Different keys need not mean independent control¶
GA-SYN-010 · Edition 2026.10.9 · Synthetic technical investigation
Proposition and operational question¶
Requiring two control groups rejects a two-voter quorum when both voters are controlled by the same group.
This matters when a continuing task reuses an earlier decision or representation. A final success label can conceal a different failure earlier in the trace. The proposition therefore identifies the exact observation that would refute it.
Definitions, trust and failure model¶
Voters a and b belong to group one; voter c belongs to group two. The threshold is two independently controlled groups. A registry of actual control is assumed accurate and current. Voters can submit distinct labels but cannot change this registry. The case examines the counting rule, not authentication or how real independence is discovered.
Worked synthetic trace¶
For voters (a,b), the label count is two but the control-group set has one member. The decision is refused. Pairs (a,c) and (b,c) each span two groups and pass the counting rule. The checker considers all three pairs. Even those accepted pairs may fail other real governance requirements that this example does not model.
Negative control¶
The mutant counts voter labels. It admits (a,b), allowing one control group to satisfy a nominal two-party threshold. The example identifies a structural counting error without publishing a real custodian roster or capture technique. A valid count cannot compensate for false control data.
Three evidence records and separate axes¶
| Record | Method and evidence | Scope and status |
|---|---|---|
| Research | Authored finite argument and explicit counterexample, 12 October 2026 | Analytical; limited to the definitions above |
| Reference | My execution, with AI assistance, of the independently designed public teaching case custody, 12 October 2026 |
Fault-tested; externally reproducible with the commands below; no independent reproduction claimed |
| Production | None for this proposition | No product conformance inferred |
Engineering maturity: reference_implemented, referring only to the disclosed teaching case. Assurance: analytical, fault_tested, externally_reproducible, each restricted to its evidence record. These labels are independent dimensions, not a score or a global verification flag. The exact machine-readable record is in the public evidence catalog. My own runs do not count as independent review.
Limitations and open obligations¶
Actual beneficial control, coercion, collusion, emergency powers, custody recovery, founder amendment powers and changing membership are outside this reference. No software test here certifies human independence. A production custody claim requires an attributable process and observation of its control boundary, as well as a faithful counter implementation.
Prior art and contribution limit¶
Separation of privilege supplies the foundation. Institutional research extends the question to who can amend the constraints and recover custody, but does not inherit an independence guarantee from cryptographic signatures. This finite counterexample claims no new threshold cryptography. See the primary reference and the comparison guide. No priority or superiority claim follows from this example.
Reproduction and falsification¶
From the successor public source package, use CPython 3.12.14 and no external packages:
python examples/investigations/check.py --case custody
python examples/investigations/check.py --case custody --mutant
The first command must exit zero, with no positive violations and at least one negative violation. The second intentionally exits one and prints an actual violated case. It is a deliberate fault result, not an installation failure. Source and reproduction contract identify the exact population and scope.
Submit a counterexample, a missing assumption or a prior-art correction through the review guide. Include GA-SYN-010, edition 2026.10.9, the command, input and output, and whether the objection concerns the model, implementation or proposed product relationship. Do not send credentials or confidential customer material.