# 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:
- Θ₇ ≅ ℤ₂₈ — exotic 7-spheres under connected sum (Kervaire-Milnor 1963)
  [NOTE: this is NOT π₀(Diff⁺(S⁶)), see `cartan_fingerprint.md` §2 for retraction]
- C(8,2) = 28 — combinatorial coupling pairs for 8 strands
- 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 C(8,2) = 28 combinatorial coupling pair configurations
  (independent of exotic sphere theory; the 28 is a triangular number)
- 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.