Source constraints behind the Covenant and cumulative root families

Preparation only. No numbered research actions have been executed. The25 questions in questions_declared.json were written before the symbolic rank and sensitivity calculations. VALIDATION.json reports19/19 checks passed. Three supplied source hashes, nine bounded excerpts and nine literal tables are retained in source_packet.json.

The strongest new result is a distinction in the cumulative/regular joins: when the clutch is derived from the maximum envelope, its Kohath→regular Levi-death landing is forced by the definition. Reusing Levi's lifespan forces the adjacent Levi-birth landing. The next Jacob→Isaac landing adds exactly the earlier147 source-rail condition. This exposes the actual source burden behind several conspicuous dates without rerunning them as independent discoveries.

What differs from C551 and C552

C551 successfully reconstructed eleven Covenant nodes, but its generating inputs included the two161 gaps that the explanation later emphasized. The new packet replaces those with source-local household timing and an independently retained terminal: call→Levi10, call→Joseph14, and1406. It also keeps Ishmael's and Levi's equal137 lifespans separately labelled. The161,299,276 and common1566 values are outputs.

C552 successfully generated42 phase/clutch coordinates from seven base dates and phase/clutch offsets. Here the seven base dates are generated from the Moses anchor and lifespan stack, and the clutch is generated from the difference between that stack and the regular biography. Neither14 nor7 is stored as a free clutch parameter. The literal household14 remains a source datum with its own role.

The source-only19-parameter packet and regenerate.py generate12 regular coordinates (the earlier eleven plus the Covenant anchor), all42 cumulative coordinates, and the old gap values. This is a more explicit dependency description, not a claim that19 is a globally minimal number of historical assumptions. The packet includes source quantities that should not be silently merged merely because their numeric values agree.

Regular source-local coordinates and rank

Use Q and T for the Covenant and terminal. Write qI,qX for Covenant→Ishmael/Isaac offsets; x for Isaac→Jacob; c for Jacob's call age; l,z for call→Levi/Joseph; and I,L,J,Z for their respective lifespans. Abraham's lifespan A is a separate scalar used by the later completion claim.

The source-local values are

For example, Levi birth is Q−qX−x−c−l; his death subtracts L. Joseph birth replaces l by z; his death subtracts Z. The full coordinate matrix for the twelve regular nodes has determinant−1 and rank12 on the first twelve parameters. It is a rooted incidence description: the complete labelled coordinates recover every selected edge quantity by subtraction. A alone is absent from these coordinates because no Abraham biography was added to this local node set.

Common translation adds the same amount to Q and T and to every regular coordinate. Passing to relative coordinates removes one parameter, leaving eleven. An absolute retained anchor restores it. This is a mathematical change of representation; it does not authorize moving the historical anchors.

The source-local packet is not a list of independent textual observations. In particular qI and qX inherit the author's Covenant placement relative to the births. A finer historical source model would resolve those dependencies against Abraham's ages and the chosen Covenant age. The present rank calculations deliberately concern this declared local parameterization.

Which Covenant claims are independent equations?

Let S be Ishmael birth minus Levi birth. Define these nine residuals, each zero at the supplied source packet:

This13-column matrix has rank8. The first eight rows form one convenient basis. Its only row circuit is

A9 = −A2 + A3 + A4 + A5.

Thus the household-middle equality is already forced by the second147 rail, equal lives, twin death arms and Joseph's double span. The four claims must not be treated as unconnected confirmations of that ninth equality.

Several familiar further relations are exact consequences:

• The first vertical60 equality is −A1; the second vertical77 equality is A2.
• The diagonal equality x+c=L is A1−A2.
• Equal birth/death displacement is A3.
• The Covenant→Levi decomposition through Isaac is A1.
• The Covenant→Levi decomposition through Ishmael is a path identity requiring no extra constraint.
• The Joseph/Levi death-gap formula lifespan difference minus birth offset is also a path identity.
• The difference between the two transformed common points is −25/299 times A6. Introducing the common-point variable therefore adds one generated coordinate; requiring both routes to agree adds exactly A6.

The rank is conditional. A7 and A8 are independent when the source lifespans A and J are allowed to vary. But

25 A8 − 2 A7 = 21A − 25J.

Once the supplied A=175 and J=147 are fixed, the right side vanishes and the two radius conditions become equivalent. This is a useful warning against assigning one universal “number of independent coincidences” to the pattern list.

Numeric normalization and identifiability

The relational claim system still leaves five formal directions in the13 source parameters. Fixing the source radius S=161 adds one constraint. Under the other conditions this fixes the second161 arm, J=147 and A=175 as consequences. Fixing Covenant→Levi299 adds another; the E/J condition then fixes Covenant→Joseph276. Fixing Q=1866 adds the absolute frame. Two formal directions remain:

1. x decreases while c increases by the same amount; Jacob's birth and death move together, while his call and the later household remain fixed.
2. qI and qX decrease together; x increases; l decreases; I and L increase together by the same amount.

