Signal8
plain-language theorem explainer
Authoritative alias for the eight-tick complex carrier type used throughout OperatorCore complex-structure forcing. Anyone citing discrete Fourier modes, shift dynamics, or mode costs on the period-8 cycle uses this name. The declaration is a one-line re-export of the Foundation definition; no new mathematics is proved.
Claim. Write $\mathrm{Signal}_8$ for the type of complex-valued signals on the eight-tick cycle: the IndisputableMonolith alias for the 8-tick complex carrier imported from Foundation complex-structure forcing.
background
Recognition Science forces an eight-tick octave (forcing step T7: period $2^3$) as the minimal closed discrete clock compatible with the Recognition Composition Law and J-cost. Complex structure on that clock is developed by treating signals as $\mathbb{C}$-valued functions on $\mathbb{Z}/8\mathbb{Z}$.
The parent Foundation module defines the carrier, the cyclic shift, its iterates, the DFT/IDFT pair, the mode inner product, and the complexified J-cost on modes. This OperatorCore module re-exports that carrier under a stable local name so downstream operator lemmas (unitary evolution, total mode cost, eigenvalue statements) share one type.
The doc-comment states the intent directly: an authoritative IM alias for the 8-tick complex carrier. Sibling names in the module (shift, dft8, idft8, inner8, JcostC, shift_period_8) all act on this type.
proof idea
Definitional abbrev only: a one-line alias equating the local name to IndisputableMonolith.Foundation.ComplexStructureForcing.Signal8. No tactics, no lemmas, no proof obligations.
why it matters
Gives OperatorCore a single stable handle on the eight-tick carrier that T7 forces. Without a shared alias, shift-periodicity, DFT mode costs, and unitary evolution lemmas would each re-import the Foundation type under divergent names. The alias sits at the base of the local sibling cluster (next index, shift iterates, eigenvalues, total mode cost) that turns the abstract eight-tick octave into concrete complex linear algebra used later in complex-structure forcing. It does not itself close a forcing step; it standardizes the carrier those steps act on.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.