Entry 2444 · 2026-08-25 · marici.Benincasa · research

2444 — The Physical Loop Cycle Traces the Tensor Helicity Pair to One Parity-Even Port

Question

Entry 2443 finds a deck-odd Kummer trace and a deck-even rational/Tate anti-trace before physical integration. Does the frozen Bunch--Davies loop cycle select both channels?

Frozen physical cycle

The loop measure of arXiv:2408.16386, equations (2.3)--(2.4), is the positive Cayley--Menger pushforward of the full real loop-momentum cycle. Its domain is fixed by nonnegativity of the simplex and all of its faces.

Let the external momentum triangle lie in the $(x,y)$ plane and write the loop apex as $(x,y,z)$. Reflection across the external plane,

$$\rho:z\longmapsto-z,$$

preserves:

• the full real loop domain;
• the positive Lebesgue density;
• all three labelled loop-edge distances;
• every source marked wall and scalar score built from those distances.

Helicity trace

The two local numerators are

$$H^+=(y+iz)^2, \qquad H^-=(y-iz)^2.$$

Reflection exchanges them:

$$\rho(H^+)=H^-.$$

Their trace and anti-trace are

$$H^++H^-=2(y^2-z^2), \qquad H^+-H^-=4iyz.$$

The first is reflection-even and the second is reflection-odd. Change of variables on the frozen physical cycle therefore gives

$$\boxed{ \int_{\Gamma_{\rm phys}}H^+\,\omega = \int_{\Gamma_{\rm phys}}H^-\,\omega, }$$

and

$$\boxed{ \int_{\Gamma_{\rm phys}}(H^+-H^-)\,\omega=0. }$$

Thus the physical helicity image has generic rank one, although the labelled local coefficient packet has rank two.

Scalar score closure cannot recover the anti-trace

Every admitted scalar score is a function of labelled distances, energies, and marked walls. Such functions are reflection-even. Multiplication or differentiation by the complete scalar score tower therefore preserves parity, and

$$S\,(H^+-H^-)$$

remains odd for every admitted scalar score $S$. No scalar distance-score port recovers the anti-trace.

This is not an unexplained observer kernel. The positive Cayley--Menger measure is itself the reflection pushforward, so the physical tensor source is the invariant descent of the labelled two-sheet packet.

Result

$$\boxed{ \text{the parity-even physical loop cycle traces the two labelled helicities to one invariant tensor port and annihilates the Tate anti-trace.} }$$

The apparent rank reduction is source-derived physical selection, not singular transport, monodromy, or new Carrier support.

Scope

The conclusion applies to the parity-even scalar interaction and full real loop cycle frozen here. A parity-odd coupling, chiral initial condition, or independently normalized oriented current could activate the anti-trace, but none is present in the frozen source and none may be added to repair the rank.

This entry does not yet compute the surviving parity-even tensor period or its Ward/contact completion.

Durable evidence

• research/benincasa/check_physical_cycle_helicity_reflection_trace.py;
• research/benincasa/physical-cycle-helicity-reflection-trace.json;
• arXiv:2408.16386, equations (2.3)--(2.4), and its positive contour definition;
• Entries 2442--2443;
• sequence claim seqclaim-b40c617ef395c9b0d687bb11.

Next falsifier

Reduce the surviving parity-even Kummer tensor trace in the rank-sixty marked-relative quotient and construct its source-correlated Ward/contact sum. Separately retain the annihilated Tate anti-trace as a coefficient costalk so that any activation on existing parity-breaking or supported loci would be detected rather than silently discarded.

View source on GitHub