Connected Boundary Residues and the Electron Prefactor in Twistor Configuration Geometry

Abstract

Paper #25 v2 [Zhang Bulk–Boundary, DOI:10.5281/zenodo.20102027] decomposes 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. Three of the four — the hard-core adjacent-pair selection $P_\partial^{\rm hc}$, the polar/complex-normal $S^1$ phase $P_\partial^{\mathbb{C}\text{-norm}}$, and the real-slice identity defect $P_e^{\rm id}$ — are physically or geometrically motivated. The fourth, $P_e^{\rm conn}$, is the connected-projection rule on the matching-sector boundary defect partition function. In Paper #25 v2 it is empirically supported within the TCG formula ledger by the existing 0.09% electron-Yukawa match, but not action-level derived.

This companion note motivates $P_e^{\rm conn}$ structurally via the standard connected-effective-action principle of quantum field theory, $W = \log Z$. The core observation: in the commutative nilpotent square-free 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 boundary-defect insertion $X_e = b_e\, \delta_0(\phi_e)$ inherits nilpotency $X_e^2 = 0$ from $b_e^2 = 0$. The Taylor series therefore truncates after one term: $\log(1 - X_e) = -X_e$ exactly. For any matching $M$ on $P_4$ with disjoint, pairwise-commuting edge defects, the connected effective action of the boundary defect partition function is

The connected boundary prefactor $B_e^{\rm conn} := 1 + W_M^{\rm def} = 1 - \widehat{\rho}_M$, with $\widehat{\rho}_M = \sum_{e \in M} b_e\, \delta_0(\phi_e)$, is therefore an exact identity in the residue algebra rather than a Taylor truncation. Averaging over $\mathrm{Match}(P_4) = \{\varnothing, 12, 23, 34, 12 \mid 34\}$ with mean dimer count $(0+1+1+1+2)/5 = 1$ and $\langle \delta_0 \rangle = 1/(2\pi)$ on the polar normal $S^1$ recovers $\langle B_e^{\rm conn} \rangle = 1 - 1/(2\pi)$, the framework’s $P_4$. The full multiplicative alternative gives $1 - 1/(2\pi) + 1/(5(2\pi)^2)$, a 0.51% disconnected correction beyond $P_4$ excluded by the existing 0.09% $y_e$ match within the TCG formula ledger (model-selection statement, not statistical exclusion).

The note does not derive the full BV–BFV boundary action that would justify the sectorwise log prescription on first principles. The applicability of the bulk QFT principle $W = \log Z$ to a sector-decomposed boundary algebra is itself part of $P_e^{\rm conn}$, recorded explicitly as failure mode F4. Sectorwise vs global log distinction made explicit (load-bearing). $\delta_0$ is Lebesgue-normalized, $\int_{S^1} \delta_0(\phi) f(\phi)\, d\phi = f(0)$. Five failure modes are listed (F1: full-boundary dominance — inherited from Paper #25 v2; F2: real-normal obstruction — inherited; F3: identity-defect ambiguity — inherited; F4: connectedness ambiguity / bulk-vs-boundary log $Z$ applicability — sharpened here; F5: trace, weight, and matching-sector measure ambiguity, including the unit-weight choice of augmentation $\epsilon$, the uniform counting measure on $\mathrm{Match}(P_4)$, and the equal phase-circle measure across edges).

The active TCG/FPA postulate ledger is unchanged: $P_0$–$P_4$, $P_{5'}$, $P_6$, $P_7$, $P_{H'}$, $P_{SO(10)}$. The four sub-postulates of the localization conjecture for $P_4$ remain sub-postulates of the conjecture, not new framework axioms. After this note, all four are structurally motivated: $P_\partial^{\rm hc}$ via the binary relevant-residue principle on FM/AS boundary geometry; $P_\partial^{\mathbb{C}\text{-norm}}$ via oriented real blow-up of a complexified collision divisor; $P_e^{\rm id}$ via the Poincaré-dual real-slice direction; and $P_e^{\rm conn}$ via the standard connected-effective-action principle of QFT, applied sectorwise and with linearity following exactly from nilpotency. The action-level BV–BFV derivation of the sectorwise log prescription itself remains the next research target.

DOI

https://doi.org/10.5281/zenodo.20102577