Finite Throughput--Group Delay Dilation Principle:\\ Unifying Bandwidth, Time Delay, and Redshift via WSIG--EBOC--RCA's ``Window--Scale--Gauge''

We establish a rigorous theory unifying ``finite throughput--queueing delay--apparent dilation (redshift)'' within windowed scattering and information geometry (WSIG) scale system. In pure theoretical language of operator--measure--function, using ``trinity'' scale identity \varphi'(E)/\pi=\rho_{\rm rel}(E)=-1{2\pi}tr\mathsf Q(E) as mother scale, define observer window's Toeplitz/Berezin compression K_{w,h} and its readout functional, characterize ``finite throughput'' abstract form via passivity and causality-preserving scattering axioms. Prove three main theorems: (T1) Throughput--Delay Dilation Theorem: Under passive load monotonicity, any windowed group delay readout monotonically non-decreasing with load; when baseline readout \Phi_w[\lambda_0]>0, induces Mellin dilation of energy and time scales scale-equivalent to redshift factor 1+z_w=\Delta_w\ge1; (T2) Group Delay--Bandwidth Product Upper Bound: Windowed energy spectrum total readout upper-bounded by product of ``group delay flux'' and ``effective bandwidth constant'', with non-asymptotic error bound from Nyquist--Poisson--Euler--Maclaurin (NPE) three-term closure; (T3) Scale Gauge Equivalence: Cosmological expansion and internal resolution enhancement scale-equivalent to gauge pairing a(t)=R(t)^{-1}, all readouts remaining invariant on same mother scale. Further provide ``front slope--group delay'' correspondence at reversible cellular automaton (RCA) dynamics layer, embedding queueing--throughput--delay discrete dynamics into EBOC static block encoding. Throughout use only finite-order Euler--Maclaurin and Poisson discipline, maintaining ``singularity non-increasing/poles = principal scales'' error control; any numerical and experimental interfacing relegated to appendices.