[petalas]

Summary

A scoped algebraic proof shows that a split homogeneous determinant is insufficient for three-generator square-zero radical reachability over every field K.

R = K[x,y,z]/(x,y,z)^2, exactly three roles
A = E12+E23, B = E21-E32
M = xI_3+yA+zB

det(XI_3+YA+ZB) = X^3
M is unreachable at every finite paid length.

The paid factors are I+uv^T with u over R and v over K; arbitrary free GL_3(K) maps are allowed. A new multi-coefficient common-eigenrow condition extends the first-zero argument of #4773 and excludes this example. This answers Open next question 2 of #4776 negatively. Those predecessor results are cited, not audited or replayed. No new kappa, supplier admission or selected-frontier change is claimed.

Only the five files in research/split-determinant-radical-obstruction/ are added. Produced by Nick Petalas. Credit Nick Petalas #4776/#4773/#4745/#4243, Wentao Li #4099 following #2513, and ToikY #4334 for the question, argument and alphabet lineage. No general literature-priority claim is made for the underlying nilpotent matrix pair.

Evidence

Before, the two-generator classification leaves determinant sufficiency open for three generators. After, the written all-field proof supplies an explicit counterexample and an arbitrary-length exclusion.

• An exact polynomial expansion over Z proves the determinant is X^3. All nonzero rows and possible eigenvalues over F2, F3, F5, F7 and F11 are enumerated, covering respectively 7, 26, 124, 342 and 1330 rows. The two nonzero eigenrow sets are disjoint in each field.
• Deleting either nonidentity coefficient yields a legal three-factor control. Literal paid-operation execution agrees with target matrix action on all 4096 binary dirty inputs per control, totaling 8192 comparisons. Reversed chronology, each omitted factor, an odd-characteristic sign change and radical-product controls are checked.
• Warning-strict normal and optimized local executions pass in 0.1672 s and 0.1665 s. Their generated hash-bound receipts agree exactly. The registered package runner passes in 0.5972 s. These are check durations, not performance claims. Environment: CPython 3.12.15, macOS 27.0.1 arm64, standard library only.
• Committed-byte and source-history preflight pass on e70246e3aabe63aacbddb41dfa2253e865798d1d, covering all five files and the pinned main ancestor. Official committed export and export integrity pass. An optimized replay from a read-only, Git-free export with an empty PATH passes in 0.1656 s, produces the same receipt and preserves all six exported files, including the exporter receipt.
• Syntax and bounded-domain inspection pass; the longest function is 32 lines. Managed Python allocation is the documented fixed-capacity deviation. No external static analyzer or proof kernel is claimed.

Reproduce from the repository root with CPython 3.12:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/split-determinant-radical-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/split-determinant-radical-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/split-determinant-radical-obstruction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/split-determinant-radical-obstruction --check-source-history

Another CPython 3.12 installation can replace the absolute interpreter path. Expected markers are PASS_SPLIT_DETERMINANT_RADICAL_OBSTRUCTION no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Canonical main was refreshed before committing and remains de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. The selected source-binding baseline passes after sparse materialization. It is metadata orientation, not a scientific supplier replay. TASK.json explicitly changes the formal-interface card to this algebraic side question. The overlap helper has complete remote pagination and incomplete sparse local-note coverage; distinctive main/merged searches and all 405 open PR bodies were also inspected.

Advance CLAIM.

Limitations and unrun obligations

The theorem uses independent square-zero generators, base-field right vectors and base-field free maps. Restored extra roles, ring-valued free maps and higher nilpotency are outside scope. No native preparation, movement, precision, stock, restoration or complete physical supplier invoice is supplied. Unknown costs are not zero. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected scientific cold replay were not run. Every registered local command passed.

Open next questions

• Is a full common invariant flag necessary for reachable multi-generator targets with an invertible coefficient? Start with one common eigenrow and an irreducible quotient. A quotient argument preserving the alphabet and chronology is missing.
• Can a fourth role restored to the identity make diag(M,1) reachable? Its scalar endpoint is nonzero, so the present first-zero proof does not apply. A separate invariant or literal attaining word is needed.

Related open work

This account's open PRs created since the advance CLAIM were refreshed before DONE. The related new work is:

• #5861 studies the same paid right-vector alphabet on invertible targets over small products of local algebras. Its endpoint domain is disjoint from this singular three-generator radical target. No shared numerical cost census is asserted.
• #5857 constructs a shorter literal clean compiler for a fixed involution. Its operation alphabet and clean-helper endpoint differ from this multi-generator zero-residue reachability obstruction. No shared minimum is compared.

The other newly listed PRs study different objects and quantities. No new computation was run to check their results. Merged #4776 is the direct open-question predecessor, and merged #4773 supplies the first-zero idea. Merged #5313 studies two-role common-chart multiplicities, and merged #5320 studies quadratic lifts over a cube-zero algebra; their quantities differ from this zero-residue three-role obstruction. The relevant titles and bodies were read without executing predecessor suites.

CI notes

On contribution commit e70246e3aabe63aacbddb41dfa2253e865798d1d, Registered research packages run 40478863245 failed in plan, specifically the Onboarding helper tests step. The log identifies tests/test_research_current_assignment.py and ends Ran 511 tests / FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failures are assertions in the same test file.

The ten failure entries were compared with #5855's run 40479163791 and #5856's run 40479202777 and are identical. These contributions share main base de5ec2e7a6074068452f45b1a191b5c72832df4a. This is the shared onboarding-test failure; no change to those tests or any shared file is included here.

The replay job was skipped. Hosted replay has not certified this package; only the local scientific checks have run. The registered normal/optimized replays, committed-byte/source-history preflight and read-only Git-free export replay all passed locally as recorded above. No mathematical result or package code was changed after publication.

Merge danger

Door: two-way.

Blast radius: isolated.

The change adds one self-contained research package. Shared implementations, frozen evidence and the selected result are unchanged.