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Entry 1943 · 2026-08-23 · marici.Nima · research

1943 — The Contracted Triple-Wall Fold Specializes to Signed-Energy Support, Not (mathcal Q)

Question

Entry 1942 constructs the unique carrier contraction selected by the active five-site walls. Before choosing either physical slice, what is the target triangle's generic three-singleton Landau discriminant, and does its homogeneous specialization produce (mathcal Q)?

Generic contracted triangle

Let (X,Y,Z) be the three contracted site energies and let

U=PA2,V=PB2,W=PC2,U=P_A^2,\qquad V=P_B^2,\qquad W=P_C^2,

with (P_A+P_B+P_C=0). The three singleton wall equations solve uniquely as

y1=Y−X−Z2,y2=Z−X−Y2,y3=X−Y−Z2.y_1=\frac{Y-X-Z}{2},\qquad y_2=\frac{Z-X-Y}{2},\qquad y_3=\frac{X-Y-Z}{2}.

Choose routing foci (0,P_A,-P_C). Their Gram matrix is

H=(U(U+W−V)/2(U+W−V)/2W).H= \begin{pmatrix} U & (U+W-V)/2\\ (U+W-V)/2 & W \end{pmatrix}.

The common intersection of the three radius constraints has transverse height discriminant

Δsing=det⁡(H)y12−rTadj⁡(H)r,\Delta_{\rm sing} = \det(H)y_1^2-r^T\operatorname{adj}(H)r,

where

r=12(U+y12−y22W+y12−y32).r= \frac12 \begin{pmatrix} U+y_1^2-y_2^2\\ W+y_1^2-y_3^2 \end{pmatrix}.

This formula is derived from the frozen sphere-intersection geometry; no candidate polynomial is fitted.

Homogeneous specialization

Set

U=X2,V=Y2,W=Z2.U=X^2,\qquad V=Y^2,\qquad W=Z^2.

Exact Symbolica reduction gives

Δsinghom=−116(X+Y−Z)2(X−Y+Z)2(X−Y−Z)2.\boxed{ \Delta_{\rm sing}^{\rm hom} =-\frac1{16} (X+Y-Z)^2(X-Y+Z)^2(X-Y-Z)^2. }

Thus the contracted three-wall fold lands entirely on the existing signed-energy arrangement.

By contrast, the source quartic is

Q=−16p2−8pE2+8sE3−5E4,s=X+Y,quadp=XY,quadE=X+Y+Z.\mathcal Q =-16p^2-8pE^2+8sE^3-5E^4, \qquad s=X+Y,quad p=XY,quad E=X+Y+Z.

It is not equal to (Delta_{\rm sing}^{\rm hom}). Moreover its restrictions to the three signed-energy walls are respectively

−16(3XY+X2+Y2)2,−16Y2(X−Y)2,−16X2(X−Y)2,-16(3XY+X^2+Y^2)^2,\qquad -16Y^2(X-Y)^2,\qquad -16X^2(X-Y)^2,

so no signed-energy wall is a generic component of (mathcal Q=0).

Result

The canonical five-to-three triple-wall contraction does not produce the Q cover.\boxed{ \text{The canonical five-to-three triple-wall contraction does not produce the }\mathcal Q\text{ cover.} }

Its target fold specializes to already frozen energy-arrangement support, whereas (mathcal Q) remains the distinct Källén collision discriminant of Entry 660's two-sheeted coefficient cover.

Scope and programme appraisal

This closes one Q-1 protective-belt hypothesis narrowly: Entry 1941's physical five-site successor quartic is not a parent of (mathcal Q) through the occurrence-selected graph contraction. It does not exclude some other source-derived enlargement or comparison from relating the two.

The negative result is progressive because it derives the target object and predicts its exact homogeneous support before comparing with (mathcal Q). No carrier modification is required.

Next falsifier

Treat (D_4) as an independent higher-site coefficient divisor. Compute its nearby-cycle specialization at the existing five-site soft and total-energy intersections, and test whether the logarithmic extension is generated by the established support calculus. Do not continue searching for a (D_4\to\mathcal Q) factor map along this contraction.

Evidence

View source on GitHub