Three-Generation Closure: Spectral Gap on T²
The T² Laplacian supports exactly three primitive (gcd=1) winding classes below the first composite mode. The ratio λ_(2,0)/λ_(1,1) = 4/(1+ι_τ⁻²) ≈ 0.4173 shows the composite (2,0) mode (λ=4) lies between the first and second primitive classes. The next primitive mode beyond (1,1
What this page is
This is a public Results-lane surface for a noteworthy Physics Registry item. It is generated from the Corpus Registry triage catalogue and keeps the generic Result catalogue unchanged.
Registry evidence
- Registry item: IV.T147
- Type: theorem
- Scope: tau-effective
- Lean status: formalized
- Book / part / chapter: Book IV · Part 5 · Chapter 35
Result summary
The T² Laplacian supports exactly three primitive (gcd=1) winding classes below the first composite mode. The ratio λ_(2,0)/λ_(1,1) = 4/(1+ι_τ⁻²) ≈ 0.4173 shows the composite (2,0) mode (λ=4) lies between the first and second primitive classes. The next primitive mode beyond (1,1) is (2,1) at λ=4+ι_τ⁻²≈12.58, far above the three primitive classes. No fourth light generation exists: any fourth primitive mode has mass scale ≥ ι_τ⁻²·m_τ ≈ 29,850·m_e, exceeding the dark-sector cutoff. (50-digit mpmath verified.)
Related Results surfaces
- No existing public Results surface is linked yet; this record is promoted as a standalone Registry-backed result.
Reading role
Read as a standalone Registry-backed noteworthy result.
Claim boundary
This page reports a Registry-backed internal result surface. It is not an external validation claim, a scientific consensus claim, or independent acceptance.
Curation rationale
- physics-facing terms: dark, mass, spectral
- theorem/proposition-class item appears externally legible enough for standalone review
Review notes
- No additional review notes recorded.