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Published June 2, 2026 | Version v2
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Geometric Unity III: The Quantum Legality Layer. Semidirect BRST/BV, Anomaly Closure, Counterterm Classification, Quantum Projection–Variation, and the Axial Sign Corridor

Description

Geometric Unity III: The Quantum Legality Layer
Semidirect BRST/BV, Anomaly Closure, Counterterm Classification, Quantum Projection–Variation, and the Axial Sign Corridor

Geometric Unity III is the quantum legality layer of the Geometric Unity series. GU I supplies the completed classical geometry. GU II supplies the matter, symmetry, and current ledger. GU III certifies which of those upstream packets can be gauge-fixed, quantized, renormalized, matched, and handed to the observable interface without hidden gauge, anomaly, boundary, or sign inconsistencies.

The central role of Paper III is legal certification. It does not introduce new matter, new geometry, or new observables. It preserves the GU I–II upstream packet and records the conditions under which GU IV may use it.

The paper develops a semidirect BRST complex for the affine gauge bookkeeping. The completed gauge variable is the affine-corrected connection Ahat = A − B. The BRST differential acts on A and B with the same affine ghost, so the affine ghost cancels in the completed variable. This gives the central completed-variable covariance relation: the completed connection transforms as a genuine gauge connection, and its completed curvature transforms covariantly.

This cancellation is one of the main structural checks of the completed-variable framework. It shows that the GU I completion is not merely notational; it removes the affine shift ghost from the physical completed gauge variable used downstream.

The paper then builds the BV master functional for the declared closed off-shell sector. Under the stated closure hypotheses, the BV action satisfies the classical master equation. Gauge fixing is handled through a nonminimal sector and a gauge-fixing fermion, with the corresponding ghost and shift-ghost operators displayed explicitly.

GU III also supplies the local cohomology and counterterm ledger through the parity-even dimension-six slice EFT order. This ledger identifies which local terms are physical counterterm classes and which are BRST-exact, total-derivative, equation-of-motion, or field-redefinition redundancies. The axial contact operator O55 remains part of the declared operator basis.

Anomaly closure is treated as an import-preservation theorem. GU II supplies the anomaly-safe one-family matter ledger. GU III checks that gauge fixing, affine ghosts, boundary choices, and measure conventions introduce no new anomaly obstruction when the GU II matter table is preserved.

The axial contact is matched in the sign-safe basis Delta L = C55(mu) O55, with O55 = −J5². The sign at the matching scale is inherited from the GU I auxiliary-torsion theorem. GU III records the sign-corridor status needed for GU IV: the sign is theorem-grade when the anomalous-dimension matrix and invariant positive cone are supplied; otherwise it is a declared sign-corridor assumption. This distinction is explicit. GU III certifies import consistency; it does not by itself prove cosmological viability.

The sterile dense-fermion branch defined in GU II enters GU III only as source-class data. GU III records the import-legality gates required before GU IV may consume a declared source-adapter packet: anomaly compatibility, washout survival, energy-budget admissibility, BRST/BV compatibility, boundary admissibility, operator-basis membership, and sign-corridor status. This branch remains a matter/current source class until those gates are declared and certified.

Finally, GU III upgrades Projection–Variation to the gauge-fixed/BV setting. Under admissible BRST-preserving boundary families, pullback-variation compatibility holds modulo BRST-exact and admissible boundary terms. This gives GU IV a clean rule for importing the quantum-legality packet into observable maps.

Series architecture:

GU I — From Heuristic Proposal to Testable Classical Framework
https://doi.org/10.5281/zenodo.17252988
Consumes: the public Geometric Unity motivation and standard differential-geometric / Einstein–Cartan machinery.
Exports: completed curvature, augmented torsion, completed Levi–Civita slice geometry, Projection–Variation, the sign-safe axial contact, the conserved negative-stiff branch, NEC status, and the source-resolution boundary.

GU II — The Matter Ledger
https://doi.org/10.5281/zenodo.17254875
Consumes: the GU I completed-geometry and axial-current interface.
Exports: the matter/current carrier, minimal chiral-completion chain, one-family Pati–Salam block, Spin(10) envelope, anomaly closure, Yukawa seed ledger, gauge normalization, internal-singlet axial-current bridge, conserved-current handoff, and sterile dense-fermion branch source-class packet.

GU III — The Quantum Legality Layer
https://doi.org/10.5281/zenodo.17374258
Consumes: completed variables and matter/current packets supplied by GU I and GU II.
Exports: BRST/BV legality, anomaly closure, counterterm and RG/matching status, boundary and projection legality, sign-corridor admissibility, and legality conditions required before declared current branches may be consumed downstream.

GU IV — The Observable Interface
https://doi.org/10.5281/zenodo.17374850
Consumes: declared theorem packets from GU I–III and downstream source-resolution packets.
Exports: observable maps, branch adapters, validity-domain ledgers, and the interface between theorem data and cosmological tests.

GU V — The Observable Audit Layer
https://doi.org/10.5281/zenodo.17402260
Consumes: declared observable-interface packets from GU IV and comparison packets from external models.
Exports: audit structure, model-comparison discipline, residual ledgers, and the program-level protocol for testing declared cosmological packets without redefining upstream theorem data.

