parity_independence
plain-language theorem explainer
The nine standard basis vectors on the recognition ledger's parity space are pairwise distinct whenever their indices differ. Anyone citing algebraic independence of the nine Z₂ parities (spacetime, color, generation) needs this fact. The proof is a one-line term wrapper that applies the sibling distinctness lemma.
Claim. For every pair of distinct indices $i,j$ among the nine parity labels $\{P_{cp},P_{B-L},P_Y,P_T,P_C^{(1)},P_C^{(2)},P_C^{(3)},P_\tau^{(1)},P_\tau^{(2)}\}$, the corresponding standard basis vectors $e_i$ and $e_j$ in $(\mathbb{Z}/2\mathbb{Z})^9$ are unequal.
background
The module counts the independent $\mathbb{Z}_2$ symmetries of the double-entry recognition ledger under tick reversal and conjugation. There are exactly nine: four spacetime parities (charge-parity, $B-L$, hypercharge, tick), three color parities from the SU(3) Cartan data, and two generation-mixing parities. Tesla's "magnificence of the 9" is read here as this exact count, not numerology.
A parity vector is a map from the nine-index type into $\mathbb{Z}/2\mathbb{Z}$. The standard basis vector at index $t$ is the function that is $1$ at $t$ and $0$ elsewhere. Spatial dimension $D=3$ (forced upstream by the T8/T9 chain) supplies the three color slots via rank-2 Cartan plus diagonal product; the generation count is likewise rank 2.
The sibling lemma already shows that distinct indices yield unequal basis vectors by evaluating at the defining coordinate.
proof idea
One-line term proof: the universal quantifier is introduced as a lambda, and the goal reduces immediately to the sibling theorem that distinct basis vectors differ at their defining index. That sibling opens the functional equality, evaluates at $i$, and simplifies using the definition of the basis vector together with $i\neq j$. No further case analysis or induction is required.
why it matters
This is item 4 of the master theorem on nine parities of the recognition ledger: the nine labels are algebraically independent and span $\mathbb{Z}_2^9$, so the configuration space has $2^9=512$ states. The master packages count, flip-under-tick-reversal, vacuum vanishing, the $4+3+2$ source decomposition, and this independence into one statement.
In the broader forcing chain, $D=3$ (T8) and the eight-tick octave (T7) fix the color and temporal slots that make the count nine rather than some other integer. Without pairwise distinctness of the basis vectors, the claim that the ledger carries nine independent $\mathbb{Z}_2$ constraints would be only a cardinality statement, not a linear-independence statement over $\mathbb{Z}_2$.
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