Walkthrough — from degree 100 to the exceptional pair

0. What was actually missing

The missing object was not a plausible inverse formula for every plane Keller map. It was much smaller: a trustworthy certificate for the only explicit degree pair left below 125 by the current Newton-polygon reduction.

At first the natural landmark looked like Moh's degree-100 computation. That was too old and too broad to be a useful finite handle. The useful question was whether the later $(72,108)$ exception was still genuinely alive.

1. Named false starts

Start from the 2026 three-variable announcement. The recent papers are mathematically suggestive, especially their tangent-line geometry, but the announcement is not community consensus and the mechanism uses an extra dimension. It could not certify anything about the plane conjecture.

Rebuild Moh's tree. Moh's author page still hosts data, but the printed argument is difficult to replay case by case. Reconstructing it would at best recover 100, below two later records.

Use an arbitrary high-degree automorphism. The shear

(x,y)\longmapsto \bigl(x+cb(ax+by)^d,\ y-ca(ax+by)^d\bigr)

works in every degree. That is a useful test fixture, but it also shows why large degree alone says nothing about the conjecture.

Commit the full certificate. The hard identity is about 89 MB and the complete replay archive is 86 MB compressed. Duplicating it would make the notebook worse to review. An immutable source commit, a whole-archive hash, internal hashes, and a one-command replay give the same byte-level pin without copying the blob.

2. The useful failure

The homogeneous-family route collapsed completely: every homogeneous plane Keller perturbation is a shear. This did not move the record, but it fixed the sign conventions, produced an exact inverse at degree 125, and exposed a general lesson. A degree search must classify possible counterexamples, not merely construct Keller maps of that degree.

The tangent-sweep route failed in a different way. Its Jacobian determinant always contains the sweep parameter. That explains, in one line, why the raw higher-dimensional picture cannot simply be drawn in the plane. It does not rule out more elaborate twists.

3. The click

The 2022 paper does more than improve Moh numerically. Its Theorem 2.1 names the entire remaining finite frontier: maximum degree at least 125, except for $(72,108)$ and its transpose. Proposition 4.3 then writes that exception as two concrete Laurent-polynomial systems.

A 2026 archive claimed exact unit-ideal certificates for precisely those two systems. The decisive observation was that this was not another open-ended search. The archive could be hashed, replayed, and connected back to the published proposition by a small independent program.

4. The argument in discovery order

First, the four source polygons were transcribed directly from Proposition 4.3. Independent enumeration found 61 and 125 lattice points in the first pair, and 25 and 47 in the second.

Second, the Laurent coordinates were recomputed rather than copied. Put

t=xy^2,\qquad z=y^{-1}.

Then $[t,z]=-1$ and $x=t z^2$ . Writing the upper bands as

P=Az^2+Bz+C, \qquad Q=Dz^3+Ez^2+Fz+G

and expanding $P_zQ_t-P_tQ_z$ gives exactly five coefficient identities, with $2AD'-3A'D=t^2$ at the top.

Third, the three normalized coefficients occur at $(1,0)$ , $(8,14)$ , and $(8,16)$ . Their exponent matrix has determinant 14. Over an algebraically closed field, the required roots exist, and a fourth output scaling preserves the right-hand side $x^2$ .

Only then was the large archive allowed to speak. Its hash and eight internal hashes matched. It regenerated the residual systems, checked the small unit ideals, evaluated the hard identity through a separate exact-arithmetic path, checked the branch symmetry, and ended with both branch markers.

Finally, Theorem 2.1 was applied once: with the exceptional pair gone, the other alternative is maximum degree at least 125.

5. Computer search and certificate

There was no new heuristic search. q1 is a certificate replay and bridge audit.

Check	Exact result
Case 1 lattice points	$61,125$
Case 2 lattice points	$25,47$
Laurent coefficient identities	5
Normalization determinant	14
First-block number-field degree	35
Number-field products in independent path	13,410
Equivalent scalar products	335,250
Hard branches	2/2 pass

The bridge was independently implemented in Python and Rust, and their output was required to agree byte for byte. The large archive was replayed under Python 3.14 with exact python-flint, sympy, and gmpy2 arithmetic. q2 used sparse integer polynomials. q3 used distinct formal symbols for every curve coefficient.

6. What is proved and what remains open

The checked finite implication is:

\boxed{\text{Every plane counterexample has } \max\{\deg P,\deg Q\}\ge125.}

That is a dent over the previously surviving maximum degree 108. It depends on the published Guccione–Guccione–Horruitiner–Valqui reduction in the ordinary way a computer-assisted corollary depends on its mathematical input; the local bridge removes the extra transcription and normalization assumptions identified by the certificate author.

The plane Jacobian conjecture is still open. No inverse was constructed for an arbitrary Keller map, and no plane counterexample was found. The 2026 three-variable announcement is neither assumed nor refuted here. The q3 obstruction covers only the raw tangent-sweep template.