Second-Order Ordering in EAS: Contextual K, Phase-Specific Recurrence, and Relational Accommodation

Abstract

This paper is archived as a speculative research work. This paper is the second foundations paper (F2) of the present Entanglement--Algebraic Spacetime (EAS) publication sequence. It depends on the formal scalar-field ontology fixed in F1 and develops only the detailed second-order-ordering (SOO) recurrence machinery deferred there. A completed SOO realization is a deterministic bijection of the common formal point set, and every nontrivial elementary action is an exact full scalar-value transposition. Contextual scalar stiffness K in [0,4] does not determine a fractional exchange. It governs a same-relational-context, same-phase second-order recurrence, d_{n+1} - 2d_n + d_{n-1} = -K_n d_n. For every admitted recurrence triplet with d_n != 0, the recurrence determines K_n = 2 - (d_{n-1} + d_{n+1}) / d_n, which is invariant under nonzero global rescaling and overall sign conjugacy. The construction and admission of a recurrence coordinate are a separate issue: F2 does not claim a universal extraction rule from the F1 relational-path abstraction alone. For constant 0 < K < 4, the recurrence is a determinant-one elliptic map with cos(omega) = 1 - K/2. The exact sampled recurrence class returns after a finite integer number of same-phase intervals if and only if omega / pi is rational. The familiar family K_m = 2[1 - cos(pi/m)] is the principal exact antireturn subfamily, not a quantization of K. The real quantity m_pr(K) = pi / arccos(1 - K/2) remains well defined for every interior K, but for generic irrational rotation number it is not an exact discrete event count. F2 therefore introduces no hidden scheduler, floor rule, phase accumulator, or tolerance trigger. A finite-resolution realization of a noninteger span is a separately declared report/model convention, not a new native SOO law. F2 also derives a phase-resolved coherence result from the F1 closure rule for derived native quantities. A physically effective same-point phase-K distinction must have native contextual support. For an individual phase SOO operation, the active association is already fixed and the scalar update is uniquely the exact transposition (u,v) |-> (v,u). If that mandatory phase operation removes the final native support for a pre-existing effective phase-K distinction, the distinction cannot persist as an independent residue. Thus every phase-resolved SOO step removes every component of same-point K-incoherence made avoidable by that step. This is a componentwise coherence-reduction theorem, not a variational principle, a new energy, a hidden selection law, or a claim that the total field-wide incoherence burden decreases monotonically. A relational path is used exactly as F1 defines it: a minimum-count modeling abstraction and equivalence class of tied minimum association connections, not an ontological route or persistent carrier. Recurrence-conditioned path accommodation is consequently required to be quotient-consistent. For a realized nontrivial path-facing phase-0 recurrence repeat, the constitutive SOO rule synchronizes exactly one atomic accommodation with |Delta L| = 1; the sign is separately typed. Accommodation changes the admitted minimum-count relational-path report as a whole and does not rewire a certified association or distinguish tied minimum representatives. The paper closes with the exact claim ceiling, negative results, and the remaining construction-level gaps.