[#P2750] Trace-indistinguishable triples in sl2(F5)

Problem. For an ordered triple \(T=(A,B,C)\in\mathfrak{sl}_2(\mathbb F_5)^3\), define its trace profile by \(\tau_T(w)=\operatorname{tr}(w(A,B,C))\) for every word \(w\) in three noncommuting letters. Among fibers of \(T\mapsto\tau_T\), determine the largest number of simultaneous \(GL_2(\mathbb F_5)\)-conjugacy classes in one fiber, and classify every fiber attaining that maximum.

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1Context

The output is a small atlas of trace failures. Word cutoffs, canonical orbit representatives, and stabilizers can be reused in finite-field invariant computations.

2Definitions

Definition 1 (The space sl_2(F_5) consists of 2 by 2 matrices over F_5 with trace zero). The space sl_2(F_5) consists of 2 by 2 matrices over F_5 with trace zero.

Definition 2 (Simultaneous conjugation sends (A,B,C) to (gAg^{-1},gBg^{-1},gCg^{-1}) using one g in GL_2(F_5)). Simultaneous conjugation sends (A,B,C) to (gAg^{-1},gBg^{-1},gCg^{-1}) using one g in GL_2(F_5). Two triples have the same trace profile when the stated traces agree for every finite word, including powers and mixed words.

3What counts as a solution

Give the maximum number of simultaneous-conjugacy classes in a trace-profile fiber and canonical representatives for every maximizing fiber.
Prove a finite trace-word cutoff sufficient for this space, or compare all words through an equivalent exact invariant-theoretic certificate, and certify the full orbit enumeration.

1Status

What counts as a solution

Current status (Current status and unresolved remainder). UNKNOWN as of 2026-08-01. Separating trace sets are studied for matrix tuples, but the exact maximum fiber and its orbit classification for traceless triples over F5 was not located.^[1]

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Record	Kind	Assessment
UNKNOWN as of 2026-08-01. Separating trace sets are studied for matrix tuples, but the exact maximum fiber and its orbit classification for traceless triples over F5 was not located.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 UNKNOWN as of 2026-08-01. Separating trace sets are studied for matrix tuples, but the exact maximum fiber and its orbit classification for traceless triples over F5 was not located.^[1] Relevance to this problem For sl2 f5 triple trace fibers, pins the dated research frontier: UNKNOWN as of 2026-08-01. Separating trace sets are studied for matrix tuples, but the exact maximum fiber and its orbit classification for traceless triples over F5 was not located. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details UNKNOWN as of 2026-08-01. Separating trace sets are studied for matrix tuples, but the exact maximum fiber and its orbit classification for traceless triples over F5 was not located. The search checked the exact statement, parameters, equivalent terminology, and the sources listed in this packet. Database silence is treated only as bounded status evidence. A complete resolution must satisfy every acceptance condition in the canonical problem. Open full record →
Give the maximum number of simultaneous-conjugacy classes in a trace-profile fiber and canonical representatives for every maximizing fiber. Prove a finite trace-word cutoff sufficient for this space, or compare all words through an equivalent exact invariant-theoretic certificate, and certify the full orbit enumeration.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give the maximum number of simultaneous-conjugacy classes in a trace-profile fiber and canonical representatives for every maximizing fiber. Prove a finite trace-word cutoff sufficient for this space, or compare all words through an equivalent exact invariant-theoretic certificate, and certify the full orbit enumeration.^[1] Relevance to this problem For Trace-indistinguishable triples in sl2(F5), record sl2-f5-triple-trace-fibers-next-research-step (“Resolve the stated acceptance conditions”) documents a concrete method, search boundary, or failed route. The record states: Give the maximum number of simultaneous-conjugacy classes in a trace-profile fiber and canonical representatives for every maximizing fiber. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work on the displayed statement under its exact conventions. Give the maximum number of simultaneous-conjugacy classes in a trace-profile fiber and canonical representatives for every maximizing fiber. Prove a finite trace-word cutoff sufficient for this space, or compare all words through an equivalent exact invariant-theoretic certificate, and certify the full orbit enumeration. Preserve source locators and any finite certificates so later work can distinguish a complete resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-08-01. UNKNOWN as of 2026-08-01. Separating trace sets are studied for matrix tuples, but the exact maximum fiber and its orbit classification for traceless triples over F5 was not located.

