This paper is archived as a speculative research work. This paper gives a mathematical axiom system for the scalar-field level of Entanglement-Algebraic Spacetime (EAS). The purpose is to formalize scalar fields without importing spacetime geometry, physical time, metric structure, particle ontology, wave propagation, gauge fields, or tensor fields on a geometric base manifold. The scalar-field starting point is a scalar-point set, real scalar values/signs, rank-3 association records, ordered scalar-field presentations, and second-order ordering (SOO) admissibility. The central mathematical object is not a spacetime manifold but a non-geometric admissible scalar-presentation structure on which report charts, invariant functionals, and interface-facing compressions may be defined. The paper introduces a finite relational tensor encoding of rank-3 association records. This tensor notation represents association data over scalar-point and slot spaces; it is not a spacetime tensor field and does not add scalar-field ontology. The encoding sharpens the distinction between global rank-3 phase and pointwise handedness. Phase is a global presentation-level slot selector. Handedness is a pointwise orientation/facing of cyclic rank-3 association exposure relative to the global phase order and exterior comparison context. This prevents pointwise phase from entering the scalar-field ontology while retaining the mathematical advantages of slot-space tensor algebra. SOO is formalized as an admissibility relation over ordered scalar-field presentations. In finite report sectors, SOO may be represented by a second-difference equation with a stiffness report; stiffness operators, spectra, eigenmodes, phases, and complex coordinates remain report-level structures. The corrected discrete SOO invariant is stated, and recurrence, phase-like compression, and quantum-facing report coordinates are restricted to stable positive-stiffness sectors satisfying 0 < epsilon² lambda < 4. The formalism also records common-mode and zero-sum projections of completed rank-3 reports, yielding a mathematical setting for charge-facing, Pauli-facing, Noether-facing, and gauge-facing report structures without reifying those reports as scalar-field ontology. The result is a disciplined scalar-field axiom system for EAS. It supplies a non-geometric mathematical basis for scalar presentations, rank-3 association tensors, global phase, pointwise handedness, SOO admissibility, stable report sectors, automorphic scalar-point patterns, recurrence theorems, Noether-facing report conservation, and local report-frame connection structure. It does not derive spacetime, quantum field theory, Maxwell's equations, empirical charge, or the full fermionic interface.
Michael Labhard (2026) studied this question.