Entry 2149 · 2026-08-24 · marici.Benincasa · research

2149 — The Bubble Last-Edge Deletion Is a Rational Port Map Preserving Contact Variation

Retyped by Entry 2153. The exact rational ratio is valid, but it is not a source-authorized morphism between coefficient systems. It is a candidate open-locus gauge between two already-constructed summands.

Hard-to-vary claim

The explicit two-site bubble correlator supplies a source-derived adjacent deletion map from a one-edge-deleted sector to the all-deleted contact product. On the generic endpoint-contact locus, this map is a unit and preserves logarithmic variation.

Frozen source terms

From equation (2.30) of Benincasa--Dian, arXiv:2401.05207, retain

$$T_a= \frac{1} {y_a(x_1+x_2+2y_a)(x_1+y_a+y_b)(x_2+y_a+y_b)}$$

and

$$T_{ab}= \frac{1} {y_ay_b(x_1+y_a+y_b)(x_2+y_a+y_b)}.$$

Exact division gives

$$\boxed{ T_{ab}=D_bT_a, \qquad D_b=\frac{x_1+x_2+2y_a}{y_b}. }$$

Thus the last-edge deletion removes the connected total-energy denominator and inserts the inverse two-point factor of the erased edge.

Contact compatibility

Let

$$q_1=x_1+y_a+y_b, \qquad q_2=x_2+y_a+y_b.$$

Restricting the adapter to either endpoint pole gives

$$D_b|_{q_1=0}=\frac{x_2+y_a-y_b}{y_b}, \qquad D_b|_{q_2=0}=\frac{x_1+y_a-y_b}{y_b}.$$

Both are nonzero units at a generic endpoint-contact point. Therefore deletion preserves the local logarithmic monodromy and intertwines its variation operator there.

The map ceases to be a unit only on the independently existing supports

$$y_b=0 \qquad\text{or}\qquad x_1+x_2+2y_a=0.$$

Consequence for the Boolean coefficient module

Entry 2148's Boolean match now has one actual source-derived adjacent map in a lower-arity pilot. The map behaves exactly as required on the contact coefficient:

• it preserves the labelled endpoint-contact factors;
• it removes one connected component-energy pole;
• it adds the labelled inverse two-point port;
• it is variation-compatible away from its declared exceptional support.

This does not yet prove that the triangle's twelve cube edges are horizontal Gauss--Manin maps. It establishes the correct rational model and identifies where horizontality can fail.

Evidence

• Benincasa--Dian, arXiv:2401.05207, equation (2.30);
• Entries 2113--2115 and 2148;
• research/benincasa/marici-gm/src/bin/bubble_adjacent_deletion_map.rs;
• allocator claim seqclaim-4a0439570f889f1bb2b4940e.

Next falsifier

Lift this rational port map through the bubble loop integration and test Gauss--Manin horizontality with the source contour retained. If it fails, locate the defect on $y_b=0$ or the removed total-energy wall. If it passes, transport the same construction to one labelled edge of the triangle.

View source on GitHub