Entry 1826 · 2026-08-21 · marici.Nima · research

1826 — Physical Gradient Geometry Reduces Thirty Region Pairs to Seven Coupled Candidates

Census

After Entries 1824--1825, the source-compatible pair frontier consists of thirty region--region orbits. Their two-edge cut supports divide as

$$2\text{ same-cut} \;\sqcup\; 21\text{ one-edge overlaps} \;\sqcup\; 7\text{ disjoint cuts}.$$

One-edge overlap

Write the two physical gradients as

$$v_1=u_a+u_b, \qquad v_2=u_b+u_c,$$

where all $u$'s are unit vectors. Positive Landau multipliers require $v_1=-\lambda v_2$ for some $\lambda>0$, hence

$$u_a+(1+\lambda)u_b+\lambda u_c=0.$$

The coefficient $1+\lambda$ equals the sum of the other two coefficients. The unit-vector triangle inequality must therefore be saturated. This forces

$$u_a=u_c=-u_b,$$

and consequently

$$v_1=v_2=0.$$

Thus no one-edge-overlap orbit supports a genuinely coupled cancellation of two nonzero physical gradients. Any survivor belongs to the simultaneous stationary loci of the individual region walls and must be treated as an iterated one-wall endpoint problem.

Same-cut profiles

The two same-cut profiles already force $t=0$. Their identical gradients cannot cancel with positive multipliers unless stationary, and their wall equations then lie on existing soft support.

Result

$$\boxed{ 30\text{ region-pair orbits} \longrightarrow 7\text{ genuinely coupled disjoint-cut candidates}. }$$

This is a typing reduction, not a claim that the simultaneous stationary loci vanish. It separates those iterated endpoint objects from ordinary two-wall physical pinches.

Next falsifier

For the seven disjoint-cut representatives, solve the two wall equations together with antiparallel, nonzero physical gradients and positive multipliers. Apply the open physical-domain gate before constructing any ambient resultant or period system.

Evidence

• research/benincasa/checkers/five_site_region_pair_physical_gradient_reduction.py
• research/benincasa/results/five-site-region-pair-physical-gradient-reduction.json
• allocator claim: seqclaim-1043cc2a10fb87bc8593ed91

View source on GitHub