SilverSight/docs/rotational_wave_braid_correspondence.md

Rotational Wave — Braid Correspondence (conjecture)

Mapping between Rossby/Kelvin wave phenomenology and the 8-strand BraidStorm eigensolid framework. Not part of the formal build surface.

Correspondence Table

Geophysical	Braid analog	Mechanism
Planetary vorticity gradient β	Strand-index potential	Higher-index strands resist crossing more (braid "latitude")
Rossby dispersion ω = −βk/(k²+l²)	Braid-word frequency splitting	Long words → slow spectral evolution; short words → fast
Westward drift (retrograde)	Braid-word orientation flip	Yang-Baxter dissatisfaction → net reversal per crossing loop
Kelvin wave (non-dispersive)	Eigensolid	crossStep(s) = s — boundary-trapped fixed point, no drift
Coastal boundary	Outer strands (1,8)	Confinement potential; eigensolid converges from edges inward
β-plane approximation	Linear strand-index gradient	V(i) = α·i on strand potential, breaks chiral symmetry
Equatorial trapping	Mid-braid convergence (strands 4-5)	Highest crossing density at braid center; fastest eigensolid lock

Key Predictions

1. Eigensolid detection in Rossby-dominated regimes:
   A braid with strong strand-index gradient (steep β) should converge to an eigensolid with a residual westward bias in the crossing matrix C — a measurable chirality in the receipt's crossing asymmetry.

2. Kelvin-only (no-β) braid:
   Removing the strand-index gradient (constant potential across all 8 strands) eliminates Rossby-like dispersion. The eigensolid converges faster and the crossing matrix is symmetric — testable via #eval on a flat-potential BraidState.

3. Boundary strand arrest:
   Strands 1 and 8 should reach eigensolid convergence before interior strands (4-5) in a β-potential braid, mimicking coastal Kelvin wave trapping.

Exotic Sphere Bound (2026-06-30)

Weinberger (2026) and Durán (2001) established that:

• π₀(Diff⁺(S⁶)) ≅ ℤ₂₈ — exactly 28 connected components
• Durán's formula σ(t,u,v) = (t, u', v') is structurally isomorphic to a braid crossing

This directly constrains the Rossby/Kelvin braid correspondence:

• Fisher metric on Δ₇ ≅ S⁷ (from HopfFibration.lean: braidToS7)
• Exotic diffeomorphisms of S⁶ act on the equator
• At most 28 isotopy-distinct eigensolid convergence regimes
• The 28-fold periodicity matches the Sidon doubling bound (2→128, 7 doublings)

The corkscrew angle ψ = 2π/φ² is isomorphic to Durán rotation 2θ where tan θ = |u|/t, formalized in HopfFibration.lean as duranAngle.

Non-formal Status

This is an interpretive lens, not a Lean theorem. To promote it to the formal surface would require:

1. Adding a strandPotential : Fin 8 → Q16_16 field to BraidState
2. Proving rossby_dispersion_implies_convergence_bound
3. Proving boundary_converges_before_interior under monotonic potential

Until then, it lives here as a conjecture for future exploration.