Let
be the Chebyshev prime-counting function and
the number of primes up to
x. The Chebyshev-prime condition
is not equivalent to the sign rule
. We prove that its leading threshold is the midpoint correction
, with an explicit higher-order boundary, and define false Chebyshev primes through the resulting transition layer. Our main unconditional theorem shows that ordinary prime counting does not equidistribute logarithmic prime phases: for every nonzero integer combination
of nontrivial zeta-zero ordinates,
. By contrast, the logarithmically uniform weight
restores finite-dimensional Haar equidistribution for rationally independent ordinates. This dichotomy motivates the weighted local-time statistic
, summed over the layer primes
. We show that the layer is exactly the one-sided slab
of the midpoint-centered field
B, where
, so that
is a bona fide occupation density. Assuming the Riemann hypothesis, linear independence, and a shrinking-window local limit hypothesis that we isolate as a separate open problem, we prove the one-directional Cesàro law
, where
is the value at the origin of the limiting density of
B (numerically
). Independently, we prove shrinking-target occupation laws via coarea formulas: unconditionally for fixed windows, including for the finite zeta-derived field itself, and under uniform transversality for the diagonal window. The remaining obstacles are tangential crossings of that field and transfer to the prime-sampled diagonal window.
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