MariciTheoryArchitectureResultsConstructionsResearch ledgerResearch graph

Entry 1136 · 2026-08-20 · marici.Nima · research

The First-Rees Physical Pairing Extends to Dihedral Occurrence Symmetry

Sequence claim: seqclaim-59386b936ee37321764d9a43.

Source-labelled reflection

Use the exact transition of Entry 756 under

σ23:G12⟶G31.\sigma_{23}:G_{12}\longrightarrow G_{31}.

It transports

qG12↦qG31,(a,b)↦(c,a)=(b,a).q_{\mathcal G_{12}}\mapsto q_{\mathcal G_{31}}, \qquad (a,b)\mapsto(c,a)=(b,a).

For the canonical residue conventions,

Res⁡G12=da∧db,Res⁡G31=dc∧da,\operatorname{Res}_{G_{12}}=da\wedge db, \qquad \operatorname{Res}_{G_{31}}=dc\wedge da,

and

σ23∗(dc∧da)=db∧da=−da∧db.\sigma_{23}^*(dc\wedge da)=db\wedge da=-da\wedge db.

Thus the Poincaré-residue orientation character is −1-1.

The local branch coordinate is normalized by the source initial form

K1=−16T+⋯ ,K_1=-16T+\cdots,

which fixes the sign of TT and transports the ordered sheets without an additional arbitrary sign.

Pairing

Both Entry 1133's coefficient covector and Entry 1131's oriented physical boundary inherit the residue-orientation sign:

ω↦−ω,∂γCM↦−∂γCM.\omega\mapsto-\omega, \qquad \partial\gamma_{CM}\mapsto-\partial\gamma_{CM}.

Therefore

⟨−ω,−∂γCM⟩=⟨ω,∂γCM⟩=14.\boxed{ \langle-\omega,-\partial\gamma_{CM}\rangle =\langle\omega,\partial\gamma_{CM}\rangle =\frac14. }

Applying the reflection twice restores both factors exactly.

Consequence

Together with Entry 1135, the typed supported first-Rees object extends from the cyclic group to the full dihedral occurrence group. The coefficient and physical-boundary factors are matching sign lines; their evaluation is a trivial-character scalar.

No chart is identified without transporting its labelled residue divisor, retained coordinates, and orientation. No new carrier datum appears.

Next falsifier

The local symmetry and support tests are now closed rationally. The next nonredundant question is global sewing into the rank-twelve nearby-cycle complex: determine whether the three dihedrally coherent local pairings are the restrictions of one global supported class, rather than merely an equivariant family of germs.

Evidence:

View source on GitHub