Variational Backbone and Regime Closures XVII: The Upper Gate (Planck-label) and the Mathematics of Summary-Quotient Opening

VBRC treats the upper gate (Planck-label) as a gate-scoped R3 time completion of the protocol-licensed summary-side quotient, not as a direct opening of raw unread interior data and not as a numerical claim about physical Planck-scale ontology. Under the Internal Invisibility Principle of Part I, raw II never enters retained laws; what the upper gate declares is an independent conservative completion of the summary FXVII = ΣHXVII,PXVII (unread content) = DXVIIII , equivalently of its summary-side quotient QXVII I = C (PXVII) I / ker DXVII, in parallel with the R3 completion of IR already permitted by Part I. Quartic dispersion, branch splitting, and critical thresholds are readouts of that licensed quotient-level completion at the retained-effective level, not of a directly observable hidden sector. This Part studies an upper-gate/UV opening—called here the upper gate (Planck-label)— within the atlas fixed in Part XV. It should be read not as a literal numerical claim about the physical Planck scale, and not as a new structural law-space, but as the gate-scoped reading in which the conservative R3 completion is applied simultaneously to the retained channel IR and to a representative of the summary-side quotient QXVII I inherited from Part I, so that the licensed summary is read on a fully time-completed window rather than a quasistatic one (as in Part XV) or a memory-frequency-dependent one (as in Part XVI). The central point is not a small correction to a single low-energy branch, and not an ontological access to a hidden sector, but an exact retained-effective closure on the inherited branch that yields quartic dispersion, genuine branch splitting, and a threshold instability of the licensed summary-quotient completion. The representative-branch conventions, the licensed summary-instance discipline, the admissible backbone class EXV, and the common reduced Schur operator Seff are inherited from Part XV and are not restated here in full. Starting from a minimal coupled action built on the inherited branch, we derive the exact representative-level coupled equations of motion and eliminate the time-completed summaryquotient representative to obtain a nonlocal effective retained-channel closure. The resulting effective restoring operator is the same common reduced Schur operator Seff of Part XV, evaluated on the upper-gate-completed unread-side window LII → □α = ∂ 2 t + αD∗D; the present Part attaches second-order time evolution to IR and to a gauge-fixed representative of the summary-side quotient QXVII I simultaneously, and reads the result. Plane-wave analysis yields the quartic dispersion relation ω 4 − (α + β)|k| 2ω 2 + αβ|k| 4 − η 2 |k| 2 = 0, with branch splitting and the critical wave number kc = η √ αβ . These are the central upper-gate outputs of the paper. Unlike representative Hořava-type linearized tensor-mode comparisons, where higher-spatial-derivative corrections are often read as deformations of a single propagating branch, the present quartic law is derived from the upper-gate summary-quotient two-channel opening (R3 simultaneously on IR and on a gauge-fixed representative of QXVII I ) and yields a genuine two-branch spectral structure. At the mode level, the instability is equivalently the loss of positive definiteness of the representative-level coupled stiffness matrix on the nonzero window 0 < |k| < kc, and