Variational Backbone and Regime Closures XVII: The Planck Gate and the Mathematics of Two-Sector Opening

This Part studies an upper-gate/UV opening—called here the Planck gate—within theatlas fixed in Part XV. It should be read not as a literal numerical claim about the physicalPlanck scale, but as the structural label for the maximal two-sector opening analyzed below.The central point is not a small correction to a single low-energy branch, but an exacttwo-sector opening on the inherited branch that yields quartic dispersion, genuine branchsplitting, and a threshold instability. The representative-branch conventions and the licensedsummary-instance discipline are therefore inherited from Part XV and are not restated herein full.Starting from a minimal coupled action built on the inherited branch, we derive the exacttwo-sector equations of motion and eliminate the closed sector to obtain a nonlocal effectiveopen-sector closure. Plane-wave analysis yields the quartic dispersion relationω4 −(α+β)|k|2ω2 +αβ|k|4 −η2|k|2 = 0,with branch splitting and the critical wave numberkc = η√αβ .These are the central upper-gate outputs of the paper. Unlike representative Hořava-typelinearized tensor-mode comparisons, where higher-spatial-derivative corrections are oftenread as deformations of a single propagating branch, the present quartic law is derivedfrom exact two-sector opening and yields a genuine two-branch spectral structure. Withinthe present symbolic linearized analysis, the lower-branch instability is interpreted as thesignal that a fully open two-sector phase cannot persist as the stable readable low-energyregime and must reorganize toward a reduced phase. In the post-instability frozen/staticexact-channel reduction considered below, one then formally recovers the static residuem2 = η2α ,so that, in general,kc η = m2αβ,which simplifies to kc η = m2 only at the symmetric point α = β. Mass is therefore alow-energy residue of the reorganization, not the primary content of the gate.