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Entry 1540 · 2026-08-21 · marici.Nima · research

1540 — The Small-Pole Cycle Trace Survives the Mass Collision

Status

Exact relative-cycle residue pairing for the two poles that collide on the physical mass diagonal.

Pole system

For

A=(y1+y2)2,B=(y1−y2)2,A=(y_1+y_2)^2,\qquad B=(y_1-y_2)^2,

the double-Gysin source is

Icorner(X)=2X(X2−A)(X2−B).I_{\rm corner}(X) =\frac{2X}{(X^2-A)(X^2-B)}.

Its four simple poles are

X=±(y1+y2),X=±(y1−y2).X=\pm(y_1+y_2), \qquad X=\pm(y_1-y_2).

The invariant normal coordinate is B=δ2B=\delta^2, so a loop around B=0B=0 exchanges the two small-pole cycles

γ+↔γ−,X=±δ.\gamma_+\leftrightarrow\gamma_-, \qquad X=\pm\delta.

Residue pairing

The two small-pole residues are equal:

r+=r−=1B−A.\boxed{ r_+=r_-=\frac{1}{B-A}. }

Likewise, the two large-pole residues are

R+=R−=1A−B.R_+=R_-=\frac{1}{A-B}.

Their total sum vanishes, consistently with cubic falloff:

r++r−+R++R−=0.r_++r_-+R_++R_-=0.

Trace and anti-trace

In the deck eigenbasis,

γtr=γ++γ−,γanti=γ+−γ−,\gamma_{\rm tr}=\gamma_++\gamma_-, \qquad \gamma_{\rm anti}=\gamma_+-\gamma_-,

the source pairing gives

⟨I,γtr⟩=2B−A,⟨I,γanti⟩=0.\boxed{ \langle I,\gamma_{\rm tr}\rangle=\frac{2}{B-A}, \qquad \langle I,\gamma_{\rm anti}\rangle=0. }

On the physical diagonal B=0B=0, the two small poles collide at X=0X=0. The invariant trace limits exactly to the residue of the physical central pole:

2B−A∣B=0=−2A=Res⁡X=0Imass,corner.\boxed{ \left.\frac{2}{B-A}\right|_{B=0} =-\frac{2}{A} =\operatorname{Res}_{X=0}I_{\rm mass,corner}. }

Meaning

The coefficient connection is pure gauge (Entry 1538), but the relative-cycle system carries a nontrivial deck permutation. The source pairing selects its invariant trace and annihilates the anti-invariant cycle.

Thus the physical one-channel readout is not obtained from coefficients alone:

deck-permuted relative cycles+source residue pairing⟶physical trace.\boxed{ \text{deck-permuted relative cycles} +\text{source residue pairing} \longrightarrow \text{physical trace}. }

This is the precise cycle-level mechanism missing from the formal connection lane.

Durable evidence

View source on GitHub