Prime Swap Curvature

Abstract

Stable prime-memory swap curvature is gauge invariant.

Theorem 1.1 (Stable prime swap curvature specification).

$$\begin{gathered} \forall K:Type,[\mathrm{Field}\left(K\right)], \\ a,b_P,{\lambda }_P,b_Q,{\lambda }_Q,\xi :K, \\ s:K\times K, \\ (a-{\lambda }_P\ne 0)\wedge (a-{\lambda }_Q\ne 0)\Rightarrow \\ ((\mathrm{fst}\left(\mathrm{stablePrimeUpdate}\left(a,b_Q,{\lambda }_Q,\mathrm{stablePrimeUpdate}\left(a,b_P,{\lambda }_P,s\right)\right)\right)-\mathrm{fst}\left(\mathrm{stablePrimeUpdate}\left(a,b_P,{\lambda }_P,\mathrm{stablePrimeUpdate}\left(a,b_Q,{\lambda }_Q,s\right)\right)\right)=\mathrm{primeSwapCurvature}\left(a,b_P,{\lambda }_P,b_Q,{\lambda }_Q\right)\times \mathrm{snd}\left(s\right))\wedge \\ (\mathrm{snd}\left(\mathrm{stablePrimeUpdate}\left(a,b_Q,{\lambda }_Q,\mathrm{stablePrimeUpdate}\left(a,b_P,{\lambda }_P,s\right)\right)\right)=\mathrm{snd}\left(\mathrm{stablePrimeUpdate}\left(a,b_P,{\lambda }_P,\mathrm{stablePrimeUpdate}\left(a,b_Q,{\lambda }_Q,s\right)\right)\right))\wedge \\ (\mathrm{primeSwapCurvature}\left(a,b_Q,{\lambda }_Q,b_P,{\lambda }_P\right)=-\mathrm{primeSwapCurvature}\left(a,b_P,{\lambda }_P,b_Q,{\lambda }_Q\right))\wedge \\ (\mathrm{primeSwapCurvature}\left(a,\mathrm{memoryGaugeShift}\left(a,{\lambda }_P,\xi ,b_P\right),{\lambda }_P,\mathrm{memoryGaugeShift}\left(a,{\lambda }_Q,\xi ,b_Q\right),{\lambda }_Q\right)=\mathrm{primeSwapCurvature}\left(a,b_P,{\lambda }_P,b_Q,{\lambda }_Q\right))\wedge \\ (\mathrm{primeSwapCurvature}\left(a,b_P,{\lambda }_P,b_Q,{\lambda }_Q\right)=(a-{\lambda }_P)\times (a-{\lambda }_Q)\times (\mathrm{observerOrigin}\left(a,{\lambda }_P,b_P\right)-\mathrm{observerOrigin}\left(a,{\lambda }_Q,b_Q\right)))\wedge \\ ((\mathrm{primeSwapCurvature}\left(a,b_P,{\lambda }_P,b_Q,{\lambda }_Q\right)=0)\iff (\mathrm{observerOrigin}\left(a,{\lambda }_P,b_P\right)=\mathrm{observerOrigin}\left(a,{\lambda }_Q,b_Q\right)))). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Observer/AgencyHolonomy/PrimeSwapCurvature.prime_swap_curvature_spec (✓ std3). ∎

Source. Repository-derived.

Commentary.

Exchanging two lifted prime updates produces a memory defect equal to the swap curvature times the scalar state, while the scalar output is unchanged. Reversing the exchange negates the curvature, and a common shift of memory origin leaves it invariant.

Under the two stated nonresonance hypotheses, the curvature factors through the difference of the observer-origin estimates. Its vanishing is therefore equivalent to agreement of those estimates; no analytic or zero-location conclusion is asserted.

References

• Truth anchor: D5/S3/Observer/AgencyHolonomy/PrimeSwapCurvature.prime_swap_curvature_spec