Appendix A. Compact Mathematics

H1(phi) = [F, Ad, U, R] at 100% each

I1(phi) = [UF, UAd, UU, UR] at 100% each

H_n = typed_union(+H_1...+H_(n-1), -I_1...-I_(n-1))

I_n = typed_union(+I_1...+I_(n-1), -H_1...-H_(n-1))

DirectWidth(H_n)=2(n-1)

S_n=3^(n-1) for the symmetric recursive state-call tree

Node(phi,h,a,s,G,B)

d_X(T,t)=1 means task T requires monitored X at time t

x_k = r_k * product(1-x_j, j<k); for MRD=k>=2 this forces r1...r_(k-1)=0 and r_k=1

MRD(X,t)=min{k>=1:x_k=1} when x0=0

PositiveActivationFailure requires d_X=1 and x0=0; MRD separately measures the anti-monitor recovery depth

Structural family contains every h; Realize(phi,h,T,t)=1 means the task-relevant H_h lobe is recruited, active, and instantiated at time t

GF(T)=(G_H, CoverageGap, MRD_profile, CrossfunctionGap, FactorDemandVector); scalar V(T) requires later weighting rules

Typed recursive basis: B_phi={tau_H,1,tau_I,1,tau_H,2,tau_I,2,...}

State equality: X=Y iff primitive components match and q_X(b)=q_Y(b) for every b in B_phi

n>m => q_Hn(tau_H,m)=+1 != 0=q_Hm(tau_H,m) => H_n != H_m

D_phi(T,t)=task-required H-orders; R_phi(T,t)=recruited H-orders; X_phi(T,t)=active H-orders

G_H(phi,T,t)=D_phi(T,t) minus X_phi(T,t)

CoverageGap(phi,T)=Omega_phi,T minus C_phi,T

Gated anti-signal g_k=d_X*x_k; raw x_k without d_X is not automatically task-diagnostic