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 T²) 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.