Geometric Unity II: The Matter Ledger. Minimal Chiral Completion, One-Family Pati–Salam, Spin(10), Anomaly Closure, and the Axial-Current Handoff

Geometric Unity I: From Heuristic Proposal to Testable Classical Framework
Completed Curvature, Canonical Torsion, Projection–Variation, and the Negative-Stiff Axial Export

Geometric Unity begins with a bold organizing idea: physics may be described on a higher-dimensional ambient space, while the world we observe appears on a four-dimensional slice. This paper develops the classical theorem layer needed to make that idea precise, testable, and usable by the rest of the Geometric Unity series.

Paper I supplies the completed-geometry foundation. It replaces raw shifted variables with completed objects that transform cleanly, project consistently to the observation slice, and can be handed downstream without changing signs or conventions. The central completed variables are the completed gauge connection, the completed curvature, the completed spin connection, and the augmented torsion. Together, these form the classical geometry packet consumed by later papers.

This paper proves the completed Levi–Civita slice theorem: when completed tangential torsion vanishes, the completed tangential connection becomes the Levi–Civita connection of the induced four-dimensional slice metric. This is a completed-connection statement. A corresponding statement for the raw uncompleted connection requires an additional tangential compensator condition.

This paper also proves fixed-embedding Projection–Variation. Instead of pulling back an unprojected higher-dimensional top form, the slice action is built from a pullback-compatible four-form. Under admissible boundary and corner conditions, variation of the slice action agrees with pulling back the ambient Euler–Lagrange forms. Moving embeddings are governed by an explicit obstruction term rather than being folded silently into the fixed-slice theorem.

The main physical export is the sign-safe axial torsion contact. Algebraic elimination of nondynamical axial torsion produces a positive coefficient in the sign-safe axial operator basis. In the standard minimally coupled Einstein–Cartan–Sciama–Kibble normalization, this recovers the coefficient 3κ/16.

On a conserved homogeneous axial-current branch, the resulting stress tensor is a negative-stiff component: its energy density and pressure are equal, negative, and scale as a^-6. This branch is null-energy-condition violating. Paper I derives the classical conserved-branch stress tensor; its bounce role, stability status, and observable activation are downstream questions handled by the legality, source-resolution, and observable-interface layers.

Paper I therefore fixes the classical law and its convention ledger. It does not determine the matter carrier supplying the axial current, the occupied species list, weak-singlet or sterile carrier status, source-current population, spin-correlation matrix, coherence class, sterile dense-fermion branch, or species-resolved parent-collapse packet. Those are downstream data supplied by later GU and KTC layers.

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, completed curvature, augmented torsion, observation slice, Projection–Variation, Einstein–Cartan theory, axial torsion, negative-stiff cosmology, NEC violation, conserved axial current, source-resolution boundary, sigma0, Kerr–Torsion Cosmology