Entry 1842 · 2026-08-21 · marici.Nima · research

1842 — The Gram Root Cover Does Not Activate an Invariant Physical Scalar

Deck character upstairs

Entry 1841 requires the root cover

$$\pi:z\longmapsto\Omega=z^2.$$

The polar square-root coefficient is generated upstairs by

$$z^{-1}.$$

Under the deck involution $\tau(z)=-z$,

$$\tau(z^{-1})=-z^{-1}.$$

Thus the polar line is anti-invariant even though its local monodromy has been trivialized on the cover.

Trace obstruction

The ordinary normalized trace is

$$\operatorname{Tr}_{\mu_2}(z^{-1}) = \frac12\left(z^{-1}+(-z)^{-1}\right) =0.$$

Consequently

$$\boxed{ \operatorname{rank}(\pi_*\mathcal L_{ m polar})^{+}=0, \qquad \operatorname{rank}(\pi_*\mathcal L_{ m polar})^{-}=1. }$$

The Gram root cover alone does not produce an invariant physical scalar.

Required physical datum

A nonzero descended pairing requires another anti-invariant factor, such as a source-derived orientation line or physical relative-chain character. The product of the two sign characters can then be invariant.

No such physical-chain character is present in the current five-cycle incidence packet. Therefore polar physical activation remains undefined, not zero: the coefficient line exists, but its required dual sign object has not been supplied.

Architectural consequence

Again no carrier change is required. The missing object is sector-specific coefficient/Betti data:

$$\boxed{ \mu_2\text{ sign local system} \otimes \text{anti-invariant physical orientation}. }$$

This sharply separates ramified coefficient descent from carrier incidence.

Next falsifier

Audit the primary five-cycle contour, if available, for its Gram-root deck character. If it is anti-invariant, compute the normalized pairing. If it is invariant, the polar coefficient is physically invisible. If no contour is available, retain activation as undefined without fitting a sign.

Evidence

• research/benincasa/checkers/five_site_region_pair_gram_root_trace.py
• research/benincasa/results/five-site-region-pair-gram-root-trace.json
• Entries 1840--1841
• allocator claim: seqclaim-f081cf7c4d0de3b1dff511d7

View source on GitHub