Channel Geometric Mismatch and the Domain Impedance Z: The Velocity-Dependent L1/L2 Framework Applied to PCTRM Channel Mechanics

Every physical interaction in the PCTRM framework is mediated by a channel. A channel extends adjacency between two solitons — two stable, self-sustaining patterns in the substrate's discrete arithmetic. Channels have endpoints, direction, throughput, and activation state. These properties are specified in the PCTRM master specification. This paper identifies one additional property that has not been made explicit: every channel has a characteristic propagation velocity, and that velocity determines the channel's geometric ratio. The geometric ratio β was established in prior work as the conversion factor between rectilinear (L1) and Euclidean (L2) measurements on circular geometry. At rest, β = π/4 ≈ 0. 785. A separate result showed that β is not a constant but a function of velocity: β (v) = E (v/c) / (1 + 1/γ), where E is the complete elliptic integral of the second kind and γ is the Lorentz factor. At v = 0, β = π/4. At v = c, β = 1. The geometry transitions continuously from circular to rectilinear as velocity increases. @HOWL-MATH-13-2026 Applying this result to channels: if a channel's content propagates at velocity vch, the channel's geometric ratio is β (vch). An electromagnetic channel propagates at c — photons advance one cell per tick by substrate construction. Its β is 1. The geometry of electromagnetic propagation is fully rectilinear. A thermal channel propagates at molecular thermal velocity — for room-temperature air, approximately 500 m/s, giving v/c ≈ 1. 7 × 10⁻⁶. Its β is indistinguishable from π/4. The geometry of thermal molecular interaction is fully circular at any achievable measurement precision.