Boundary Superselection Obstruction for the Electron Prefactor in Twistor Configuration Geometry

The bulk-boundary localization program of Twistor Configuration Geometry (TCG) decomposed the electron-boundary postulate P_4, ⟨B_e⟩ = 1 - 1/(2π), into four sub-postulates of a logarithmic BV-BFV bulk-boundary sector localization conjecture (Zhang 2026, DOI:10.5281/zenodo.20102027). The companion connected-boundary-residue note (Zhang 2026, DOI:10.5281/zenodo.20102577) reduced the residual sub-postulate P_e^{conn} from "arbitrary linear-operator selection" to "sectorwise importation of the standard W = log Z structure of QFT," with linearity following exactly from the nilpotency b_e^2 = 0 of single-edge boundary defects. The remaining question — whether the sectorwise prescription is forced by a boundary action principle, equivalently whether the hard-core matching sectors M ∈ Match(P_4) are forced to be BFV superselection sectors — was named as failure mode F4 of the connected-log derivation.

This closure note reports the F4 attempt and its outcome. The verdict is negative-conditional. Two obstructions are identified.

First, an algebraic obstruction (Theorem 1): the square-free hard-core residue algebra A_∂^{hc}(r) = C[b_1,...,b_{r-1}]/(b_i^2, b_i b_{i+1}) labels matchings of P_r by nilpotent monomials b_S = b_{i_1} ⋯ b_{i_k}, but every nonzero matching monomial satisfies b_S^2 = 0 (since b_{i_j}^2 = 0 for each generator). Therefore no nonempty matching monomial is idempotent. A direct-sum decomposition of a finite state space requires central orthogonal idempotents e_M satisfying e_M e_{M'} = δ_{MM'} e_M and Σ_M e_M = 1; the nilpotent generators do not have this property. The matching monomial basis is a vector-space basis of A_∂^{hc}, not a semisimple direct-sum decomposition. To pass from one to the other requires an additional quantization or semisimplification rule that the current FPA/FM-AS construction does not supply.

Second, a structural (corner-theory) obstruction: FM/AS-type corner-aware boundary theories naturally encode incidence relations among strata. In CMR-style BV-BFV theory, the boundary differential is induced by the variational boundary terms of the bulk theory, and block-diagonality is a property of a particular boundary condition rather than a consequence of the existence of a boundary. The natural default is Q_∂: H_M → ⊕_{M'} H_{M'}, with possible incidence maps between matching sectors, rather than the block-diagonal Q_∂ Π_M = Π_M Q_∂ required for sectorwise factorization of the connected effective action.

A consistent sectorwise construction can be built by hand: declare H_∂,4^{pol} = ⊕_{M ∈ Match(P_4)} L^2((S^1)^M) with sector-diagonal Ω_∂ and Q_∂. Theorem 2 then shows that this construction implies the sectorwise connected-log prescription W_∂ = ⊕_M W_M with W_M = log Z_M, recovering the value ⟨B_e⟩ = 1 - 1/(2π) of the connected-boundary residue note. But this is a declared structure, not a derivation. The block-diagonal form is the superselection input, exactly what must be justified.

The residual postulate is named P_{BFV}^{sec} and bundles four pieces: (i) sector orthogonality of the boundary state space; (ii) BRST/BFV preservation Q_∂ Π_M = Π_M Q_∂; (iii) unit augmentation ε(b_{i_1} ⋯ b_{i_k}) = 1 on matching monomials; (iv) uniform counting measure on Match(P_4) with normalized Haar measure dφ/(2π) on each polar-normal phase circle. Postulate-burden accounting: P_{BFV}^{sec} is NOT WEAKER than the existing P_e^{conn} sub-postulate; it makes the burden more explicit by identifying the exact mathematical structure a future derivation must supply, but does not reduce it.

The literature gap is identified: the corner-extended logarithmic BV-BFV theory required to formulate the question precisely — whose boundary phase space is canonically a sector-decomposed hard-core polar-normal matching state space and whose transgression is block-diagonal in matching sectors — is not currently supplied by Cattaneo-Mnev-Reshetikhin 2014 + Costello-Gwilliam factorization algebras + FM/AS / wonderful compactification corner combinatorics + Stanley-Reisner machinery, taken either separately or in combination as published. The honest conclusion is not that such a theory is impossible, but that it is not currently available and should not be assumed silently.

Five failure modes are listed (E1: future corner-extended BV-BFV superselection; E2: trace and measure derivation; E3: global-log alternative; E4: different boundary algebra; E5: full-boundary dominance — the F1 obstruction of the bulk-boundary localization paper). The active TCG/FPA postulate ledger is unchanged: P_0–P_4, P_{5'}, P_6, P_7, P_H', P_{SO(10)}. P_{BFV}^{sec} is the structural content of the existing P_e^{conn} sub-postulate of the localization conjecture for P_4, not a new framework axiom.

Arc (3) of the current unification map — the action-level derivation of P_4 from a single boundary BV-BFV principle — is converted from open to closed-conditional. There is an explicit algebraic obstruction (matching monomials are nilpotent, not idempotent), an explicit structural (corner-theory) obstruction (natural BV-BFV theories on FM/AS-type compactifications encode incidence relations, not block-diagonal sector decompositions), and an explicit residual postulate (P_{BFV}^{sec}). Future progress would require a genuine corner-extended logarithmic BV-BFV construction with sector-decomposed transgression, or a different route to the electron boundary prefactor.