Forces: Gauge Structure from Winding
All four Standard Model interactions emerge from Z₇ winding number conservation — not as separate inputs, but as facets of the same polynomial symmetry.
Prerequisites: Complete Path 1 (Foundations), Path 2 (Selection), and Path 3 (Particles) first — they establish the polynomial, the MDL selection mechanism, and the particle ontology that this path builds on.
What this path covers
All four interactions — strong, electromagnetic, weak, and gravity — emerge from Z₇ winding number conservation on the GTE polynomial. The gauge group SU(3)×SU(2)×U(1) is a theorem, not a postulate.
The Weinberg angle and fine-structure constant are derived purely from orbit arithmetic: sin²θ_W = 3/13 and α⁻¹ = 2⁷ + 3² = 137. Both machine-certified in Lean 4 with zero free parameters.
θ_QCD = 0 exactly, by three independent machine-certified proofs from the algebraic structure of the Frobenius group F₂₁. No axion required or predicted — a testable structural fact.
The diagonal of the GTE polynomial reveals x²+x−1: its discriminant is 5 = N_fam (fermion families). Over ℝ it gives the golden ratio and Higgs VEV; over GF(7) it is rootless, forcing Rule 110 universality.
Lessons
The Four Forces
How SU(3)×SU(2)×U(1) emerges from Z₇ winding conservation — the Frobenius group F₂₁ as the symmetry skeleton of the polynomial.
Lesson 1 of 4 · 25% of path
The Weinberg Angle and Fine-Structure Constant
sin²θ_W = 3/13 and α⁻¹ = 2⁷ + 3² = 137 — both derived from orbit arithmetic with zero free parameters, both machine-certified.
Lesson 2 of 4 · 50% of path
Why the Strong CP Problem Is Solved
θ_QCD = 0 exactly from F₂₁ group theory — three independent machine-certified proofs, no axion needed or predicted.
Lesson 3 of 4 · 75% of path
The Master Quadratic
p(x,x,x)−x = −x(x²+x−1): one equation with two crown jewels — the Higgs VEV over ℝ and computational universality over GF(7).
Lesson 4 of 4 · Path complete
Connections to other paths
The Z₇ winding numbers that assign particles to sectors (L15) are the same winding numbers conserved at every force vertex here.
The polynomial p(L,C,R) whose symmetry group is F₂₁ was introduced in L03. Path 4 extracts the force structure from that symmetry.
Gravity is previewed in L18 as emergent from CMCA tape geometry. Path 5 develops this into a full treatment of GR from MDL.