Turán graphs, C₅ blow-ups, and the finite-order threshold for average quantum transport on triangle-free graphs

T₂(n) is the unique maximizer of F_G(tau/n) over all labelled triangle-free graphs on n = 5,6,7,8 vertices, exactly, at pinned rational tau

The finite-order threshold sits strictly above the graphon threshold: T₂(n) is still the unique maximizer at tau = 19/5 > tau_c = 3.790989 for n = 5,6,7, and at tau = 9/2 for n = 6

Above the finite threshold the maximizer is a C₅ blow-up: explicit witnesses beat T₂(n) at tau = 9/2 (n = 5,7) and tau = 5 (n = 5,6)

Edge-count monotonicity, strictly stronger than the source's Problem: every triangle-free graph on [0,n) below the Mantel bound is strictly beaten, in the exact order-K truncation F^(K) of the transport value, by one with exactly one more edge (n = 5,6,7; tau = 1/2 and 19/5). The sweep pins no margin, so the reading at the level of F itself costs an additive 2 eta <= 7.1e-8

Beyond every exhaustion: at tau = 5 the balanced C₅ blow-up strictly beats T₂(n) for every 5 <= n <= 18, and at tau = 9/2 for n = 5, 7 and every 9 <= n <= 18 – each with a pinned margin, hence at the level of F itself – with n = 6, 8 explicit exceptions; the n = 6 exception carries a margin too, but the n = 8 one is a margin-free comparison of the order-24 truncations F²⁴

Identification of the finite maximizer above the threshold: at (n,tau) = (5,9/2), (5,5), (6,5) and (7,9/2) the maximum over ALL triangle-free graphs on [0,n) of the exact order-K truncation F^(K) of the transport value is attained by a C₅ blow-up. The sweep pins no margin, so at the level of F itself this says only that the blow-up comes within 2 eta <= 1.1e-7 of the maximum

The source's six-vertex flag-algebra inequality 881 - 2400 t(K₂) + 2856 t(C₄) - 1216 t(C₆) >= 0, kernel-checked at every triangle-free graph on at most 7 labelled vertices, with equality at T₂(n) for even n

Mantel's theorem with its equality case, by the same exhaustion, for n <= 7

The number of labelled triangle-free graphs on [0,n): 1, 2, 7, 41, 388, 5789, 133501 for n = 1..7

Instrument controls: coefficient table against its specification, recogniser non-vacuity, constancy of the score on the T₂ isomorphism class, margin not slack by two orders, and refutation of the too-large 'for every tau > 0' claim

Explicit rate for the source's own prop:fixed-n: epsₙ >= 2.97 n^(-5/2), i.e. T₂(n) is the unique maximizer of F_G(tau/n) for all 0 < tau < 2.97 n^(-3/2), where the source proves only that some epsₙ > 0 exists

Cost of the labelled route, and where it dies: n <= 8 landed, n >= 9 needs isomorphism reduction

Closure of the coefficient-table control: coefOK verified at all 51 parameter sets (K,p,q,n) the development evaluates — the 28 sets the 23-set coefₜablesₒk missed (n = 4, the n = 8 exhaustion, and every witness sweep) included — and over the whole box K ≤ 30, p ≤ 40, q ≤ 8, n ≤ 40 (468,999 sets), with the used list proved to lie inside the box