefs beside its own (Genesis 36:20-30, Deuteronomy 2:12), and Moab and Ammon
displaced the Emim and the Zamzummim (Deuteronomy 2:10-11, 20-21). On this site's
own reading of how nationhood spreads, that means their biological share of their
own nations is lower than Israel's, and 2.5 is the discount that implies.

**T, Terah's cluster** = A1 x A2 x (1 + A3):

| | |
|---|---|
| low | 80,200 |
| central | **187,100** |
| high | 378,900 |

## 4. A4: the rest of Noah's line

This is the term the first draft omitted entirely, and it is the largest.

The trunk from Noah down to Terah forks ten times with male issue outside the
trunk: at Noah, Shem, Arphaxad, Kainan, Shelah, Eber, Peleg, Reu, Serug and
Nahor. Genesis 10 names 44 founders in Japheth's and Ham's houses, 8 more among
Shem's other sons and Aram's four, and 14 in Joktan's line. Genesis 11 adds
"other sons and daughters" at seven further forks. That is **about 80 founder
lines outside Terah's**, forking six hundred to a thousand years earlier than
Terah's did.

They are not nations on this site's reading. Genesis 10 describes nationhood
spreading through peoples already here, so these men founded nations without
their descendants being those nations. But they had descendants, and the
descendants carry the chromosome. The first draft gave all eighty of them zero.

**How fast did they grow?** Slowly, and the reason is worth stating. Terah's
cluster reaches 187,000 men in six centuries, which is a surviving line doubling
every 39 years. Israel's own rate from Jacob is faster still. If the eighty side
lines had grown at anything like that, Noah's descendants would outnumber the
world before Moses. So the tracked line is tracked because it flourished, and its
rate cannot be lent to the others.

S is bracketed between a floor and a ceiling:

| | S | what sets it |
|---|---|---|
| low | 13,000 | eighty lines doubling every 150 years, the slowest rate that still counts as spreading out |
| central | **479,000** | doubling every 121 years |
| high | 1,060,000 | doubling every 108 years |

The central figure is a choice, not a calculation, and `flags.md` §3 logs it with
the direction that would flatter the argument.

## 5. A5: the cap, and the check at Babel

**The cap.** Adam's descendants in our world cannot outnumber the people among
whom they lived. The Genesis 10 horizon, meaning Egypt, Nubia, the Levant,
Mesopotamia, Anatolia, Iran, Arabia, the Aegean and the Caucasus, held about
**13.8 million people at 1266 BC, so about 6.9 million men** (HYDE 3.3; see
`sources.md` 1.3). That is the hard cap on N, and no row of this record comes
near it: the largest N is 1.44 million.

An earlier version of this section set a lower ceiling, a quarter of the
horizon, on the ground that Noah's lines had to stay a minority among the
peoples nationhood spread through. Mike withdrew it on 4 September 2026: the
site's reading of Genesis 10 is a claim about the founding of nations among
peoples already there, about 2800 BC, and says nothing about proportions six
centuries later (`flags.md` 27). The side-branch bracket of section 4 keeps
its high value of 1.06 million as a judgment about growth rates, no longer as a
consequence of a ceiling.

**The check at Babel.** What the reading does fix is where Adam's descendants
were before the dispersion. Genesis 11 puts the people of one language on the
plain of Shinar until Babel, which this site dates to the end of Peleg's life,
about 2033 BC (`pages/babel-and-sumerian.html`), and identifies the language as
Sumerian, carried into our world by Cain about 5000 BC and again by Noah's
household after 2903 BC. On that reading the people of Sumer at Babel are the
site's best estimate of the numerical extent of Adam's descendants in our world, Cain's
line and Noah's together, along with whatever remained of the peoples who were
there before Cain came. So Adam's line at Babel is at most the population of
Sumer, and the site does not know by how much less; in Mike's words, an upper
bound and "the best we have", not an equality.

The surveys of southern Mesopotamian settlement give that population. Adams,
*Heartland of Cities* (1981), pp. 147-48, puts the Tigris-Euphrates alluvium in
the Third Dynasty of Ur, 2112-2004 BC, at "probably closer to a half million
than to a million" people, and at p. 251 says it had "crept upward past the
half-million mark" by the end of the third millennium; for the Uruk period
before 3000 BC he gives fewer than a hundred thousand. Taking **550,000 people,
275,000 men**, Sumer was about 5 per cent of the 10.29 million people HYDE
gives the horizon at 2000 BC (`sources.md` 4). Khuzestan, where the site places
Cain's early line, lies outside Adams's survey and adds a little.

