A Taxonomy of Results in the Emergence Canvas Model: Pillars, Theorems, Derived Formulas, Rules, and Open Problems

What This Paper Does The Emergence Canvas Model is a proposed unified framework for fundamental physics derived from eight primitive concepts and four dynamical pillars. The programme has produced a large body of results spanning combinatorics, gauge theory, particle physics, mixing matrices, cosmology, quantum foundations, gravity, and detachment dynamics. This paper presents a complete taxonomy of every significant statement in the programme, classifying each into one of five categories: · Pillars (4): The irreducible axioms—Unified Wave Equation, Threshold Condition, Eigenvalue Equation, Feed Equation· Theorems (24): Results proved from the pillars (19 directly proved, 3 algebraic/tautological, 2 conditional on construction)· Derived Formulas (15): Exact expressions for physical quantities (9 fully concrete, 6 partially concrete)· Derived Rules (8): Consequences, organizing principles, and identities (all concrete)· Schematic/Open Items (14): Results specified at the conceptual level but not yet quantitatively determined For each item, the paper provides a precise statement, its status within its category, its dependencies on other items, and a pointer to the location of its derivation in the collected works. The Dependency Graph The taxonomy reveals a clear dependency structure. Every theorem depends on at least one pillar (P3 supports the most, with 10 theorems). The derived formulas follow from the theorems and primitive values. The open items are largely downstream of four constitutive coefficients (S1–S4): loop tension \mu_s, twist stiffness \kappa_{\theta,s}, threshold-to-loop susceptibility \chi_s, and source normalization g_s. Of the 14 primary open items, 10 are directly blocked by these coefficients. The Bottleneck The paper identifies a sharp bottleneck: deriving S1–S4 from the primitive action or from the finite spectral operator (T23) would convert the majority of open items into computable predictions. The fermion mass spectrum, CKM subleading parameters, PMNS \delta_{\text{CP}}, and new particle masses all become determinable once these coefficients are known. Why This Matters The taxonomy serves multiple purposes: · A snapshot of the programme at the August 2026 research freeze point· A map for new researchers showing exactly what has been done and what remains· A constraint preventing the programme from claiming derived status for Conditional results· A guide identifying the highest-priority target (S1–S4)· An immune system against the blurring of derivational provenance The paper distinguishes what has been established from what remains conditional. It identifies the exact remaining task. And it provides a framework for evaluating future progress: as items move from the Schematic to the Derived categories, the taxonomy is updated, and the predictive scope of the model expands. The Provenance System The paper includes a provenance checklist (Appendix B) for evaluating any output of the Canvas Engine. Only results passing all ten checks should be called first-principles predictions. The checklist ensures that no calibrated constants, fitted parameters, or post-hoc selections are mistaken for derived predictions. Complete Cross-Reference A complete cross-reference table (Appendix A) maps every item to its location in the collected works, making the programme's internal logic transparent and verifiable. Keywords: Canvas Model, taxonomy, pillars, theorems, derived formulas, derived rules, schematic items, open problems, dependency graph, constitutive coefficients, provenance, unified framework, fundamental physics