Lorentz Report-Chart Characterization in EAS: Non-Primitive Simultaneity, Observer Provenance, and the Non-Abelian 3+1 Extension – Foundations Paper F3

This paper is archived as a speculative research work. Entanglement–Algebraic Spacetime (EAS) does not take spacetime, a Lorentz metric, inertial frames, physical time, or physical simultaneity as primitive scalar-field structure. Foundations Paper F1 fixes instead a formal scalar-field ontology and a strict ontology–report–interface firewall. Foundations Paper F2 adds contextual same-path same-phase SOO recurrence and recurrence–accommodation synchrony: for a selected relational-path abstraction, phase 0 is the path-facing recurrence role, and every realized nontrivial phase-0 repeat is synchronized one-for-one with one path accommodation. F2 also supplies the exact canonical free photon-like K = 4 recurrence control, which repeats at every successive path-facing phase occurrence. The first part of F3 retains the previously established Lorentz report-chart characterization. For a completed trace, T_raw counts completed rank-3 cycles and N_raw counts realized path-facing accommodation events, so 0 ≤ N_raw ≤ T_raw. Recurrence–accommodation synchrony makes the K = 4 free control saturate the bound. After a single global report-axis orientation, symmetric boundary completion, algebraic real completion, continuous homogeneous charting, individual preservation of the two saturation rays, a moving-origin condition, reciprocal chart equivalence, and global report-axis orientation covariance, the completed one-direction report family is t′ = γ(t − βx),x′ = γ(x − βt),γ = (1 − β²)^−1/2, and preserves t² − x². This remains a report/interface characterization; it does not promote a Lorentz metric or physical simultaneity into primitive SF ontology. The present revision adds the observer and spatial-provenance step that was previously left implicit. The frozen relational observer scope treats an observer as the role of a point in one defined binary relationship, not as a new SF object. One actual relationship has one direct presentation/association provenance, while the rank-3 organization belongs to the point across the three distinct SF/phase contexts. Observer roles whose direct relationships lie in different SF contexts are therefore not observationally identical: which relationship is direct and which information must be synthesized differs. A relational-path report may inherit this provenance without becoming a persistent path object, and the geometry interface preserves directed source–slot–target provenance when separately obtained binary records are represented geometrically. On the independently established nondegenerate three-dimensional EAS geometry sector, three separately certified relational witnesses from one focal source are represented by linearly independent tangent vectors. Distinct SF labels alone do not imply nonparallel spatial directions; nonparallelity follows from the positive-definite Gram certificate of the reconstructed local geometry. For any selected unit relation direction n, the existing F3 single-direction 3+1 embedding, retaining its declared transverse-invariance premise, gives the directional report boost t′ = cosh(η)t − sinh(η)h(n,x), x′ = x + (cosh(η) − 1)h(n,x)n − sinh(η)tn. Its infinitesimal generator is K_n(t,x) = (−h(n,x), −tn). For two directions n and m, K_n,K_m = (0, h(m,x)n − h(n,x)m), which is a nonzero h-antisymmetric spatial rotation generator whenever n and m are nonparallel. On a certified nondegenerate rank-3 geometry cell, the three boost generators and their three rotational commutators close on a six-dimensional Lie algebra isomorphic to so(1,3). Exponentiation therefore generates the proper identity-connected Lorentz report group on this local interface sector. The exact product of two non-collinear nonzero boosts is not a pure boost; after factoring out the unique net pure boost, the remaining spatial factor is nontrivial, giving the Wigner-rotation corollary. The infinitesimal Baker–Campbell–Hausdorff residue is the corresponding boost commutator. The extension is scope-bounded. Observer inequivalence alone does not imply nonparallel spatial directions; degenerate or collinear geometry is a negative control. Rank-3 SF/slot labels are not Cartesian axes. The reconstructed metric is higher-scope interface structure, not data contained in one slot. The Lorentz boost commutator is not identified with the SOO overlap commutator or the NCG differential commutator, and the Wigner rotation is not identified without a separate theorem with charge handedness, spin, or photon polarization. Nor does the completed SO+(1,3) report group prove that every group element is physically realized by an EAS trace. The new result is narrower and stronger: on the certified nondegenerate geometry sector, the frozen observer provenance plus the existing F3 directional boost machinery derive the non-Abelian local 3+1 Lorentz report algebra rather than admitting it as an external Lorentz-group premise.