Holland's $\Lambda$ (Hayman-Lingham 4.26), round 2: a proved rotation-averaging inequality, certified feasible-point lower bounds $M_{50},M_{100},M_{240}$ in exact arithmetic, and the Fekete route to $\Lambda=\lim M_n/n$ (existence still open)

The rotation-averaging inequality is NON-SHARP and does NOT establish superadditivity. Its clean universal form gives only M_{m+n} >= max(M_m,M_n) (trivial, since K_m is a subset of K_{m+n}). With the actual optimizer coefficients, Gamma_{m,n} > AB/(1+A+B) (A=M_m-1, B=M_n-1) for every tested pair except (1,1) where it is equality; hence M_m M_n/(1+Gamma) < 1+A+B = M_m+M_n-1, i.e. the averaged bound falls short of the superadditivity target for all nontrivial pairs.

CERTIFIED feasible-point lower bounds in exact arithmetic: M_50 >= 35.0407732538, M_100 >= 69.3874975384, M_240 >= 165.563491944 (values rounded DOWN from exact rationals). Each V_n = <g^2>/<g>^2 for an explicit reflection-symmetric extreme point g = prod_j (1-cos(theta-theta_j))(1-cos(theta+theta_j)) (times (1-cos theta) if n odd) with rational cosines c_j=cos(theta_j), computed via integer Fourier kernels [Q^2,-4pQ,2Q^2+4p^2,-4pQ,Q^2] per +-pair (no floating point in the bound). Since g/<g> is a bona fide nonnegative mean-1 trig polynomial of degree <=n, M_n >= V_n rigorously. Each V_n is verified as an EXACT rational to exceed 1+2n/3 (so (V_n-1)/n > 2/3). Machinery self-checks: it reproduces M_2=15/7, M_3=2.808840..., M_4=3.483450.... Consistent with the proved upper bound M_n <= n+1.

CONDITIONAL lower bound on Lambda. IF a_n=M_n-1 is superadditive (a_{m+n} >= a_m + a_n; conjectured, strongly supported numerically but NOT proved), then by Fekete's lemma Lambda = lim M_n/n exists and equals sup_n a_n/n, so Lambda >= (M_240 - 1)/240 >= (V_240 - 1)/240 = 0.685681 > 2/3, improving the classical lower bound Lambda >= 2/3. The Fekete implication is rigorous; the hypothesis is not established here. (Unconditionally only Lambda_n = M_n >= V_n holds, for n=50,100,240.)

Asymptotics update (numerical, not proved): (M_n-1)/n increases 0.6808 (n=50) -> 0.6819 (60) -> 0.6839 (100) -> 0.6844 (120) -> 0.6857 (240); a Lambda - c'/n fit gives c' approx 0.31 (matching the defect limit) and Lambda approx 0.687, inside [2/3,1]. If M_n is subadditive (delta<1, strongly supported) then additionally Lambda = inf_n M_n/n <= M_120/120 = 0.69272, improving the classical Lambda <= 1; this upper bound is conditional on the unproved subadditivity.

Rotation-averaging inequality (PROVED). For any nonnegative mean-1 trig polynomials u_m (deg<=m), u_n (deg<=n), and w_phi(theta)=u_m(theta)u_n(theta-phi): w_phi>=0 has degree<=m+n and positive mean, so w_phi/<w_phi> is feasible for M_{m+n}, giving M_{m+n}<w_phi>^2 >= <w_phi^2> for all phi. Integrating over phi and using the exact identities avg_phi<w_phi^2>=<u_m^2><u_n^2> and avg_phi<w_phi>^2 = sum_k |uhat_m(k)|^2|uhat_n(k)|^2 =: 1+Gamma (Parseval in phi) yields M_{m+n}(1+Gamma) >= <u_m^2><u_n^2>; at optimizers M_{m+n} >= M_m M_n/(1+Gamma_{m,n}) with 0 <= Gamma_{m,n} <= min(M_m,M_n)-1 (since |uhat(k)|<=uhat(0)=1 and sum_{k!=0}|uhat_m(k)|^2 = M_m-1).

Superadditivity of a_n=M_n-1 (a_{m+n} >= a_m + a_n) holds on all 23 tested pairs up to n=120, with defect delta=a_{m+n}-a_m-a_n in [0.198, 0.307], increasing monotonically toward ~0.31. Since delta < 1 throughout, M_n is simultaneously subadditive (M_{m+n} <= M_m + M_n). Either one-sided inequality would, by Fekete, prove the limit Lambda = lim M_n/n exists; combined they say M_n is additive up to O(1) with a_n = Lambda*n - c + o(1), c approx 0.31. NEITHER is proved: existence of Lambda remains OPEN. The natural (and essentially only degree-respecting) gluing of two extreme points into a degree-(m+n) extreme point is the product/rotation construction, and its best rotation max_phi <w_phi^2>/<w_phi>^2 undershoots the target 1+A+B by an amount growing ~0.16n for m,n>~4 (not summable), so it yields neither exact nor approximate-Fekete superadditivity.