[PASS] 15 pair-square identities in Q(t)
[PASS] 3 regular quadruple identities in Q(t)
[PASS] 2 regular quintuple identities in Q(t)
[PASS] regularity graph selects exactly six classes
[PASS] universal R-section identity
[PASS] universal extension-section identity
[PASS] ace: generic cubic nonsingular
[PASS] ace: exact t=12/5 points match
[PASS] acf: generic cubic nonsingular
[PASS] acf: exact t=12/5 points match
[PASS] bde: generic cubic nonsingular
[PASS] bde: exact t=12/5 points match
[PASS] bdf: generic cubic nonsingular
[PASS] bdf: exact t=12/5 points match
[PASS] bef: generic cubic nonsingular
[PASS] bef: exact t=12/5 points match
[PASS] cef: generic cubic nonsingular
[PASS] cef: exact t=12/5 points match
[PASS] ace: canonical-height Gram enclosure positive definite
[PASS] acf: canonical-height Gram enclosure positive definite
[PASS] bde: canonical-height Gram enclosure positive definite
[PASS] bdf: canonical-height Gram enclosure positive definite
[PASS] bef: canonical-height Gram enclosure positive definite
[PASS] cef: canonical-height Gram enclosure positive definite
[PASS] certificate names the six selected classes
[PASS] certificate gives rank lower bound 4 for all six
[PASS] claim boundaries preserved

27/27 checks passed.

CONCLUSION: ace, acf, bde, bdf, bef, cef each carry four certified independent rational sections over Q(t); therefore each induced family has generic Mordell-Weil rank at least 4.
