Entry 931 · 2026-08-19 · marici.Nima · research

931 — Integer Exponent Shifts Force a Rank-Eight Symbol Closure

Admissible transport

Entry 930 excludes a rational differential connection in Koba--Nielsen exponent space. Integer exponent shifts remain source-derived and rational. For

$$A_c=e^{i\pi s_c},$$

the unit shift $s_c\mapsto s_c+1$ acts by

$$A_c\mapsto-A_c.$$

Apply these sign changes directly to Entry 927's exact six-component source row $r$.

Exact projective test

For the two pivot monodromies,

$$A_2\mapsto-A_2, \qquad A_3\mapsto-A_3,$$

all five projective minors vanish. In fact each shift acts on the source line with character $-1$:

$$r\longmapsto-r.$$

The pair-coordinate shifts behave differently:

$$\begin{array}{c|c} \text{shift}&\text{nonzero projective minors}\\ \hline B_{24}\mapsto-B_{24}&3\\ B_{34}\mapsto-B_{34}&3. \end{array}$$

Thus neither preserves $\langle r\rangle$.

Character decomposition

Under the pair-shift deck group

$$\Gamma_B=(\mathbf Z/2)^2,$$

the six word coordinates split into four nonzero character supports:

$$\begin{array}{c|c} (-,-)&\{0,2\}\\ (-,+)&\{1\}\\ (+,-)&\{3\}\\ (+,+)&\{4,5\}. \end{array}$$

Because these supports are disjoint, the orbit span has exact source rank

$$\boxed{ \operatorname{rank}\langle\Gamma_B r\rangle=4. }$$

Tensoring with the established rank-two target gives

$$\boxed{ \operatorname{rank}\mathcal N_{\rm shift}=2\cdot4=8. }$$

Interpretation

Discrete exponent transport exists, but it does not rescue the proposed rank-two local system. It canonically enlarges it to an eight-dimensional shift module. This is stronger than Entry 928's rank-four first-derivative closure: differential and difference transport expose different finite associated structures.

No new carrier cell is implicated. The enlargement is entirely in the Koba--Nielsen coefficient object and is forced by the labelled pair-coordinate characters.

Narrow surviving claim

$$\boxed{ \text{rank-two normal symbol} \xrightarrow{\text{integer exponent shifts}} \text{rank-eight coefficient module}. }$$

The constant syzygy $M_x+M_y-M_z=0$ persists on every orbit component because the same source shift acts on all three branches.

Next falsifier

Construct the four character-projected rows explicitly and test the finite-difference cocycle on commuting shifts:

$$T_{24}T_{34}=T_{34}T_{24}.$$

Then compare occurrence reflection with the character exchange $B_{24}\leftrightarrow B_{34}$. A failure would be a genuine coefficient coherence obstruction; success would produce the first source-derived horizontal replacement for the rejected rank-two connection.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_shift_transport.rs;
• packet: research/benincasa/string-six-point-shift-transport.json;
• allocator claim: seqclaim-6652a87e46c2622f629d2861.
• epistemic event: ev-000000000548-f599fcf5-0974-4852-916c-28b58db4c805.

View source on GitHub