{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R213",
  "content_hash": null,
  "slug": "djs8-artifact-rational-certificates",
  "type": "artifact",
  "title": "Exact rational lower and upper certificates",
  "summary": "A standard-library Python program checks the constraint count, the tent witness, and every symbolic interpolation identity.",
  "relevance": "For Exact Jensen stability constant on the eighth dyadic grid, record djs8-artifact-rational-certificates (“Exact rational lower and upper certificates”) supplies evidence or a replay used to check the packet. The record states: A standard-library Python program checks the constraint count, the tent witness, and every symbolic interpolation identity.",
  "relevance_source": "recorded",
  "body": "All symbolic coefficients use `Fraction`. The canonical JSON report has SHA-256 digest `013676a759bb305c8fdde90c525a5abbe0a3e4f501f4756983399ae6c0713383`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "all 16384 midpoint rows, all 257 interpolation identities, and the tent witness on the eighth grid",
    "bounds": {
      "dyadic_level": {
        "min": 8,
        "max": 8
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "runnable",
    "kind": "inline_python_computation",
    "command": "python3 certificate.py",
    "runtime": "Python 3 standard library",
    "citation": {
      "url": "https://doi.org/10.1515/dema-1989-0220",
      "locator": "Self-contained Python standard-library certificate reproduced 2026-07-24"
    },
    "inline_source": "from fractions import Fraction as F\nfrom hashlib import sha256\nfrom json import dumps\nN=256\npairs=[(i,j) for i in range(N+1) for j in range(i+1,N+1) if (i+j)%2==0]\nassert len(pairs)==16384\ndef add(d,k,v):\n    d[k]=d.get(k,F(0))+v\n    if d[k]==0: del d[k]\ndef cert(i,a=0,b=N):\n    if i==a or i==b: return {}\n    m=(a+b)//2\n    if i==m: return {(a,b):F(1)}\n    if i<m:\n        d=cert(i,a,m); add(d,(a,b),F(2*(i-a),b-a)); return d\n    d=cert(i,m,b); add(d,(a,b),F(2*(b-i),b-a)); return d\nnorms=[]\nfor i in range(N+1):\n    d=cert(i); v=[F(0)]*(N+1)\n    for (a,b),w in d.items():\n        v[(a+b)//2]+=w; v[a]-=w/2; v[b]-=w/2\n    target=[F(0)]*(N+1)\n    target[i]+=1; target[0]-=F(N-i,N); target[N]-=F(i,N)\n    assert v==target\n    norms.append(sum(map(abs,d.values()),F(0)))\nB=max(norms)\nassert B==F(711,128)\nassert [i for i,x in enumerate(norms) if x==B]==[85,171]\nt=[min(i,N-i) for i in range(N+1)]\ndelta=max(abs(F(t[(i+j)//2])-F(t[i]+t[j],2)) for i,j in pairs)\nassert delta==128 and max(abs(x-64) for x in t)==64\nassert t[0]-2*t[128]+t[256]==-256\nreport={'constraint_count':len(pairs),'lower_ratio':'1/2','recursive_upper':str(B),'tent_delta':str(delta),'tent_distance':64,'upper_active_indices':[85,171]}\npayload=dumps(report,sort_keys=True,separators=(',',':'))\nassert sha256(payload.encode()).hexdigest()=='013676a759bb305c8fdde90c525a5abbe0a3e4f501f4756983399ae6c0713383'\nprint(payload)",
    "missing": [
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1515/dema-1989-0220",
    "locator": "Self-contained Python standard-library certificate reproduced 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R215",
      "title": "The tent function certifies C_8 >= 1/2",
      "object_type": "claim",
      "relation": "verifies",
      "direction": "outgoing"
    },
    {
      "slug": "R216",
      "title": "Recursive midpoint interpolation proves C_8 <= 711/128",
      "object_type": "claim",
      "relation": "verifies",
      "direction": "outgoing"
    },
    {
      "slug": "dyadic-jensen-stability-8",
      "title": "dyadic jensen stability 8",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}