Higher-order pole pencils with compressed low-coefficient keys
- Model
- gpt-6.1-sol
- Turns
- 26
- Duration
- 11 min
The attempt explored higher-order pole pencils with compressed low-coefficient keys. It produced a kernel-checked `HigherPole.lean` proving divisibility, factorization, challenge separation, and root-agreement identities, and exact integer screens of sparse profiles found no improvement: best agreement remained 139,775, with ordinary keys providing 30.617 bits against 57.932 required. No candidate reached the gate, so the incumbent, score, and unsafe index were unchanged.
Lesson for the next attempt. Before extending the pole-pencil proofs, reject key designs whose exact fibre entropy is below the threshold; the tested ordinary and sparse profiles did not provide enough compression to improve agreement.
Ended because: gave up: No strictly improving attack found; incumbent Solution.lean, score and unsafe index remain unchanged. Delivered kernel-checked ProximityPrize/SubmissionUpper/HigherPole.lean: vanishing/shared low coefficients imply X^s divisibility; the numerator D-C(D(a)/a^s)X^s factors as X^s(X-C a)Q with deg Q<=q when deg D<=q+s+1; challenge separation and root-agreement identities are proved, plus ordinary-coordinate cancellation. scratch/pole_profiles.py performs exact integer counting for s=1..12, h=1,2,4,8, fibre sizes 128,256,512,1024,2048, fixed-factor degrees 0..12 and 24 core sizes. It counts the union of high/low coordinates and the constant-product key jointly, using the exact density threshold floor(p^6/2^128)=274980728111395087. Best agreement remains 139775. In the ordinary 273-column family, all pole orders retain 15 field coordinates and a 512-valued product key, giving only 30.617 bits versus 57.932 required. Sparse h=2 and h=4 profiles reach at most 138751 and 137727 agreemThought process
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started on "Higher-order pole pencils with compressed low-coefficient keys" with GPT 6.1 Sol (medium effort)
+00:03 · 2.4s
turn 1: 1 tool call, 29 output tokens
+00:03 · list_files
listed repo/ProximityPrize/SubmissionUpper (7 entries)
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turn 2: 2 tool calls, 88 output tokens
+00:05 · read_file