{
  "ranking": [
    {
      "id": "R01",
      "rank": 1,
      "tier": "pass",
      "score": 96,
      "one_line": "Most faithful package: exact main theorem/proof, exact arbitrary-repetition characterization, and essentially correct negative results with proper index-based success notion.",
      "budget": 400,
      "condition": "off",
      "run": 2,
      "trial": "400_off_run2"
    },
    {
      "id": "R06",
      "rank": 2,
      "tier": "pass",
      "score": 94,
      "one_line": "Very strong and rigorous on all target statements; only minor modeling/presentation blemishes keep it just behind R01.",
      "budget": 400,
      "condition": "on",
      "run": 4,
      "trial": "400_on_run4"
    },
    {
      "id": "R11",
      "rank": 3,
      "tier": "pass",
      "score": 93,
      "one_line": "Clean exact reconstruction of the positive theorem and arbitrary-text iff, with a solid three-case element-generator proof and correct index impossibility.",
      "budget": 400,
      "condition": "on",
      "run": 3,
      "trial": "400_on_run3"
    },
    {
      "id": "R12",
      "rank": 4,
      "tier": "pass",
      "score": 92,
      "one_line": "Accurate statements and a rigorous main proof; companion boundary theorems are well recovered and negatives are close to the intended package.",
      "budget": 400,
      "condition": "off",
      "run": 3,
      "trial": "400_off_run3"
    },
    {
      "id": "R15",
      "rank": 5,
      "tier": "pass",
      "score": 91,
      "one_line": "Faithful overall reconstruction with correct main construction/proof and good companion results; only small proof-style looseness remains.",
      "budget": 400,
      "condition": "off",
      "run": 4,
      "trial": "400_off_run4"
    },
    {
      "id": "R16",
      "rank": 6,
      "tier": "pass",
      "score": 90,
      "one_line": "Gets the core package right and proves it well; negative results are strong enough though not always closest to the audited proof.",
      "budget": 400,
      "condition": "off",
      "run": 5,
      "trial": "400_off_run5"
    },
    {
      "id": "R05",
      "rank": 7,
      "tier": "pass",
      "score": 89,
      "one_line": "Excellent on the main theorem and arbitrary-text characterization; the element-generator impossibility is proved by a stronger diagonal argument.",
      "budget": 200,
      "condition": "on",
      "run": 3,
      "trial": "200_on_run3"
    },
    {
      "id": "R07",
      "rank": 8,
      "tier": "pass",
      "score": 88,
      "one_line": "Rigorous and faithful on the positive and iff results, with acceptable strengthened negatives and the right overall boundary picture.",
      "budget": 200,
      "condition": "on",
      "run": 5,
      "trial": "200_on_run5"
    },
    {
      "id": "R08",
      "rank": 9,
      "tier": "pass",
      "score": 87,
      "one_line": "Main theorem and arbitrary-repetition theorem are reconstructed well; negatives are mostly right, though slightly less polished.",
      "budget": 200,
      "condition": "off",
      "run": 1,
      "trial": "200_off_run1"
    },
    {
      "id": "R02",
      "rank": 10,
      "tier": "borderline",
      "score": 84,
      "one_line": "Very good main theorem, but the arbitrary-text necessity proof wrongly uses a constant text as if it were an admissible text enumerating an infinite language.",
      "budget": 200,
      "condition": "on",
      "run": 1,
      "trial": "200_on_run1"
    },
    {
      "id": "R10",
      "rank": 11,
      "tier": "borderline",
      "score": 83,
      "one_line": "Strong main theorem and correct companion negatives, but the arbitrary-text necessity direction relies on a one-point-full-memory style argument stronger than the text model stated.",
      "budget": 200,
      "condition": "on",
      "run": 4,
      "trial": "200_on_run4"
    },
    {
      "id": "R03",
      "rank": 12,
      "tier": "borderline",
      "score": 81,
      "one_line": "Accurate main theorem and arbitrary-text theorem, but it uses an incorrect exact-identification notion for index-generators and has a shaky element-generator proof.",
      "budget": 300,
      "condition": "on",
      "run": 1,
      "trial": "300_on_run1"
    },
    {
      "id": "R18",
      "rank": 13,
      "tier": "borderline",
      "score": 80,
      "one_line": "Positive and arbitrary-text results are mostly right, but the necessity proof again leans on constant texts inconsistent with the stated enumeration definition.",
      "budget": 200,
      "condition": "on",
      "run": 2,
      "trial": "200_on_run2"
    },
    {
      "id": "R13",
      "rank": 14,
      "tier": "borderline",
      "score": 78,
      "one_line": "Good main theorem, but it defines index-generator success as eventual exact identification rather than eventual subset containment, weakening fidelity to the target package.",
      "budget": 400,
      "condition": "on",
      "run": 2,
      "trial": "400_on_run2"
    },
    {
      "id": "R17",
      "rank": 15,
      "tier": "borderline",
      "score": 77,
      "one_line": "Main theorem and arbitrary-text characterization are solid, but the element-generator impossibility proof has real gaps despite the correct high-level conclusion.",
      "budget": 300,
      "condition": "on",
      "run": 5,
      "trial": "300_on_run5"
    },
    {
      "id": "R04",
      "rank": 16,
      "tier": "borderline",
      "score": 75,
      "one_line": "The positive theorem is basically right, but the reconstruction has a broken theorem reference and a substantially incomplete element-generator proof.",
      "budget": 400,
      "condition": "on",
      "run": 5,
      "trial": "400_on_run5"
    },
    {
      "id": "R09",
      "rank": 17,
      "tier": "borderline",
      "score": 74,
