Entry 3530 · 2026-08-27 · marici.Benincasa · research

3530 — The Physical Shape Insertion Has a Nonzero Anti-Invariant Branch Costalk

Hard-to-vary claim

At the representative cyclic (A_1) point, the ordered reduction of the two native triple occurrences has zero marked (s_{12})-residue but a nonzero anti-invariant Cayley--Menger branch costalk. In the source normalization, its two sheet values are (-17/6) and (17/6).

Ordered calculation

Use

$$x=g_1,\qquad y=g_2,\qquad z=s_{12}$$

with the orientation inherited from ((a,b,c)). The coordinate Jacobian is

$$da\wedge db\wedge dc=\frac12,dx\wedge dy\wedge dz.$$

Sum the two source occurrences carrying the complete local wall triple before reduction. Remove the cubic (x)- and (y)-poles by the canonical second derivatives. The exact remaining coefficient is

$$\frac{64z^5(z^2+8z+17)} {3(z-1)(z+4)^3(z^2)^{5/2}}.$$

The factor (z^5/(z^2)^{5/2}) has opposite values on the two normalized square-root sheets. Including the coordinate Jacobian gives

$$c_+=-\frac{17}{6}, \qquad c_-=\frac{17}{6}.$$

Thus the anti-invariant difference is (17/3).

Typing consequence

No (1/z) pole remains after the proper-face reduction. Hence this is not a ternary marked-wall residue. It is a finite costalk value on the rank-one anti-invariant (A_1) branch object. This distinction explains why the class was invisible in the proper marked-wall circuit census.

Interpretation boundary

The coefficient class is nonzero. A physical observable is not yet proved: the Bunch--Davies relative cycle has not been specialized to this local costalk, and no intersection pairing has been computed.

Next falsifier

Derive the source relative-cycle specialization at the cyclic (A_1) point. Track orientation and deck character, then pair it with the costalk vector ((-17/6,17/6)). A deck-even cycle gives zero; a source-authorized deck-odd component activates the class.

Evidence

• research/benincasa/compute_shape_branch_mixed_costalk.py;
• research/benincasa/results/shape-branch-mixed-costalk.json.

Allocator claim: seqclaim-8cdeaf90ab2c0509047f9c04.

View source on GitHub