KT Foundations IV: The Observable Interface The sigma_0 Export Packet, Negative-Stiff Backgrounds, Source Adapters, Real-Branch Admissibility, Sign-Locked Predictions, and Data-Facing Model Maps KT Foundations IV establishes the observable-interface layer for the KT Foundations and Kerr Torsion research program. It converts upstream theorem packets into observable maps, domain gates, likelihood-ready prediction vectors, and export packets that can be audited without reopening the classical, matter, or quantum-legality ledgers. KT Foundations I supplies the completed classical geometry, the sign-safe axial contact, the total-current-square law, and the conserved negative-stiff stress tensor. KT Foundations II supplies the matter/current carrier, admissible current branches, and source-class gates. KT Foundations III supplies BRST/BV legality, anomaly preservation, counterterm status, boundary admissibility, and axial sign-corridor status. KT Foundations IV builds the data-facing interface for declared packets that pass those upstream gates. The central export is the sigma_0 stiffness packet. On the conserved homogeneous axial branch, the packet produces a negative-stiff contribution with rho_5 = p_5 = -sigma_0^2 a^-6. The raw conserved branch is treated as a persistent-background control branch. Its observable use is restricted to real background domains, where the expansion function remains positive throughout the declared observational interval. The main interface result is the real-branch ceiling. A persistent raw negative-stiff branch cannot be freely extrapolated through the early universe. If the raw branch is required to remain real through an early observational domain, its allowed stiffness is tightly bounded, and the 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. KT Foundations IV separates the control branch from the live observable branch. The raw branch is the persistent-background control branch. The live branch is a KT Foundations II-gated, KT Foundations III-legal, KT Foundations IV-real source/window packet. The live branch is defined by three gates. The source branch must satisfy the matter/current gates of KT Foundations II. It must preserve the quantum-legality and sign-corridor status certified in KT Foundations III. It must also remain real on the KT Foundations IV background domain. This turns the source/window branch into a controlled observable packet rather than a free phenomenological function. The paper exports the standard background and distance observables: H(z), D_H, D_C, D_M, D_A, D_L, D_V, D_M/r_d, D_H/r_d, and D_V/r_d. It also supplies the sound-horizon ledger, including the fixed-domain raw-branch sign and the required caveats for windowed branches, moving endpoints, moving drag epochs, and moving real-domain boundaries. The sign-locked observable package ties multiple outputs to the same sigma_0/window data. The same stiffness/window packet controls the bounce/root temperature, the radiation-era effective-species shift, and the growth response. When the conserved negative-stiff branch is active during radiation domination, the effective-species shift is negative. A declared source/window branch produces a computable growth response through the standard GR growth equation. The default perturbation packet is kept clean: mu = eta = Xi_0 = 1. Growth changes enter through H(a) and Omega_m(a), not through hidden modified-gravity functions. This keeps background response, growth response, and modified-gravity response separate. The sign-locked package exports four linked quantities: Bounce/root temperature. Negative Delta N_eff when the branch is active during radiation domination. Local growth-driving response through H(a) and Omega_m(a). A single-stiffness correlation tying those outputs to the same sigma_0/window data. KT Foundations IV defines observable packets. It does not claim detection and does not perform the final audit. KT Foundations V, or an external likelihood engine, can audit the exported packets against data while preserving the upstream theorem ledgers. The exported invariants of KT Foundations IV are observable maps, source/window branch adapters, real-domain gates, sound-horizon ledgers, sign-locked prediction packets, background-distance observables, growth-response maps, likelihood-ready prediction vectors, and the interface rules required to test declared cosmological packets without redefining upstream theorem data. Series architecture: KT Foundations I: The Classical Completion Layerhttps://doi.org/10.5281/zenodo.17252988 KT Foundations II: The Matter Ledgerhttps://doi.org/10.5281/zenodo.17254875 KT Foundations III: The Quantum Legality Layerhttps://doi.org/10.5281/zenodo.17374258 KT Foundations IV: The Observable Interfacehttps://doi.org/10.5281/zenodo.17374850 KT Foundations V: The Audit Protocolhttps://doi.org/10.5281/zenodo.17402260 Kerr Torsion 0: The Dense-Fermion Creation Enginehttps://doi.org/10.5281/zenodo.20277423 Kerr Torsion 0.5: The Cartan Sealhttps://doi.org/10.5281/zenodo.21134177 Kerr Torsion I: The Spin Data That Survived the Big Banghttps://doi.org/10.5281/zenodo.19756639 Kerr Torsion II: A Detectable Deci-Hz Tensor Corridor from Dense Fermionshttps://doi.org/10.5281/zenodo.20221826 Kerr Torsion III: The Directional Spin-Memory Channelhttps://doi.org/10.5281/zenodo.20218504 Kerr Torsion IV: Inherited Helicity as the Source of Matterhttps://doi.org/10.5281/zenodo.20636020 Proof bridge: Minimal Chiral Completion and the Axial Einstein--Cartan Contact: Pati--Salam, Spin(10), and the Positive C_55 Sign Corridorhttps://doi.org/10.5281/zenodo.20450613 Keywords: KT Foundations; Kerr Torsion; observable interface; sigma_0 export packet; negative-stiff cosmology; real-branch admissibility; real-branch ceiling; source adapter; windowed branch; background observables; distance observables; H(z); D_H; D_C; D_M; D_A; D_L; D_V; BAO ratios; sound horizon; Delta N_eff; growth response; sign-locked prediction; likelihood-ready prediction vector; sterile dense-fermion branch; source-window packet; audit-ready cosmology.

Additional Notes Geometric Unity IV functions as the observable-interface and viability-gate layer of the Geometric Unity series. It converts the upstream GU I–III theorem packets into real-domain background maps, source/window packets, sign-locked observable handles, and likelihood-ready export structures. Real-branch admissibility is a central theorem. Every observable branch must remain on a real background domain before it enters a distance, BAO, sound-horizon, growth, or likelihood calculation. For the raw negative-stiff branch, this produces a dataset-dependent ceiling on the allowed stiffness. The raw branch serves as the persistent-background control branch. When this branch is required to remain real through early-universe domains, the real-domain ceiling tightly bounds the stiffness and makes the raw branch background-distance silent at later epochs. The live observable lane is the gated source/window packet. A source/window branch enters the GU IV observable packet only when the GU II source-class gates, GU III legality status, and GU IV real-domain condition are simultaneously satisfied. The sign-locked observable packet supplies three correlated handles: the bounce/root temperature, the negative radiation-era (\Delta N_{\rm eff}) sign, and the growth-response adapter. These are controlled by the same stiffness/window data rather than by independent observable amplitudes. The growth sector remains conservative. GU IV exports (\mu=\eta=\Xi_0=1) by default. A declared source/window branch may still affect (D_+) and (f\sigma_8) through (H(a)), but this is a background-growth response, not hidden modified gravity. The sound-horizon and BAO packet is domain-controlled. The fixed-domain raw-branch sound-horizon response has a definite sign, while windowed branches, moving endpoints, moving drag epochs, and moving real-domain boundaries remain declared-domain calculations. The export protocol is machine-auditable. The GU IV packet records real-domain status, source/window status, sign-locked observable fields, growth-response requirements, likelihood fields, artifact checksums, and claim-status tags. Analysis Geometric Unity IV gives the GU series its observable face. It translates upstream theorem packets into background, distance, BAO, source/window, sign-locked, growth-response, and likelihood-ready outputs while preserving strict scope discipline. It defines what a valid observable packet is, what gates it must pass, and how GU V or an external audit engine can test it without reopening the GU I–III theorem ledgers.