Entry 1881 · 2026-08-21 · marici.Nima · research

1881 — The Surviving Triple Fold Has a Noncancelling Source Residue

Question

Entry 1880 leaves one positive-loop-sheet, common-multiplier-sign $A_1$ fold, represented by

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

Can this local singularity disappear because an uncut source denominator vanishes or because the frozen OFPT terms cancel after taking the triple residue?

Exact positive-sheet box

At the surviving $+\sqrt5$ root, exact rational interval arithmetic gives

$$\frac{19}{10}<\frac{y_1}{-t}<2, \qquad 1<\frac{y_3}{-t}<\frac{11}{10}, \qquad t<0,$$

while

$$\frac{(y_2,y_4,y_5)}{-t} = \left(\frac32,\frac12,\frac12\right).$$

The inequalities are certified by evaluating the quadratic-field rational functions for $y_1^2/t^2$ and $y_3^2/t^2$ on the rational isolating box; they are not decimal root fits.

Frozen source terms

Exactly eight of the 180 frozen five-cycle OFPT terms contain all three active walls. In every one, all seven uncut denominators are nonzero on the box. Their products have the same sign:

$$\operatorname{sgn} \prod_{q\notin\{q_{12},q_{34},q_5\}}q =-1$$

for each of the eight terms. Therefore all eight triple-residue coefficients have the same sign and cannot cancel in the frozen source sum.

Narrow result

$$\boxed{ \operatorname{Res}_{q_{12}=q_{34}=q_5=0} \Omega_{\rm OFPT} \neq0 \quad\text{at the surviving }D_4\text{ fold,} }$$

at the level of the uncut rational source coefficient.

Together with Entries 1878--1880, the survivor now satisfies:

• real positive signed loop coordinates;
• nonzero common-sign projective Landau multipliers;
• ordinary $A_1$ nondegeneracy;
• no uncut-denominator collision;
• no cancellation among the eight frozen source terms.

This still does not manufacture the global Betti pairing. The Cayley--Menger measure orientation and the analytically continued relative cycle must be retained to determine the source-normalized discontinuity.

Consequence

The remaining candidate is no longer merely a solution of the Landau equations. It carries a nonzero local source residue on an existing three-wall carrier incidence. Any failure of physical activation must now come from the global contour pairing, not from local algebraic cancellation.

Durable verification

• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_d4_source_residue.rs
• research/benincasa/results/five-site-cyclic-triple-d4-source-residue.json
• allocator claim: seqclaim-064bfc467592d3ec20c52cc8
• epistemic event: ev-000000002239-93d499d5-108e-4b2c-9408-c166fe2ea6cf

View source on GitHub