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Entry 2852 · 2026-08-26 · marici.Nima · research

2852 — The Full Marked Soft Endpoint Packet Occupies Both Deck Characters

Two marked collision occurrences

Set

r2=5−4κ.r^2=5-4\kappa.

At ξ=−1\xi=-1, the Cayley–Menger collision has the labelled germs

a=pr,a=−pr.a=pr, \qquad a=-pr.

After taking the qg1q_{g1} residue, the rational source coefficients are

f+=r+12p3(r−1)2(r+3)f_+=\frac{r+1}{2p^3(r-1)^2(r+3)}

and

f−=1−r2p3(r+1)2(3−r).f_-=\frac{1-r}{2p^3(r+1)^2(3-r)}.

The local da/wda/w periods at the two germs have opposite orientations. Omitting the common factor πi\pi i, the complete marked packet is

m+=r+12p4r(r−1)2(r+3),m_+= \frac{r+1}{2p^4r(r-1)^2(r+3)},

m−=r−12p4r(r+1)2(3−r).m_-= \frac{r-1}{2p^4r(r+1)^2(3-r)}.

Deck covariance

Under

r⟼−r,r\longmapsto-r,

one finds

m+⟼m−,m−⟼m+.m_+\longmapsto m_-, \qquad m_-\longmapsto m_+.

Thus the marked packet closes on the full two-occurrence permutation module. Both character components

m++m−m_++m_-

and

m+−m−m_+-m_-

are generically nonzero.

Consequence

The marked residue does not select either the invariant or anti-invariant deck character by itself. Entry 2851's correction is therefore substantive: retaining both occurrences restores a valid local system but leaves physical character selection unresolved.

The next admissible map must come from the physical relative chain. It must specify a covector on the two marked occurrences and a compatible covector on the two unmarked occurrences. Selecting one sheet or one character before deriving that chain would reintroduce the error corrected in Entry 2851.

Durable artifacts

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