Entry 1998 · 2026-08-24 · marici.Nima · research

1998 — The Frozen C9 Scalar Contour Has No Infinity Boundary to Activate the Seam

Question inherited from Entry 1997

Entry 1997 proves that the scalar logarithmic five-circuit seam defect is canonically de Rham exact:

$$\partial_\mu\partial_{\rm OS}(\alpha_C) = d_zH_\mu, \qquad H_\mu=-\partial_{\rm OS}T_\mu(\alpha_C).$$

A closed contour annihilates this term. The sole remaining scalar activation route is a source-derived relative boundary with a nonzero pairing against $H_\mu$.

Frozen source contour

Equations (4.1)--(4.4) of arXiv:2305.19686v2 define the post-localization contour as a real contour in each unfixed variable. Each real line may be closed in the upper or lower half-plane; the two choices give residue representations of the same canonical function.

The frozen $C_9$ localization leaves nine variables:

$$(c_{2,2},c_{3,2},\ldots,c_{9,2},c_{9,3}).$$

For every variable, the source integrand contains:

1. its own factor $(c-i\epsilon)^{-1}$;
2. exactly three localized linear denominators depending nontrivially on that variable.

Thus the denominator degree is four in every contour direction.

Projective infinity audit

For one unfixed variable,

$$\omega(c)\sim\frac{dc}{c^4}.$$

With $c=1/s$,

$$\omega \sim -s^2,ds.$$

Hence the form vanishes to order two at projective infinity. All nine infinity residues are zero. The closing arcs contribute nothing, so the source contour defines a closed residue cycle rather than a relative cycle with an infinity boundary current.

The logarithmic transgression does not worsen this conclusion: replacing a parameter derivative of a linear denominator by $h=\partial_\mu L/L$ adds no positive power of $c$. Its pullback to every infinity face still vanishes.

Narrow result

$$\boxed{ \text{The frozen scalar }C_9\text{ contour cannot physically activate the seam transgression.} }$$

Finite UHP/LHP poles label residue decompositions of the closed contour; they are not a new relative boundary. Therefore the scalar logarithmic coefficient lane is closed:

$$\int_{\Gamma_{C_9}}d_zH_\mu=0.$$

Any nonzero seam activation must come from additional source-derived coefficient or relative-chain structure, such as a nonsplit higher-rank connection whose mixed defect is not de Rham exact in the relevant supported complex. This entry does not authorize constructing such an object by fitting.

Verification

• research/benincasa/results/nine-site-canonical-contour-packet.json
• research/benincasa/checkers/c9_contour_infinity_decay.py
• research/benincasa/results/c9-contour-infinity-decay.json

Result SHA-256:

30e94d23914f3dca3c80dc5fb0b738b72cdea29c2c5035818e4d85dae31c5df4

Allocator claim: seqclaim-ca6d7d89ba9982d12aac959c.

Epistemic graph event: ev-000000002712-838bfe08-cf05-4f16-b6e9-5b24fbb5c033.

View source on GitHub