Geometric Unity IV: The Observable Interface. The σ0 Export Packet, Negative-Stiff Backgrounds, Source Adapters, Real-Branch Admissibility, Sign-Locked Predictions, and Data-Facing Model Maps

Geometric Unity IV: The Observable Interface
The σ₀ Export Packet, Negative-Stiff Backgrounds, Source Adapters, Real-Branch Admissibility, Sign-Locked Predictions, and Data-Facing Model Maps

Geometric Unity IV is the observable-interface layer of the Geometric Unity series.

GU I supplies the completed classical geometry, the sign-safe axial contact, and the conserved negative-stiff stress tensor. GU II supplies the matter/current ledger, admissible current branches, and source-class gates. GU III supplies BRST/BV legality, anomaly closure, counterterm status, and the axial sign-corridor ledger. GU IV turns those upstream theorem packets into observable maps, domain gates, likelihood-ready prediction vectors, and machine-readable export packets.

The central object of GU IV is the σ₀ stiffness packet. On the conserved homogeneous axial branch, this packet produces a negative-stiff contribution with density and pressure scaling as a⁻⁶. The raw branch is treated as a persistent-background control branch. GU IV proves that this branch is observable only on real background domains, where the expansion function remains positive throughout the declared observational interval.

The central observable-interface result is the real-branch ceiling. A persistent raw negative-stiff branch cannot be freely extrapolated through the early universe. If it is required to remain real through an early observational domain, the allowed stiffness is tightly bounded, and the raw branch becomes background-distance silent at later epochs. Observable nontriviality must therefore enter through a declared windowed branch, a declared source-adapter branch, or another separately activated observable lane.

This release makes that distinction explicit:

The raw branch is a persistent-background control branch.
The live observable branch is a GU II-gated, GU III-legal, GU IV-real source/window packet.

GU IV defines the live branch through a three-gate structure: the source branch must satisfy the GU II matter/current gates, preserve GU III legality and sign-corridor status, and remain real under the GU IV background-domain condition. This turns the source/window branch into a controlled observable packet rather than a free phenomenological function.

The paper exports standard background and distance observables, including H(z), D_H, D_C, D_M, D_A, D_L, D_V, and the BAO ratios D_M/r_d, D_H/r_d, and D_V/r_d. It also supplies a sound-horizon ledger with the fixed-domain raw-branch sign, while preserving the required caveats for windowed branches, moving endpoints, moving drag epochs, and moving real-domain boundaries.

GU IV also adds a sign-locked observable prediction package. The same stiffness/window packet controls the radiation-era bounce temperature, the effective-species shift, and the growth response. In particular, the conserved negative-stiff branch predicts a negative radiation-era effective-species shift when active, and a declared source/window branch produces a computable growth response through the standard GR growth equation. The paper keeps this distinction precise: the default perturbation packet has μ = η = Ξ₀ = 1, so any growth response comes through H(a) and Ω_m(a), not through hidden modified gravity.

The sign-locked package ties together:

Bounce/root temperature
Negative ΔN_eff when active during radiation domination
A local growth-driving response through H(a)
A single-stiffness correlation linking these outputs to the same σ₀/window data

GU IV does not claim a detection and does not replace the audit layer. It defines the observable packet. GU V, or any external likelihood engine, may then audit that packet against data without reopening the GU I–III theorem ledgers.

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