Prime-Golden Third-Order Free-Lie Bridge

Abstract

A nonzero degree-three free-Lie primitive strictly refines an explicit prime-golden step-two chronology fiber.

Theorem 1.1 (A concrete degree-three observer escapes a full step-two fiber).

$$\begin{array}{c}\mathrm{primeGoldenBidegree}([A,B,B,A])=\mathrm{primeGoldenBidegree}([B,A,A,B])\wedge \mathrm{sameScalarTrajectory}([A,B,B,A],[B,A,A,B])\wedge \\ \mathrm{chronologicalSignature}(g,[A,B,B,A])=\mathrm{chronologicalSignature}(g,[B,A,A,B])\wedge \\ \mathrm{thirdOrderReadout}(g,[A,B,B,A])\ne \mathrm{thirdOrderReadout}(g,[B,A,A,B]).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Observer/Chronology/PrimeGoldenThirdOrderFreeLieBridge.explicit_degree_three_strict_refinement (✓ std3). ∎

Source. Repository-derived.

Commentary.

The ABBA and BAAB histories have the same prime-golden bidegree, the same scalar Euler trajectory at every time, and the same complete step-two chronological signature under the explicit integer-matrix observation.

Their cubic difference is the represented free-Lie primitive minus the bracket of the sum with the first commutator. The E12 and E21 representation evaluates it to a concrete nonzero integer matrix.

This proves strict refinement for one genuine residual fiber. It does not assert that degree three separates every finite chronology.

References

• Truth anchor: D5/S3/Observer/Chronology/PrimeGoldenThirdOrderFreeLieBridge.explicit_degree_three_strict_refinement
• Dependency: D5/S3/Observer/Chronology/PrimeGoldenThirdOrderChronologyEscape
• Dependency: D5/S3/Observer/Chronology/StepTwoFreeLieBridge