# C330 independent exact dependency review

24 September 2026. This audit uses only the two frozen C329 modes, with ten endpoint variables ordered SP's five roles followed by MT's five roles. The mixed G2 Noah/Flood–G1 Shem binding retains its source labels and is not relabeled as a coherent single-Gear biography.

**Result:** the13 named duration equations have exact rational rank9, four independent cycle dependencies and one common-translation null direction. All25 cross-context residual equations are already forced by them; appending those equations adds rank0 in either binding.

| Object | Equations / variables | Rank | Consequence |
|---|---|---:|---|
|Four SP adjacent spans, four MT adjacent spans, five same-role460 legs|13 /10|9|Four row dependencies; one common-translation freedom|
|Chosen basis: four SP adjacent spans plus five460 legs|9 /10|9|All other displayed durations follow|
|All13 plus25 cross-context residuals|38 /10|9|Every added residual has rank increment0|

For adjacent role index i, write the SP span as `s_i`, the MT span as `m_i`, and the same-role SP−MT span as `c_i`. The four independent cycle relations are

`s_i−m_i−c_i+c_(i+1)=0`, for i=0,1,2,3.

They close in both modes. The independently computed left nullspace has dimension4; these named cycles span it. The right nullspace is exactly the all-ones vector, so duration equations alone leave common absolute translation undetermined. The source dates remain external admitted inputs; no absolute anchor follows from this rank calculation.

For any MT anchor role a and SP terminal role c, the residual is the signed SP adjacent path from a to c, minus the anchor's context leg. The JSON supplies all25 explicit coefficient vectors in the selected nine-edge basis, checks their incidence rows and forced values, and reproduces the C329 residuals. Each added equation individually and all25 together add no rank. The augmented systems remain consistent with rank9.

## The paired fields and exact phases

The birth field `(SP Noah, MT Noah, SP Shem birth, MT Shem birth)` and Flood/death field `(SP Flood, MT Flood, SP Shem death, MT Shem death)` differ componentwise by600 in each named binding. The same600 comparison holds at both inherited like-phase components.

This transport follows from the source's Noah→Flood600 and Shem-lifespan600 relations; their MT copies follow through the same-role460 constraints. It identifies a repeated metric configuration within each binding. It does not make Noah and Flood the same event, merge their distinct cumulative source branches, or equate the Shem terminal's inherited ancestry with the anchor's. The500/40/960 configuration and502/42/962 configuration remain separately bound as established in C329.

All1024 independent assignments of the held two physical phases to the ten endpoints were checked per mode. Exactly two preserve all13 constraints: all-minus and all-plus. Thus the complete enumeration has2048 assignments and four successes. This is the phase consequence of a connected difference system: every required edge forces equal output offsets at its ends. It is not evidence of shared cumulative ancestry.

The ranks, nullspaces and phase counts describe dependence among admitted equations. They do not count independent historical witnesses, prove source intention or establish a preferred chronology.

## Reproduction

`C330_Independent_Audit.py --sources DIR --out DIR` reads frozen `PREVIOUS_DATA.json` (C329) and uses exact `Fraction` arithmetic with a standalone row-reduction implementation. Its JSON contains the incidence matrix, both nullspaces, named cycle certificates, the nine-edge basis, all25 residual certificates per mode, all2048 phase assignments and the source-qualified600 comparison. **51 checks passed.** The script, results and review are frozen for handoff. No new endpoint, phase, context, expansion, fitted anchor or C331 calculation was performed.