At the (75, 125) Frontier Exact Reductions and Open Branches for the Planar Jacobian Conjecture

We examine the proof obligations surrounding a hypothetical Keller pair (𝑃 , 𝑄) of total degrees75 and 125 over a field of characteristic zero. The highest homogeneous parts satisfy a commonpowerrelation 𝑃75 = 𝑎𝐻3, 𝑄125 = 𝑏𝐻5, with deg 𝐻 = 25. Factoring 𝐻 gives an exhaustive splitby its number of distinct linear factors; the at-most-two-points-at-infinity theorem excludes thestratum with at least three. The ordinary two-factor stratum remains 𝐻 = 𝑐𝐿𝑒𝑀25−𝑒, 1 ≤ 𝑒 ≤ 12,under the results used here; Moskowicz’s separate multiplicity-gcd result is not transferred tothe original carrier. A separate characteristic-pair sieve leaves the signature ℎ = 3, 𝑑3 = 5 asits only uncovered case. In the normalized 𝐹2 chart, the top edge reduces to a quartic carrierℎ, and the terminal Newton chain forces a repeated root. Subsequent coefficient bands imposedivisibility identities and local root-jet conditions; exact screens remove selected local profiles, butno screen excludes either quartic branch as a whole. An independent Makar-Limanov–Trakhtenbergendpoint transport reproduces a terminal half-space already present in the 𝐹2 model, and a separateMagnus-IV reduction supplies another finite support system. We record these reductions, exactcomputation summaries, and their proof-scope limits. Finite-field Macaulay screens are reportedonly as reconnaissance; a source audit shows that one recorded weight-30 runner does not implementthe normalization stated in its attached selection manifest. The global attachment from everyhypothetical Keller pair to the later specialized systems remains a proof obligation, so (75, 125) isnot excluded by the results documented here.Keywords: planar Jacobian conjecture; Keller maps; Newton polygons; common powers; exactcomputation; Gröbner bases.