A Growth-Rate Reformulation of Regime-Dependent Spectral Residual Analysis for the Riemann Hypothesis

This paper presents a conditional transform-side framework for residual spectral analysis related to
the Riemann zeta function. The starting point is a windowed residual field motivated by the classical
explicit formula for
psi(x) − x. From this field we define a normalized low-frequency band-energy, its X-average over
a horizon H, and a Laplace transform whose abscissa of convergence is proposed as the principal
large-scale observable. The central idea is that the decisive coordinate for distinguishing critical-line
zeros from hypothetical off-critical-line zeros is not the frequency variable alone but the macroscopic
growth rate in the parameter X.
The paper makes four contributions. First, it derives a zero-side decomposition of the normalized
band-energy and identifies a hierarchy of growth exponents for critical-critical, top-block–critical, and
subtop contributions. Second, it proves positivity and positive-definiteness of finite top-block Gram
matrices, thereby isolating a robust nondegeneracy mechanism for a hypothetical rightmost off-line
block. Third, it formulates explicit minimal weighted interaction hypotheses controlling the forward
and reverse directions of the program. Fourth, it introduces a second-difference transform whose kernel
annihilates the micro-diagonal obstruction to second order and reduces the forward bottleneck to a dyadic
mesoscopic interaction condition.
The paper does not prove the Riemann Hypothesis. Its contribution is a rigorous reduction framework:
under clearly stated interaction and nondegeneracy assumptions, the question is reformulated as a statement
about the abscissa of convergence of an averaged Laplace transform. This makes explicit which parts of
the argument are unconditional, which parts are conditional, and where future analytic progress would
be required.
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