Signature and Ordered-Moment Semantics

Abstract

For every finite word, doubled degree two is its event-square sum plus twice its ordered-pair moment.

Theorem 1.1 (The stored coordinate for all finite words).

$$\forall A,E,\mathrm{Semiring}\left(A\right),o:E\to A,w:\mathrm{List}\left(E\right),\mathrm{doubledDegreeTwo}\left(\mathrm{chronologicalSignature}\left(o,w\right)\right)=\mathrm{sum}\left(\mathrm{map}\left(e\mapsto o\left(e\right)\cdot o\left(e\right),w\right)\right)+2\cdot \mathrm{orderedPairMoment}\left(\mathrm{map}\left(o,w\right)\right)$$

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

Source. Repository-derived.

Commentary.

A and E are arbitrary types, A is a possibly noncommutative semiring, o maps E to A, and w is a finite list over E. The diagonal term sums o(e)o(e) over the occurrences in w; M2 is the existing orderedPairMoment on the value list. The formula accounts for both repeated events and ordered cross terms.

Theorem 1.2 (Magnus is the difference between the two orientations).

$$\forall A,E,\mathrm{Ring}\left(A\right),o:E\to A,w:\mathrm{List}\left(E\right),\mathrm{doubledMagnusDegreeTwo}\left(\mathrm{chronologicalSignature}\left(o,w\right)\right)=\mathrm{orderedPairMoment}\left(\mathrm{map}\left(o,w\right)\right)-\mathrm{orderedPairMoment}\left(\mathrm{reverse}\left(\mathrm{map}\left(o,w\right)\right)\right)$$

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

Source. Repository-derived.

Commentary.

For arbitrary types A and E, a ring A, o : E to A, and w : List E, the corrected coordinate is the forward ordered-pair moment minus the moment of the reversed value list. The proof uses the all-word square decomposition, which separates event squares from both ordered orientations without assuming commutativity or dividing by two.

References

• Truth anchor: D5/S3/Observer/Chronology/SignatureOrderedMoment.chronological_signature_doubledDegreeTwo_eq
• Truth anchor: D5/S3/Observer/Chronology/SignatureOrderedMoment.doubledMagnusDegreeTwo_eq_orderedPairMoment_sub_reverse
• Dependency: D5/S3/Observer/Chronology/PrimeGoldenThirdOrderChronologyEscape