Ramified Conjugate Jet

Abstract

A repeated residue eigenvalue retains the infinite power jet of its nilpotent part.

Definition 1.1 (The ramified jet records every positive residual power).

$$\mathrm{RamJet}\left(lambdaZero,T\right)=(lambdaZero,(k\mapsto (T-lambdaZeroI{)}^{k+1})).$$

Formalization. D5/S1/Eigenstructure/RamifiedConjugateJet.ramifiedConjugateJet (✓ std3).

Source. Repository-derived.

Commentary.

The scalar center is followed by a natural-number-indexed sequence. Index zero stores the first residual power, so the definition is an infinite sequence rather than a truncated tuple.

Theorem 1.2 (A nonzero square-zero direction realizes the golden ramified jet).

$$\begin{array}{r}\exists N,T\in \mathrm{Matrix}\left(2,\mathrm{ZMod}\left(5\right)\right),\\ N=\mathrm{single}\left(0,1,1\right)\wedge T=3I+N,\\ N\ne 0\wedge N^2=0\wedge \mathrm{rank}\left(N\right)=1,\\ \mathrm{charpoly}\left(T\right)=(X-3{)}^2\wedge \mathrm{rootMultiplicity}\left(3,\mathrm{charpoly}\left(T\right)\right)=2,\\ \mathrm{RamJet}\left(3,T\right)=(3,(k\mapsto N^{k+1})),\\ \mathrm{tail}\left(\mathrm{RamJet}\left(3,T\right),0\right)=N\wedge \forall k\ge 1,\mathrm{tail}\left(\mathrm{RamJet}\left(3,T\right),k\right)=0.\end{array}$$

Proof. Machine-checked in Lean as D5/S1/Eigenstructure/RamifiedConjugateJet.exists_golden_ramified_conjugate_jet (✓ std3). ∎

Source. Repository-derived.

Commentary.

Over ZMod 5, the upper off-diagonal matrix N is nonzero, has rank one, and satisfies N squared equals zero. Translating it by three times the identity gives a matrix whose characteristic polynomial has three as a root of multiplicity two.

The resulting jet has N as its index-zero term and N to the power k+1 at every index k. Square-zero nilpotence makes every term after index zero vanish, while preserving a nontrivial first direction.

The proof reuses the repository’s standard rank-one nilpotent matrix witness and Mathlib’s two-by-two characteristic-polynomial and root-multiplicity theorems.

References

• Truth anchor: D5/S1/Eigenstructure/RamifiedConjugateJet.exists_golden_ramified_conjugate_jet
• Truth anchor: D5/S1/Eigenstructure/RamifiedConjugateJet.ramifiedConjugateJet
• Dependency: D5/S0/Observation/PowerTraceSimilarityCountermodel