Element Reduction and Coarse Invariants

Abstract

Entrywise reduction separates two integral matrices at every prime while trace and characteristic polynomial merge the same reductions.

Theorem 1.1 (Prime reduction can separate what coarse invariants merge).

$$\begin{array}{r}A=\mathrm{zeroMatrix}(2,Z),N=\mathrm{single}(2,0,1,1,Z),\\ A\ne N\wedge \\ \forall p\in Primes,\\ \mathrm{reduction}(p,A)\ne \mathrm{reduction}(p,N)\wedge \\ \mathrm{trace}(\mathrm{reduction}(p,A))=\mathrm{trace}(\mathrm{reduction}(p,N))\wedge \\ \mathrm{charpoly}(\mathrm{reduction}(p,A))=\mathrm{charpoly}(\mathrm{reduction}(p,N)).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/ResidueCoding/ElementReductionCoarseInvariant.element_reduction_coarse_invariant_fork (✓ std3). ∎

Source. Repository-derived.

Commentary.

Take the two-by-two integral zero matrix A and the matrix N whose only nonzero entry is one in row zero and column one. These are distinct global integral objects.

For every prime p, entrywise reduction modulo p still distinguishes A from N because the distinguished entry remains one.

On those same reduced matrices, both traces are zero and both characteristic polynomials are X squared. The positive separation and the coarse collision therefore use one shared construction at every prime.

References

• Truth anchor: D5/S3/ConceptDynamics/ResidueCoding/ElementReductionCoarseInvariant.element_reduction_coarse_invariant_fork
• Dependency: D5/S0/Observation/PowerTraceSimilarityCountermodel