490d C330 — Constraint rank and600 transport

24 September2026. Step7 of10 in the C324–C333 authorization.

The residual ledger and paired equal-interval fields are dependent consequences of one source-labelled ten-endpoint configuration in each named mode. Adding their equations does not add independent coordinate constraints.

The ten vertices are five SP and five MT endpoints. The fixed edge inventory is four adjacent biography intervals in each tradition plus five same-role460 context differences:13 equations. Exact rational elimination gives rank9, with four cycle dependencies and one common-translation null direction. This is a linear-algebra count at fixed source values, not a count of historical witnesses or distinct source constants.

A basis consists of the four SP biography edges and the five context edges. The MT biography follows from that basis. For each adjacent role pair, SP interval minus MT interval minus the first context difference plus the second context difference is zero. These four identities account for all four redundant edge equations. Appending all25 cross-context residual edges leaves rank9.

The complete source coordinates are recoverable from that basis once one already admitted coordinate is retained. This verifies consistency; the free common-translation direction does not authorize a new anchor or chronology. In particular, Noah950=460+490 is the subtraction of its already held lifespan and context leg. It supplies no additional condition once those are fixed.

The apparent repetition of the two equal fields also has a precise dependency:

Each coordinate in the second row is600 lower than its corresponding coordinate in the first. Two distinct source relations produce that correspondence: Noah's age600 at the Flood, and Shem's lifespan600. Equal numbers do not equate these event meanings. The correspondence holds in both the500/100 and502/98 binding, so the second field repeats the first field's geometry rather than supplying an independent fit.

All2048 independent ten-vertex quarter-phase assignments were tested across the two modes. Only the all-minus and all-plus assignments preserve all13 edges in each mode: four exact complete instances. Equal physical output phases do not merge different cumulative ancestors. In particular, the mixed502 field is not relabelled as C314's exact500 reversal. The600 correspondence here is a finite relation among named endpoints, not a newly declared global translation operator.

Verification and limits

19 main checks and 14 frozen-input/predecessor authentication checks passed. A second main replay produced byte-identical DATA/RESULTS. Independent review and its scope are recorded in the evidence. All frozen originals and standing controls remain unchanged. Verification counts are not independent-witness counts.

Rank, phase and endpoint dependence only. No numerical witness multiplication, source-operator promotion, fitted anchor or canonical change.

Mirror/529 focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction January AD2; no civil year zero. Regular Gears are Noah/Shem/Flood only. SP minus-one person companions remain local to SP Noah/Shem; Flood close is a separate boundary. No upstream/cumulative Gear, new endpoint or phase, fitted anchor, global axis, primer reopening, canonical or graph edit. Capture now; canonize later. Source identity, context, modal binding, generated coordinates and phase ancestry stay distinct.

Decision after this result

C331, proposed and unexecuted: Insert only the existing SP Noah-death and same-Gear MT Noah-birth records into C287’s four complete2K edges, retaining SP primary/companion and MT Flood start/close identities. Which950/490 partitions are source-admitted?

The residual dependence is now explicit. The next bounded step reconnects the qualified death endpoint to the existing529 construction and tests the known companion boundary instead of assuming the primary490 relation transfers to every row.