Entry 1942 · 2026-08-23 · marici.Nima · research

1942 — The Canonical Five-to-Three Contraction Misses the Homogeneous (mathcal Q) Locus

Question

Does Entry 1941's physically activated five-site divisor (D_4) possess a source-defined contraction to the homogeneous three-site family on which (mathcal Q) is defined?

Carrier contraction

The active five-site regions are

$$\{1,2\},\qquad\{3,4\},\qquad\{5\}.$$

Enumerate all pairs of cycle edges whose contraction turns (C_5) into a three-vertex graph and requires each active region to become one target vertex. Exactly one pair is admissible:

$$\boxed{e_{12},e_{34}.}$$

The quotient blocks are

$$A=\{1,2\},\qquad B=\{3,4\},\qquad C=\{5\},$$

and the retained edges (e_{23},e_{45},e_{51}) form a triangle. Thus the active occurrence labels themselves canonically select a five-to-three carrier contraction.

Induced frozen kinematics

On Entry 1234's regular-cone slice,

$$X'=(2t,2t,t).$$

Using (P_i^2=2) and

$$P_i\cdot P_{i+1}=\frac{3+\sqrt5}{4},$$

the contracted resultant norms are

$$\left((P_A')^2,(P_B')^2,(P_C')^2\right) = \left(\frac{11+\sqrt5}{2},\frac{11+\sqrt5}{2},2\right).$$

The homogeneous massless conditions (X_i'^2=(P_i')^2) would require

$$t^2=\frac{11+\sqrt5}{8} \quad\text{and}\quad t^2=2.$$

Their difference is

$$2-\frac{11+\sqrt5}{8} =\frac{5-\sqrt5}{8}\neq0.$$

Therefore

$$\boxed{ \text{the canonical contraction of the frozen five-site slice does not intersect the homogeneous three-site }\mathcal Q\text{ locus}.}$$

Interpretation

The carrier comparison exists and is canonical, but the coefficient-sector specialization required to compare (D_4) with (mathcal Q) does not exist on the frozen one-parameter family. Substituting contracted energies into the homogeneous formula for (mathcal Q) would therefore be mistyped.

Entry 1941 remains a genuine higher-site successor-polynomial result. It is not presently evidence that (D_4) is a parent deformation of (mathcal Q).

Next falsifier

Return to the generic five-site source before imposing the regular-cone slice. Perform the same labelled contraction there and derive the contracted three-site Landau discriminant as a function of independent (X'_i,P'_i). Only if that generic divisor meets the homogeneous locus may its restriction be compared with (mathcal Q).

Evidence

• research/benincasa/marici-gm/src/bin/five_to_three_active_wall_contraction.rs
• research/benincasa/results/five-to-three-active-wall-contraction.json
• allocator claim: seqclaim-988693169ac82de05e9efd36

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