recognitionUpdate
plain-language theorem explainer
The recognition update on eight-tick complex signals is the linear map that first applies the bare cyclic one-tick shift, then projects onto a chosen structured sector (all odd Fourier modes, no DC, optional extra non-DC modes). Operator-core and T7/T8-bridge work cite it as the concrete one-step evolution law. The body is a one-line composition of the sector projector with the shift.
Claim. For each structured sector $S$ (a finite set of DFT modes on $\mathrm{Fin}\,8$ that contains every odd index and excludes the DC mode $0$), the recognition update is the $\mathbb{C}$-linear map $\mathrm{Signal}_8\to\mathrm{Signal}_8$ given by $f\mapsto P_S(\sigma f)$, where $\sigma$ is the cyclic one-tick shift and $P_S$ is the orthogonal projector that retains exactly the Fourier modes in $S$.
background
The ambient carrier is the eight-tick complex signal space $\mathrm{Signal}_8=\mathrm{Fin},8\to\mathbb{C}$, the canonical recognition register forced by the eight-tick octave (T7) once spatial dimension is fixed at $D=3$ (T8). Evolution is written in the DFT-8 eigenbasis of the cyclic shift.
A structured sector packages a mode set that always keeps the odd quarter-turn indices ${1,3,5,7}$ and never the DC mode. The sector projector rebuilds a signal from only those DFT coefficients. The bare one-tick propagator is the linear cyclic shift on $\mathrm{Signal}_8$.
The module presents the analytic 8-tick operator as the canonical recognition operator, with a thin legacy ledger surface kept only for older bridges. The update law is the concrete projector-after-shift composition used throughout the operator core.
proof idea
Definitional one-liner: the update is the $\mathbb{C}$-linear composition of the sector projector after the bare shift, written as (sectorProject S).comp shiftLinear. No further proof obligations; linearity is inherited from the two factors. The companion simp lemma unfolds it to $P_S(\sigma f)$ on points.
why it matters
This is the concrete one-tick recognition law that the public RecognitionOperator record evolves by, and the identity used when the update collapses to bare shift on the quarter-turn core. Downstream, the Unified Forcing Chain packages it into OperatorCore_Forced and the T7/T8→Operator-Core bridge: dimension forcing gives $D=3$, the eight-tick cadence supplies $\mathrm{Signal}_8$, and the odd-mode core is neutral under this update.
On the gravity/quantum-channel side it is the matter and channel factor in the canonical recognition-coupled factorization (both sides equal to this update, joint map the tensor of cyclic shifts). It therefore sits at the join of the forcing chain (T7 eight-tick, T8 dimension) and the amplitude-linear substrate certificates.
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