Paper XI visual guide · Visual 40 of 47 · PART IV / REALIZATION AND COMPARISON

Why nontrivial stabilizer actions need another step

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Original source locator: Block 166, stabilizer contract

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For the current totally ramified degree-\(p^r\) maximal-type host, character-preserving normalizers have

\[ N/(F^\times J^1)\simeq\mathbb Z/p^r\mathbb Z. \]

A native action of a finite prime-to-\(p\) group \(C\) on the quotient therefore factors through a \(p\)-group and is trivial.

The kernel of the action of center-fixing scalar-extension automorphisms on \(W\) is \(\operatorname{Hom}(W,{\mathbb C}^\times)\), an elementary abelian \(p\)-group. Hence the full native scalar-extension action is trivial as well. Honest implementers on an irreducible Heisenberg carrier are scalar by Schur's lemma.

Scope

This is an obstruction to the native inner-normalizer route for this host. It does not refute the original finite contract or Conjecture 66. Original breakwise H2 may supply a genuinely nontrivial symplectic action on separate actual type targets. Transporting abstract operators does not by itself identify a common arithmetic action.

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