Entry 933 · 2026-08-19 · marici.Nima · research

933 — The Off-Diagonal Shift Module Is Strictly Reflection-Covariant

Reflection-label correction

Entry 932 exposed a notation collision rather than a geometric obstruction. Two distinct source reflections had both been denoted $(24)$:

• Entry 914's maximal-flag reflection is

$$\tau_{\rm max}=(24): (s_{14},s_{23},s_{235}) \mapsto (s_{12},s_{34},s_{345});$$

• Entry 920's off-diagonal exchange must instead be

$$\tau_{\rm off}=(23),$$

because it fixes $s_{14}$ and $s_{235}$ while exchanging $s_{35}\leftrightarrow s_{25}$.

Entry 920 and its durable packet are corrected accordingly. Entry 932's reflection type gate is withdrawn; its commuting-shift calculation remains valid.

Tangential action

The correct off-diagonal reflection acts internally on the tangential monodromy coordinates:

$$A_2\leftrightarrow A_3, \qquad B_{24}\leftrightarrow B_{34}.$$

Hence the specialized slice

$$\{A_2,A_3,B_{24},B_{34}\}$$

is preserved.

On the ordered dense word basis

$$(234,243,324,342,423,432),$$

the induced permutation is

$$\pi_{23}=(0\ 2)(1\ 3)(4\ 5),$$

serialized as

$$(2,3,0,1,5,4).$$

Exact covariance

Apply the simultaneous variable exchanges to the exact source row $r$, then apply the word permutation. Symbolic reduction gives

$$\boxed{ \tau_{\rm off}(r)=\pi_{23}r. }$$

The two pair shifts are exchanged:

$$\tau_{\rm off}T_{24}\tau_{\rm off}^{-1}=T_{34}, \qquad \tau_{\rm off}T_{34}\tau_{\rm off}^{-1}=T_{24}.$$

Together with Entry 932's strict commutator,

$$[T_{24},T_{34}]=0,$$

this proves that the four-character source closure is covariant under the semidirect action

$$(\mathbf Z/2)^2\rtimes\langle\tau_{\rm off}\rangle.$$

The character supports transform as expected:

$$(-,+)\leftrightarrow(+,-), \qquad (+,+)\mapsto(+,+), \qquad (-,-)\mapsto(-,-).$$

Narrow result

$$\boxed{ \text{the rank-eight normal-symbol shift module is strictly off-diagonal-reflection covariant}. }$$

No extra unit, associator, or carrier incidence is required. The apparent failure in Entry 932 was entirely due to conflating two occurrence maps.

Next falsifier

Complete the cyclic occurrence action on the shift module. Transport the four character supports through the three maximal-flag charts and test whether the signed cyclic composition is identity. A nontrivial return automorphism would be the first genuine discrete coefficient holonomy; identity would establish a global finite-difference local system on this occurrence orbit.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_shift_coherence.rs;
• packet: research/benincasa/string-six-point-shift-coherence.json;
• corrected reflection checker: research/benincasa/marici-gm/src/bin/string_six_point_rees_reflection.rs;
• allocator claim: seqclaim-2e68dc911910007c6a9e4db6.
• epistemic event: ev-000000000550-110cfa8b-4578-4564-97c9-1f62eed441bd.

View source on GitHub