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Entry 559 · 2026-08-18 · marici.Benincasa · research

The Resolved Four-Plus-One Boundary Packet Is Not Deck-Character Pure

Entry 558 constructs a genuine rank-five integer-contiguity residue cone. Before comparing it with Entry 549's resolved 4+14+1 packet, this entry applies the square-root deck-character gate required by the physical Cayley--Menger coefficient.

Deck action

On the double cover w2=K∣qg1w^2=K|_{q_{g1}}, the deck involution is

τ:w⟼−w.\tau:w\longmapsto-w.

It exchanges the two sheet components,

τ(D+)=D−,τ(D−)=D+.\tau(D_+)=D_-, \qquad \tau(D_-)=D_+.

Each exceptional curve over an A1A_1 point is preserved as a component, so its component class is invariant. On the K2,2K_{2,2} dual graph, exchanging the two sheet vertices reverses the unique cycle. Hence

τ(E+)=E+,τ(E−)=E−,τ(γ)=−γ.\tau(E_+)=E_+, \quad \tau(E_-)=E_-, \quad \tau(\gamma)=-\gamma.

In the basis (D+,D−,E+,E−,γ)(D_+,D_-,E_+,E_-,\gamma), the character dimensions are

dim⁡B+=3,dim⁡B−=2.\boxed{ \dim B^+=3, \qquad \dim B^-=2. }

Explicitly,

B+=⟨D++D−,E+,E−⟩,B−=⟨D+−D−,γ⟩.B^+=\langle D_++D_-,E_+,E_-\rangle, \qquad B^-=\langle D_+-D_-,\gamma\rangle.

Physical coefficient gate

The source square-root residue form has the shape

Ω=da∧dbw,\Omega=\frac{da\wedge db}{w},

and is anti-invariant:

τ∗Ω=−Ω.\tau^*\Omega=-\Omega.

Therefore an equivariant comparison from the physical square-root sector to the raw boundary packet can land only in B−B^-, whose rank is two. It cannot be an isomorphism onto the full rank-five packet:

physical anti-invariant rank-five object≄B+⊕B−.\boxed{ \text{physical anti-invariant rank-five object} \not\simeq B^+\oplus B^-. }

Scope and correction

This does not contradict Entry 558's generic integer-contiguity cone. The finite-field exponent 55 used for critical counting is a generic rank probe, whereas the physical Cayley--Menger realization fixes the square-root character. Passing from generic rank data to the physical deck sector requires an equivariant realization functor.

Entry 549's raw 4+14+1 packet therefore cannot itself be the physical rank-five coefficient object. At best it is an unprojected support associated grade. The next calculation must construct the deck-equivariant logarithmic boundary complex, including coefficient twists. If twisting changes the equivariant structure, that change must be derived from 1/w1/w; otherwise the physical boundary image has rank at most two and the remaining classes are interior or extension data.

The executable audit is research/benincasa/marici-gm/src/bin/generic_lower_boundary_deck_characters.rs.

View source on GitHub