# Exodus-era paternal descendants: first-stage rebuild

Working analysis for Mike Russell, 7 September 2026. Not website copy and not a replacement for the archived model. The calculation is reproducible with `python3 calculate.py`; `results.json` contains the outputs. No new population distribution has been fitted. Values in sensitivity tables are not confidence limits.

## Question and accepted premises

Accepting Mike's reading of the census and genealogies, what can be inferred about Noah's paternal descendants alive at the firstborn census?

- Accept 22,273 as a count of actual firstborn males at least one month old.
- Use Mike's household interpretation, controlled by Exodus 12:30. Do not substitute a maternal-first-child interpretation as the baseline.
- Exclude both military ledger totals and the 22,000 Levite ledger total from population calibration. Mike explicitly confirmed that 22,273 means actual firstborn males and 22,000 is nominal. The mechanism of redemption through nominal units remains an explanatory question, not a population input.
- Accept the recorded paternal relationships and the site's Henry Smith-based reconstruction of Genesis 5 and 11. Do not substitute either blanket MT or blanket LXX values.
- Noah's paternal descendants are a subset of Adam's in the hypothesis. Cain's line must not be counted as Noah's. Matching chromosomes would not erase this genealogical distinction.
- The immediate target is the census date, not departure day. Numbers 1 locates its census in the second year after departure. Moving a count to the Exodus itself requires a births/deaths/membership adjustment. Use 'Exodus-era' until that adjustment is made.
- All calculations below concern males aged at least one month. Newborns below the threshold and Levites, where outside the counted population, need separate treatment.

## 1. A direct calculation that needs no household multiplier

Let F = 22,273 and p_F be the fraction of these particular counted firstborn whose paternal ancestry reaches Noah. Then F*p_F is their number. It is a lower contribution to the total living Noahic paternal population, before any other males are added.

| Assumed Noahic fraction of counted firstborn | Noahic males already in the count, approximately |
|---|---:|
| 10% | 2,227 |
| 25% | 5,568 |
| 50% | 11,137 |
| 60% | 13,364 |
| 80% | 17,818 |
| 90% | 20,046 |
| 100% | 22,273 |

These are conditional calculations, not estimates of the fraction. Fractional people in the source output reflect percentage arithmetic; a realised count is an integer.

To establish at least 10,000 from this group alone requires at least 10,000 of the 22,273, about 44.9%. To establish 5,000 requires about 22.4%. We can therefore investigate a concrete threshold instead of needing a precise ancestry percentage immediately.

This does not assign Noahic status to a random sample. The count supplies F, not observations of paternal ancestry. There is no binomial likelihood for p_F from F alone.

## 2. The household calculation

For the population to which the count applies, let H be its number of households, m its mean number of eligible males per household, and f its mean number of counted firstborn per household. Then:

F = H*f; M = H*m = F*m/f.

These are identities for pooled totals, not expectations of household-by-household ratios. Household variation and correlation do not invalidate the identities; they do prevent casually replacing m/f by an average of individual household ratios.

Mike's controlling verse supports household coverage under his reading. Coverage means at least one, not automatically exactly one. Also the transfer from Egyptian households at the plague to Israelite households at the later census is an interpretive assumption to record. The count itself does not fix H.

Exactly one eligible firstborn per household is a useful stronger scenario: f = 1. If m = 3, it gives 66,819 eligible males. If households average 1.5 counted firstborn at the same m, it gives 44,546. At f = 2, it gives about 33,410. None of these illustrative household structures has been empirically established for Israel.

A father, son and grandson can each be firstborn in their respective families. Household membership and the inheritance of firstborn status through household formation must therefore be specified before a generational simulation can calibrate f. 'Sons per household' is not necessarily the number of all living males per household.

Let p_M be the Noahic fraction of all eligible males. Their contribution is F*(m/f)*p_M. Do not silently assume p_M = p_F: household composition or age structure could differ by paternal origin.

| Males per counted firstborn | 25% Noahic | 50% Noahic | 80% Noahic | 100% Noahic |
|---|---:|---:|---:|---:|
| 2 | 11,137 | 22,273 | 35,637 | 44,546 |
| 3 | 16,705 | 33,410 | 53,455 | 66,819 |
| 4 | 22,273 | 44,546 | 71,274 | 89,092 |

The rows are sensitivity examples, not population bounds. The old central Israel figure, about 53,455, is reproduced exactly from its stated assumptions. Its arithmetic survives. Its interpretation as a probable estimate still requires support for the assumptions.

## 3. What the genealogies contribute

The recorded chain through Jacob establishes paternal ancestry within the accepted reading. It supports identifying particular lines, including the named Levite families of Exodus 6. A population-wide percentage also requires a rule connecting membership to biological descent.

Genesis 17 explicitly includes purchased males in the covenant household. Exodus 12 includes other people accompanying Israel. Neither supplies the proportion of non-Noahic fathers in the census. Outsiders can themselves be Noahic; females entering a household do not change its sons' paternal ancestry when the father is Noahic. Cainite males require their own category.

A mechanism can make the uncertainty explicit. In a deliberately simplified generation model, if a fraction a of each new male generation comes from non-Noahic paternal origins and all resident lines otherwise reproduce equally, p_next = (1-a)*p. Starting from a wholly Noahic population gives p_g = (1-a)^g. This is an identity under that model, not a description established for Israel.

To reach only 10% Noahic ancestry requires repeated outside contributions of about 31.9%, 25.0%, or 20.6% per generation over 6, 8, or 10 generations respectively. Those generation counts are illustrative and are NOT substitutions for the named chronology. A single large addition, unequal reproduction, or earlier household incorporation produces different histories. The calculation therefore cannot justify a prior by itself, but it makes claims about extensive replacement explicit and examinable.

