Boundary Superselection Obstruction for the Electron Prefactor in Twistor Configuration Geometry

Abstract

The bulk—boundary localization program of Twistor Configuration Geometry [Zhang Bulk-Boundary, DOI:10.5281/zenodo.20102027] decomposed the framework’s electron-boundary postulate $P_4$, $\langle B_e \rangle = 1 - 1/(2\pi)$, into four sub-postulates of a logarithmic BV—BFV bulk—boundary sector localization conjecture. The companion connected-boundary residue note [Zhang Connected Bdry, DOI:10.5281/zenodo.20102577] sharpened the residual sub-postulate $P_e^{\rm conn}$ 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 in the hard-core residue algebra $\mathcal{A}_\partial^{\rm hc}(r) = \mathbb{C}[b_1,\ldots,b_{r-1}] / (b_i^2,\, b_i b_{i+1})$. The remaining question was whether the sectorwise prescription $W_M = \log Z_M$ is forced by a boundary BV—BFV action principle, equivalently whether the hard-core matching sectors $M \in \mathrm{Match}(P_4)$ are forced to be BFV superselection sectors of the boundary theory.

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

Theorem 1 (algebraic obstruction). For any nonempty matching $S = \{i_1, \ldots, i_k\}$ of $P_r$, the product $b_S = b_{i_1} \cdots b_{i_k}$ satisfies $b_S^2 = 0$ (since $b_{i_j}^2 = 0$ for each generator). Therefore no nonempty matching monomial is idempotent, and the matching monomial basis $\{1, b_1, b_2, b_3, b_1 b_3\}$ of $\mathcal{A}_\partial^{\rm hc}(4)$ does not supply central orthogonal projectors onto matching sectors. A direct-sum decomposition of a finite state space requires central orthogonal idempotents $e_M$ with $e_M e_{M'} = \delta_{MM'} e_M$ and $\sum_M e_M = 1$; the nilpotent generators $b_i$ do not have this property. The hard-core residue algebra is a square-free incidence/residue ring (Stanley—Reisner-type for the matching complex of $P_r$), not a semisimple direct-sum algebra.

Structural (corner-theory) obstruction. FM/AS-type corner-aware boundary theories naturally encode incidence relations among strata. In Cattaneo—Mnev—Reshetikhin BV—BFV theory [CattaneoMnevReshetikhin], the boundary phase space and boundary differential are induced by the variational boundary terms of the bulk action, and block-diagonality is a property of a particular boundary condition rather than a consequence of the existence of a boundary. In Costello—Gwilliam factorization-algebra language [CostelloGwilliam], local-to-global maps are gluing maps, not superselection projectors. The natural default differential reads as $Q_\partial : \mathcal{H}_M \to \bigoplus_{M'} \mathcal{H}_{M'}$, with possible incidence maps between matching sectors, rather than the block-diagonal $Q_\partial \Pi_M = \Pi_M Q_\partial$ required for sectorwise factorization. Forbidding $M \to M'$ transitions is structural input, not derivation.

Theorem 2 (conditional closure). Assume the residual postulate $P_{\rm BFV}^{\rm sec}$, which bundles four pieces: (i) sector orthogonality of the boundary state space $\mathcal{H}_{\partial,4}^{\rm pol} = \bigoplus_{M \in \mathrm{Match}(P_4)} L^2((S^1)^M)$ with projectors $\Pi_M$; (ii) BRST/BFV preservation $Q_\partial \Pi_M = \Pi_M Q_\partial$ with sector-diagonal $\Omega_\partial$ and $Q_\partial$; (iii) unit augmentation $\epsilon(b_{i_1} \cdots b_{i_k}) = 1$ on matching monomials; (iv) uniform counting measure on $\mathrm{Match}(P_4)$ with normalized Haar measure $d\phi/(2\pi)$ on each polar-normal phase circle. Then the boundary connected functional factorizes sectorwise $W_\partial = \bigoplus_M W_M$ with $W_M = \log Z_M$, and the connected-log prescription of [Zhang Connected Bdry] follows.

Postulate-burden accounting. $P_{\rm BFV}^{\rm sec}$ is NOT WEAKER than the existing $P_e^{\rm conn}$ sub-postulate. It bundles four specific assumptions, all of which are implicit in the sectorwise connected-log prescription. Replacing one by the other clarifies but does not reduce the residual burden; its value is as a more precise research handle.

Literature gap. The corner-extended logarithmic BV—BFV theory required to formulate the question precisely --- whose boundary phase space is canonically the hard-core polar-normal matching-sector direct sum and whose transgression is block-diagonal in matching sectors --- is not currently supplied by Cattaneo—Mnev—Reshetikhin 2014 + Costello—Gwilliam factorization algebras + Fulton—MacPherson 1994 / Axelrod—Singer 1994 / wonderful compactification corner combinatorics + Stanley—Reisner-style square-free quotient conventions, 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 $\log \langle Z \rangle_M$ rather than $\langle \log Z \rangle_M$; E4: different boundary algebra ansatz; E5: full-boundary dominance --- the F1 obstruction of [Zhang Bulk-Boundary]).

The active TCG/FPA postulate ledger is unchanged: $P_0$—$P_4$, $P_{5'}$, $P_6$, $P_7$, $P_{H'}$, $P_{SO(10)}$. $P_{\rm BFV}^{\rm sec}$ is the structural content of the existing $P_e^{\rm conn}$ sub-postulate of the localization conjecture for $P_4$, not a new framework axiom. Arc (3) of the unification map (action-level derivation of $P_4$ from a single boundary BV—BFV principle) is converted from open to closed-conditional with explicit obstruction and named residual postulate. 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.

DOI

https://doi.org/10.5281/zenodo.20110780