[1337isnot1337]

Result

Adds research/centre-braid-triangle-audit, an exact audit of the distance-2 triangle proxy used in the degree-capacity screen in PR #3147.

For B = D = Z/13, join D_j to B_j, B_(j+1), and B_(j+3) modulo 13. This is a 3-regular C4-free incidence graph. Its distance-2 graph has 39 triangles, of which 13 are incidence claws and 26 correspond to six-cycles. The audited formula gives 52 triangles and, after subtracting claws, a value of 39 for the six-cycle lower bound. The complement has 13 triangles; the exact identity including that term gives 39 and 26 respectively.

Scope and limits

This graph exceeds the degree-3 braid-cycle capacity of four six-cycles, so it is not a feasible braid-labelled graph and does not refute the block-floor conclusion. It shows that this proxy is not a valid general lower bound on the stated C4-free incidence class, so it cannot be extended to |B| = 13 without a repaired argument. This example does not by itself disprove the reported 7 <= |B| <= 12 result. No labelled m = 13 CSP is solved and no new κ bound is claimed.

Verification

• python3.12 scripts/research_package.py --package research/centre-braid-triangle-audit — passed on Python 3.12.15.
• The registered verifier exactly checks all incidence, C4, triangle, claw, six-cycle, wedge, and complement-triangle counts; the package uses only the Python standard library.