A more general model must retain Noahic males N and other males U separately:

N_next = R_N*N + I_N; U_next = R_U*U + I_U.

R denotes surviving male contribution over the interval; I denotes incoming males, with a stated entry time. A chronological individual model would also track age, deaths and births. The named sons' birth dates constrain particular events, not the entire fertility schedule. We have not filled these unknowns with fitted distributions.

## 4. Other descendants and scope

Noah's total is the disjoint sum of his descendants inside the counted group, excluded Israelite groups such as Levites if appropriate, and those outside Israel. Adding these groups cannot reduce the total. Their existence does not supply their headcounts at this date. We leave them unquantified rather than assign them zero or borrow Israel's growth rate.

The direct firstborn contribution is therefore useful even before the rest can be estimated. It may later support a survival calculation. It does not itself establish survival, modern numbers, detection, identity, or a requirement that Adam's chromosome was copied.

## 5. Textual issues needing Mike's consideration

**Levites.** The site's census argument treats the 273 redeemed firstborn as a numerical surplus over 22,000 Levites. An actual person-for-person surplus supports taking the corresponding replacement number seriously, even though it is round. If the 22,000 is merely nominal, the mapping from ledger units to persons redeemed needs explaining. Mike has now selected the nominal-ledger interpretation. Do not add 22,000 people. The literal-Levite sensitivity calculations in results.json are retained only as an explicitly excluded alternative, not part of the adopted calculation. Treating the redemption as an account can preserve the subtraction arithmetically, but an account unit must be distinguished from an actual substitute person; this explanation is not yet established. Numbers 3 also has a subtotal issue: 7,500 + 8,600 + 6,200 = 22,300; its NIV footnote records 8,300 in some Greek manuscripts, giving 22,000. Do not resolve this by assuming all Greek readings are Mike's choices.

**Named enumeration.** The site's assertion that the firstborn count uniquely uses named enumeration is incorrect: Numbers 1:2 and 1:18 also use that language. Accepting the ledger interpretation for this calculation does not require retaining that supporting argument. We do not use the wording contrast to validate F; F is an explicit premise accepted from Mike.

**Initial population.** The old 64-male seed comes from an interpretation of the Genesis 46 travelling list. It must not be silently equated to everyone in the paternal population at settlement: Joseph's household was already in Egypt, Jacob himself is a separate counting question, and the textual totals require reconciliation. No growth rate in this rebuild uses that seed. Smith's Genesis 5/11 choices do not settle every other textual census issue automatically.

## 6. Historic evidence and what has actually been obtained

Bagnall and Frier's study of Roman Egyptian census declarations is relevant comparative evidence for household composition, age and survival. It concerns a much later population, not Exodus-era Israel. The publisher's household chapter summary and the book catalogue were inspected; the full numerical tables were not available without access. No claimed Egyptian household mean or mortality distribution has been extracted from the restricted book or imported into this calculation.

Historical household counts would help constrain m. They will not automatically constrain f under Mike's particular firstborn definition, nor p_F or p_M. For those we need an explicit household rule and evidence about paternal incorporation. HYDE world totals cannot supply these quantities. We therefore have not replaced missing information by world-average growth or arbitrary Monte Carlo priors.

## 7. Current conclusion and next discriminating work

The count can support thousands or tens of thousands of Noahic males under explicit ancestry assumptions. It already gives a simple threshold: a Noahic share of approximately 45% among the counted firstborn suffices for 10,000, without any household multiplier. The original central Israel calculation remains a permissible conditional scenario.

Neither a best estimate nor a probability interval is identified by the information supplied so far. Many household structures and paternal fractions yield the same firstborn count. Counting simulation draws across an arbitrary grid would not turn them into historical probabilities.

The next work is to operationalise household firstborn status, obtain usable comparative household data, and examine what textual membership evidence can constrain about the paternal fraction. The nominal Levite total is excluded by Mike's decision; actual Levite population and the scope of the firstborn count still require care. If these cannot constrain the inputs, the defensible output remains conditional thresholds rather than a statistical certainty claim. Modern survival and sampling remain later, separate questions.

## Sources inspected

- Local `numbers/census.html`, entire extracted prose; archived `research/adam-population-model/model.md`, `flags.md`, `sources.md`; chronology metadata in `data/patriarchs-ages.json` and `data/patriarchs.json`.
- [Numbers 3, NIV](https://www.biblegateway.com/passage/?search=Numbers+3&version=NIV): count, replacement arithmetic, age threshold and variant note.
- [Numbers 1, NIV](https://www.biblegateway.com/passage/?search=Numbers+1&version=NIV): date, registration wording and separate treatment of Levi.
- [Exodus 12, NIV](https://www.biblegateway.com/passage/?search=Exodus+12&version=NIV): household premise and accompanying population.
- [Genesis 17:12–13, NIV](https://www.biblegateway.com/passage/?search=Genesis+17%3A12-13&version=NIV): covenant household inclusion.
- [Exodus 6:14–27, NIV](https://www.biblegateway.com/passage/?search=Exodus+6%3A14-27&version=NIV): named family relationships.
- [Genesis 46, NIV](https://www.biblegateway.com/passage/?search=Genesis+46&version=NIV): different scopes of the travelling family and Joseph's household; not adopted as a blanket textual choice.
- [Bagnall and Frier, Households](https://www.cambridge.org/core/books/abs/demography-of-roman-egypt/households/B87EA3F18CEF3E1CD373D97C47BEF93F), chapter summary only.
- [The Demography of Roman Egypt, book access record](https://www.fulcrum.org/concern/monographs/pg15bf06h), full text restricted.