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# Rotational Wave — Braid Correspondence (conjecture)
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Mapping between Rossby/Kelvin wave phenomenology and the 8-strand
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BraidStorm eigensolid framework. Not part of the formal build surface.
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## Correspondence Table
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| Geophysical | Braid analog | Mechanism |
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|-------------|-------------|-----------|
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| Planetary vorticity gradient β | Strand-index potential | Higher-index strands resist crossing more (braid "latitude") |
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| Rossby dispersion ω = −βk/(k²+l²) | Braid-word frequency splitting | Long words → slow spectral evolution; short words → fast |
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| Westward drift (retrograde) | Braid-word orientation flip | Yang-Baxter dissatisfaction → net reversal per crossing loop |
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| Kelvin wave (non-dispersive) | Eigensolid | `crossStep(s) = s` — boundary-trapped fixed point, no drift |
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| Coastal boundary | Outer strands (1,8) | Confinement potential; eigensolid converges from edges inward |
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| β-plane approximation | Linear strand-index gradient | `V(i) = α·i` on strand potential, breaks chiral symmetry |
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| Equatorial trapping | Mid-braid convergence (strands 4-5) | Highest crossing density at braid center; fastest eigensolid lock |
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## Key Predictions
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1. **Eigensolid detection in Rossby-dominated regimes:**
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A braid with strong strand-index gradient (steep β) should converge to an
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eigensolid with a residual westward bias in the crossing matrix C — a
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measurable chirality in the receipt's crossing asymmetry.
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2. **Kelvin-only (no-β) braid:**
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Removing the strand-index gradient (constant potential across all 8 strands)
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eliminates Rossby-like dispersion. The eigensolid converges faster and the
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crossing matrix is symmetric — testable via `#eval` on a flat-potential
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`BraidState`.
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3. **Boundary strand arrest:**
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Strands 1 and 8 should reach eigensolid convergence before interior
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strands (4-5) in a β-potential braid, mimicking coastal Kelvin wave
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trapping.
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## Exotic Sphere Bound (2026-06-30)
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Weinberger (2026) and Durán (2001) established that:
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- Θ₇ ≅ ℤ₂₈ — exotic 7-spheres under connected sum (Kervaire-Milnor 1963)
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[NOTE: this is NOT π₀(Diff⁺(S⁶)), see `cartan_fingerprint.md` §2 for retraction]
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- C(8,2) = 28 — combinatorial coupling pairs for 8 strands
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- Durán's formula σ(t,u,v) = (t, u', v') is structurally isomorphic to a braid crossing
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This **directly constrains** the Rossby/Kelvin braid correspondence:
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- Fisher metric on Δ₇ ≅ S⁷ (from HopfFibration.lean: braidToS7)
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- Exotic diffeomorphisms of S⁶ act on the equator
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- At most C(8,2) = 28 combinatorial coupling pair configurations
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(independent of exotic sphere theory; the 28 is a triangular number)
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- The 28-fold periodicity matches the Sidon doubling bound (2→128, 7 doublings)
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The corkscrew angle ψ = 2π/φ² is isomorphic to Durán rotation 2θ where tan θ = |u|/t, formalized in `HopfFibration.lean` as `duranAngle`.
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## Non-formal Status
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This is an interpretive lens, not a Lean theorem. To promote it to the
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formal surface would require:
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1. Adding a `strandPotential : Fin 8 → Q16_16` field to `BraidState`
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2. Proving `rossby_dispersion_implies_convergence_bound`
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3. Proving `boundary_converges_before_interior` under monotonic potential
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Until then, it lives here as a conjecture for future exploration.
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