Two-Sided Closure Theory: A Self-Contained Overview

v4 update (May 2026): Binary Screen concept throughout. A₃ derived from minimality principle (not assumed). Readout obligations updated to reflect Papers IV and V. Three readout lemmas now the complete statement of open obligations. Related papers: Paper I: 10.5281/zenodo.20340356 Paper II: 10.5281/zenodo.20340692 Paper IV: 10.5281/zenodo.20340084 Paper V: 10.5281/zenodo.20329022 This document is the front door to the Two-Sided Closure Theory (TSCT) programme. It is written for a theoretical physicist with no prior exposure to TSCT. One reading provides enough context to engage critically with any paper in the series. TSCT proposes that the observable sector of a relativistic quantum field theory is the subalgebra selected by an algebraic closure condition. From this single principle and the standard axioms of algebraic quantum field theory we derive or motivate: the Einstein field equations (conditional on the Bisognano–Wichmann identification for interacting fields), the Koide formula for charged-lepton masses (conditional on two named lemmas), and a candidate value for the solar neutrino mixing angle (pending one structural lemma). Every result is labelled Proved, Conditional (on a named assumption), or Open (pending a specific lemma). Nothing is presented as stronger than its actual status. The primer contains:- A self-contained derivation of the conditional Koide theorem (K = 2/3 from the A₃ Jones index M:N = 2, conditional on Yukawa democracy and Pimsner–Popa saturation of the oscillatory mode)- The full gravity chain from Jones projection mismatch to Einstein equations, with explicit thermodynamic assembly- The two-sided A₃ closure formula (δ + δ⁻¹)²/(‖d‖²‖d⁻¹‖²) = 9/20 giving the candidate solar angle arctan√(9/20) ≈ 33.87° (observed: 33.4° ± 0.75°)- Three named, bounded proof obligations (O1, O2a, O2b)- A section "How to Attack It" listing four specific falsification routes This is Paper III of the TSCT series. Papers I and II on Zenodo are the starting point for detailed engagement. Companion papers:Paper I: Einstein's Field Equations from Algebraic Closure (doi: 10.5281/zenodo.20282969)Paper II: Towards Standard Model Mixing Parameters from the A₃ Jones Subfactor (doi: 10.5281/zenodo.20283404) Keywords: algebraic quantum field theory, Jones subfactors, A₃ subfactor, Koide formula, solar mixing angle, Einstein equations, Bisognano–Wichmann, relative modular operator, Araki relative entropy, Pimsner–Popa theorem, neutrino mixing, Two-Sided Closure Theory.