Finite Spectral Shadows for the Collatz Valuation Cocycle

We study residue equidistribution for the accelerated Collatz/Syracuse map$S(x)=(3x+1)/2^{v_2(3x+1)}$ through a pathwise twisted Birkhoff functional---the\emph{deterministic defect cocycle} $W_n(x;\chi,s)$---and the finiteresidue--height transfer operators $\Lcal_{s,Q,L,\chi}$ that govern it. Our aimis not to prove the Collatz conjecture but to identify, rigorously, the correctfinite spectral object and the correct quotient on which its gap should besought. We prove two clean finite identities: a \emph{sliding-window congruence}expressing $x_n\bmod 3^r$ as a finite window of the valuation word, and the factthat the quadratic character $\chi_2$ of $(\Z/3^r)^\times$ satisfies$\chi_2(S(x))=(-1)^{v_2(3x+1)}$, i.e.\ $\chi_2$ is \emph{valuation parity indisguise} and is an exact coboundary of the faithful residue--valuation skewproduct. Consequently the right form of the finite spectral hypothesis (``H23a'')is a gap for all \emph{non-coboundary} characters of the \emph{faithful}(synchronized) operator, not for all nontrivial characters. We then prove threetheorems about that operator: a \emph{classification of coboundaries}($\operatorname{Cob}=\langle\chi_2\rangle$, for arbitrary positive branchweights), whence by Wielandt's theorem a \emph{strict} spectral gap for everynon-coboundary character at every finite level and tilt; a \emph{conductorcollapse} (the nonzero spectrum of a conductor-$3^c$ sector is computed at level$3^c$), which reduces uniformity of the gap in $Q$ to the boundedness of asingle sequence of computable constants $\gamma_c(s)$; and a \emph{rotationidentity} showing the quadratic sector of the killed height-strip operator is arigid rotation of the principal one. The uniformity question is then\emph{resolved}: a threshold ($m=c$) Schur analysis of the adjoint kernelyields bounded Plancherel shells for the renewal constants($\tilde Q_L\le3C_1<5.16$), cascade row contraction at rate $(\sqrt3/2)^c$,a level-cost law with the exact constant $\tfrac47<3^{-1/2}$, and aroot-ball analysis closing the degenerate budget---together the finiteH23a theorem $\sup_c\gamma_c(0)\le(\sqrt3/2)^{1/2}+o(1)<1$ at $s=0$. Onthe global side we prove the \emph{keystone arrow} (every bad orbit castsa finite non-coboundary two-point shadow), reduce the remaining quenchedexclusion to a joint $(2,3)$-adic max-plus rigidity, and prove twounconditional exclusion theorems: periodic traps via linear forms inlogarithms, and---by a new \emph{repetition rigidity} argument in whichrepeated valuation blocks are exact cycles---all bounded-discrepancyvaluation words of subexponential complexity, in particular everySturmian word. A counterexample must therefore mimic randomness at everybanded scale. The elementary core (the $2$-adic coding lemma andrepetition rigidity) is machine-verified in Lean~4. A companioncomputational report companion} documents the numerical suite.