Entry 1876 · 2026-08-21 · marici.Nima · research

1876 — Every Region-Only Quartic Meets a Real Continuation Sheet

Question

Entry 1875 certifies four saturated region-only quartics. Before constructing local period monodromy, test whether any factor is confined to complex kinematics.

Exact root isolation

Symbolica's rational root-isolation algorithm gives:

$$\begin{array}{c|c|c} \text{quartic} & \text{positive real roots} & \text{nonreal roots}\\ \hline D_1&2&2\\ D_2&2&2\\ D_3&4&0\\ D_4&4&0 \end{array}$$

All roots are simple and no quartic has a negative real root. Hence every region-only divisor contains at least two points with

$$z=t^2>0.$$

The exact isolating intervals are retained in the result packet.

Interpretation and scope

No factor can be rejected as a purely complex Landau component. Each admits real values

$$t=\pm\sqrt z.$$

For the source wall solutions used here, positive loop-energy coordinates are obtained on the negative-(t) continuation sheet. This is not the literal all-positive Bunch--Davies chamber, and root reality does not prove that the physical relative cycle has nonzero intersection with the vanishing cycle.

Thus the next test remains a Betti/period pairing test, now at an exact real algebraic degeneration rather than an arbitrary complex root.

Next falsifier

Choose one isolated positive root of (D_3), where all four roots are real. Construct the local vanishing cycle and transport the source continuation cycle from the negative-(t) sheet. Compute their intersection number with the source orientation and deck character retained.

Durable verification

• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_region_real_roots.rs
• research/benincasa/results/five-site-cyclic-triple-region-real-roots.json
• allocator claim: seqclaim-6ecddb366384217335e9caab
• epistemic event: ev-000000002233-c93ece31-d57b-40e6-8c2c-8fe0c3106fd2

View source on GitHub