Entry 934 · 2026-08-19 · marici.Nima · research

934 — The Rank-Eight Shift Module Has Trivial Cyclic Holonomy

Global occurrence test

Entries 931 and 933 establish a rank-eight finite-difference module in one maximal-flag chart and its strict off-diagonal reflection covariance. The remaining descent condition is cyclic transport through the three occurrence charts.

Use the source cycle

$$\sigma=(234).$$

It transports the ordered pair-shift generators as

$$(B_{24},B_{34}) \longmapsto (B_{23},B_{24}) \longmapsto (B_{34},B_{23}) \longmapsto (B_{24},B_{34}).$$

The generator order is preserved at every step, so the local character labels

$$(--),\quad(-+),\quad(+-),\quad(++)$$

transport without an additional permutation.

Source-row return

Apply the full cyclic label action

$$A_2\to A_3\to A_4\to A_2,$$

$$B_{23}\to B_{34}\to B_{24}\to B_{23}$$

to every exact component of the source row. Symbolic reduction gives

$$\boxed{ \sigma^3r=r. }$$

Serialized-basis return

The independently frozen dense and sparse transition permutations are

$$J,\qquad J,\qquad I.$$

Therefore

$$JJI=I$$

in both variances. Their orientation characters cancel stepwise:

$$(+1,+1,+1).$$

No residual sign or unit remains.

Result

The complete cyclic return on the four-character source closure is identity:

$$\boxed{ \operatorname{Hol}_{C_3}(\mathcal N_{\rm shift})=I_8. }$$

Together with Entry 933, this constructs a global source-derived finite-difference coefficient system on the tested maximal-flag occurrence orbit:

$$\boxed{ \operatorname{rank}\mathcal N_{\rm shift}=8, \qquad D_3\text{-covariant}, \qquad \text{trivial cyclic holonomy}. }$$

No carrier enlargement, higher associator, or fitted transition is needed.

Scope

This is a difference local system in integer Koba--Nielsen exponent directions. It is not a differential Gauss--Manin connection, and it does not turn the original rank-two associated grade into a rank-two local system. Entry 930's digamma obstruction remains in force.

Next falsifier

Test compatibility between integer shifts and Cartier specialization itself. For each normal coordinate $N$, compare

$$\operatorname{gr}_{N=1}\circ T_c \quad\text{with}\quad T_c\circ\operatorname{gr}_{N=1}.$$

A nonzero commutator would locate a genuine difference/Rees Beck--Chevalley class. Vanishing would show that the rank-eight coefficient system descends through the same associated-grade calculus as the original normal symbols.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_shift_cyclic_atlas.rs;
• packet: research/benincasa/string-six-point-shift-cyclic-atlas.json;
• allocator claim: seqclaim-6db4fde4c4854928540b1d0c.
• epistemic event: ev-000000000551-303511f8-e9a8-4fff-9c29-326b8f01024f.

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