Complement-factor syzygies for a 273-column pencil
- Model
- gpt-6.1-sol
- Turns
- 14
- Duration
- 5 min
The attempt investigated whether complement-factor syzygies could create a rank defect for the 273-column pencil. Exact symbolic elimination and exhaustive 16×16 tests over F257 and F769 found no viable construction: with fourteen coordinates fixed, changing the fifteenth makes the quotient degree 256, above the allowed 255; slack terms and a third denominator factor introduce corresponding agreement or independent-value constraints. No candidate reached the gate, and the submission remained unchanged; the computation and analysis were saved in `scratch/complement_syzygy.py` and `scratch/complement_syzygy.md`.
Lesson for the next attempt. For this strategy, track the independent constraints introduced by each proposed slack or denominator factor: saving a top-degree condition does not help if it restores a full-rank condition elsewhere.
Ended because: gave up: No rank defect found. Saved exact computation and symbolic elimination in scratch/complement_syzygy.py and scratch/complement_syzygy.md; submission unchanged. From VW=A, the first fifteen complement coefficients are a triangular reparameterization of the fifteen selected-factor coefficients, with determinant -1. With fourteen fixed, a differing fifteenth gives quotient degree 256, exceeding 255. Adding AT slack eliminates to the original identity; low-degree WB corrections cannot preserve agreement at all 273 roots. Adding a third denominator factor saves a top condition but requires an independent V(beta) condition (rank 15 again). Exhaustive 16x16 tests over F257 and F769 check factorization, selected/core agreement, pole values, and explicit dropped-coordinate degree failures; the third-root test also exhibits nonzero remainders. Exact counting passes with fourteen coordinates (ratio floor 6), fails with fifteen. Hypothetical index 121857 scores 11546, but no valid attack eThought process
+00:00
started on "Complement-factor syzygies for a 273-column pencil" with GPT 6.1 Sol (medium effort)
+00:03 · 2.3s
turn 1: 1 tool call, 39 output tokens
+00:03 · read_file
read repo/ProximityPrize/SubmissionUpper/OrbitPencil.lean lines 1-240
+00:06 · 3.0s
turn 2: 3 tool calls, 106 output tokens