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BraidStateN.lean: fix π₀(Diff⁺(S⁶)) → Θ₇, note retraction HopfFibration.lean: fix comment, remove diffeomorphism claim CLAIMS_STATUS.md: move π₀ claim to retracted, mark Noether as dead Retraction headers added to: - hopf_portability_criterion.md: ⛔ RETRACTED header - hopf_ingest_bridge.md: ⛔ RETRACTED header (depends on retracted criterion) - noether_route.md: ⛔ DEAD header (3 fatal math errors) rotational_wave_braid_correspondence.md: fix 28 = C(8,2), remove π₀ claim rossby_e8_completion_roadmap.md: fix coupling pairs language Cleanup: no file still claims π₀(Diff⁺(S⁶)) ≅ ℤ₂₈ as true.
3.2 KiB
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
-
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. -
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#evalon a flat-potentialBraidState. -
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:
- Adding a
strandPotential : Fin 8 → Q16_16field toBraidState - Proving
rossby_dispersion_implies_convergence_bound - Proving
boundary_converges_before_interiorunder monotonic potential
Until then, it lives here as a conjecture for future exploration.