slug: bradshaw-arithmetic-derivative-collatz-squarefree-refutation bibkey: bradshaw2025collatz doi: null url: https://cs.uwaterloo.ca/journals/JIS/VOL28/Bradshaw/bradshaw3.pdf triage: theorem motivation_gids:
- D5/S0/Certificates/BradshawConjectureTwentyRefutation.result
Bradshaw’s Conjecture 20
Problem
Z. P. Bradshaw, On a Family of Solutions to Arithmetic Differential Equations Involving the Collatz Map, Journal of Integer Sequences 28 (2025), Article 25.1.8, Conjecture 20, printed page 16:
The solutions to the commutation problem D(C_{a,b}(n)) = C_{a,b}(D(n)) are squarefree for every a ≡ b (mod 2).
The same page defines C(a,b,n) = (a*n+b)/2 for odd n and n/2 for
even n. Printed page 3 defines D : Nat -> Nat by D(0)=D(1)=0,
D(p)=1 at each prime, and D(m*n)=D(m)*n+m*D(n) for all m,n.
The closed formal claim quantifies over such D and all natural a,b,n,
with equal parity of a,b and 1 <= n, and concludes Squarefree n
from the displayed commutation equality.
Motivation
The frozen BradshawConjectureTwentyRefutation.result : Not claim
refutes the positive-input restriction of the source assertion.
At (a,b,n)=(17,7,125), both sides equal 641, while 125=5^3
is not squarefree. The source’s five tested parameter pairs do not
include this pair.
Gap
The bounded literature screen reported in issue 8643 located no prior resolution of Conjecture 20. The orchestrator’s arXiv conjunction query returned zero entries, with eleven entries for the arithmetic-derivative positive control. Its OEIS query returned A247583 and A383619, neither stating this resolution. The citation-index gap and the unidentified citing-item lead remain unresolved; this is not a claim of exhaustive literature coverage.
Route
Construct an arithmetic derivative internally as
D0(n) = sum (p,k) in n.factorization, k * NatDiv(n,p).
The empty factorizations give zero at zero and one. Prime
factorization of a product supplies Leibniz’s rule, treating zero
factors separately. The four defining properties then give
D(25)=10, D(125)=75, D(533)=54, and D(1066)=641.
Since C(17,7,125)=1066 and C(17,7,75)=641, the equality holds.
The prime square 5^2 divides 125, contradicting squarefreeness.
Falsifier
A source restriction excluding positive odd parameters with b<a, an incorrect interpretation of the two branches, or failure of the arithmetic-derivative properties would invalidate source fidelity. A prior published resolution of this exact conjecture would invalidate open-problem-resolution eligibility without changing the counterexample.
Evidence
The formal carrier is
D5/S0/Certificates/BradshawConjectureTwentyRefutation.lean.
Its public surface consists of IsArithmeticDerivative, C, claim,
and the sole theorem result : Not claim. The existence construction
and numerical deductions stay inside that proof. The Scribe result
node binds this dossier with OpenProblemResolutionClaim(Refuted).
The kernel proof uses the standard three axioms and no sorry or
additional axiom. The source identity and printed locators are in
Library/Certificates/bradshaw2025collatz.md.
Triage
Tier 1 external named conjecture, preregistered in issue 8643 before
the probe. The result has proof_shape: bind-only,
escape_witness: null, and admission_basis: open-problem-resolution.
Its computational use is a certified-instance with basis=refutes:
the result negates the closed claim. This dossier concerns only
Conjecture 20; Conjectures 17–19 about the standard Collatz map are
outside its scope.
ASSUMED-UNVERIFIED
Source fidelity is a semantic review obligation, separate from kernel
verification. The extra hypothesis 1 <= n makes claim weaker than
the printed conjecture, so its refutation refutes the source and
excludes the trivial n=0 solution. The intended parameter domain is
read as naturals of equal parity; the witness has positive odd a,b
with b<a, as in the source’s five tested pairs.
Literature completeness is ASSUMED-UNVERIFIED. No citation index was retrieved in full. A ChatGPT Pro literature seat reported a Google Scholar search-index excerpt showing a citation-count-like value of 1 beside the article, but could not identify the citing item. That unresolved lead does not establish either zero citations or a prior resolution. The same seat reported that the author’s earlier ResearchGate preprint, publication/384084846, states the same assertion as Conjecture 6 and gives no counterexample; these are seat-reported literature readings.