E1P in Algebra: Phase Space and Operators for Coherence Calculation

Energetic First Principles (E1P) is defined energetically. Its core assertion is that energy is the ratio of Active to Connective components, resonant to Unity: E ≡ A/C ⇋ 1, with normalized states satisfying A + C = 1. From this relation E1P develops the four phase characters AA, CC, CA, AC, the six dyadic engagements, canonical phase rotation, entropy-adaptive sequencing, and the AC three-stream partition into closure, spectra, and remainder. This paper introduces an instrumental algebraic notation for using E1P in calculational settings. The purpose is to define the operators and symbols used when E1P phase dynamics are represented algebraically. The central object is Primary Energetic Wheelspace, denoted 𝕎_E: the structured phase-domain in which E1P states move through phase positions, dyadic teeth, and coherence transitions. The central calculus instruments are the CA integral ∫_CA and the AC derivative d_AC/dx. The central output operator is the AC partition Π_AC(x) = (κ(x), σ(x), r(x)), where κ denotes closure, σ denotes spectra, and r denotes remainder. The central operational comparison is the entropy-dependent ordering of CA and AC, written as the sequence-order contrast Ω_CA,AC. The compact one-step coherence update is x_{n+1} = Tr_AA(Π_AC(𝒮_Θ(x_n)(x_n))): select sequence, apply, partition, return through closure or propagate remainder. The paper sits in a three-document scaffold: Primer → Wheelspace → Wheelgebra. The Primer gives the grammar; Wheelspace gives the grammar in motion; Wheelgebra gives the operating notation when the grammar must calculate. Public wheelgebra and wheelspace literature (Ginzburg–Schedler wheelgebras, generalized wheelspaces, wheeled PROPs, TRAPs) is a formal neighbour; Wheelgebra in this paper is the instrumental algebraic handling of operations in E1P's Primary Energetic Wheelspace.