evalAt_apply
plain-language theorem explainer
Coordinate evaluation of an eight-slot complex signal at index i recovers the i-th component. Anyone simplifying joint-substrate or channel-response calculations on Signal8 will cite this simp rule. The proof is pure definitional equality: evalAt is LinearMap.proj, so the identity holds by rfl.
Claim. For every index $i\in\{0,\ldots,7\}$ and every signal $v:\{0,\ldots,7\}\to\mathbb{C}$, the coordinate-evaluation linear map at $i$ applied to $v$ equals the component $v(i)$.
background
Track 2.C works on the joint matter-plus-channel substrate
JointSubstrate := Signal8 \otimes[\mathbb{C}] Signal8, where each factor is an eight-slot complex ledger Signal8 = Fin 8 \to \mathbb{C} (the eight-tick octave). Session 85 already closed the single-factor dichotomy: no nontrivial channel response is both amplitude-linear and density-only. This module lifts that dichotomy to pure-tensor factorizations of joint operators.
The helper evalAt i is the coordinate-evaluation functional Signal8 \to_l[\mathbb{C}] \mathbb{C}, defined as LinearMap.proj i. It is the standard way to read a single ledger slot, and appears whenever one tests nontriviality of a channel or matter response by asking whether some coordinate is nonzero.
proof idea
Term-mode proof by rfl. Because evalAt i is definitionally LinearMap.proj i, applying it to a signal v is definitionally the component v i. No lemmas are invoked.
why it matters
Local simp infrastructure for the joint-substrate lift of amplitude-linear forcing. Downstream pure-tensor factorization theorems (matter-side and channel-side amplitude-linearity, and the Track 2.C closure that a density-only channel response collapses under nontrivial matter coupling) repeatedly evaluate coordinates on Signal8 factors; this lemma keeps those calculations definitionally clean.
It does not itself close Track 2.C. Full upgrade of paper IV T2 from MODEL to THEOREM still needs the joint recognition operator to be shown $\mathbb{C}$-linear via the single-factor Schrödinger lift and PiTensorProduct.map. The eight-tick structure (Fin 8) is the T7 octave already fixed in the forcing chain.
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