# Blinded comparative ranking of reconstructions

Judge: `openai/azure/gpt-5.4` — one call, 30 candidates, shuffled (seed 7), OFF/ON hidden until after ranking.

| rank | budget | condition | run | tier | score | reason |
|---:|---:|:--|---:|:--|---:|:--|
| 1 | 400 | off | 2 | pass | 96 | Most faithful package: exact main theorem/proof, exact arbitrary-repetition characterization, and essentially correct negative results with proper index-based success notion. |
| 2 | 400 | on | 4 | pass | 94 | Very strong and rigorous on all target statements; only minor modeling/presentation blemishes keep it just behind R01. |
| 3 | 400 | on | 3 | pass | 93 | Clean exact reconstruction of the positive theorem and arbitrary-text iff, with a solid three-case element-generator proof and correct index impossibility. |
| 4 | 400 | off | 3 | pass | 92 | Accurate statements and a rigorous main proof; companion boundary theorems are well recovered and negatives are close to the intended package. |
| 5 | 400 | off | 4 | pass | 91 | Faithful overall reconstruction with correct main construction/proof and good companion results; only small proof-style looseness remains. |
| 6 | 400 | off | 5 | pass | 90 | Gets the core package right and proves it well; negative results are strong enough though not always closest to the audited proof. |
| 7 | 200 | on | 3 | pass | 89 | Excellent on the main theorem and arbitrary-text characterization; the element-generator impossibility is proved by a stronger diagonal argument. |
| 8 | 200 | on | 5 | pass | 88 | Rigorous and faithful on the positive and iff results, with acceptable strengthened negatives and the right overall boundary picture. |
| 9 | 200 | off | 1 | pass | 87 | Main theorem and arbitrary-repetition theorem are reconstructed well; negatives are mostly right, though slightly less polished. |
| 10 | 200 | on | 1 | borderline | 84 | 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. |
| 11 | 200 | on | 4 | borderline | 83 | 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. |
| 12 | 300 | on | 1 | borderline | 81 | 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. |
| 13 | 200 | on | 2 | borderline | 80 | Positive and arbitrary-text results are mostly right, but the necessity proof again leans on constant texts inconsistent with the stated enumeration definition. |
| 14 | 400 | on | 2 | borderline | 78 | Good main theorem, but it defines index-generator success as eventual exact identification rather than eventual subset containment, weakening fidelity to the target package. |
| 15 | 300 | on | 5 | borderline | 77 | Main theorem and arbitrary-text characterization are solid, but the element-generator impossibility proof has real gaps despite the correct high-level conclusion. |
| 16 | 400 | on | 5 | borderline | 75 | The positive theorem is basically right, but the reconstruction has a broken theorem reference and a substantially incomplete element-generator proof. |
| 17 | 300 | off | 4 | borderline | 74 | 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. |
| 18 | 300 | on | 2 | fail | 68 | Gets the main theorem, but cites external context despite instructions and gives a seriously incomplete/handwavy proof for the element-generator impossibility. |
| 19 | 300 | off | 5 | fail | 66 | 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. |
| 20 | 400 | on | 1 | fail | 65 | 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. |
| 21 | 400 | off | 1 | fail | 63 | 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. |
| 22 | 200 | off | 2 | fail | 61 | 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. |
| 23 | 200 | off | 4 | fail | 59 | 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. |
| 24 | 300 | off | 1 | fail | 57 | 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. |
| 25 | 300 | off | 3 | fail | 55 | Main theorem and arbitrary-text characterization are decent, but index-generator success is changed to exact identification and the index proof is correspondingly misaligned. |
| 26 | 300 | off | 2 | fail | 53 | 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. |
| 27 | 200 | off | 5 | fail | 51 | 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. |
| 28 | 200 | off | 3 | fail | 49 | Strong main theorem and arbitrary-text result, but the element-generator impossibility proof is plainly wrong (a two-occurrence trick cannot defeat eventual success). |
| 29 | 300 | on | 3 | fail | 45 | 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. |
| 30 | 300 | on | 4 | fail | 41 | 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. |

## Mean rank by cell (lower = better)

| budget | OFF mean-rank (tiers) | ON mean-rank (tiers) |
|---:|:--|:--|
| 200 | 21.8 (f,f,f,f,p) | 9.8 (b,b,b,p,p) |
| 300 | 22.2 (b,f,f,f,f) | 20.8 (b,b,f,f,f) |
| 400 | 7.4 (f,p,p,p,p) | 11.0 (b,b,f,p,p) |

## Judge 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.