From Nothingness to EAS Scalar-Field Ontology

This paper is archived as a speculative research work.Entanglement-Algebraic Spacetime (EAS) is developed from a foundational physical stance in which no primitive spacetime, geometry, matter, field medium, or other physical background is assumed. The EAS scalar-field (SF) construction is therefore not itself a primordial physical substrate. It is a formal mathematical apparatus for stating the minimal relational ontology from which later report structures and physical interfaces may be constructed. This revision removes a residual global-state assumption from the earlier F1 formulation without weakening its universal structural ontology. There remains one common nonempty formal scalar-point domain X and one common scalar-field succession ℓ . Apart from formal identity itself, the SF-native point data by which scalar points may differ are their scalar datum in an admitted relational context and their rank–3 association record. That record carries an intrinsic cyclic order; its two conjugate cyclic orientations are the two point hands. A sign h(x)∈+1,-1 may encode those two orientation classes, but it is not an additional independent point datum. No point-local phase origin, phase offset, clock, cadence, scheduler, coordinate, or other native discriminator is admitted. Every point has exactly three locally labelled association roles, each with either a defined target or explicit undefinedness. A point that is defined in the canonical G1 sense is association-complete and therefore has three distinct defined binary associations. The numerical entry labels are local representatives only; they are not an absolute first phase and do not define handedness. At each common rank–3 SF occurrence exactly one association role of each point is active. A defined bilateral association may occupy different local entry labels at its two endpoints, but the common association anchors a reciprocal structural correspondence between the two ordered rank–3 descriptions. Path-facingness is structure-specific rather than universal; when local notation labels a path-facing reference entry by 0 , the numeral is a convention rather than ontology. A coherent bounded support possesses no path-facing association. Ordinary defined points additionally carry exactly one S and two O association characters, independently of endpoint hand comparison. The ontology also admits a scalar binary bit (SBB) only as a four-point structural carrier. At a registered cycle-boundary occurrence, one active binary carrier context has a minimally conjugate relational scalar datum and the disjoint companion carrier context has an exact zero–zero relational scalar datum. These are two distinct binary-context statements, not one globally defined four-point scalar tuple. A bare (+q,-q) bilateral pair is not an SBB; without the two additional carrier points and their fixed rank–3 association organization it is only an ordinary binary datum subject to native SOO. ER3C is the constitutive absolute four-cycle/internal-role association rule for the four-point carrier across the one common rank–3 succession, leaving one residual third association role at each carrier point for exterior embedding or explicit undefinedness. ER3C is not a scalar-transport rule: neither SOO nor point identity carries a scalar value from one relational context into another. Universal association reachability remains fundamental: every ordered point pair is connected by a finite association succession. This universal ontological law does not imply that all relational scalar states are simultaneously co-given to a privileged observer. Scalar value is strictly relative to relational context. For an ordered presentation x→ y of a bilateral binary relation, the endpoint values define the derived relational scalar difference ^rel_ℓ(x→ y) = _ℓ^\\,x|y(y)- _ℓ^\\,x|y(x). Native second-order ordering acts only on the active binary relationship by exact full scalar transposition, (u,v) (v,u) . It preserves the pair scalar multiset, signed total, and sign-separated scalar magnitudes. A simultaneous scalar table or completed successor involving several binary relations is a scope-relative representation and is not thereby a primitive observer-independent SF state. Relational actuality also has two exhaustive organization-defining domains: Relationships and the Related. A Relationship-domain organization is exhausted as the relational carrier it constitutes; a Related-domain organization retains relational organization that can remain identifiable while relationships involving it differ, and can therefore hold relationships. These domains are organizational roles over the same point set, not two point populations and not a fourth point datum. Their numerical recurrence representation is not part of F1. Likewise, an ``observer'' in F1 is only an existing point considered in its participant role within one direct binary relation. Direct source-relative provenance records the common occurrence together with the source identity, its endpoint-local association entry, and the target; it creates no external observer object or global viewpoint. Relational paths remain modeling abstractions rather than ontological route objects. A path is the minimum-count abstraction between the relevant SF edges; tied minimum realizations remain path-indistinguishable. Every path extends between the edges of the SFs, and every count between its endpoints is relational to the count beyond. The paper closes with an expanded ontology–representation–interface firewall and a sharply bounded foundational conjecture: quantum probability may be required when a higher-scope description predicts prospective relational actuality, but neither the Born rule nor quantum mechanics is derived here.