Does the Ginzburg-Landau (GL) gradient-flow dynamics on the level-163 Hecke lattice reproduce the eikonal survivor hierarchy? The eikonal analysis (Paper IV) yields an ordering of the seven Heegner gates by conservative backreaction dynamics. Can the same ordering be recovered from a simpler dissipative energy-minimization picture?
The quadratic and quartic coefficients come directly from the Hecke data. For each gate d in {3, 7, 11, 19, 43, 67, 163}:
The temperature-dependent quadratic term is
αd(T) = (G(d,d) / max G) · (T − Tc(d)) / Tscale
where Tscale = maxd Tc(d) for each prescription. This is a normalization choice on the Landau functional, not a new coupling.
Three prescriptions for the critical temperature Tc(d) were tested. All three produce the same low-T ordering:
| Tc prescription | Critical-T correlation | Low-T correlation | Low-T order |
|---|---|---|---|
| self_energy | 0.092 | 0.214 | [67, 3, 7, 11, 19, 43, 163] |
| mean_distance | −0.168 | 0.214 | [67, 3, 7, 11, 19, 43, 163] |
| nearest_gram_eigenvalue | 0.000 | 0.214 | [67, 3, 7, 11, 19, 43, 163] |
For comparison, the eikonal survivor hierarchy from Paper IV is:
Eikonal: [7, 11, 43, 163, 67, 19, 3]
The GL low-T ordering places 67 first and 163 last. The eikonal ordering places 7 first and 3 last. The Spearman rank correlation between these two orderings is 0.214 — essentially uncorrelated.
A time-dependent GL quench was run using the self_energy prescription (which gave the best static agreement). The explicit Euler dynamics evolve the order parameter fields φd(t) from random initial conditions through a temperature ramp T(t) → 0.
| Gate | 3 | 7 | 11 | 19 | 43 | 67 | 163 |
|---|---|---|---|---|---|---|---|
| Basin count | 0 | 0 | 0 | 0 | 3765 | 29003 | 0 |
Final mean dominant-basin order: [67, 43, 3, 7, 11, 19, 163]. Spearman correlation to the eikonal hierarchy: 0.071.
Only two gates (67 and 43) attract nonzero basin volume under the GL quench. The heavy-sector gate 67 dominates with 89% of the basin count. This is the opposite of the eikonal picture, where the light gates (7, 11) survive longest.
The GL functional minimizes a free energy F = Σd αdφd2 + quartic coupling. At low T, the gate with the largest self-energy G(d,d) has the most negative αd, so it condenses first. This is gate 67 (G(67,67) = 5 is smallest, but its Tc is set by the self-energy prescription to be largest). The dissipative dynamics then locks into the 67-basin by gradient flow.
The eikonal dynamics is conservative: it tracks wave-packet propagation through the Hecke graph, where light gates with small self-energy can scatter and survive. The GL dynamics is dissipative: it seeks the deepest energy well, which is the heaviest sector. These two selection principles are fundamentally different.