Quadratic-cover words and polynomial Pell agreement lists
- Model
- deepseek-v4-pro-0813
- Turns
- 55
- Duration
- 28 min
The attempt investigated whether quadratic-cover words and polynomial Pell-style agreement lists could extend the existing 272-column OrbitPencil by the nine agreement positions needed to reach index 122360. Raising the pencil to 273 columns imposed a degree constraint that reduces the options to highly constrained subset-polynomial classes; the estimated number of subsets per class was far below what the required density would need. Coset-based constructions ran into a root-count obstruction, and small finite-field searches found no high-agreement examples for the tested quadratic-cover construction. No candidate reached the gate, so there was no verified attack, and the agent gave up without producing a kernel-checked result strictly below 11613.
Lesson for the next attempt. Before extending the OrbitPencil by another column, find a construction that meets the degree constraint while producing a sufficiently large, explicitly countable family of agreement subsets; the tested arbitrary-subset, coset, and branch-interpolation approaches did not.
Ended because: gave up: I could not produce a kernel-checked attack strictly below 11613. The needed index is <=122360, i.e. nine more agreement positions than the current 272-column OrbitPencil. Raising the pencil to 273 columns hits a hard degree constraint: the numerator B - V_U - gamma Y must have degree 273, so the 273 selected subset polynomials V_U must lie on one affine line modulo the denominator; for arbitrary 273-subsets this is a p^16 constraint class, while the expected number of subsets in a class is only about 2^8 (C(512,273)/p^16), far below the 2^58 distinct challenges needed for density > 2^-128. Coset-structured families fail by number theory: t=273 is coprime to 512, so a full-coset part leaves a partial subset whose sparse polynomial is Y^r+c; such a polynomial cannot have r distinct roots in the 512-th roots when gcd(r,512)=1. Small finite-field searches for the proposed quadratic-cover (branch interpolation, A_gamma^2 - Y(B0+gamma B1)^2) found no high-agreement line. A next attThought process
+00:00
started on "Quadratic-cover words and polynomial Pell agreement lists" with DeepSeek V4 Pro (high effort)
+00:06 · 3.9s
turn 1: 3 tool calls, 285 output tokens
+00:06 · list_files
listed repo/ProximityPrize/SubmissionUpper (7 entries)
+00:06 · read_file
read repo/ProximityPrize/SubmissionUpper/Solution.lean lines 1-83