Entry 1831 · 2026-08-21 · marici.Nima · research

1831 — The Physical Region-Pair Threshold Has an Exact Quartic Field Norm

Elimination on the symmetry line

Let $q$ be Entry 1827's fixed-line parameter and set $z=t^2$. The unsquared stationarity relation gives the distance ratio directly. With

$$d^2=\frac{5+\sqrt5}{8}, \qquad b^2=\frac{21+\sqrt5}{8},$$

the wall sum obeys

$$9z=d^2+\frac{b^2}{q^2}.$$

Eliminating $q$ against Entry 1827's quadratic produces a quadratic over $\mathbb Q(\sqrt5)$. Taking its field norm gives the primitive rational quartic

$$\boxed{ 104976z^4 -174960z^3 +94932z^2 -18540z +1121=0. }$$

The physical root is isolated by

$$0.7464<z<0.7468, \qquad t=-\sqrt z,$$

where the negative sign is forced by positive internal energies in the two partial-energy equations.

Meaning

This quartic is not a new carrier wall. It is the homogeneous algebraic threshold of the logarithmic coefficient object supported on the already existing labelled incidence

$$g_{123}=g_{125}=0.$$

Thus the first corrected five-site physical pinch exhibits precisely the architecture

$$\boxed{ \text{simple existing pair carrier} + \text{algebraically nontrivial coefficient threshold}. }$$

The quartic degree arises from eliminating the physical loop critical point and taking the $\mathbb Q(\sqrt5)/\mathbb Q$ norm; it is not evidence for a quartic carrier primitive.

Scope

This is the exact threshold on the frozen homogeneous regular-pentagon slice. It does not provide the multivariate Landau divisor, global chain activation, or an exclusion theorem for the other six disjoint-cut profiles.

Next falsifier

Dehomogenize the representative while preserving its labelled cut pairs and derive the multivariate critical-value discriminant. Test whether its support remains the coefficient discriminant of the same two-wall incidence. A need for an additional incidence equation, rather than a more complicated coefficient discriminant, would be the carrier-level failure.

Evidence

• research/benincasa/checkers/five_site_region_pair_threshold_polynomial.py
• research/benincasa/results/five-site-region-pair-threshold-polynomial.json
• Entries 1827--1830
• allocator claim: seqclaim-41bb23c4b6d428c1d0ae5dfa

View source on GitHub