Holland's $\Lambda$ (Hayman-Lingham 4.26), round 3: dilation-tensor supermultiplicativity $M_{m+kn}\ge M_mM_n$ (proved, exact), a Fejer multi-scale constraint (proved), extremal zeros equidistribute (measure-variational route degenerate), and the zero-partition route to existence quantified

Float-validity audit: the exact-Fourier-convolution evaluator used across rounds 2-3 for M-values suffers catastrophic cancellation for n >~ 35, producing values that violate the PROVEN bounds M_n <= n+1 and |uhat(k)| <= 1 (e.g. 'M_40' = 67.5 > 41 on a near-uniform config, demonstrated in verify3.py part 4). All affected round-3 raw outputs are quarantined (partition2 rows N>=40; profile.py coefficient data at N=120; round3.py arc-family entries n>=40). Round-2 caveat: superadditivity-defect entries (30,30) and (60,60) used the same evaluator and should be re-verified in high-precision arithmetic; they are smoothly consistent with the exact-integer certified V_50, V_100, V_240 (immune - pure integer arithmetic), so corruption is unlikely but unverified.

Leave-one-out root deletion cannot prove monotonicity of M_n/n (which itself holds numerically on all n<=24, as does monotone increase of (M_n-1)/n): for every 4<=n<=23, even the best single-root deletion from the degree-(n+1) extremal gives a K_n element with energy strictly below (n/(n+1))M_{n+1} (undershoot growing to ~8% by n=23; average deletion ~30% low). Extremizers at consecutive degrees are not root-nested. Documented dead end.

Frozen-band additive splicing costs Theta(n), not O(1): defining X(m,n) = max{<v^2> : v in K_{m+n}, vhat(k) = uhat_m*(k) for |k|<=m} (freeze the low band to the degree-m extremal; any additive-splicing proof of approximate superadditivity would need X ~ M_m + a_n - O(1)), penalty-solver feasible points give X - M_m growing at only ~0.30 per degree of n (m=2,3; n up to 16) versus the free rate a_n/n ~ 0.66. Rigorous easy bounds: X <= M_{m+n}, and X >= (3/2)M_m for n >= 2m+1 via v = u_m*(1+cos Lt); pointwise-bounded modulations u_m*T (|T|<=1) cap at 2*M_m. Solver values are one-sided (lower bounds on X), so this disfavors rather than closes the route.

THEOREM (dilation-tensor supermultiplicativity): for any u_m in K_m, u_n in K_n and integer k>=2m+1, w(t)=u_m(t)u_n(kt) is in K_{m+kn} and <w^2> = <u_m^2><u_n^2> exactly; hence M_{m+kn} >= M_m M_n for all k>=2m+1, e.g. M_{(2m+1)n+m} >= M_m M_n. The condition is sharp: at k=2m the identity fails. Proof: (a,b)->a+kb is injective on |a|<=m,|b|<=n when k>=2m+1, so each Fourier coefficient of w is a single product term; Parseval.

LEMMA (Fejer multi-scale constraint): for every u in K_N (any N) and every m>=1, sum_{|k|<=m} (1-|k|/(m+1))^2 |uhat(k)|^2 <= M_m, since the Fejer mean sigma_m u = F_m * u lies in K_m. This gives infinitely many rigorous cross-scale constraints on extremal coefficient profiles, but is one-way: no smoothing/truncation argument can upper-bound M_N by smaller scales, because convolution can only discard (never relocate) the high-frequency energy sum_{|k|>m}|uhat(k)|^2, and modulation destroys nonnegativity. The truncation route to subadditivity is closed.

The dilation-tensor route cannot prove existence of Lambda: the construction multiplies energy but multiplies degree by >=2m+1, so its iteration slope obeys x -> x^2/2-type maps whose fixed point 2 exceeds every M_n/n <= (n+1)/n, and lacunary/Riesz-product iterations give only sublinear growth N^(log M_m / log(2m+1)). Its only unconditional consequence for the limit is M_m <= 2m+1, strictly weaker than the known M_m <= m+1. Documented dead end.

The extremal zero configurations equidistribute: wrapped-angle std approaches pi/sqrt(3)=1.8138 (uniform value; measured 1.834/1.808/1.820 at n=32/64/128), arc-occupation fractions match the uniform measure, and nearest-neighbor gaps approach 2pi/n (normalized min gap 0.96-0.99; at n=120 normalized gaps lie in [0.94,2.19] with std 0.11). Since the exact lattice configuration has energy exactly 3/2, the Theta(n) extremal energy lives in O(1/n)-scale fluctuations invisible to the weak limit of the zero measure: the fixed-measure variational route (rounds 1-2 next-direction, this round's route 2) is degenerate as stated, and the correct limit object is a translation-invariant unit-intensity point process on R. Supporting heuristic (not a theorem): Laplace asymptotics give only O(sqrt n) energy for any smooth non-uniform limiting density. Angle statistics only (no exact-convolution evaluation); numerical, optimizer-based.

Zero-partition (Fejer-Riesz factor split) route, the surviving route to existence: partitioning the N circle-roots of the degree-N extremal as q = q_m q_n and normalizing, the deficit D(N) = M_N - max_partition[<v_m^2>+<v_n^2>] is NEGATIVE up to N~16 (best split carries more energy than M_N: D(6)=-0.59, D(12)=-0.20, D(16)=-0.06), crosses zero at N~18, then D(24)=+0.19, D(32)=+0.90 (balanced splits; interleaved/arc/random seeds + swap hill-climb). If some explicit partition rule achieves D(N)=O(log N), then M_{m+n} <= M_m+M_n+O(log) and existence of Lambda = lim M_n/n follows from the de Bruijn-Erdos approximate-Fekete lemma. The observed growth (increments +0.25 for 16->24, +0.72 for 24->32) cannot yet separate O(log N) from Theta(N): the partition search is a heuristic whose quality degrades with N (measured D is only an upper estimate of the true deficit), and N=32 sits at the float-validity edge. Unresolved; the sharpest computational question this round leaves.