Principal minors of Fourier matrices at square-free order: the Caragea–Lee–Malikiosis–Pfander lifting hypothesis, decided at 452 splits

F₇0: the CLMP lifting hypothesis holds at (p, N') = (7, 10)

F₁43: the hypothesis holds at (13, 11)

F₁43 again through the split (11, 13), which Gu does not use

the six three-factor orders 130, 154, 170, 190, 238, 255

the four two-prime orders 187 = 11*17, 209 = 11*19, 221 = 13*17, 247 = 13*19

the lifting route provably fails at (7,15), (11,10), (11,15), (13,14), (13,15), with counts 30, 240, 1080, 210, 60

named first witnesses for the five failures

N' = 6: the hypothesis holds at every prime 5 <= p <= 59

N' = 10: the whole row of primes p <= 59 – 12 orders settled, 3 refuted

N' = 11: the whole row – 14 settled, 2 refuted

N' = 13: the whole row – 12 settled, 4 refuted

N' = 14: the whole row – 9 settled, 6 refuted

N' = 15: the whole row – 6 settled, 9 refuted

negative control, kernel-clean over C: if p² divides N then F_N has a zero principal minor

control: a known-good order can have a failing route (70 via (5,14), 30 via (3,10))

control: the checker detects singularity, and every cyclotomic constant is derived and verified

control: the four known-good positive anchors 30, 42, 66, 78 (= 6p) reproduced one at a time; five more (22, 55, 77, 26, 91) appear inside the n=11/n=13 sweep rows

N = 210: the norm hypothesis (H) fails at all four splits, with named witnesses, and the four are all of them

N = 210: the per-prime hypothesis (H') fails at all four splits too, with witnesses covering every prime of Z[zeta_N'] above p

(H') holds where (H) fails, at (29,13) and (59,15): the orders 377 = 13*29 and 885 = 3*5*59, from the bases 13 and 15 that no fixed-base classification covers

(H') at (31,10): N = 310 re-derived, and Wang-Zhang's two safe charts recovered as the roots of a polynomial

control: (H) and (H') are genuinely different, in both directions

the affine-orbit reduction (CLMP prop:affinenorminvariance) implemented, and checked against the exhaustive engine at 73 splits

the two parked moduli run: N = 357 = 3*7*17 via (17,21) and N = 682 = 2*11*31 via (31,22)

N' = 10 above the old prime bound: every prime 61 <= p <= 499 passes, 78 orders 610... 4990

N' = 14 above the old prime bound (no fixed-base classification exists for base 14): 69 settled, 9 refuted (71, 83, 113, 127, 197, 281, 379, 421, 491), 69 new orders 854... 6986

N' = 15 above the old prime bound (no fixed-base classification exists for base 15): 68 settled, 10 refuted (61, 79, 89, 151, 181, 211, 241, 271, 331, 421), 68 new orders 1005... 7485

the affine-orbit sweep made bitset-free and complement-folded: a local canonicity test replaces the 2ⁿ "seen" array, and CLMP prop:unitarycomplementarity folds each complementary orbit pair into its lighter member

N = 253 = 11*23: the CLMP lifting hypothesis holds at (11,23), the only split available, since (23,11) is one of the family's landed refutations

the base 23 swept whole at every prime p <= 59: 13 settled, 3 refuted; ten orders 253, 391, 437, 667, 713, 851, 943, 989, 1219, 1357 that no published theorem reaches

named witnesses for the three base-23 failures, and at p = 47 the complete answer in size 3: 0,1,3 fails while the other three affine orbits of 3-subsets of Z₂3 do not

the base 21 swept whole at every prime p <= 59: 5 settled, 10 refuted; four new three-factor orders 483, 987, 1113, 1239, with five named witnesses

the base 22 swept whole at every prime p <= 59: 11 settled, 4 refuted; nine new three-factor orders 286, 374, 418, 638, 814, 902, 1034, 1166, 1298, with four named witnesses

the base 17, which this family had never run, swept whole at every prime p <= 59: 14 settled, 2 refuted; ten new orders 323, 391, 493, 527, 629, 697, 731, 799, 901, 1003

the base 19, which this family had never run, swept whole at every prime p <= 59: 11 settled, 5 refuted; nine new orders 323, 437, 551, 589, 779, 817, 893, 1007, 1121

the divisor rule accounts for one of the ten base-21 failures and two of the four base-22 ones, and nothing else: below p = 60 the bases 3 and 5 never fail, 7 fails only at p = 2, 11 only at p = 3, 23

every square-free order N = pq with p, q distinct ODD primes <= 23 is settled — all 28 such pairs, nine of them by exactly one of the two directions (label clarified 2026-08-30: "all 28 pairs" is the 28 pairs with both primes odd, which is what TwoPrime.lean decides; the eight remaining orders 2q, q <= 23 odd, are CLMP Thm 1.3 instances and are not decided by this row, so "every pq with p, q <= 23" is 36 orders = 28 here + 8 cited)

CLMP's four published "bad pairs" tested at the principal-minor level: (11,89) is not an obstruction and settles the new order N = 979 = 11*89; (13,53), (17,953), (19,457) are obstructions

what decides whether the route works is the splitting type of the characteristic, not its size: the failure rate is monotone in g = phi(n)/ordₙ(p), the number of primes of Z[zetaₙ] above p — an inert characteristic fails in 4 of 238 cells (1.7%), a totally split one in 40 of 71 (56%), and at the base 15 the README's question is actually about, all nine totally split primes below 500 fail, 9 for 9

the lifting route fails at (11,30), (13,30), (17,30), (19,30), (23,30) — the only reachable splits of the four-factor orders 330, 390, 510, 570, 690 — with named witnesses; with the landed (7,30) that is six for six at the base 30, and no square-free order with four prime factors below 870 is settled by this route: each of the TWELVE such orders 210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858 is either unreachable (no base under 2⁴2 masks) or refuted. Label corrected 2026-08-30: Base30.lean's docstring listed only eleven, omitting 770 = 2*5*7*11; 770's splits are (2,385), (5,154), (7,110), (11,70), smallest base 70, so it joins the unreachable group and the conclusion is unchanged. The Lean asserts the five base-30 refutations only; the completeness over all twelve orders is a prose claim, checked by hand.