Exact certificate atlas
Generic fiber degrees 3 through 100
A prompt-driven mathematical investigation by Alexis Gallagher
For every integer n from 3 through 100, this page specifies a polynomial map Fn: ℂ³ → ℂ³ with det JFn = 1, one rational target, and all n rational points in its fiber. The same inverse equation proves that its generic fiber degree is exactly n.
This is one uniform construction instantiated 98 times, not 98 numerical experiments.
The table contains 5,047 exact preimages. Every integer and fraction is recomputed in the browser;
the independent repository verifier uses Python's standard-library Fraction class.
Checking all 98 maps and 5,047 rational preimages exactly in this browser…
What one row proves
It is a counterexample
The universal calculation gives det JFn=1. The displayed target has n≥3 distinct preimages, so the map is not injective and cannot have an inverse.
Its generic degree is n
A target fiber reduces to one equation of degree n in an auxiliary variable w, with a unique reconstruction from each admissible root. The displayed target realizes all n roots simply.
The common mapping formula
Each row supplies a degree-n polynomial Hn. From it define
qn(w)=wH′n(w)−Hn(w),
an=−(1+H″n(1))/(2+H″n(1)).
For an input (x,y,z), put
α=u+qn(w)/γ², β=1+pn(w)/γ,
Fn(x,y,z)=(α/x², β/x, xγ).
The quotients are only a compact presentation: the endpoint identities make them cancel, so all three coordinates of Fn are polynomials. The short determinant calculation and the cancellation proof appear below the table.
The atlas
Open any row to see the row-specific map and all of its exact rational preimages. The factorization of Hn is displayed rather than an expanded coefficient list because it simultaneously exhibits all n inverse roots.
| n | Mapping Fn | Target | Complete fiber |
|---|
Why the construction works
1. Integer inverse roots
Each row begins with Gn(w)=∏(w−r) for the displayed distinct integer roots. They are chosen so that Gn(1)=G′n(0).
2. The seed conditions
Set λn=G′n(0)−G′n(1) and Hn=Gn/λn. Then p(0)=0, p(1)=−1, and ∫₀¹p=0.
3. Polynomiality and determinant
The seed conditions remove the apparent divisions. With P=βγ=γ+p(w) and Q=αγ²=wγ+q(w), direct differentiation gives ∂(P,Q)/∂(w,γ)=−γ. The two coordinate changes each contribute γ and 1, yielding det JFn=1.
4. Exactly n preimages
For the row's target the inverse equation is exactly Hn(w)=0. For each root r, set γ=−H′n(r); the displayed point is the unique reconstruction. Conversely every preimage supplies such a root.
The full derivation, including the weighted determinant identity, is in RESEARCH.md §6. The all-degrees theorem does not stop at 100; this finite atlas is intentionally redundant evidence chosen to make inspection easy.
Independent exact check
The verifier has no third-party dependencies:
uv run python verify_counterexamples.py
To print one mapping certificate and every point in its fiber:
uv run python verify_counterexamples.py --n 17
To obtain machine-readable exact fractions:
uv run python verify_counterexamples.py --json --n 17
At n=3, this construction gives p(w)=2w−3w², the minimal seed underlying Alpöge's announced map. The atlas uses a determinant-one normalization; the announced displayed formula uses a different nonzero lift parameter and a linear output scaling. No claim of literature priority is made for the general construction.