Signal8
plain-language theorem explainer
Local type alias for the eight-tick analytic signal space: maps from the discrete octave `Fin 8` into `ℂ`. Gravity Track 2.C cites it as the single-factor carrier on which amplitude-linear versus density-only channel responses are compared. The body is a pure re-export of the canonical foundation carrier, so no new mathematics is proved here.
Claim. Write $\mathrm{Signal}_8$ for the type of eight-tick analytic signals, i.e. functions $\mathrm{Fin}\,8\to\mathbb{C}$, identified with the canonical carrier from complex-structure forcing.
background
Track 2.C of the quantum-gravity plan upgrades paper IV's T2 from a modeling assumption to a theorem: amplitude-linearity of the gravitational channel must be forced from substrate linearity. The module begins with a single-factor dichotomy on one channel copy.
The carrier is the eight-tick analytic signal space. In Recognition Science the forcing chain fixes an eight-tick octave (T7, period $2^3$), so ledger states on one factor are complex-valued functions on $\mathrm{Fin},8$. The foundation module ComplexStructureForcing already names this type; the present abbrev only imports that name into the gravity quantum-channel namespace.
On this space a candidate response $R$ is amplitude-linear when it agrees with some $\mathbb{C}$-linear map (preserving coherent superpositions), and density-only when it is invariant under unit-modulus phase multiplications (depending only on $|\psi\rangle\langle\psi|$).
proof idea
Definitional re-export only. The abbrev equates the local name with Foundation.ComplexStructureForcing.Signal8; there is no tactic proof, no lemma application, and no computational content beyond type identity.
why it matters
Gives Track 2.C a stable local name for the single-factor state space on which the substrate dichotomy is stated. Sibling predicates (IsAmplitudeLinear, IsDensityOnly) and the forcing lemmas (eq_zero_of_isAmplitudeLinear_isDensityOnly, not_isDensityOnly_of_isAmplitudeLinear_of_ne_zero, not_exists_nontrivial_isAmplitudeLinear_and_isDensityOnly) all quantify over maps on this carrier.
The eight-tick structure ties directly to T7 in the unified forcing chain. Later Track 2.C work is planned to lift the single-factor dichotomy to the joint matter-plus-channel macroscopic ledger, composing factor-wise with the linear recognition operator, so this alias is the base type for that upgrade path. Zero sorry and no new RS axioms at this layer.
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