Wave4

The module freezes the independent continuum target, mesh carrier, normalized TT data, pure-gauge family, and honesty decoys for the QG full-theory 4D continuum closure plan, before any continuum recovery is proved. Contract 1 is a canonical periodic Freudenthal 4-torus of side $N\ge 3$.

In that setting a wave mode is a real assignment to the four lattice directions. Sibling objects include integer modes, real-mode embeddings, squared wave and momentum norms, and the Frobenius norm used to normalize Euclidean TT polarizations (Gate A0 analog). The continuum object later compared to the frozen Einstein-Hilbert quadratic is the exact flat cross-term symbol, $|k|^2$-normalized on TT and vanishing on pure gauge.

Nothing in this layer proves continuum recovery: Tendsto targets and $S_{\mathrm{RS}}\to\mathrm{EH}$ remain open, and the EH quadratic is frozen independently of lattice weights.

Pure definition: a one-line abbreviation equating the wave-mode type with functions from the four-element finite type into the reals. No lemmas, tactics, or proof obligations.

Gives the ambient type for Bloch/momentum data on the frozen 4-torus mesh that the continuum preflight must not reverse-engineer from the EH answer. Downstream symbol uniqueness, Frobenius pin lemmas, and decoy discriminators (module tier tags) all speak in this language when they compare the exact flat cross-term symbol to the independently frozen linearized EH quadratic on TT versus pure gauge.

It sits at the first binding increment of the 4D continuum closure plan: mesh and mode carriers are named before any Tendsto or gap-action recovery is claimed. Framework context is discrete Regge gravity aiming at continuum Einstein-Hilbert in $D=3+1$, not the T0-T8 forcing chain itself. No used-by edges are recorded yet; the type is infrastructure for later gates.