  Using NOEMA and ALMA 3mm line scans, we measure spectroscopic redshifts of
six new dusty galaxies at 3.5<z<4.2 by solidly detecting [CI](1-0) and CO
transitions. The sample was selected from the COSMOS and GOODS-North
super-deblended catalogs with FIR photometric redshifts z>6, based on template
IR spectrum energy distribution (SED) from known submillimeter galaxies at
z=4--6. Dust SED analyses explain the photo-z overestimate from seemingly cold
dust temperatures (Td) and steep Rayleigh-Jeans (RJ) slopes, providing
additional examples of cold dusty galaxies impacted by the Cosmic Microwave
Background (CMB). We therefore study the general properties of the enlarged
sample of 10 ``cold" dusty galaxies over 3.5<z<6. We conclude that these
galaxies are deceivingly cold at the surface but actually warm in their
starbursting cores. Several lines of evidence support this scenario: (1) The
high infrared surface density and cold Td from optically thin models appear to
violate the Stefan-Boltzmann law; (2) the gas masses derived from optically
thin dust masses are inconsistent with estimates from dynamics and CI
luminosities; (3) the implied high star formation efficiencies would conflict
with cold Td; (4) high FIR optical depth is implied even using the lower,
optically-thick dust masses. This work confirms the existence of a substantial
population of deceivingly cold, compact dusty starburst galaxies at z>~4,
together with the severe impact of the CMB on their RJ observables, paving the
way for the diagnostics of optically thick dust in the early universe.
Conventional gas mass estimates based on RJ dust continuum luminosities
implicitly assume an optically thin case, overestimating gas masses by a factor
of 2--3 on average in compact dusty star-forming galaxies.
