GENERATOR PHYSICS IV From Admissibility to Worldhood Generators, Representations, Fields, and Spacetime Closure Architecture and the Realization of Physical Freedom

Closure admits; generators organize; representations carry; fields realize; connections compare; dynamics evolves; observables disclose; reclosure validates. A generator is an infinitesimal direction of continuous transformation: rotation, translation, phase, boost, or another continuous freedom. A field is not merely a carrier for a representation; it is locally realized content distributed across a localization domain with rules for comparison and global compatibility. The paper asks what must be in place before either structure is physically meaningful. The proposed answer is closure admissibility: a prior constraint on which relational and transformational structures can coexist coherently. This does not mean every later physical structure has already been derived from closure. Much of the paper is devoted to making precisely those still-open transitions visible. A reader should therefore distinguish two kinds of statements throughout. Some are standard mathematics: Lie groups give Lie algebras, connections give curvature, and suitable causal cones determine conformal Lorentzian information. Other statements are hypotheses of Generator Physics: that primitive difference is transferable, that spatial realization obeys vector–bivector co-closure, or that closure can ultimately select physical dynamics. The notation [E], [I], [H], [O], and [D] marks these differences explicitly. Generator Physics IV develops a closure-prior architecture connecting admissibility, generators, representations, fields, geometry, dynamics, and empirical realization. The motivating question—whether fields or generators are more fundamental—is reframed by distinguishing mature reciprocal physical structures from their common admissibility conditions. Within the proposed architecture, neither generator algebra nor physical fieldhood is primitive relative to closure admissibility. C_adm ≺ {𝔤, Φ}, while after realization C_adm → (𝔤 ↔ρ Φ). The manuscript develops four conditional reconstruction programs. First, transferable relational difference is organized through heap/torsor structure and, under an additional exchange condition, yields Abelian translation and its infinitesimal generators. Second, the proposed vector–bivector spatial co-closure condition dim V_sp = dim Λ²V_sp selects three spatial dimensions as the unique nonzero finite solution. Third, a one-dimensional causal-order sector combined with three-dimensional isotropic spatial structure and a nondegenerate null boundary yields Lorentzian signature and the standard Lorentz/Poincaré generator grammar. Fourth, representation theory is extended to fieldhood through localization, bundle structure, compatible local realization, connections, and curvature. The paper does not claim that symmetry uniquely determines dynamics. Instead it distinguishes kinematic admissibility from dynamical selection and introduces a thirty-four-bridge ledger identifying where standard mathematics ends, where conditional derivation applies, and where physical-selection problems remain open. The result is a modular architecture from mathematical possibility to empirical worldhood. Its principal contribution is not a complete derivation of known physics, but an explicit dependency structure in which assumptions, theorems, hypotheses, no-go statements, failure conditions, and empirical obligations are separated rather than conflated. The central question of this paper begins simply: Which is more fundamental—fields or generators? In mature physics the answer is not straightforward. Fields transform under generators; generators become physically meaningful through their action on states or fields. Generator Physics IV argues that this apparent circularity results from beginning the analysis too late. Closure admissibility is prior to both generator grammar and mature physical fieldhood. The paper therefore replaces a binary choice with a layered dependency: admissibility → transformation → representation → localization → fieldhood → dynamics → observables → empirical reclosure. Four constructive tests then ask whether translation, three-dimensional spatial rank, Lorentzian spacetime, and fieldhood can be reconstructed conditionally without smuggling their downstream structure into the premises. • Under H1 and H2, transferable relational difference can be represented by heap/torsor structure and, in the flat benchmark, Abelian translation. • Under H3, dim V_sp = dim Λ²V_sp gives the unique nonzero finite solution n = 3; metric and orientation are still separate inputs before Hodge duality is invoked. • Under H4–H6, a one-dimensional succession sector, three-dimensional spatial sector, and nondegenerate isotropic null boundary imply Lorentzian signature. • Representation is distinguished from fieldhood: ρ does not by itself produce E_ρ, Φ, a connection 𝒜, or a physical state. • Symmetry and field content constrain candidate dynamics but do not generally select a unique physical law; the dynamics-selection bridge remains open. • Mathematical closure does not establish empirical truth.