Problem 60 (no 5 points on a sphere): improved lower bound C(13) >= 36, explicit certificate

The repository's lower bounds for Problem 60 end at n = 12 ("C_6.60(12) >= 33");
nothing is listed for n = 13, so the best previously known bound was
C(13) >= C(12) >= 33. Below is an explicit 36-point subset of {0,...,12}^3
with no 5 points on a common sphere or plane, giving **C(13) >= 36**.

**The point set** (36 points, 0-indexed coordinates in {0,...,12}):

```json
[[0,1,12],[0,3,7],[0,9,10],[0,12,0],[1,4,1],[1,6,4],[1,12,11],[2,6,12],
[2,8,1],[3,1,10],[3,5,7],[3,7,10],[4,3,5],[4,8,3],[5,0,0],[5,2,2],
[5,11,3],[6,0,4],[6,12,8],[7,1,9],[7,10,10],[7,12,12],[8,4,9],[8,9,7],
[9,5,2],[9,7,5],[9,11,2],[10,4,11],[10,6,0],[11,0,1],[11,6,8],[11,8,11],
[12,0,12],[12,3,2],[12,9,5],[12,11,0]]
```

**Verification** is a finite exact-integer computation: for every 5-subset,
the 5x5 lifted determinant det[x, y, z, x^2+y^2+z^2, 1] must be nonzero
(this excludes spheres and planes simultaneously, planes being the
degenerate case — the same convention as the repository's evaluator, but in
exact arithmetic rather than floats). All C(36,5) = 376,992 determinants are
nonzero; the minimum absolute value is 2. Self-contained checker (Python,
stdlib only; runs in ~2 s):

```python
import json, sys
from itertools import combinations

def check(points, n=13):
pts = [tuple(p) for p in points]
if len(set(pts)) != len(pts):
return False, "duplicate points"
if any(len(p) != 3 or any(not isinstance(c, int) or not 0 <= c < n for c in p) for p in pts):
return False, "non-integer or out-of-range coordinate"
L = [(x, y, z, x * x + y * y + z * z) for (x, y, z) in pts]
for idx in combinations(range(len(pts)), 5):
p = L[idx[0]]
(a0, a1, a2, a3), (b0, b1, b2, b3), (c0, c1, c2, c3), (d0, d1, d2, d3) = \
[tuple(L[i][k] - p[k] for k in range(4)) for i in idx[1:]]
det = ((a0*b1 - a1*b0) * (c2*d3 - c3*d2) - (a0*b2 - a2*b0) * (c1*d3 - c3*d1)
+ (a0*b3 - a3*b0) * (c1*d2 - c2*d1) + (a1*b2 - a2*b1) * (c0*d3 - c3*d0)
- (a1*b3 - a3*b1) * (c0*d2 - c2*d0) + (a2*b3 - a3*b2) * (c0*d1 - c1*d0))
if det == 0:
return False, [pts[i] for i in idx]
return True, None

pts = json.load(open(sys.argv[1]))
ok, bad = check(pts)
print("VALID m=%d n=13" % len(pts) if ok else ("INVALID: %s" % (bad,)))
```

To confirm convention compatibility: the same checker (and two further
independently written checkers, plus one written by an adversarial referee
agent from a different model family) accepts all six record sets for
n = 7..12 published in this repository's notebook, and rejects mutated
certificates (7-mutant suite). Exact arithmetic matters here: the minimum
|det| over the 376,992 subsets is 2, i.e. the set sits at the edge of
degeneracy, like the published records.

**Structure.** The set is invariant under the central inversion
p -> (12,12,12) - p: 18 antipodal pairs whose squared distances from the
cube center are pairwise distinct. It is saturated (no 37th point of the
grid can be added). It was found in 841 s on 8 CPU threads by an
exact-arithmetic iterated local search restricted to centrally symmetric
configurations; en route, four independent search pilots produced thousands
of distinct 34-point sets and dozens of 35-point sets, suggesting C(13) may
exceed 36. No improvement is claimed for any other n, and the trivial upper
bound C(13) <= 52 is untouched.

**Disclosure.** This construction was produced in Demonstrandum, a
verification-first AI-assisted workflow
(Anthropic's Claude agents for the search and checkers; OpenAI's GPT-5.5 as
an independent adversarial verifier), directed by a human who takes
responsibility for the claim. Verification is fully mechanical and
independent of any model: the certificate plus the checker above.

A short note with the construction, the search method, and the verification
details is available at https://doi.org/10.5281/zenodo.20673865 (deposit
copy also at https://github.com/demonstrandum-research/artifacts). I am happy to
send a CLA-signed pull request adding the point set and the n = 13 entry to
the problem page/notebook in whatever format you prefer.

John Erlbacher | Independent Researcher | erlbacher.research@gmail.com | ORCID 0009-0003-6851-4139