Time-Ordered Memory Chronology Signature Bridge

Abstract

The step-two logarithm of the frozen time-ordered memory matrices recovers the oriented swap curvature and specializes the finite Hopf reversal law.

Theorem 1.1 (Nonzero swap curvature detects event order).

$$\begin{gathered} \mathrm{primeSwapCurvature}\ne 0\Rightarrow \\ \mathrm{doubledMagnusDegreeTwo}\ne 0. \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Observer/MemoryChronology/TimeOrderedMemoryChronologySignatureBridge.timed_matrix_two_event_order_detected (✓ std3). ∎

Source. Repository-derived.

Commentary.

The matrix-valued step-two chronological signature has commutator logarithm. Its upper-right entry is the negative of the already frozen two-event memory swap curvature, due to the reverse matrix-product convention for earlier-first evolution.

The same adapter specializes the chronological Hopf antipode: reverse the timed event word and negate each event matrix. No infinite signature or continuous Magnus convergence is asserted.

References

• Truth anchor: D5/S3/Observer/MemoryChronology/TimeOrderedMemoryChronologySignatureBridge.timed_matrix_two_event_order_detected
• Dependency: D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation
• Dependency: D5/S3/Observer/Chronology/ChronologicalSignatureHopf