Kirchhoff's Law of Thermal Emission with Built-in UV/IR Cutoffs from HQIV's Discrete Null Lattice

This is the third paper in the HQIV Tier-1 foundation-extension trilogy. It derives Kirchhoff’s law of thermal emission directly from two HQIV axioms: a discrete null lattice with Planck cell size and informational monogamy fixing α = 3/5. Key results: The mode count per shell is finite by construction: ( N(m) = (m+2)(m+1) ), with closed-form cumulative count given by the hockey-stick identity. The resulting blackbody spectrum is a finite sum between an explicit ultraviolet cutoff (Planck pole at ( m = 0 )) and an infrared cutoff at the lock-in reference shell. No ultraviolet catastrophe arises and no renormalisation is performed. Under the proton lock-in scale-witness convention, the framework yields a quantitative prediction for isotropic cosmic birefringence at the present epoch: 0.379 degrees, which agrees with Planck PR4 EB data within −0.40σ. Substituting HQIV’s apparent age (13.8 Gyr) instead of the wall-clock age (51.2 Gyr) yields 0.103° (−2.55σ), providing an internal discriminator. Friedmann radiation-era scaling ( H proportional to T^2 ) is recovered as a consequence of the propagation-shell identification, not imported. All core theorems are machine-checked (sorry-free) in Lean 4. Numerical results are reproduced by the supplementary `scripts.zip` bundle. The work follows the HQIV patch-theory ontology: the discrete patch layer is primary; continuum expressions are used only as calculation-approximation translations. This is Preprint v1 (27 May 2026) and forms part of the HQIV publication series.