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 theprotocol-licensed summary-side quotient, not as a direct opening of raw unread interiordata and not as a numerical claim about physical Planck-scale ontology. Under the InternalInvisibility Principle of Part I, raw II never enters retained laws; what the upper gate declaresis an independent conservative completion of the summary FXVII = ΣHXVII,PXVII (II ) =DXVIIII , equivalently of its summary-side quotient QXVIII = C(PXVII)I/ ker DXVII, in parallelwith the R3 completion of IR already permitted by Part I. Quartic dispersion, branchsplitting, and critical thresholds are readouts of that licensed quotient-level completion atthe 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 aboutthe physical Planck scale, and not as a new structural law-space, but as the gate-scopedreading in which the conservative R3 completion is applied simultaneously to the retainedchannel IR and to a representative of the summary-side quotient QXVIIIinherited fromPart I, so that the licensed summary is read on a fully time-completed window rather than aquasistatic one (as in Part XV) or a memory-frequency-dependent one (as in Part XVI). Thecentral point is not a small correction to a single low-energy branch, and not an ontologicalaccess to a hidden sector, but an exact retained-effective closure on the inherited branchthat yields quartic dispersion, genuine branch splitting, and a threshold instability of thelicensed summary-quotient completion. The representative-branch conventions, the licensedsummary-instance discipline, the admissible backbone class EXV, and the common reducedSchur 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 exactrepresentative-level coupled equations of motion and eliminate the time-completed summaryquotient representative to obtain a nonlocal effective retained-channel closure. The resultingeffective restoring operator is the same common reduced Schur operator Seff of Part XV,evaluated on the upper-gate-completed unread-side window LII → □α = ∂2t + αD∗D; thepresent Part attaches second-order time evolution to IR and to a gauge-fixed representative ofthe summary-side quotient QXVIIIsimultaneously, and reads the result. Plane-wave analysisyields 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 derived fromthe upper-gate summary-quotient two-channel opening (R3 simultaneously on IR and ona gauge-fixed representative of QXVIII) and yields a genuine two-branch spectral structure.At the mode level, the instability is equivalently the loss of positive definiteness of therepresentative-level coupled stiffness matrix on the nonzero window 0 < |k| < kc, and thelower branch selects a unique fastest-growing scale|k|2max =η2√αβ (√α +√β)2.Within the present symbolic linearized analysis, the lower-branch instability is interpretedas the signal that the licensed summary-quotient completion cannot persist as the stablereadable low-energy regime and must reorganize toward a reduced phase in which thesummary is again read on a quasistatic window. The threshold η2 − αβ|k|2continues, onegate higher, the same threshold grammar that organized Part XV’s low-energy threshold ∆cand Part XVI’s mixed discriminant ∆mix. In the post-instability frozen/static exact-channelreduction considered below, the upper gate fixes the magnitude of the exact-channel Schurresiduem2res :=η2α,so that, in general,kc η = m2resrαβ,which simplifies to kc η = m2res only at the symmetric point α = β. The quantity m2res = η2/αis not a universal mass term; it is a channel-selective retained residue, restricted to the exactchannel and absent on the coexact and harmonic channels. The sign at which η2/α entersthe direct linear frozen calculation is the negative shift − η2/α on the exact channel, whichis itself a linearized Schur signature of the lower-branch instability; its positive mass/gapinterpretation belongs to the stable reduced branch selected after the gate-handoff mechanism.Mass is therefore not the primary content of the gate but the residue of the reduced stablephase reached through gate handoff. The passage from instability to reduced stability isconditional on a supercritical saturation scenario and on a frozen-handoff condition; the linearanalysis alone establishes the instability of the upper-gate completion of the summary-sidequotient, while the stable reduced phase is the admissible post-instability reading consideredhere.