On this record's own curves, Noah's line at 2033 BC is only about 2,383 men
at the central N, because the curve runs at a constant rate from four men at the
ark to N in Moses' day and so puts most of its growth after Babel (section 6).
Cain's line at the same date is about 5,417 men at the central K and about
31,045 at the high K (section 9). Together Adam's line at Babel is under ten
thousand men on the central settings, a few per cent of Sumer's 275,000, and the
check passes on every setting the companion page allows; the page reports it as
a live line, and it would fail only for a Cainite line above about four million
men in Moses' day.

That figure carries a consequence the record should not hide. If Adam's line at
Babel was a few thousand men, then the half-million people of Sumer were
overwhelmingly not Adam's descendants by the male line, and "the sons of the
Adam" who built at Babel (Genesis 11:5, as the Babel page reads it) were a
small group among Sumerian speakers, or Sumerian was a language the peoples of
the plain had adopted. A curve that put more of Noah's growth before Babel would
raise the figure, but only at the cost of the slow generations the longevity
model requires. The tension is recorded, not resolved.

Two things the check does not do. It does not fix the share of Sumer that was
Adam's line rather than the peoples there before Cain, which the site leaves
unstated. And it does not bound Terah's house, which grew after Babel.

## 6. N: men carrying Adam's line at 1266 BC

**N = min(T + S, cap)**, the cap being the 6.9 million men of section 5

| | Terah's cluster | side branches | N |
|---|---|---|---|
| low | 80,200 | 13,000 | **93,200** |
| central | 187,100 | 479,000 | **666,100** |
| high | 378,900 | 1,060,000 | **1,438,900** |

Terah's whole cluster is 28% of the central N. Israel's own share is **8%**.

That last figure is the answer to the question that prompted the rebuild. The
size of Israel is not what determines how large Adam's footprint should be today.

---

## 7. Should any of it survive?

Write s for the probability that one man's male line reaches the present, and
take the N men as independent. Then P(no line survives) is e^(-sN).

s is read off the measured tree, not assumed: 1,091 ancestral houses alive at
about 5100 BC still have living male-line descendants, against about 9.2 million
men alive then, giving s = 1.2 x 10^-4, or 2.2 x 10^-4 allowing 2,000 surviving
houses for those the sample missed.

| | N | s = 1.2e-4 | s = 2.2e-4 |
|---|---|---|---|
| low | 93,200 | 1.4e-5 | 1.3e-9 |
| central | 666,100 | 1.9e-35 | 2.3e-64 |
| high | 1,438,900 | 1.0e-75 | 3.3e-138 |
| Terah's line only, low | 80,200 | 6.6e-5 | 2.2e-8 |

**P(at least one of Adam's male lines is alive today) is indistinguishable from 1
on every row.** The corner where extinction was a live possibility in the first
draft is gone: reaching it now requires deleting not only Israel's sister peoples
and Cain's line but every one of Noah's eighty other founder lines, which
contradicts Genesis 10:32 and 11:11-25.

Two features of the anchor push the true s higher than the figures used, and
neither is corrected for: it is measured over 7,120 years and applied over 3,200,
and 1,091 is a count of houses visible in a sample, so it is a floor.

## 8. How large, and would the tree have caught it?

Multiply N by G, the number of times more men alive now than in Moses' day. Two
standard reconstructions of ancient population disagree about this by a factor of
two and a half, so both are carried. `sources.md` §1 explains the choice.

One sampled leaf on the frozen tree stands for about **230,000 living men**, but
only if testing were spread evenly across the world's men, and it is not. Testing
is project-driven, and Near Eastern and Jewish lineages are heavily
over-represented while much of Asia and Africa is thin. So leaf counts are
compared with leaf counts below, and men-counts are not printed for any named
branch. `flags.md` §7a records what this does to the comparison. The measured
houses run: median 1 leaf, p75 = 4, p90 = 14, p95 = 31, p99 = 155,
largest 3,599. A branch of 10^8 men is 435 leaves, which is the size at which one
would be hard to miss.

| | G = 82 (HYDE) | | G = 207 (compilation) | |
|---|---|---|---|---|
| | men today | leaves | men today | leaves |
| low | 7.6 million | 33 | 19.3 million | 84 |
| **central** | **54.6 million** | **238** | **137.9 million** | **600** |
| high | 118.0 million | 513 | 297.8 million | 1,295 |

**This is where the reading has become harder, and the record says so.** The
central prediction sits above p99 on either reconstruction, meaning among the
ten or so largest branches out of 1,091. On the second reconstruction it passes
the point where a branch should have been noticed, and the high row passes it on
both.

The first draft said the tree's silence was what this reading predicts. On the
rebuilt numbers that holds for the low row only. For the central and high the
claim inverts: **this reading now predicts a large branch, forking about 2900 BC,
sitting inside a house of Adam's era.** That is a harder claim and a more
checkable one.

