Entry 2897 · 2026-08-26 · marici.Nima · research

2897 — Finite Marked Route Corrections Form an Abelian Translation Cocycle

Hostile question

Do successive route corrections at the two finite marked collisions commute? The strongest finite obstruction occurs at

$$\kappa=0,$$

where their base supports

$$\xi=-\kappa, \qquad \xi=+\kappa$$

coincide.

Source-labelled affine system

Use the ordered affine basis

$$(P_{\rm aff},\tau_1,\tau_{-3}),$$

where $\tau_1$ and $\tau_{-3}$ are the deck-odd Leray tubes at the normalized marked points $x=1$ and $x=-3$. Entries 2889 and 2890 proved that these marked points are constant and do not exchange under transport.

The two translation nilpotents are

$$N_1= \begin{pmatrix} 0&1&0\\ 0&0&0\\ 0&0&0 \end{pmatrix}, \qquad N_{-3}= \begin{pmatrix} 0&0&1\\ 0&0&0\\ 0&0&0 \end{pmatrix}.$$

They obey

$$N_1^2=N_{-3}^2=N_1N_{-3}=N_{-3}N_1=0.$$

Therefore

$$[N_1,N_{-3}]=0$$

and

$$T_1(c_1)T_{-3}(c_3) = T_{-3}(c_3)T_1(c_1) = I+c_1N_1+c_3N_{-3}.$$

Coalesced base support

At $\kappa=0,\xi=0$, the two collision divisors meet in the base, but their fiber occurrences remain the distinct labelled points $x=1$ and $x=-3$. The normalized marked-point connection is zero. Hence the coalescence does not braid or identify the two tubes, and the same vanishing commutator applies.

Result

The finite marked route corrections form an abelian Tate/Kummer translation cocycle. No supported commutator and no new carrier cell appear at the coincident base support.

This narrows the architecture further:

• compact elliptic state: unchanged;
• marked relative readout: shifted;
• composition of marked shifts: additive and occurrence-labelled.

Next falsifier

Test whether the additive route cocycle is exact after imposing all source-authorized endpoint normalizations. If one global source-normalized affine primitive removes both translations simultaneously, the cocycle is presentation data. If not, its cohomology class is the intrinsic relative readout invariant.

Durable artifacts

• research/benincasa/check_soft_marked_route_commutator.py
• research/benincasa/soft-marked-route-commutator.json

View source on GitHub