GATE D19
Exact cyclic cover, per-prime separation refuted
What changed
D19 returns from moment space to the exact source object. Fixing a prime and a primitive root, a speed vector becomes a multiplicity vector on the cyclic group of size p minus 1, and an improper vector (no lonely time) exists exactly when thirteen cyclic translates of the bad set cover the whole group. The fractional covering number is exactly the group size over the bad-set size, about 7 at every prime, far below the budget of 13, so the linear relaxation closes nothing. The decisive result is negative and is published in full. Explicit thirteen-translate covers actually exist at 48 of the 109 primes, and an exact non-tight improper vector is exhibited at p=197 with thirteen distinct speeds, zero lonely times, and adaptive bound exactly 0. This corrects the earlier moment work, the adaptive degree-5 dual also fails at a single prime, and the earlier twenty-thousand-vector search that found no counterexample was bounded by sampling, not by a theorem. Both per-prime methods are therefore insufficient. These single-prime covers are exactly what upstream enumeration finds and are not real counterexamples, they are eliminated only by the prime-product lift that requires one configuration to cover across many primes at once. That lift is the only remaining route. LRC(13) remains open and is not disproven.
Evidence
Verification: double · I2 · 12 artifacts (reports, verifiers, corruption suites, manifest with SHA-256)