Prime-Golden Chronology Fiber Separation

Abstract

Scalar prime-golden observation is constant on a bidegree fiber, while a noncommutative second-Magnus readout can separate chronology inside that fiber.

Theorem 1.1 (Magnus separates swapped histories hidden by scalar observation).

$$\begin{array}{c}\forall f,\forall u,\forall w,\mathrm{commutator}(f(u),f(w))\ne \mathrm{commutator}(f(w),f(u))\Rightarrow \\ \mathrm{primeGoldenBidegree}([u,w])=\mathrm{primeGoldenBidegree}([w,u])\wedge \\ \mathrm{sameScalarTrajectory}([u,w],[w,u])\wedge \\ \mathrm{doubledMagnusDegreeTwo}(\mathrm{chronologicalSignature}(f,[u,w]))\ne \mathrm{doubledMagnusDegreeTwo}(\mathrm{chronologicalSignature}(f,[w,u])).\end{array}$$

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

Source. Repository-derived.

Commentary.

A fixed-prime scalar endpoint factors through the prime-event and short-step bidegree, so every word in one bidegree fiber has the same complete scalar trajectory.

A two-event word and its reversal share that bidegree and scalar trajectory.

When the two oriented commutators differ, the degree-two Magnus coordinate distinguishes the reversed histories inside the same scalar fiber.

References

• Truth anchor: D5/S3/Observer/Chronology/PrimeGoldenChronologyFiberSeparation.prime_golden_chronology_fiber_separation
• Dependency: D5/S3/Observer/Chronology/PrimeGoldenBidegreePhaseSeparation
• Dependency: D5/S3/Observer/Chronology/StepTwoChronologicalSignature