A Global Characteristic Obstruction for Carlitz Five-Orbits

Abstract

A constructed integer in the closed-five-orbit ideal confines every prime characteristic.

The residual is the actual nested Carlitz residual already defined in CarlitzFiveOrbit. A ring endomorphism sigma is iterated by composition. The theorem keeps every field, every prime characteristic and every closed five-orbit in its quantified domain.

Theorem 1.1 (The finite characteristic support forced by all five equations).

$$\forall K\in \mathrm{Type}\left(\right),(\mathrm{Field}\left(K\right))\Rightarrow (\forall p\in \mathbb{N},((\mathrm{CharP}\left(K,p\right))\wedge (\mathrm{NatPrime}\left(p\right)))\Rightarrow (\forall sigma\in \mathrm{RingHom}\left(K,K\right),\forall theta\in K,((\mathrm{sigma}\left(\mathrm{sigma}\left(\mathrm{sigma}\left(\mathrm{sigma}\left(\mathrm{sigma}\left(theta\right)\right)\right)\right)\right)=theta)\wedge (\mathrm{residual}\left(\mathrm{sigma}\left(theta\right)-theta,\mathrm{sigma}\left(\mathrm{sigma}\left(theta\right)\right)-theta,\mathrm{sigma}\left(\mathrm{sigma}\left(\mathrm{sigma}\left(theta\right)\right)\right)-theta,\mathrm{sigma}\left(\mathrm{sigma}\left(\mathrm{sigma}\left(\mathrm{sigma}\left(theta\right)\right)\right)\right)-theta\right)=0))\Rightarrow (p\mid 8\cdot 25\cdot 19\cdot 263\cdot 519555805809266011)))$$

Proof. Machine-checked in Lean as D5/S3/Arith/FunctionField/CarlitzFiveCharacteristic.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every field K and natural prime p with CharP K p, every ring endomorphism sigma of K, and every theta in K, assume sigma^5(theta)=theta and the residual at sigma^i(theta)-theta, i=1,2,3,4, is zero. Then p divides 82519263519555805809266011. Primality of the last explicit factor is not a premise or conclusion of this theorem.

Applying sigma supplies the other four cyclic-origin residual equations. An explicitly authored degree-ten integer polynomial with 508 terms has a cyclic weighted residual sum equal to 4673196650932024062540600. Ring normalization verifies the integer identity. Its factorization has an extra factor nine, which is removed by a second 340-term cyclic certificate equal to one in characteristic three.

Both certificates enter the deduction. Neither a Groebner-basis result, a list of tested characteristics, nor the desired characteristic support is assumed. The conclusion is a global exclusion statement for the literal orbit equations, not an extrapolation from a scan.

The unified Wieferich dossier gives additional ordinary proofs of the two newly constructed characteristic families and their exact extension-degree classes. Those conclusions are separate from this theorem. The integer Wall-Sun-Sun existence problem remains a separate unresolved subproblem within the same lifting-and-elimination family.

References

• Truth anchor: D5/S3/Arith/FunctionField/CarlitzFiveCharacteristic.result
• Dependency: D5/S3/Arith/FunctionField/CarlitzFiveOrbit