These are formal sensitivity directions, not admitted chronology variants. They show that the selected patterns and prominent numeric totals still do not determine all the supplied biological/household data. The source qI=1 and x=60 each remove one remaining direction, restoring rank13. The known age and event data therefore retain explanatory work after the pattern reconstruction.

diagnostics.json stores the full source sensitivity matrix, five null vectors of the relational system, the two residual vectors after normalization, and an independently checked nonzero8×8 minor. None of these counts is a probability calculation.

The42-coordinate clutch field as a constrained array

The exact family is

X(i,j,k) = b(i) + j p − k h,

with seven rows i, two phase states j=0,1, and three clutch positions k=0,1,2. Before imposing further source equalities, the seven b values, p and h form nine parameters. The42-coordinate field has rank9 and33 independent linear output relations.

A smallest observation set is the seven unshifted Nisan coordinates, one unshifted Aaron coordinate, and one half-clutch Nisan coordinate. Its observation determinant is−1. This gives a concrete identifiability result beyond merely regenerating all42 entries. Without any Aaron observations, any shifted observations, or one entire source row, the unconstrained model has rank8.

Two additional equalities can be imposed separately:

• h=2p, using the exact source7 and3.5, lowers the dimension to8.
• The repeated137 source lifespan, b(Levi)−b(Kohath)=b(Amram)−b(Moses), also lowers it to8.
• Imposing both lowers it to7.

The first is a relation between the supplied numeric values, not a universal identity defining every display branch. Under the whole-year phase3 with h=7, h−2p=1. The exact and whole-year fields require separately stated models. The repeated137 is also a labelled equality of two biographies; it is not permission to conflate Levi and Amram.

The endogenous clutch removes apparent extra joins

Let M be Moses' fixed birth, E the Exodus, B and D Levi's regular full-state birth and death, and L,K,A_m the selected Levi, Kohath and Amram lifespans. The source maximum-envelope construction gives

d = [L+K+A_m+(M−E)] − (B−E) = K+A_m+M−D.

The Exodus anchor cancels. Cumulative Kohath is C_K=M+A_m+K, so

d=C_K−D.

Consequently C_K−d=D is a defining consequence of the source envelope. Cumulative Levi is C_L=C_K+L, and the regular Levi biography has B=D+L. Hence C_L−d=B follows from the same lifespan value. It adds no separate landing condition.

The entire translated seven-node root, measured from D, is

Every original row contains M once, and d contains M once, so M cancels from all seven translated coordinates when the source-derived clutch is recomputed. This is a dependency statement. It does not move Moses' historical birth or claim that the original cumulative path is anchor-free.

The translated Jacob boundary is B+J. Requiring it to equal regular Isaac birth X adds the residual

B+J−X = −A1.

Thus the Jacob→Isaac join contributes the earlier source147 rail condition, not a new unrelated chronological law. The Amram and Moses-side landings are then the extremal capacity path below Levi death; they remain generated interfaces rather than historical successor births.

The exact Aaron field retains the same Nisan-derived d and adds p, preserving its3.5 displacement. Recomputing a separate clutch from each phase to the same fixed regular D would cancel that phase displacement. That is a different rule and is not the source execution. The declared anchor/phase policy therefore carries actual information.

Coupled source-only regeneration

The combined regular and cumulative model has19 declared parameters: the13 listed above plus Moses birth, the215 comparison shift, Isaac/Kohath/Amram lifespans and exact phase. The complete55-output array—twelve regular nodes, Abraham's lifespan scalar and42 cumulative coordinates—has rank19 before imposing the equalities selected for particular patterns. No independent clutch parameter is added.

Abraham, Jacob and Levi lifespans are reused across the regular and cumulative paths as the same source values. Ishmael's and Amram's137 lives remain separately labelled even though they equal Levi's number. This distinction separates shared-source consequences from equal-valued observations concerning different people.

Twenty-five bounded candidate steps

The earlier question file is authoritative for the pre-calculation wording. These are candidate research units for root reassessment, not executed numbered steps. Adjacent candidates should be combined if the result would otherwise be mere bookkeeping.

This module deliberately leaves the active Actual/Rounded inverse-junction analysis to the root's parallel work. Its transferable result is how to distinguish a generated equality from a further source constraint, using a complete source path rather than isolated endpoint agreement. It does not repeat the C890–906 endpoint-coordinate results or add a second inversion.

Files and checks

• questions_declared.json:25 questions written before calculations.
• source_packet.json: exact local sources, source excerpts/tables,19 source-local inputs and their roles.
• regenerate.py and source_only_reproduction.json: complete source-only reconstruction.
• analyze.py, exact.py, diagnostics.json: standard-library exact affine/rational calculations; no external algebra package required.
• validate.py, VALIDATION.json:19 checks, including all42 literal coordinate values, source hashes, a nonzero modular8×8 minor, all33 clutch relation vectors, null vectors and forced joins.

No canonical source was edited. No primer,1486 calculation, second inverse, upstream Gear transport, remote extension or statistical tally was introduced.