| { |
| "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." |
| } |
