Development Status
Paper I is the original working document in which the closure-class framework, assignment rules, and mass law were developed. It contains the full derivation chain but was written as an internal record, not as a standalone presentation.
Paper IV extracts and formalises the key results from this paper into a self-contained, externally readable form. Paper I remains the development record: it shows how the results were found, including dead ends and intermediate steps.
Closure Classes
Particles are proposed as equivalence classes of bounded field configurations on a φ-structured manifold. Each class is characterised by three structural labels:
- Shell support S — which nested φ-scaled shells the configuration occupies
- Winding number w — the topological winding around the shell
- Symmetry sector σ — charged vs neutral, boundary vs interior
Five assignment rules (R1–R5) constrain admissible classes:
- R1: Boundary closure requires minimal shell support
- R2: The electron is the unique minimal boundary class: shell {1}
- R3: Interior composites require at least three shells
- R4: The proton is the unique minimal interior composite: shells {2, 3, 4}
- R5: Winding number w = 1 for ground-state particles
The Combinatorial Invariant
The key structural quantity is the combinatorial invariant ΔC, computed from the graph structure of the closure class:
The electron {1} and proton {2, 3, 4} are uniquely determined as the minimal admissible classes in their respective sectors. This is not a choice — no other shell assignments satisfy R1–R5 with fewer shells. The invariant ΔC = 15 follows with no freedom.
The Three-Order Law
Mass factorises into independent structural sectors. The cumulative result:
Exponent 1265/81 is in lowest terms. Zero fitted continuous parameters.
| Order | Term | Exponent | Predicted | Error |
|---|---|---|---|---|
| 0th | φΔC | 15 | 1364 | 25.7% |
| 1st | + |E|/|V| | 47/3 | 1880 | 2.4% |
| 2nd | − Var(deg)·|E|/|V|² | 1265/81 | 1835.8 | 2 × 10⊃−&sup4; |
Validation
| Ratio | Predicted | Observed | Status |
|---|---|---|---|
| mp/me | 1835.8 | 1836.15 | Structural agreement (2×10⊃−&sup4;) |
| mn/mp | (compatible) | 1.00138 | Compatible at correct scale; not derived |
| mμ/me | — | 206.8 | Fails under shell extension (no-go) |
| mτ/me | — | 3477 | Fails under shell extension (no-go) |
The neutron mass is compatible at the correct scale but is not derived. The muon and tau explicitly fail under shell-extension — this is acknowledged and addressed in Paper II, which proves the failure is structural (no-go theorem) and introduces the winding excitation operator.
What This Does Not Claim
- That this paper is a standalone presentation — it is an internal development record; see Paper IV for the externally readable version
- A first-principles mass derivation — the normalisation is structurally selected, not axiomatically derived
- That the φ-structured manifold is established physics — it is a modelling assumption
- That lepton generations are addressed here — they explicitly fail and are treated in Paper II
- That the assignment rules R1–R5 are derived from deeper axioms — they are structural postulates
The development record behind Paper IV.