technical 07 counts · 490d Research

7. Counts, grouping and symmetry

An ordered prefix map with its full selected basis is an invertible integer encoding of the source counts. For Esau, the six clean prefixes, third and fifth female prefixes, and grand total recover all nine counts by addition and subtraction. This is a complete alternative description of nine numerical inputs, not their derivation from fewer independent values.

A grouping $g$ carries an operation $r$ to the grouped object precisely when

g(i)=g(j)\ \Longrightarrow\ g(r(i))=g(r(j)).

The condition says that one group cannot acquire two different images. For Toledot, occurrences 9 and 10 share Esau’s section. Reflection $r(i)=12-i$ sends them to 3 and 2, distinct sections. Reflection therefore does not descend to the ten-section grouping. Giving Esau multiplicity two still preserves the eleven-occurrence measure.

The NT primary path has 77 edges of seventy years. Grouping them into eleven complete seven-edge blocks preserves duration and reverses block order under reflection, retaining twelve block boundaries. The separately displayed AD65–135 extension is outside that primary object.

For the NT Key knot, if $u$ is slot width and $w$ the selected radius, the two comparison equations reduce to the same relation $w=69u$ . The separate placement $35u=2450$ fixes $u=70$ , then $w=4830$ . Thus two displayed Key landings do not provide two independent scale conditions.

On the shared-name birth comparison, define $D_i=NT_i-S_i$ . With coordinates decreasing toward later births,

D_{i+1}-D_i=(S_i-S_{i+1})-(NT_i-NT_{i+1}).

All twenty-one edges follow this recurrence. Omitting Cainan from the shared-name display leaves an NT gap of 140 between Arphaxad and Shelah. Native LXX retains intervals 135 and 130 and changes the successive comparison values by 65 and 60.

Source/proof route: inherited structural-count and NT-knot models; C1266–C1272; grouping relation C1309.