vacuum_is_zero_vector
plain-language theorem explainer
The vacuum parity assignment on the recognition ledger is identically the zero vector in (ℤ/2ℤ)⁹: every one of the nine independent parities evaluates to 0. Anyone working the nine-parity ledger, tick-reversal conjugation, or vacuum-page balance cites this. The proof is pure definitional equality (rfl).
Claim. The vacuum parity vector equals the zero map on the nine parity indices: for every index $i$ among $\{P_{cp}, P_{B-L}, P_Y, P_T, P_C^{(1)}, P_C^{(2)}, P_C^{(3)}, P_\tau^{(1)}, P_\tau^{(2)}\}$, the vacuum value is $0 \in \mathbb{Z}/2\mathbb{Z}$.
background
The module formalizes nine independent $\mathbb{Z}2$ parities that govern the double-entry recognition ledger under tick reversal and conjugation. They split by origin into four spacetime parities ($P{cp}$, $P_{B-L}$, $P_Y$, $P_T$), three color parities from the SU(3) Cartan, and two generation-mixing parities. The count nine is the exact dimension of this discrete symmetry space, not numerology.
A parity vector is a map from the nine-index type into $\mathbb{Z}/2\mathbb{Z}$. The vacuum parity vector is defined pointwise as the constant-zero assignment: every index is sent to $0$. In the language of the module, this is the vacuum page of the ledger, the unique zero element of the $(\mathbb{Z}/2\mathbb{Z})^9$ vector space of parity configurations.
Upstream, the fundamental tick $\tau_0 = 1$ sets the discrete time quantum against which tick reversal acts; the present statement only needs the index type and the vacuum definition, not the dynamics of the tick itself.
proof idea
One-line term proof by reflexivity. The vacuum parity vector is defined as fun _ => 0, so it is definitionally equal to the constant-zero function on the parity-index type valued in $\mathbb{Z}/2\mathbb{Z}$. No lemmas are applied; rfl closes the goal.
why it matters
This pins the vacuum page as the algebraic origin of the nine-parity space: the unique zero of $(\mathbb{Z}/2\mathbb{Z})^9$. The module overview lists the companion claims that all nine parities flip under conjugation plus tick reversal, that the scalar vacuum page has vanishing parities, and that the nine are algebraically independent over $\mathbb{Z}_2$. Establishing that vacuum is literally the zero vector is the baseline those statements sit on.
In the broader Recognition framework the nine parities constrain which ledger configurations are admissible. The vacuum being zero means the empty page carries no parity charge, so any nonzero parity vector is a genuine excitation relative to vacuum. The doc-comment further identifies this zero as the unique fixed point of parity-preserving operations. No downstream dependents are recorded yet; the lemma is infrastructure for the parity-count, independence, and tick-reversal flip theorems in the same module, and for the Tesla "magnificence of 9" decoding (exact independent $\mathbb{Z}_2$ count on the vacuum page).
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