A dependency cycle is a construction obligation¶
GA-SYN-006 · Edition 2026.10.9 · Synthetic technical investigation
Proposition and operational question¶
For flip, identity and constant on {0,1}, accepting a proposed fixed point only after checking x=f(x) admits no non-solution. Enumeration distinguishes zero, one and multiple solutions; evaluating total components alone establishes neither existence nor uniqueness.
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¶
A fixed point is a value x satisfying x=f(x). Work only over the set {0,1}. The three functions are flip(x)=1-x, identity(x)=x and constant(x)=0. All functions terminate on all their inputs. The question is existence and uniqueness of a value that satisfies the whole equation. This is elementary algebra, not a cryptographic search.
Worked synthetic trace¶
For flip, test zero: f(0)=1, so zero is not a solution. Test one: f(1)=0, so one is not a solution. There are no fixed points. Identity has both zero and one as solutions. Constant has exactly zero. The checker computes all six evaluations. Identical component-level test coverage produces three different construction outcomes.
Negative control¶
The consumer proposes a value and must check the fixed-point equation before accepting it. Removing this equation check makes the flip function accept 0 even though flip(0)=1. The printed counterexample shows the proposed value and its image. Enumerating the true roots separately yields zero for flip, two for identity, and one for constant; this distinguishes existence from uniqueness. The negative control violates actual acceptance soundness, rather than merely printing a false-inference label.
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 construction, 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¶
The public AD8 paper identifies an unresolved cyclic identity construction. This example explains the logical burden without reproducing its protected composition. A construction needs an agreed staging rule or other justified semantics, existence, uniqueness where required, and an implementable procedure. A proposed weaker identity definition is a design change, not a proof of the old claim. References to already committed history must also be distinguished from a candidate referring to its own future inclusion. Migration must preserve old verification rules instead of reinterpreting old bytes.
Prior art and contribution limit¶
Provenance records can name relations among artifacts, but recording a relation does not construct a mutually dependent artifact pair. This is an obligation analysis, not a new fixed-point theorem. A reviewer can improve it by providing an explicit construction and stating exactly which original semantics it preserves. 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 construction
python examples/investigations/check.py --case construction --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-006, 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.
For the wider architecture, see the AD8 paper, architecture synthesis, claim register, and AD8 investigation overview. Native reference results and open constructions have separate scope; these examples do not implement them.