      "one_line": "Main theorem is okay and arbitrary-text theorem is acceptable, but the element-generator proof is notably incomplete and the index-success notion is slightly misstated.",
      "budget": 300,
      "condition": "off",
      "run": 4,
      "trial": "300_off_run4"
    },
    {
      "id": "R19",
      "rank": 18,
      "tier": "fail",
      "score": 68,
      "one_line": "Gets the main theorem, but cites external context despite instructions and gives a seriously incomplete/handwavy proof for the element-generator impossibility.",
      "budget": 300,
      "condition": "on",
      "run": 2,
      "trial": "300_on_run2"
    },
    {
      "id": "R14",
      "rank": 19,
      "tier": "fail",
      "score": 66,
      "one_line": "Positive and arbitrary-text theorems are fine, but the element-generator impossibility is only sketched and the index proof is logically flawed on late-occurrence reasoning.",
      "budget": 300,
      "condition": "off",
      "run": 5,
      "trial": "300_off_run5"
    },
    {
      "id": "R20",
      "rank": 20,
      "tier": "fail",
      "score": 65,
      "one_line": "Main positive theorem and iff theorem are good, but the element-generator impossibility proof is only partial and does not rigorously establish the stated result.",
      "budget": 400,
      "condition": "on",
      "run": 1,
      "trial": "400_on_run1"
    },
    {
      "id": "R27",
      "rank": 21,
      "tier": "fail",
      "score": 63,
      "one_line": "Strong on the main and arbitrary-text statements, but the index impossibility is misstated for all positive texts and the element proof contains significant reconstruction uncertainty.",
      "budget": 400,
      "condition": "off",
      "run": 1,
      "trial": "400_off_run1"
    },
    {
      "id": "R26",
      "rank": 22,
      "tier": "fail",
      "score": 61,
      "one_line": "Main theorem is fine, but it defines index-generator success as exact identification rather than subset containment and so misses part of the target theorem package.",
      "budget": 200,
      "condition": "off",
      "run": 2,
      "trial": "200_off_run2"
    },
    {
      "id": "R23",
      "rank": 23,
      "tier": "fail",
      "score": 59,
      "one_line": "Main theorem and arbitrary-text theorem are serviceable, but the element-generator proof is very weak and the index proof is built on delaying finitely many special points rather than the common-core argument.",
      "budget": 200,
      "condition": "off",
      "run": 4,
      "trial": "200_off_run4"
    },
    {
      "id": "R30",
      "rank": 24,
      "tier": "fail",
      "score": 57,
      "one_line": "Positive theorem is correct, but the index-generator notion is changed to exact equality and the element-generator proof is too handwavy to count as rigorous.",
      "budget": 300,
      "condition": "off",
      "run": 1,
      "trial": "300_off_run1"
    },
    {
      "id": "R28",
      "rank": 25,
      "tier": "fail",
      "score": 55,
      "one_line": "Main theorem and arbitrary-text characterization are decent, but index-generator success is changed to exact identification and the index proof is correspondingly misaligned.",
      "budget": 300,
      "condition": "off",
      "run": 3,
      "trial": "300_off_run3"
    },
    {
      "id": "R22",
      "rank": 26,
      "tier": "fail",
      "score": 53,
      "one_line": "Positive theorem is okay, but index-generator success is incorrectly defined as exact equality and the element-generator proof does not establish the claimed impossibility.",
      "budget": 300,
      "condition": "off",
      "run": 2,
      "trial": "300_off_run2"
    },
    {
      "id": "R29",
      "rank": 27,
      "tier": "fail",
      "score": 51,
      "one_line": "Good main theorem, but the element-generator impossibility proof is invalid (the constructed sequence need not enumerate the target), and the index notion is misstated.",
      "budget": 200,
      "condition": "off",
      "run": 5,
      "trial": "200_off_run5"
    },
    {
      "id": "R25",
      "rank": 28,
      "tier": "fail",
      "score": 49,
      "one_line": "Strong main theorem and arbitrary-text result, but the element-generator impossibility proof is plainly wrong (a two-occurrence trick cannot defeat eventual success).",
      "budget": 200,
      "condition": "off",
      "run": 3,
      "trial": "200_off_run3"
    },
    {
      "id": "R24",
      "rank": 29,
      "tier": "fail",
      "score": 45,
      "one_line": "Main theorem is recovered, but the element-generator proof is only a sketch and the index-generator notion is incorrectly strengthened to eventual exact equality.",
      "budget": 300,
      "condition": "on",
      "run": 3,
      "trial": "300_on_run3"
    },
    {
      "id": "R21",
      "rank": 30,
      "tier": "fail",
      "score": 41,
      "one_line": "The arbitrary-text necessity proof incorrectly uses constant texts that are not valid under its own text definition, and both negative proofs are seriously flawed.",
      "budget": 300,
      "condition": "on",
      "run": 4,
      "trial": "300_on_run4"
    }
  ],
  "notes": "The top candidates all recovered the exact positive theorem with the explicit construction G(x)=J_{n(x)}(x) and the audited finite-bad-set proof, and they also matched the arbitrary-repetition iff boundary and the two negative boundary statements with the right success notions. The biggest separators lower down were changing the index-generator task from eventual subset containment to exact identification, using invalid constant texts for the arbitrary-text necessity direction under the stated enumeration model, and giving only sketchy or incorrect proofs for the element-generator impossibility. Bottom candidates typically had the main positive theorem but lost rigor or fidelity on one or both companion boundary theorems."
}