A free-chain counterexample to a Nash-intersection conjecture

This is the authors' abstract. We don't add key points for this paper.

For the complete family of prime divisors obtained by n successive free point blowups over the smooth complex surface germ, fixing a prefix of k centers with 2 <= k < n gives the closed Nash-set intersection {ord x >= ceil(n/k), ord y >= n}. Five blowups give thresholds (3,5) and (2,5) when two or three centers are fixed. Neither is q(1,j) for any positive integers q,j, so no positive contact multiple of any reference prefix divisor equals the intersection. The proof exhausts all complex free-center choices and treats both charts, satellite limits, identically zero x, constant arcs, exact/at-least contact closures and all integer multiplicities. Explicit separating arcs are nonconstant convergent polynomials. This is a negative answer to the specified-reference-multiple assertion of Conjecture 1 in Fernandez de Bobadilla, Pe Pereira and Popescu-Pampu, arXiv1501.06898v3 p.31, and Pe Pereira's joint OWR46/2016 contribution p.2679, under every contact/index convention explicitly mapped in the paper. In particular fixing the genuinely free selected c3 still leaves c4,c5 variable. The source uses an intersection; a catalogue union is a transcription discrepancy. Its bare equivalence symbol is undefined and its global indexing inconsistent, so no unique author-intended correction or unspecified replacement is asserted. The intersections ARE maximal divisorial sets for other primitive toric weights; no arbitrary-divisor negative, nonreduced-scheme equality, Conjecture 2 or general Nash-problem solution is claimed. The standard divisorial/contact and toric framework, and the original paper's combinatorial invariance and cylindrical intersection results, are explicitly credited. The contribution is the complete free-family calculation and reference-ray obstruction. A bounded extensive primary-source investigation found no authenticated earlier exact negative calculation; no exhaustive novelty, historical firstness or continuous-open-status certificate is claimed. Actual arXiv/workshop bodies and selected older/new primary theorems were read; the publisher-typeset original body remains inaccessible403. The proof was already public in PR1332 before this note; the present version is dated22October2026 and is not backdated. The proposed deposit contains the five-page paper and complete editable-source/verification ZIP. New standard-library exact diagnostics supplement the universal written proof and named standard background; no independent replay of unavailable claimant programs or proof-assistant certification is claimed. AI tools were extensively involved in mathematical research, proof development and reconstruction, exact and symbolic computation, source/literature investigation, manuscript preparation, editing and adversarial auditing. Alec Kriebel is the named human author. This is an unrefereed preprint without formal independent human peer review or journal acceptance. Authored prose is CC BY4.0 and authored code is MIT licensed.