OPTIMAL AFFINITY: EXACT EXAMPLES AND SAVED FIGURES
A_w = sum w_i ln(j_i_plus/j_i_minus); it is not multiplied by N a second time.
Recycling m=2, r=3/2: exp(A_w*)=7/4; gross ratio=2; J''(log x)=-33/10; exact checks PASS
Default: k+=[3, 7], k-=[2, 6], f=[1, 3], h=[3/2, 7/2].
Default: P=[[0, 1], [4/7, 3/7]], stationary weights=[4/11, 7/11], curvature weights=[4/5, 1/5].
A*=ln(7/4)=0.5596157879 < ln(2)=0.6931471806; detailed balance at (7/4,21/8).
Two-reaction global certificate: polynomial identity (3.1) PASS.
For J>1, both terms on its right are nonnegative and the second is positive.
For J=1, its only positive steady state is x=y=1. This proves the global maximum.
Recycling: general identities (4.1)-(4.2) PASS using the coefficient-sum induction;
literal sums for m=2..8 with symbolic r,y,J also PASS. Positive coefficients prove J<=1.
Capacity: L=1+1/m, approached as r tends to 1 from above; never attained at finite interior r.
Three-reaction global certificate (4.3) PASS; y>0 gives J<6u, making the contradiction strict.
Power family: literal source elimination and positive global gap PASS for d=2..8.
Recycling m=100, r=101/100: exp(A_w*)=10199/10000; gross ratio=2; J''(log x)=-102000001/29900; exact checks PASS
Three-reaction circuit: exp(A_w*)=9/5; gross ratio=2; J''(log x)=-16/5; exact checks PASS
Power d=2: exp(A_w*)=4; gross ratio=4/3; J''(log x)=-1/2; exact checks PASS
Power d=3: exp(A_w*)=9/4; gross ratio=6/5; J''(log x)=-3/2; exact checks PASS
Power d=4: exp(A_w*)=16/9; gross ratio=6/5; J''(log x)=-3; exact checks PASS
Power d=8: exp(A_w*)=64/49; gross ratio=10/9; J''(log x)=-14; exact checks PASS
Additional interior-minimum example: exp(A_w*)=sqrt(2) + 3/2; gross ratio=3; J''(log x)=-16/7 - 4*sqrt(2)/7; exact checks PASS
Interior example: exact minimum exp(A_w)=3/2+sqrt(2), t=2*sqrt(2), N=2; balancing PASS.
Interacting cores: both supplied witnesses and contradiction identities PASS.
In 8.3, forward directions force 0<a<1, but joint productivity would force a>1.
Numerical branch check: J(-0.001)=0.9999983520, J(+0.001)=0.9999983480;
finite-difference J''=-3.30000238; exact J''=-33/10; absolute error=2.38e-06.
Maximum residual in the plotted steady states: 1.78e-14.
All requested script checks PASS. Plots are numerical illustrations; global claims use the exact positive-gap arguments.