Padé slack factors instead of monic subset-polynomial collisions
- Model
- deepseek-v4-pro-0813
- Turns
- 39
- Duration
- 5 min
The attempt investigated whether Padé slack factors could improve the upper-track construction, using symbolic analysis of the existing rational pencil and its coefficient constraints. It found that agreement on an extra column forces a polynomial factor with that column as a root, adding a top-coefficient key coordinate; the resulting entropy ratio is about 2^-28 below the requirement of 2^58 winning challenges. Enlarging the core or adding generic auxiliary factors does not remove this deficit. No improved candidate was submitted, and the gate verified no result.
Lesson for the next attempt. Do not pursue Padé slack for this construction; the next attempt should seek a genuine rank defect in the top-coefficient map that improves the entropy ratio.
Ended because: gave up: No improved candidate was reached. The proposed Pade-slack direction was analysed symbolically and shown not to relax the real bottleneck: in the rational pencil, agreement on an extra column is equivalent to multiplying the fixed quotient by a polynomial whose root is that column, and this adds one top-coefficient key coordinate. For 272 columns plus one extra column (or for any usable 273-column/core-expanded variant) the entropy ratio becomes C(511,272)/(p^15*512), which is about 2^-28 too small for the required 2^58 winning challenges. Generic auxiliary factors merely move coordinates into an enlarged quotient and cancel out of the entropy ratio. The current construction already optimises the core-degree trade for r=14 (D=258, Q deg<=255, core<=511). To extend the core to 512+ one must drop Q degree to 254, forcing r=15 and the same 28-bit deficit. The next attempt should not chase Padé slack: it should look for an actual rank defect in the top-coefficient map. Two concreThought process
+00:00
started on "Padé slack factors instead of monic subset-polynomial collisions" with DeepSeek V4 Pro (high effort)
+00:04 · 3.6s
turn 1: 2 tool calls, 233 output tokens
+00:04 · list_files
listed repo/ProximityPrize/SubmissionUpper (7 entries)
+00:04 · read_file
read repo/ProximityPrize/SubmissionUpper/Solution.lean lines 1-83