We study covert block-activity information transmission over a thermal-loss bosonic channel, where messages are encoded in weight-constrained activity patterns that Bob must recover through block-local decisions by prescribed deadlines. Shared circular Gaussian displacement modulation makes Willie's averaged lost-light state exactly thermal, with a strictly convex relative-entropy cost in signal energy, whereas a fixed Gaussian receiver at Bob yields a Gaussian mean-shift divergence linear in energy. This asymmetry produces an exact finite-block energy--information frontier and a detector-independent latency converse, matched in order by a block-reset cumulative-sum detector. For block covertness budget $δ_b$ and error target $ε_b$, the required active length scales as $δ_b^{-1}\log^2(1/ε_b)$ up to an explicit channel--receiver factor. Lifting this local law to communication yields the matching transmission limit $\log M=Θ(\sqrt n/\log n)$ for the symmetric coordinatewise architecture under a fixed total covertness budget, maximal-message covertness, vanishing maximal error, and local deadlines. A relaxed full-horizon reference with the same modulation family and fixed measurement supports $Θ(\sqrt n)$, showing that covertness and the selected measurement alone do not impose the extra logarithmic factor. Finally, public quadrature phase-shift keying codebooks selected by $O(\sqrt n)$ secret bits remove ideal continuous shared randomness without changing the payload order.

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