Claims and risk register

Claim	Evidence	Status	Main risk
\(\det JF=-2\)	Hand proof; SymPy; Sage; Magma	Proved	Formula transcription
Announced collision	Exact substitution in three systems	Proved	None after formula freeze
Fiber is exactly three reduced points	Ideal equality and radicality	Proved	Coordinate-order mistakes in reuse
Normalized \(H\) has determinant \(1\)	Linear conjugation and CAS	Proved	Quoting an older normalization
Generic degree is \(3\)	Marked simple-root bijection	Proved	Saying every fiber has size three
Set-theoretic image omits exactly triple-root curve	Complete projective-root classification	Proved over \(\mathbb C\)	Scheme-theoretic image is still all target space
Nonproperness set is the discriminant	Explicit escape plus bounded inverse branches off discriminant	Proved over \(\mathbb C\)	Definition of nonproperness over other fields
First public occurrence	Exact-phrase search; timestamped post	Provisional	Unindexed or private prior work
Community acceptance	Same-day online checks only	Not yet established	No peer review as of cutoff
Dimension two remains open	Current refereed literature	Current as of 2026-07-20	Future developments

Frozen notation

• Source: \((x,y,z)\).

• Output functions: \((A,B,C)=(F_1,F_2,F_3)\).

• Fixed target: \((a,b,c)\).

• Binary cubic: \(Q_{a,b,c}(U,V)\).

• Affine root: \(w=U/V=y+1/x\).

• Discriminant factor: \(\Delta\), with \(\operatorname{Disc}(p)=4\Delta\).

Any later edit should preserve these conventions or state the change prominently.