Problem 60 (no 5 points on a sphere): two chiral 36-point witnesses at n = 13

This is a structural observation about Problem 60 at n = 13, not a new bound.

The lower bound `C(13) >= 36` is already established (issue #4, and the
Demonstrandum artifact bundle, DOI 10.5281/zenodo.20673864). I am **not**
claiming an improvement on it, and I did not find a valid 37-point set.

## What is new

Every record-grade construction reported for this problem family so far is
**centrosymmetric** — the n = 13 witness in issue #4 (18 antipodal pairs), and
the n = 14..17 certificates in issue #6 (the 44-point set is described as
22 antipodal pairs about (8,8,8)). Central symmetry halves the search dimension
and pre-saturates the 4-points-per-plane budget, so it is the natural template,
and it is what the successful searches have used.

I am reporting **two 36-point subsets of {0,...,12}^3 that are not
centrosymmetric**, and in fact have no non-trivial symmetry at all:

- Stabilizer under all 48 isometries of the cubic grid: **trivial** (identity only),
by direct enumeration.
- All 36 points have **pairwise distinct multisets** of squared distances to the
other 35. Any Euclidean isometry preserving the set must therefore fix every
point, so the full Euclidean automorphism group is trivial and both sets are
chiral.
- **Zero points in common** with the issue #4 witness, and zero in common with
each other.

So size 36 at n = 13 is reachable from at least three structurally independent
configurations, two of which lie entirely outside the centrosymmetric subspace.
The practical consequence: a search restricted to that subspace is provably
discarding valid solutions at this size. Whether the same holds at n >= 14 is
untested.

## Artifacts

Repository: https://github.com/Ganador1/lazarus-no-five-sphere
Archived: https://doi.org/10.5281/zenodo.21988176

Canonical hashes under the 48 grid isometries (SHA-256 of the lexicographically
minimal image):

```
issue #4 witness f2f0b02c5df67822759f82e560bf00afdb775ba43d44ef8f31b524b7546e20d6
witness A 4167d6adbaa06343786981873803b646063760c4956f91873d52c8381d634286
witness B 717ebbc2d54641d3c878b638f942001eca50fe2f0db21bc023964aa52d30af9b
```

The bundle also includes 19 representative 35-point configurations, selected
deterministically across structural invariants, and a canonical-hash index of
all 1,685 distinct valid 35-point classes the search produced. Notably, none of
the 1,685 is centrosymmetric — consistent with the parity constraint, since a
centrosymmetric set has even size unless it uses the grid centre.

## Verification

Two independently written exact-integer checkers, in different languages, using
different determinant algorithms:

- `verification/verify_all.py` — Python 3, stdlib only, Bareiss fraction-free
elimination. Also computes the stabilizer and distance signatures, and runs
mutation tests (four deliberate corruptions per witness, all must be rejected).
- `verification/verify_independent.cpp` — C++17, Leibniz expansion over all 120
permutations.

Both report, for each 36-point witness: 376,992 five-subsets checked, zero
degenerate, minimum |det| = 2. No floating point anywhere in either path.

```
python3 verification/verify_all.py # expect: TOTAL: 50 PASS, 0 FAIL
```

## What this does not claim

- No improvement to any bound; `C(13) >= 36` already held.
- No 37-point set was found.
- `C(13) = 36` is not claimed, nor is 36 claimed maximal. The upper bound
remains the trivial `C(13) <= 4n = 52`.
- Priority is stated as "no public chiral 36-point construction for n = 13 was
found at the time of writing", not as a guarantee against unpublished or
concurrent work.

## Disclosure

The witnesses were found by exact-arithmetic stochastic local search
(ruin-and-recreate with integer-exact incremental validity checking), run in a
lane deliberately seeded from algebraic constructions unrelated to the known
record, after diagnostics showed the other search lanes were confined to a
single basin. The search was AI-assisted; the mathematical claim rests entirely
on the published coordinates and the two reproducible exact checkers, not on
model output. Search provenance, including a bug that briefly produced invalid
GPU-lane outputs and how it was caught, is documented in `provenance/SEARCH.md`.

Happy to provide the material in whatever format the maintainers prefer.

Giovanni Vicenzo (ORCID 0009-0005-9759-1154)

Start by reading the issue's verification/verify_all.py, verification/verify_independent.cpp, and provenance/SEARCH.md, then run the documented Python checker. The issue does not specify an implementation change; completion would require maintainer direction on whether to incorporate or archive the reported witnesses and verification artifacts.

Written by the indexing model from the issue text.