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 finite
residue--height transfer operators $\Lcal_{s,Q,L,\chi}$ that govern it. Our aim
is not to prove the Collatz conjecture but to identify, rigorously, the correct
finite spectral object and the correct quotient on which its gap should be
sought. 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 fact
that 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 in
disguise} and is an exact coboundary of the faithful residue--valuation skew
product. 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 three
theorems about that operator: a \emph{classification of coboundaries}
($\operatorname{Cob}=\langle\chi_2\rangle$, for arbitrary positive branch
weights), whence by Wielandt's theorem a \emph{strict} spectral gap for every
non-coboundary character at every finite level and tilt; a \emph{conductor
collapse} (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 a
single sequence of computable constants $\gamma_c(s)$; and a \emph{rotation
identity} showing the quadratic sector of the killed height-strip operator is a
rigid rotation of the principal one. The uniformity question is then
\emph{resolved}: a threshold ($m=c$) Schur analysis of the adjoint kernel
yields 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 a
root-ball analysis closing the degenerate budget---together the finite
H23a theorem $\sup_c\gamma_c(0)\le(\sqrt3/2)^{1/2}+o(1)<1$ at $s=0$. On
the global side we prove the \emph{keystone arrow} (every bad orbit casts
a finite non-coboundary two-point shadow), reduce the remaining quenched
exclusion to a joint $(2,3)$-adic max-plus rigidity, and prove two
unconditional exclusion theorems: periodic traps via linear forms in
logarithms, and---by a new \emph{repetition rigidity} argument in which
repeated valuation blocks are exact cycles---all bounded-discrepancy
valuation words of subexponential complexity, in particular every
Sturmian word. A counterexample must therefore mimic randomness at every
banded scale. The elementary core (the $2$-adic coding lemma and
repetition rigidity) is machine-verified in Lean~4. A companion
computational report companion} documents the numerical suite.