Hinge Theorem: Books IV–VII Are Sector Instantiations of Book III
All content of Books IV–VII is a sector instantiation of the Book III template; all inter-sector couplings are rational functions of ι_τ with zero free parameters.
Overview
The Hinge Theorem (III.T41) proves that Books IV–VII are sector instantiations of the 4+1 sector template derived in Book III: the four primitive sectors (D=gravity, A=weak, B=EM, C=strong) plus one derived sector (ω=Higgs) generate all physical content. The No Knobs Theorem (III.T42) proves that all 10 inter-sector couplings are rational functions of ι_τ = 2/(π+e) — zero free parameters. Together these two theorems are the structural explanation for why physics books exist and why their predictions are testable without tuning.
Detail
Book III derives the 4+1 sector template from the Canonical Ladder: at enrichment level E₁, exactly four primitive sectors emerge from the four ρ-orbit classes (α=gravity sector D, π=weak sector A, γ=EM sector B, η=strong sector C), plus one derived sector from the ω generator (Higgs sector). The Hinge Theorem III.T41 establishes that every physical object in Books IV–V is a sector-instantiated object: particles are E₁-objects in specific sector positions, forces are inter-sector morphisms, and cosmological observables are E₁-level data on the base τ¹. The No Knobs Theorem III.T42 counts the C(5,2) = 10 pairwise inter-sector coupling constants and proves that each is a rational function of ι_τ alone — no continuous free parameters. This makes all predictions in Books IV–V testable and falsifiable from a single master constant.
Result Statement
III.T41 (Hinge Theorem): Books IV–VII are sector instantiations of the 4+1 sector template derived in Book III. III.T42 (No Knobs Theorem): all 10 inter-sector couplings are rational functions of ι_τ = 2/(π+e) — zero free parameters.