RarePriorFullFields

Abstract

RarePriorFullFields

The deterministic acquired row and complete renderer use the original partial native update. Counts and payload-return counters remain in the source. Finite fields retain every write, latch, held record and ordered event block. Factorization holds for every stream, including distinct infinite seed paths.

Theorem 1.1 (deterministic actual rows).

$$\forall (\mathrm{ops}:\mathrm{List}\mathrm{Operation}),(\forall (c:\mathrm{AcquiredNativeState}),((\mathrm{run}\mathrm{ops}=\mathrm{some}c)\Rightarrow (\exists z:\mathrm{Runtime},\mathrm{runtimeRun}\mathrm{ops}=\mathrm{some}z\wedge z.\mathrm{fields}=c.\mathrm{source}.\mathrm{finiteFields}\wedge \mathrm{NativeObserverJointLaw}.\mathrm{row}\mathrm{actualObserver}\mathrm{ops}=\mathrm{PMF}.\mathrm{pure}z\wedge (\forall \mathrm{op}:\mathrm{Operation},\mathrm{NativeObserverJointLaw}.\mathrm{row}\mathrm{actualObserver}(\mathrm{ops}++[\mathrm{op}])=\mathrm{actualUpdate}\mathrm{op}z))))$$

Proof. Machine-checked in Lean as D5/S3/Observer/ProbabilisticClosure/TrajectoryLaws/RarePriorFullFields.deterministic_actual_rows (✓ std3). ∎

Source. Repository-derived.

Commentary.

The statement retains the complete original telescope and conditions.

Theorem 1.2 (full fields factorization).

$$\forall (c:\mathrm{AcquiredNativeState}),(\forall (\mathrm{omega}:\mathrm{Stream}),(\mathrm{fullTranscript}c\mathrm{omega}=\mathrm{fieldsTranscript}c.\mathrm{source}.\mathrm{finiteFields}\mathrm{omega}\wedge (\forall (d:\mathrm{AcquiredNativeState}),c.\mathrm{source}.\mathrm{finiteFields}=d.\mathrm{source}.\mathrm{finiteFields}\Rightarrow \mathrm{fullTranscript}c\mathrm{omega}=\mathrm{fullTranscript}d\mathrm{omega})\wedge (\forall (s:\mathrm{ActivePhase})(t:\mathrm{ValidTail}s),\mathrm{fullRenderer}cst=\mathrm{fieldsRenderer}c.\mathrm{source}.\mathrm{finiteFields}st)))$$

Proof. Machine-checked in Lean as D5/S3/Observer/ProbabilisticClosure/TrajectoryLaws/RarePriorFullFields.full_fields_factorization (✓ std3). ∎

Source. Repository-derived.

Commentary.

The statement retains the complete original telescope and conditions.

References

• Truth anchor: D5/S3/Observer/ProbabilisticClosure/TrajectoryLaws/RarePriorFullFields.deterministic_actual_rows
• Truth anchor: D5/S3/Observer/ProbabilisticClosure/TrajectoryLaws/RarePriorFullFields.full_fields_factorization
• Dependency: D5/S3/Observer/ProbabilisticClosure/TrajectoryLaws/NativeObserverJointLaw
• Dependency: D5/S3/Observer/ProbabilisticClosure/TrajectoryLaws/RarePriorResidentCode