A 2026-07-27 search checked Lopatin and Reimers, arXiv:2202.05717, Drensky, arXiv:math/0506614, and exact parameter searches. No matching F5 triple-fiber table was found.
The field F5 avoids the characteristic-two trace collapse while keeping the full set at 125^3 triples. Three matrices are the first tuple size where genuinely mixed trace data proliferate.
Use Cayley-Hamilton and trace identities to derive a finite word-length cutoff, then enumerate simultaneous conjugacy orbits and hash their certified profiles.
Trap: trace words generally see semisimplification rather than every nonclosed orbit. Finding one separating word for sampled pairs does not prove a fiber complete.
Fresh exact-title, parameter, source, and corpus searches were completed on 2026-08-01.

Recorded example 1. Let N=E_12. The triples (0,0,0) and (N,0,0) are not conjugate, but they have the same trace profile because N^2=0 and tr(N)=0.

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ProblemTrace-indistinguishable triples in sl2(F5)

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Resolve the stated acceptance conditions addresses Current status and unresolved remainder

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Plain text

“Trace-indistinguishable triples in sl2(F5).” TheoremDB. P2750. Problem statement; statement text SHA-256 91e7876a14faa169947ff703cf2c016c315da6fd4d883539bfe38754efd150c9. https://theoremdb.org/statement/?ref=P2750

BibTeX

@misc{theoremdb-problem-91e7876a14faa169947ff703cf2c016c315da6fd4d883539bfe38754efd150c9,
  title = {{Trace-indistinguishable triples in sl2(F5)}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 91e7876a14faa169947ff703cf2c016c315da6fd4d883539bfe38754efd150c9},
  url = {https://theoremdb.org/statement/?ref=P2750}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from arxiv.org^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. Jonathan Elmer, “The separating variety for 2x2 matrix invariants,” Linear and Multilinear Algebra 72(3) (2024), 389-411. DOI 10.1080/03081087.2022.2158300; arXiv:2202.05717v3. The simultaneous-conjugation trace-invariant setting for tuples of 2 by 2 matrices. ↗preprint · primary source · arXiv:2202.05717v3 · checked 2026-08-01Source use: original summary.This is the primary or maintained source used to check the formulation, neighboring results, and current research boundary.Also cited at Source identified in the candidate’s dated status assessment; exact page or record linked above.Also cited at The contributor formulated the finite-field fiber question after reviewing separating trace sets for matrix tuples.Also cited at Editorial research route recorded 2026-08-01.Supports the dated status review or a neighboring result for “Trace-indistinguishable triples in sl2(F5).”Source used to assess the problem's recorded status.For Trace-indistinguishable triples in sl2(F5), the reviewed source scope is The contributor formulated the finite-field fiber question after reviewing separating trace sets for matrix tuples.. The packet makes no inference beyond that cited scope.Source named by the research packet.
Vesselin Drensky, “Computing with matrix invariants,” Mathematica Balkanica (N.S.) 21(1-2) (2007), 141-172. arXiv:math/0506614v2. Survey of simultaneous-conjugation invariants for matrix tuples. ↗preprint · secondary source · arXiv:math/0506614v2 · checked 2026-08-01Source use: original summary.This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.Also cited at Source identified in the candidate’s dated status assessment; exact page or record linked above.Also cited at survey of simultaneous-conjugation matrix invariants over characteristic zero.Supports the dated status review or a neighboring result for “Trace-indistinguishable triples in sl2(F5).”Source used to assess the problem's recorded status.For Trace-indistinguishable triples in sl2(F5), this source supplies adjacent invariant-theory methods and does not settle the finite-field triple-trace target.

CC0 finite classification of the failure of trace words to separate nonsemisimple matrix-tuple orbits.

[#P2750] Trace-indistinguishable triples in sl2(F5)

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