source_decomposition

The Nine Parities module counts the independent $\mathbb{Z}_2$ signs that constrain the double-entry recognition ledger under conjugation and tick reversal. The full set is

${P_{cp}, P_{B-L}, P_Y, P_T, P_C^{(1)}, P_C^{(2)}, P_C^{(3)}, P_\tau^{(1)}, P_\tau^{(2)}}$.

These nine signs are not an arbitrary list. They arise from three independent structural sources: four spacetime parities (charge-parity, $B-L$, hypercharge parity, and tick reversal), three color-charge sign flips from the SU(3) Cartan, and two generation-mixing signs from the rank-2 structure of three generations.

The module's overview states that Tesla's "magnificence of the 9" is exactly this independent parity count on the vacuum page of the ledger, not numerology. The present identity is the bare arithmetic of that source split.

One-line term proof: norm_num discharges the ground arithmetic $4+3+2=9$ in $\mathbb{N}$. No lemmas from the tick, Clifford, or holography imports are required; those edges are ambient module context only.

Feeds the master theorem nine_parities_master, whose third clause records that the nine parities "Decompose as 4 (spacetime) + 3 (color) + 2 (generation)". That master result also packages the flip-under-conjugation+tick-reversal law, vacuum vanishing, algebraic independence over $\mathbb{Z}_2$, and the $2^9=512$ configuration count.

In the Recognition framework this is the bookkeeping step that makes the nine-parity story a forced source decomposition rather than a bare cardinality claim. It sits beside parity_count_eq_nine and the trichotomy classifiers for spacetime, color, and generation indices. The eight-tick octave and $D=3$ enter only as background for why spacetime and color ranks look as they do; this lemma itself is pure addition.