The Dynamics of the Calibration Field: Transition from Order Two to Order Four in the Triadic Structure, and the External Scale

The triad (Δ, Λ, Ψ) is an algebraic structure carrying three operations at once: Δ, dimension; Λ, relation; and Ψ, orientation. This paper constructs one of the two open fronts named by the guide to the series — the dynamics of the calibration field (relation Λ). When the calibration takes a single value at each point of spacetime, forming a field λ(x), its behavior is written using only the operations of the triad. The central result is that the calibration field, across a critical point, undergoes a transition from a phase of order two (a single well, the half-loop) to a phase of order four (a double well, the full-loop), realizing at the level of field dynamics the "establishment up to order two and the closure of order four by reversal" that the series has established algebraically. The transition occurs at the fixed point of Ψ (unit calibration λ = 1). The calibration field receives a single external input L that supplies dimension, but the dimensionless calibration ratio does not depend on the value of L — the protection of axiom T2, which this paper preserves rather than resolves. As a cross-check, the paper proceeds from the general to the particular: the single well ↔ double well ↔ critical point structure is common to the Landau universality class (ferromagnet, ferroelectric, superconductor), with two features specific to the calibration field (the critical point seated at the fixed point of Ψ, and the T2 demarcation of the external scale); within that class, the two local structures of liquid water reported by Li et al. (Nature Physics, 2026) give a measurable instance. This correspondence is a structural agreement within a universality class, not a claim of identity. No new dynamical principle is introduced. The theorem-level claims are machine-verified by the accompanying script (29 checks, all passing).