cpt_phase_zero
plain-language theorem explainer
The raw CP-violating phase and its time-reversed counterpart cancel: their sum is identically zero, so the combined CPT phase vanishes. Cite this when checking that Recognition Science keeps CPT while allowing weak CP violation from the directed Gray-code Berry phase. Proof is a one-line ring identity on the reals.
Claim. If $\delta$ denotes the raw CP phase extracted from the directed cycle, then $\delta + (-\delta) = 0$. Equivalently, the CP contribution and the opposite T contribution cancel, so the total CPT phase is zero.
background
The module derives the CKM CP phase from geometric (Berry) phase accumulated by generation eigenstates on the directed 8-tick Gray-code cycle. Consecutive overlaps contribute arguments; the generation-dependent sum is nonzero because the Gray code is chiral and the traversal is oriented, breaking time-reversal while leaving a topological phase that cannot be minimized away.
By contrast, the QCD vacuum angle $\theta_{\mathrm{QCD}}$ is treated as an energetic J-cost parameter and is forced to zero. The present statement sits on the CPT side of that split: forward (CP) chirality and backward (T) chirality are opposite phases of the same raw quantity.
Upstream infrastructure includes the discrete 8-tick phases $k\pi/4$ and the recognition cost apparatus, but this lemma itself only needs the additive group law on $\mathbb{R}$.
proof idea
One-line tactic proof: ring closes $\delta + (-\delta) = 0$ from the ring structure of the reals. No Recognition-specific lemmas are invoked; the content is pure cancellation once the T phase is identified with the additive inverse of the raw CP phase.
why it matters
In the module narrative, weak CP violation is topological (Berry phase on the chiral directed cycle) while strong CP is energetic (J-cost minimum at $\theta_{\mathrm{QCD}}=0$). This identity records that CPT still holds: CP phase and T phase cancel exactly. It pairs with the nonzero CP-phase result and the strong-CP resolution listed in the module header, and sits next to generation-dependent Berry phases and sign change under cycle reversal. No downstream theorems currently depend on it; it is a local CPT bookkeeping step inside the CKM-from-cube derivation, consistent with the eight-tick octave (T7) setting of the cycle.
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