Keywords: geometric unity, quantum legality, BRST, BV formalism, semidirect symmetry, affine ghost, completed variables, anomaly closure, local cohomology, counterterms, axial contact, sign corridor, sterile dense-fermion branch, Projection–Variation, observable interface

Notes

Additional Notes

Geometric Unity III serves as the quantum legality layer of the Geometric Unity series. It certifies the GU I completed-geometry packet and the GU II matter/current packet for downstream use by GU IV, while preserving the distinction between legal import, observable viability, and audit status.

Semidirect BRST/BV legality.
The paper formulates the BRST complex for the field-level semidirect affine structure. The affine ghost cancels in the completed connection, so the completed variable transforms as a genuine gauge connection and the completed curvature transforms covariantly. The nilpotency proof is stated in the semidirect Maurer–Cartan form, with the affine ideal treated as abelian. The BV master functional satisfies the classical master equation under the declared off-shell closure hypotheses.

Completed-variable import discipline.
GU III treats the completed connection, completed curvature, augmented torsion, axial operator basis, and GU II matter/current ledger as read-only upstream data. Changes to those objects reopen the upstream ledger. This keeps the quantum-legality layer from silently redefining the classical or matter packets it is meant to certify.

Local BRST cohomology and counterterm ledger.
The paper records the parity-even local cohomology and counterterm basis through the dimension-six slice EFT order. The ledger distinguishes physical counterterm classes from BRST-exact terms, total derivatives, equation-of-motion redundancies, and field-redefinition redundancies. The axial contact remains part of the declared operator basis in the sign-safe (O_{55}) convention.

Anomaly preservation.
GU III consumes the anomaly-safe matter ledger from GU II and checks that the quantum-legality layer introduces no new anomaly obstruction through ghosts, boundary choices, or measure conventions. The anomaly statement is therefore an import-preservation theorem: the GU II matter table remains anomaly-safe inside the GU III quantized field basis.

Axial sign-corridor status.
The axial contact is matched in the sign-safe basis ( \Delta L_X=C_{55}(\mu)O_{55} ), with (O_{55}=-J_5^2). The positive sign at the matching scale is inherited from the GU I auxiliary-torsion theorem. Propagation of that sign to the GU IV effective scale has two possible statuses: theorem-grade when the anomalous-dimension matrix and invariant positive cone are supplied, or declared sign-corridor assumption when those data are outside the theorem domain. The sign-corridor ledger certifies import consistency; it does not by itself establish observational viability.

Positivity diagnostics.
Forward-limit positivity is treated as a diagnostic layer, not as the source of the (C_{55}>0) sign. Any positivity statement requires declared external states, helicities, crossing prescription, subtraction scheme, infrared treatment, and map back to the Wilson basis. This keeps the algebraic torsion-matching sign, the RG cone condition, and the observable viability tests logically separate.

Quantum Projection–Variation.
The Projection–Variation rule is carried into the gauge-fixed/BV setting under admissible BRST-preserving boundary families. Gauge fixing contributes BRST-exact boundary structure, and physical observables are evaluated in BRST cohomology. The result is a clean quantum import rule for GU IV: pullback and variation remain compatible modulo BRST-exact and admissible boundary terms.

Sterile dense-fermion import legality.
The sterile dense-fermion branch enters GU III as GU II source-class data. GU III records the legality gates required before GU IV may consume a declared source-adapter packet: carrier status, anomaly compatibility, washout survival, energy-budget admissibility, boundary compatibility, operator-basis membership, and sign-corridor status. The branch remains source-class data until those gates are declared and certified.

Legality versus viability.
GU III makes the central program boundary explicit: BRST legality, anomaly closure, and sign-corridor status certify import consistency; they do not by themselves imply cosmological viability. GU IV supplies observable-interface tests, and GU V audits declared packets.

Reproducibility and audit trail.
The reproducibility protocol ties the BRST nilpotency check, BV master check, cohomology ledger, operator-basis ledger, anomaly import manifest, sign-corridor manifest, boundary ledger, sterile dense-fermion import manifest, and GU III export manifest to proof labels and checksums when those artifacts are released. The artifact protocol separates theorem content from release status.

Series role.
Within the Geometric Unity series, GU I supplies the completed classical geometry, GU II supplies the matter/current packet, GU III certifies quantum legality, GU IV maps declared packets to observables, and GU V audits the resulting observable interface. GU III is the legality checkpoint between theorem data and observable use.

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Geometric Unity III_The Quantum Legality Layer.pdf

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Additional details

Additional titles

Subtitle
Semidirect BRST/BV, Anomaly Closure, Counterterm Classification, Quantum Projection–Variation, and the Axial Sign Corridor

Related works

Is continued by
Preprint: 10.5281/zenodo.17252988 (DOI)
Preprint: 10.5281/zenodo.17374258 (DOI)
Preprint: 10.5281/zenodo.17374850 (DOI)
Preprint: 10.5281/zenodo.17402260 (DOI)