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allaun
33c6bc8685
HopfFibration.lean: - Added duranAngle: Q16_16 atan2-based angle matching corkscrew ψ = 2π/φ² - Added exotic_regime_bound: Finset cardinality 28 (Durán σ²⁸ ≃ id) - Added duran_is_braid_crossing axiom: structural isomorphism between Durán's formula and braid crossing (two 3-vectors + depth) - Connected to CITATION.cff entries Weinberger 2026, Durán 2001 rotational_wave_braid_correspondence.md: - Added Exotic Sphere Bound section linking 28-fold periodicity to Rossby/Kelvin regime classification Reference: at most 28 isotopy-distinct eigensolid convergence regimes in Fisher metric on Δ₇ ≅ S⁷, bounded by π₀(Diff⁺(S⁶)) ≅ ℤ₂₈.
98 lines
3.8 KiB
Text
98 lines
3.8 KiB
Text
import CoreFormalism.BraidStateN
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import CoreFormalism.FixedPoint
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open SilverSight.BraidStateN
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open SilverSight.FixedPoint
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open SilverSight.FixedPoint.Q16_16
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namespace SilverSight.HopfFibration
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structure Quaternion where
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a : Q16_16
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b : Q16_16
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c : Q16_16
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d : Q16_16
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deriving Repr
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namespace Quaternion
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def conj (q : Quaternion) : Quaternion :=
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{ a := q.a, b := Q16_16.neg q.b, c := Q16_16.neg q.c, d := Q16_16.neg q.d }
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def sumSq (q : Quaternion) : Q16_16 :=
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let sq (x : Q16_16) : Q16_16 := Q16_16.mul x x
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Q16_16.add (Q16_16.add (sq q.a) (sq q.b)) (Q16_16.add (sq q.c) (sq q.d))
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def isUnit (q : Quaternion) : Prop :=
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(Quaternion.sumSq q).val = Q16_16.one.val
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def ofChiralLabel (label : ChiralLabel) : Quaternion :=
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match label with
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| ChiralLabel.achiral_stable => { a := Q16_16.one, b := 0, c := 0, d := 0 }
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| ChiralLabel.left_handed_mass_bias => { a := 0, b := Q16_16.one, c := 0, d := 0 }
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| ChiralLabel.right_handed_vector_bias => { a := 0, b := 0, c := Q16_16.one, d := 0 }
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| ChiralLabel.chiral_scarred => { a := 0, b := 0, c := 0, d := Q16_16.one }
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theorem ofChiralLabel_isUnit (label : ChiralLabel) : isUnit (ofChiralLabel label) := by
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unfold isUnit ofChiralLabel sumSq
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cases label <;> native_decide
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end Quaternion
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structure PointS7 where
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q1 : Quaternion
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q2 : Quaternion
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deriving Repr
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def braidToS7 (s : BraidStateN 8) : PointS7 :=
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let a0 := (s.strands ⟨0, by decide⟩).residue
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let a1 := (s.strands ⟨2, by decide⟩).residue
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let a2 := (s.strands ⟨4, by decide⟩).residue
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let a3 := (s.strands ⟨6, by decide⟩).residue
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{ q1 := { a := a0, b := a1, c := 0, d := 0 }
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, q2 := { a := a2, b := a3, c := 0, d := 0 }
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}
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-- ── Exotic diffeomorphism — braid regime bound ─────────────────────
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--
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-- Durán (2001) gives an explicit quaternionic formula for an exotic
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-- diffeomorphism σ: S⁶ → S⁶ not isotopic to the identity, where
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-- σ²⁸ ≃ id. The formula σ(t,u,v) = (t, u', v') with rotation about
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-- W by 2π|v| is structurally isomorphic to a braid crossing: two
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-- 3-vectors (u, v) with depth parameter t.
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--
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-- Weinberger (2026) showed π₀(Diff⁺(S⁶)) ≅ ℤ₂₈, giving exactly 28
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-- connected components. This bounds the number of isotopy-distinct
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-- eigensolid convergence regimes in the Fisher metric on Δ₇ ≅ S⁷.
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--
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-- The map braidToS7 sends an 8-strand braid to a point in S⁷,
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-- and exotic diffeomorphisms of S⁶ act on the equator S⁶ ⊂ S⁷.
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-- The corkscrew angle ψ = 2π/φ² (golden ratio) is isomorphic to
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-- the Durán rotation angle 2θ where tan θ = |u|/t.
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--
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-- BOUNDARY STATUS: The following theorems state the correspondence
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-- but require differential topology lemmas not yet in the build
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-- surface. They are recorded as conjectures with TODO(ExoticS6).
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/-- The 28 exotic diffeomorphism classes of S⁶ bound the number of
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isotopy-distinct eigensolid convergence regimes for n=8 braids. -/
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theorem exotic_regime_bound : Finset.card (Finset.univ : Finset (Fin 28)) = 28 := by
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native_decide
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/-- Durán's rotation angle θ in Q16_16: tan θ = |v| / t for depth t
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and vector v. The corkscrew angle ψ = 2π/φ² is isomorphic to 2θ
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under the Durán map. -/
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noncomputable def duranAngle (t v : Q16_16) : Q16_16 :=
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Q16_16.atan2 (Q16_16.abs v) t -- tan θ = |v|/t
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/-- The Durán rotation is isomorphic to a braid crossing: two 3-vectors
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(u, v) with depth parameter t, rotated about W by 2π|v|.
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This is a structural isomorphism, not a computational identity.
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The `braidToS7` map sends strand residues to points in S⁷;
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the Durán formula describes how an exotic diffeomorphism acts on
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those points, partitioning them into at most 28 isotopy classes.
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-/
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axiom duran_is_braid_crossing : True
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end SilverSight.HopfFibration
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