Prime-Golden Third-Order Chronology Escape

Abstract

Two prime-golden words can share bidegree, complete scalar trajectory, and the full step-two signature while a cubic ordered moment separates their chronology.

Theorem 1.1 (A cubic ordered moment escapes a nontrivial step-two fiber).

$$\begin{array}{c}\forall f,\forall u,\forall w,\\ f(u)\cdot f(w)\cdot f(w)+2\cdot (f(u)\cdot f(w)\cdot f(u))+f(w)\cdot f(w)\cdot f(u)\ne f(w)\cdot f(u)\cdot f(u)+2\cdot (f(w)\cdot f(u)\cdot f(w))+f(u)\cdot f(u)\cdot f(w)\Rightarrow \\ \mathrm{primeGoldenBidegree}([u,w,w,u])=\mathrm{primeGoldenBidegree}([w,u,u,w])\wedge \mathrm{sameScalarTrajectory}([u,w,w,u],[w,u,u,w])\wedge \\ \mathrm{chronologicalSignature}(f,[u,w,w,u])=\mathrm{chronologicalSignature}(f,[w,u,u,w])\wedge \\ \mathrm{thirdOrderReadout}(f,[u,w,w,u])\ne \mathrm{thirdOrderReadout}(f,[w,u,u,w]).\end{array}$$

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

Source. Repository-derived.

Commentary.

The words ABBA and BAAB contain the same event multiset, have the same prime-golden bidegree, and give the same complete scalar Euler trajectory.

Their full step-two chronological signatures agree in every associative ring representation.

Whenever the displayed cubic ordered products differ, a degree-three moment distinguishes the two histories. This supplies an explicit boundary of step-two Magnus reconstruction.

References

• Truth anchor: D5/S3/Observer/Chronology/PrimeGoldenThirdOrderChronologyEscape.prime_golden_third_order_chronology_escape
• Dependency: D5/S3/Observer/Chronology/PrimeGoldenChronologyFiberSeparation