Golden Busemann Coordinate

Abstract

Golden null coefficients carry a nontrivial Busemann rapidity coordinate.

Theorem 1.1 (The golden null basis exposes Busemann rapidity).

$l e t Q_{ϕ} (x, y) = x^{2} - x \cdot y - y^{2}; l e t v (a, b) = pair (a \cdot ϕ + b \cdot p hi P r im e, a + b); l e t e t a (a, b) = \frac{1}{2} \cdot log (\frac{a}{- b}); l e t b e t a_{1} (a, b, c, d) = e t a (c, d) - e t a (a, b); (((((((Q_{ϕ}) (ϕ, 1) = 0 \land (Q_{ϕ}) (p hi P r im e, 1) = 0) \land (\forall a \in Real (), b \in Real (), (Q_{ϕ}) (pair (a \cdot ϕ + b \cdot p hi P r im e, a + b)) = - 5 \cdot a \cdot b)) \land (\forall a \in Real (), b \in Real (), (0 < a \land b < 0) \Rightarrow 0 < \frac{a}{- b})) \land (\forall a \in Real (), b \in Real (), ((0 < a \land b < 0) \land (Q_{ϕ}) (pair (a \cdot ϕ + b \cdot p hi P r im e, a + b)) = 1) \Rightarrow e t a (a, b) = log (a \cdot sqrt (5)))) \land (\forall a \in Real (), b \in Real (), c \in Real (), d \in Real (), e \in Real (), f \in Real (), (b e t a_{1}) (a, b, c, d) + (b e t a_{1}) (c, d, e, f) = (b e t a_{1}) (a, b, e, f))) \land (\forall a \in Real (), b \in Real (), (0 < a \land b < 0) \Rightarrow e t a (ϕ^{2} \cdot a, ϕ^{- 2} \cdot b) = e t a (a, b) + 2 \cdot log (ϕ))) \land (((((Q_{ϕ}) (pair (\frac{1}{sqrt ( 5 )} \cdot ϕ + - \frac{1}{sqrt ( 5 )} \cdot p hi P r im e, \frac{1}{sqrt ( 5 )} + - \frac{1}{sqrt ( 5 )})) = 1 \land e t a (\frac{1}{sqrt ( 5 )}, - \frac{1}{sqrt ( 5 )}) = 0) \land (Q_{ϕ}) (pair (\frac{2}{sqrt ( 5 )} \cdot ϕ + - \frac{1}{2 \cdot sqrt ( 5 )} \cdot p hi P r im e, \frac{2}{sqrt ( 5 )} + - \frac{1}{2 \cdot sqrt ( 5 )})) = 1) \land e t a (\frac{2}{sqrt ( 5 )}, - \frac{1}{2 \cdot sqrt ( 5 )}) = log (2)) \land (\neg e t a (\frac{1}{sqrt ( 5 )}, - \frac{1}{sqrt ( 5 )}) = e t a (\frac{2}{sqrt ( 5 )}, - \frac{1}{2 \cdot sqrt ( 5 )})))$

Proof. Machine-checked in Lean as D5/S3/Observer/GoldenCoding/GoldenBusemannCoordinate.golden_busemann_coordinate (✓ std3). ∎

Source. Repository-derived.

Commentary.

The real Lorentz form uses the existing sign convention Q_phi(x,y)=x^2-xy-y^2. The golden ratio and its negative conjugate give two null vectors, and direct polarization reduces the form of a v_plus+b v_minus to -5ab.

On the branch a>0 and b<0, the ratio a/(-b) is positive. At unit Lorentz level the coefficient product is a(-b)=1/5, so the half-log definition agrees exactly with log(a sqrt(5)). Differences of this coordinate satisfy the Busemann cocycle law by telescoping.

Reciprocal golden-square scaling preserves the branch and adds 2 log(phi) to rapidity. The points with coefficients (1/sqrt(5),-1/sqrt(5)) and (2/sqrt(5),-1/(2sqrt(5))) both have Lorentz value one, but their rapidities are zero and log(2).

References

Truth anchor: D5/S3/Observer/GoldenCoding/GoldenBusemannCoordinate.golden_busemann_coordinate
Dependency: D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix
Dependency: D5/S3/Observer/GoldenCoding/GoldenLorentzUpdate

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Golden Busemann Coordinate

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