Appendix A.1 Zig-Zag Purification and Emergence Mathematics

Z_n = -I_n

Core(H_(n+1)) contains Core(H_n) plus the new recursively typed coordinate Z_n and the newly inherited +H_n coordinate

Pi_n = PurificationContribution(H_n, Z_n)

Gamma_n = EmergenceContribution(H_n, Z_n)

Expanded(H_(n+1)) = Cl_Phi(Core(H_(n+1)))

A_star,1=A; A_star,n+1 in Gamma_A(H_n,Z_n); A_star,n+1 ~_A A; A_star,n+1 != A_star,j for j<=n

CoreWidth(H_n)=2(n-1) for n>=2; total ExpandedWidth additionally includes emergent dimensions and is not fixed by CoreWidth alone

Canonical generativity: Gamma_n contains at least one new positive-directed emergent dimension; the Adeptness-like lineage supplies an explicit example

The exact number of emergent dimensions at each order is intentionally not forced by the current mathematics. The logic fixes their distinct typed existence and generative role; a more detailed phenotype/output law would be needed to enumerate every emergent property.

Expanded(H_h(phi)) = Core(H_h(phi)) union Emergent(H_h(phi))

E*_phi,h(t) subseteq Emergent(H_h(phi)) = task-time emergent dimensions actually used

HEC(C)=1 iff exists e in ActiveInputs(C): e in E*_phi,h(t), h>=2, and C_without_e != C

HLC(C)=1 iff at least one participating positive lobe has h>=2; HEC subseteq HLC

SameOrderHEC: Cross(L_phi,h, L_psi,h) -> C*

CrossOrderHEC: Cross(L_phi,h1, L_psi,h2) -> C*, h1!=h2

HC_1=Cross(X1,...,Xm); HC_(c+1)=Cross(HC_c,Y1,...,Yr)