### 8.1 What the tree contains at that size

Large young branches are not absent from the tree. R-L151 alone holds 3,143 of
the 17,611 sampled leaves, about 720 million living men, with a common ancestor
at 2850 BC. What is nearly absent is that size **with the predicted structure**:

| filter | count |
|---|---|
| branches of 100 leaves or more, forking between 3300 and 2050 BC | 114 |
| of those, having an ancestor node whose formation date falls between 5450 and 4850 BC | **13** |
| distinct houses containing them | **3** |

The three are J-Z1853 (formed 5350 BC), N-M2126 (5450 BC) and E-M4694
(6350 BC). An earlier version of this record named the third as E-FGC91110,
a younger node under E-M4694 that formed about 4850 BC; it is not a family by
the rule the count uses, since it was not yet alive at 5170 BC. Corrected
3 September 2026; `candidates.md` records the check and the size sweep. The second filter is looser than the structure this reading predicts,
which is an immediate parent near 5100 BC with an empty stem below it. Nobody has
tested for that shape at this size. R-L151 fails the second filter cleanly: its ancestry runs from R-L23
at 6,400 years before present straight to R-M269 at 13,300, with nothing in the
window. Ninety-nine in every hundred large young branches are excluded by the
nesting condition alone.

Against the rebuilt central prediction of 238 leaves at G = 82:

| | leaves | against the central prediction of 238 |
|---|---|---|
| J-Z1884 | 2,259 | 9.5 times |
| J-Y2919 | 124 | half |
| J-Z640 | 121 | half |

Men-counts are deliberately absent. Converting 2,259 leaves at the uniform rate
would give 520 million men, one man in eight on earth, which is what the sampling
bias above produces when it is taken literally.

No size criterion was frozen before those figures were read, and none is set
here. `flags.md` §16 records why, and what would restore one.

### 8.2 The undersampling escape does not work any more

A small clade can hide if its region is undersampled. That cover is gone. Noah's
side branches are placed by Genesis 10 in Anatolia, the Aegean, Iran and Egypt,
which are among the better-sampled populations on the tree. And absorption spread
these lines widely: Edom's men were circumcised into Judea about 125 BC, Midian's
Kenite branch settled inside Judah, and Aramean and Arab populations absorbed the
rest. The prediction is a line spread across many present-day peoples, not
concentrated in one.

---

## 9. Cain's line

Genesis 4 describes a line that multiplied into city-building, herding, metalwork
and music. Cain is Adam's son, so his male line carries the same chromosome, and
this site puts him in our world from about 5000 BC while the flood ended the
other one. So his line runs alongside Noah's, untouched by the ark.

Three surviving lines at the ark is the narrowest point of **Noah's** channel. It
is not the narrowest point of Adam's line in our world, which has been here
through Cain for two thousand years by then.

K, the number of Cainite males at 1266 BC, is bracketed 5,000 to 450,000 and
constrained by nothing: there is no ancient DNA from Mesopotamia in the window.
It is carried on the same footing as the side branches of section 4, since both
are lines the text says are there and neither is counted anywhere. At G = 82 it
would add between 0.4 and 37 million further men.

---

## 10. What the tree cannot settle

No ancient DNA has been recovered and attributed to Edom, Moab, Ammon, Aram or
Midian. That is a limit of the evidence, not a gap that more work closes, and it
means every claim about those lines rests on the modern tree alone.

---

## 11. Does the Bible's own account of Israel hold together?

This asks a different question from sections 6 to 8, and the numbers that decide
it barely move the footprint. Two kinds of evidence bear on it. The first is
arithmetic and needs no genetics. The second is genetic, and it is the sharper of
the two.

### 11.1 The arithmetic

Israel enters Egypt as about 64 male descendants (Genesis 46:26) and leaves 215
years later on this site's chronology. The fastest sustained natural increase ever
recorded in a human population is **4.12% a year**, the Hutterites from 1880 to
1950 (Eaton and Mayer 1953).

| | Jacob's line at the Exodus | growth needed |
|---|---|---|
| ledger, low | 33,400 | 2.91% |
| ledger, central | 53,500 | 3.13% |
| ledger, high | 70,200 | 3.26% |
| **literal, 603,550** | 603,550 | **4.26%, above the record** |
| **literal, all ages** | 1,207,100 | **4.58%, above the record** |

**This table depends on the 215-year sojourn**, which this site takes from the
Greek and Samaritan reading of Exodus 12:40. On the Hebrew text read as 430 years
in Egypt, the literal census needs 2.13% a year and clears the record
comfortably. The growth check tells against the literal reading only for a reader
who accepts the short sojourn.

