Time-Ordered Memory Matrix Representation
Abstract
The frozen timed memory cocycle is the upper-right entry of a matrix word.
Definition 1.1 (Memory state column).
Lean statement: D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.memoryStateVector
Formalization. D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.memoryStateVector (✓ std3).
Source. Repository-derived.
Commentary.
The memory and scalar coordinates are placed in a two-component complex column vector.
Definition 1.2 (Timed event matrix).
Lean statement: D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.timedEventMatrix
Formalization. D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.timedEventMatrix (✓ std3).
Source. Repository-derived.
Commentary.
One frozen affine event update is represented by an upper-triangular two-by-two complex matrix.
Definition 1.3 (Chronological word matrix).
Lean statement: D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.timeOrderedWordMatrix
Formalization. D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.timeOrderedWordMatrix (✓ std3).
Source. Repository-derived.
Commentary.
A finite word matrix stores the stable power, memory cocycle, zero lower-left entry, and scalar cocycle.
Definition 1.4 (Reverse-ordered matrix product).
Lean statement: D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.chronologicalMatrixProduct
Formalization. D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.chronologicalMatrixProduct (✓ std3).
Source. Repository-derived.
Commentary.
The head event acts first, so later event matrices multiply on the left of earlier event matrices.
Theorem 1.5 (Memory cocycle is the upper-right entry).
Proof. Machine-checked in Lean as D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.time_ordered_word_matrix_upper_right (✓ std3). ∎
Source. Repository-derived.
Commentary.
Matrix-vector multiplication reproduces the existing timed affine update exactly, so the closed word matrix acts as the frozen list evolution on every memory/scalar state; a one-event word summary is exactly the corresponding event matrix, and concatenation is represented by the later word matrix multiplied by the earlier word matrix.
The event-by-event matrix product equals the matrix assembled from the existing scalar and memory cocycles, identifying the complete finite memory summary with the upper-right coefficient and the scalar word cocycle with the lower-right coefficient.
Swapping two timed events changes the upper-right coefficient by the already frozen prime swap curvature.
References
- Truth anchor:
D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.chronologicalMatrixProduct - Truth anchor:
D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.memoryStateVector - Truth anchor:
D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.timeOrderedWordMatrix - Truth anchor:
D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.time_ordered_word_matrix_upper_right - Truth anchor:
D5/S3/Observer/AgencyHolonomy/TimeOrderedMemoryMatrixRepresentation.timedEventMatrix - Dependency: D5/S3/Observer/AgencyHolonomy/TimeOrderedPrimeMemoryCocycle