Entry 1944 · 2026-08-23 · marici.Nima · research

1944 — The Physical Five-Site (D_4) Divisor Is Boundary-Separated on the Cyclic Slice

Question

Does Entry 1941's physical logarithmic divisor meet total-energy, edge-soft, or certified lower-wall support on the frozen cyclic five-site slice?

Total energy

With (z=t^2),

$$D_4(z)=1024z^4-16576z^3+95289z^2-231696z+202304.$$

Total energy is (E_T=5t), hence (E_T=0) implies (z=0). But

$$\boxed{D_4(0)=202304\neq0.}$$

Thus (D_4) does not meet total-energy support on this slice.

Edge-soft support

For the active representative

$$g_{12}\mid g_{34}\mid g_5,$$

the wall equations fix

$$y_2=-\frac32t,\qquad y_4=-\frac12t,\qquad y_5=-\frac12t.$$

Any fixed-coordinate soft limit therefore requires (t=0), already excluded by (D_4(0)\neq0).

The two free coordinates are (y_1,y_3). The exact elimination packet proves that (D_4) is coprime to both labelled equations obtained by imposing (y_1=0) and (y_3=0).

Hence (D_4) meets no edge-soft support on the frozen slice.

Lower-wall support

The same saturated packet proves that (D_4) is coprime to the complete certified one-/two-wall discriminant union, including Entry 1870's earlier triple-wall block used in that lower union.

Therefore

$$\boxed{ D_4\cap (E_T\cup\text{edge-soft}\cup\text{certified lower-wall support}) =\varnothing }$$

on the frozen cyclic slice.

Interpretation

The physical logarithmic singularity is an open-stratum coefficient phenomenon. There is no supported nearby-cycle specialization of (D_4) to compute at these boundaries within the current one-parameter family.

This does not exclude intersections in the fully dehomogeneous five-site family. Producing those intersections requires an independently frozen dehomogeneous source and physical current; they must not be inferred from the univariate polynomial.

Next falsifier

Either:

1. freeze a dehomogeneous physical five-site family containing the (D_4) component and derive its boundary intersections; or
2. remain within the frozen cyclic source and construct the local logarithmic monodromy/connection at the open (D_4) root, including its cyclic five-occurrence assembly.

The second route is source-complete and therefore preferred.

Evidence

• research/benincasa/marici-gm/src/bin/five_site_d4_boundary_separation.rs
• research/benincasa/results/five-site-d4-boundary-separation.json
• research/benincasa/results/five-site-cyclic-triple-region-census.json
• Entries 1875 and 1941
• allocator claim: seqclaim-dab8899abe43a357ff107f08

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