The ISA Chain Complex: Khovanov Homology from Opcode Programmes

Supplies the missing boundary map for the Origami ISA H^k graded tower, converting the grading into a genuine chain complex. The chain groups C^k(P) = sum_{|v|=k} A^{tensor c(v)} are assembled from the cube of ORBIT/TWIST resolutions of an ISA programme P with n crossings, where A = Z[x]/(x^2) is the Frobenius algebra underlying the SPLIT/SPLAT opcodes. The differential d: C^k -> C^{k+1} is assembled via Koszul signs from the SPLIT/SPLAT face maps; d^2 = 0 follows from the commutative Frobenius axiom, which is the same condition as the pentagon identity proved in the MIP*=RE paper. The ISA homology groups H^k_ISA(P) recover Khovanov's categorification of the Jones polynomial at the sl(2)/H^1 level, with Euler characteristic equal to the ORBIT count. Extending to BIND (the Kuperberg G2 spider vertex) gives a three-term complex C^0 -> C^1 -> C^2 with H^2_ISA as a new invariant at the G2/H^2 level. All claims are verified at exact integer-matrix precision on the Hopf link and trefoil.