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Entry 2902 · 2026-08-26 · marici.Nima · research

2902 — The Additive Marked Route Cocycle Is Globally Nonexact

Exactness question

Entry 2897 proved that the two marked route corrections commute. The next question is whether one source-normalized, single-valued meromorphic affine primitive can remove both corrections globally.

Route one-forms

For the x=1x=1 occurrence, let

B=3−κ(1−κ)2.B=\frac{3-\kappa}{(1-\kappa)^2}.

Entry 2892 gives the simple-pole residues

Res⁡ξ=−κ(R1 dξ)=B64p4,Res⁡ξ=−1(R1 dξ)=−B64p4.\operatorname{Res}_{\xi=-\kappa}(R_1\,d\xi) =\frac{B}{64p^4}, \qquad \operatorname{Res}_{\xi=-1}(R_1\,d\xi) =-\frac{B}{64p^4}.

For the x=−3x=-3 occurrence,

R−3=−164p4(ξ+1)(κ−ξ),R_{-3} = -\frac{1}{64p^4(\xi+1)(\kappa-\xi)},

so

Res⁡ξ=κ(R−3 dξ)=164p4(1+κ),Res⁡ξ=−1(R−3 dξ)=−164p4(1+κ).\operatorname{Res}_{\xi=\kappa}(R_{-3}\,d\xi) =\frac{1}{64p^4(1+\kappa)}, \qquad \operatorname{Res}_{\xi=-1}(R_{-3}\,d\xi) =-\frac{1}{64p^4(1+\kappa)}.

Obstruction

The derivative of a single-valued rational or meromorphic affine gauge has zero residue at every pole. Both labelled route directions instead have nonzero generic residues. Their local primitives necessarily contain logarithms.

For generic κ\kappa, the marked supports ξ=−κ\xi=-\kappa and ξ=+κ\xi=+\kappa are distinct. Consequently the two labelled Kummer classes cannot cancel each other under one scalar gauge.

Result

The additive marked route cocycle is globally nonexact in the admitted single-valued meromorphic category. Local logarithmic primitives exist, but no global source-normalized affine splitting removes both translations.

Thus the route packet is not presentation data. It is intrinsic relative readout information:

No new carrier divisor is required. The obstruction lives in the coefficient and readout layer over existing marked support.

Next falsifier

Compute the source-selected monodromy representation of this rank-two Kummer class and test whether the physical soft observable depends only on its cohomology class or also on the chosen pointed normalization.

Durable artifacts

View source on GitHub