The ledger reading itself is not comfortable either. Sustaining 3.13% a year for
215 years exceeds anything any population has held for that long; the Hutterite
record of 4.12% ran for seventy years with modern medicine behind it. Exodus 1:7
claims exceptional fruitfulness and the arithmetic requires it, at the edge of
what has ever been observed.

**The firstborn ratio.** With 22,273 firstborn males and 1,207,100 males of every
age, there would be one firstborn for every 54 males, which is 54 sons in a
household. No fertility regime reaches it.

**The horizon.** On the literal reading Terah's descendants alone come to about
3.4 million men, which is half of every man in the world Genesis 10 describes,
before one of Noah's other eighty lines is counted.

### 11.2 What the male family tree of Jewish men shows

Genesis and Numbers describe Israel as a set of nested male lines: twelve tribes
from Jacob's twelve sons, Levi's descendants set apart, and within Levi the
priesthood running through Kohath and Amram to Aaron. Read as strict biological
descent, that structure makes a prediction the tree can check. Kohanim must be a
branch inside the Levites, because Aaron is Levi's great-grandson.

**The tree does not show that.** The best-studied lineage among Kohanim, on
haplogroup J-P58, accounts for 46% of sampled Kohanim and no more; the rest are
spread across twenty other haplogroups. The lineage most common among Ashkenazi
Levites belongs to a different haplogroup altogether. Two groups that must share a
male line if the text is read as biology do not share one in the data.

What follows is narrower than it first looks, and I had it wider in an earlier
draft.

The discordance bounds the **total** admixture across three thousand three hundred
years. It does not reach back to the composition of Israel in Moses' day.
Unrecorded paternity alone, at well under one per cent a generation across a
hundred and ten generations, leaves about 46 per cent of founding lineages intact,
which is close to what the priestly data shows. So a reader who takes every
Israelite at the Exodus to be Jacob's biological descendant, and allows nothing
afterwards but ordinary leakage, predicts the same modern picture this site does.

**What the data does rule out** is a patriline held closed at every point from
Jacob to the present, with tribal and priestly membership passing only by blood
and nothing entering at any stage. The tree would then show one Israelite line
branching into twelve with the priesthood nested inside Levi, and it shows nothing
of the kind.

**What the data cannot do** is separate admixture at the Exodus from admixture in
the two thousand years since. So it neither confirms A2 nor constrains it, and no
figure in section 2 can be read back from it.

Genetics therefore bears on the Bible's account of Israel, and it does not bear on
the number this model needs.

### 11.3 What that case also demonstrates

The search for a priestly lineage is the closest precedent for what this site
attempts: a claimed patrilineal descent from a named man in Scripture, checked
against the measured tree by mainstream geneticists.

Its history is worth reading before trusting any result of the same kind. The 1997
claim rested on two markers. The signature it identified was later found across
the Near East and among the Lemba of southern Africa, and its specificity
collapsed. A narrower signature published in 2009 recovered part of the finding,
with two coalescence estimates that do not agree: 3,190 years give or take 1,090
from one method, and a median of 4,415 years from another whose 95 per cent
interval runs from 1,130 to 21,530. A separate published estimate puts it near
1,075 years. The dating is disputed. At no point did the authors put Aaron's name to it.

The same caution applies to every date-based filter used on this site.

## 12. What is load-bearing, and what is not

**Survival holds on every reading and depends on nothing contested.** From the
ledger floor to the literal ceiling, Adam's male line reaches the present with
probability indistinguishable from 1. So the question the copy proposal answers is
forced on any reading of the numbers, given that a branch of the predicted size
would be visible in what has been sampled.

**Footprint size is decided by the reading of Genesis 10, not by the census.**
Israel is 8% of N. What sets the footprint is the ceiling in section 5: what
share of the Genesis 10 horizon Noah's lines were by 1266 BC, and how fast the
side branches grew. The first draft named the census as the load-bearing term.
That was wrong, and `flags.md` §1 and §2 record it.

**The census reading is load-bearing elsewhere, and in two ways.** It calibrates
the one countable Noahic house, which is what shows how fast a flourishing line
grows and therefore why the others cannot have grown that fast. And it carries
the whole of section 11.

## 13. Reproducing every number

    A1  = 22,273 x c
    T   = A1 x A2 x (1 + A3)
    N   = min(T + S, ceiling)
    P(extinct)  = exp(-s N)
    carriers    = N x G
    leaves      = carriers / 230,000

`verify.py` in this directory recomputes every table above from the constants in
sections 1 to 6 and prints them. `build_model_json.py` writes
`data/adam-model.json` from the same constants.