Toward Full Quantum Consistency for TEBAC 9D/9D+: A Fixed-Sector Theorem Program from Gauge Closure to BRST/BV and Physical-State Construction

This preprint presents a theorem-bearing fixed-sector program toward quantum consistency for TEBAC 9D/9D+. It should be read as an honest intermediate step: the manuscript does not claim final nonperturbative closure of the unfrozen full theory, but instead develops a rigorous bridge from the previously established anomaly/topological consistency package to a local full-sector quantum-consistency assembly theorem in an exact local perturbative, branch-fixed regime. The upstream TEBAC 9D/9D+ program already provides substantial anomaly/topological input, including local anomaly cancellation, global-anomaly and cobordism language, topological/TQFT protection, and spectral/Kaluza–Klein structure. In parallel, the companion TEBAC 9D EFT work fixes a minimal 9D gauge–matter sector, explicit gauge and trace conventions, compactification data, KK spectra, and a reproducible phenomenology pipeline, while explicitly stating that it is an EFT construction rather than a unique ultraviolet completion or a final theory of everything. Against that background, the present manuscript advances the dynamical theorem chain through the QC-I–QC-X architecture. In its current form, it develops theorem-bearing local full-sector layers for symmetry and constraint closure, gauge fixing and local measure control, BRST/BV structure and local cohomology, local gauge-independence of physical observables, physical-state-space construction, ghost decoupling, conditional physical-unitarity descent, EFT-style UV-consistent closure, compactification/branch-interface consistency, and a conditional final assembly theorem. The result is a coherent local full-sector quantum-consistency package in the exact perturbative branch-fixed regime, written in a referee-facing and status-explicit manner. At the same time, the manuscript is deliberately explicit about what still remains before one may honestly claim genuinely full quantum consistency in a stronger sense. The most important next mathematical step lies beyond the current local QC-II–QC-X chain: the QC-VII positivity/completion upgrade. At present, the manuscript establishes full-sector ghost decoupling and a physical-unitarity descent theorem, but this still rests on a positivity/completion hypothesis. A stronger next-stage result would replace that hypothesis by a theorem-level positivity/completion theorem for the induced physical form and the completed physical sector. A second major remaining task is a stronger QC-VIII upgrade beyond the present EFT-style closure. The current manuscript proves an honest EFT-style UV-consistent closure statement, but a stronger final claim would still require full RG running, explicit operator matching, full spectral threshold computations beyond proxy sums, and a stronger UV statement beyond EFT-consistent closure. This is fully in line with the companion EFT paper itself, which explicitly notes that a stronger predictive treatment requires fixing the gauge embedding and trace normalization, the descent normalization, an operator basis for EFT coefficients, full spectral threshold computations, and, where relevant, cosmological production history. A third remaining task is a stronger QC-IX upgrade. The present compactification/interface theorem is formulated in the exact local perturbative, branch-fixed regime. A stronger version would require backreaction-sensitive control, genuinely dynamical compactification, and non-frozen handling of moduli/radion sectors. The companion EFT text itself points in this direction by explicitly highlighting the need for moduli/radion safety and broader cosmological viability conditions. Finally, the deepest remaining burden is a global and nonperturbative upgrade of the current local package. This includes Gribov/global-slice issues, global measure control, and a nonperturbative completion of the present local theorem chain. Accordingly, this preprint should be read neither as a purely formal sketch nor as a final claim of ultimate closure, but as a substantial theorem-bearing intermediate step whose explicit aim is to continue toward genuinely full quantum consistency for TEBAC 9D/9D+ in a stronger mathematical sense.