Local Euler Frame-History Non-Reconstruction

Abstract

The finite local Euler shadow does not determine frame history or cross-prime transitions.

Theorem 1.1 (Local determinants cannot clone frame history).

$$\forall chi\in Nat.Primes\to \mathrm{Complex}\left(\right),\mathrm{let}{p}_2:Nat.Primes=\mathrm{prime}\left(2\right),\mathrm{let}{p}_3:Nat.Primes=\mathrm{prime}\left(3\right),\mathrm{let}localOperator:Nat.Primes\to \mathrm{Matrix}\left(\mathrm{Fin}\left(2\right),\mathrm{Fin}\left(2\right),\mathrm{Complex}\left(\right)\right)=(p:Nat.Primes\mapsto \mathrm{diagonal}\left((branch:\mathrm{Fin}\left(2\right)\mapsto \mathrm{ite}\left(branch=0,1,chi\left(p\right)\right))\right)),\exists firstFrame\in Nat.Primes\to \mathrm{GL}\left(\mathrm{Fin}\left(2\right),\mathrm{Complex}\left(\right)\right),secondFrame\in Nat.Primes\to \mathrm{GL}\left(\mathrm{Fin}\left(2\right),\mathrm{Complex}\left(\right)\right),\left(\forall p\in Nat.Primes,x\in \mathrm{Complex}\left(\right),\det \left(\mathrm{identityMatrix}\left(\mathrm{Fin}\left(2\right),\mathrm{Complex}\left(\right)\right)-\mathrm{smul}\left(x,firstFrame\left(p\right)\cdot localOperator\left(p\right)\cdot \mathrm{inverse}\left(firstFrame\left(p\right)\right)\right)\right)=\left(1-x\right)\cdot \left(1-x\cdot chi\left(p\right)\right)\right)\wedge \left(\left(\forall p\in Nat.Primes,x\in \mathrm{Complex}\left(\right),\det \left(\mathrm{identityMatrix}\left(\mathrm{Fin}\left(2\right),\mathrm{Complex}\left(\right)\right)-\mathrm{smul}\left(x,secondFrame\left(p\right)\cdot localOperator\left(p\right)\cdot \mathrm{inverse}\left(secondFrame\left(p\right)\right)\right)\right)=\left(1-x\right)\cdot \left(1-x\cdot chi\left(p\right)\right)\right)\wedge \left(firstFrame\ne secondFrame\wedge \left(\mathrm{inverse}\left(firstFrame\left(p_3\right)\right)\cdot firstFrame\left(p_2\right)\ne \mathrm{inverse}\left(secondFrame\left(p_3\right)\right)\cdot secondFrame\left(p_2\right)\wedge \left(\neg \left(\exists R\in \left(Nat.Primes\to \left(\mathrm{Complex}\left(\right)\to \mathrm{Complex}\left(\right)\right)\right)\to \left(Nat.Primes\to \mathrm{GL}\left(\mathrm{Fin}\left(2\right),\mathrm{Complex}\left(\right)\right)\right),R\left(((p,x)\mapsto \det \left(\mathrm{identityMatrix}\left(\mathrm{Fin}\left(2\right),\mathrm{Complex}\left(\right)\right)-\mathrm{smul}\left(x,firstFrame\left(p\right)\cdot localOperator\left(p\right)\cdot \mathrm{inverse}\left(firstFrame\left(p\right)\right)\right)\right))\right)=firstFrame\wedge R\left(((p,x)\mapsto \det \left(\mathrm{identityMatrix}\left(\mathrm{Fin}\left(2\right),\mathrm{Complex}\left(\right)\right)-\mathrm{smul}\left(x,secondFrame\left(p\right)\cdot localOperator\left(p\right)\cdot \mathrm{inverse}\left(secondFrame\left(p\right)\right)\right)\right))\right)=secondFrame\right)\right)\right)\right)\right)$$

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

Source. Repository-derived.

Commentary.

The local operator at each prime is the canonical diagonal two-branch operator with eigenvalues one and chi at that prime. Its framed Euler determinant is the finite spectral shadow retained by the scalar observation.

The frozen local-transition owner supplies two frame histories whose determinants both equal the same explicit Euler polynomial at every prime and scalar. Their histories are distinct and their transitions from prime two to prime three differ.

No decoder of the complete local determinant family can return both histories correctly. This is stronger input than two global scalar functions, so those functions cannot clone the framed observer or its formation history.

References

• Truth anchor: D5/S3/Observer/AgencyHolonomy/LocalEulerFrameHistoryNonreconstruction.local_euler_determinants_do_not_reconstruct_frame_history
• Dependency: D5/S3/Observer/AgencyHolonomy/LocalEulerTransitionNonreconstruction