Bulk–Boundary Localization in Twistor Configuration Geometry: A Conjecture for the Coupling/Mass Sector Split

Abstract

The FPA realization of Twistor Configuration Geometry [Zhang FPA, DOI:10.5281/zenodo.20076012] supplies two theorem-level combinatorial outputs at each stratum $n$ with line-deformation rank $r_n = 2n - 2$: a bulk chamber count $Z_{\rm bulk}(r_n) = r_n!$ and a hard-core adjacent-pair matching count $Z_\partial(r_n) = F_{r_n + 1}$. The framework currently assigns gauge couplings to the bulk count and lepton masses / Yukawas to the boundary count. The combinatorics on each side is theorem-level. The assignment is not. The framework reference paper itself flags this, and the DAEDALUS Review [Zhang Review, DOI:10.5281/zenodo.19984246] and Methodology audit [Zhang Methodology, DOI:10.5281/zenodo.20062462] record the sector-assignment as the framework’s deepest open structural question, distinct from individual numerical gaps such as $g_{2,W}^2 = 4/(3\pi)$ [Zhang EW, DOI:10.5281/zenodo.20075926] or $m_p / m_e \approx 6\pi^5$ [Zhang Hadronic, DOI:10.5281/zenodo.20077931], and distinct from the gauge-algebraic arc [Zhang Wall Deletion, Parabolic, Spin10, DOIs 10.5281/zenodo.20045987, 20090828, 20091562] that closes the framework’s $\mathfrak{su}(2)_R$ gap at the postulate-equivalent level.

This note begins a derivation. We separate three layers: (i) a theorem-level bulk chamber algebra $\mathcal{A}_{\rm bulk}(r) = \mathbb{C}[\pi_0(\operatorname{Conf}^{\rm lab}_r(I))]$ of dimension $r!$; (ii) a theorem-level hard-core adjacent-residue algebra $\mathcal{A}_\partial^{\rm hc}(r) = \mathbb{C}[b_1, \ldots, b_{r-1}] / (b_i^2,\, b_i b_{i+1})$ of dimension $F_{r+1}$ after the adjacent-pair residue sector is selected from the full FM/AS boundary; (iii) a physical localization conjecture identifying marginal dimensionless couplings with interior classes and masses / Yukawas with hard-core boundary residues. The boundary algebra is the square-free / exterior-face Stanley–Reisner-type incidence algebra of the path-matching complex, not the ordinary polynomial Stanley–Reisner ring; the relations $b_i^2 = 0$ and $b_i b_{i+1} = 0$ encode the geometric incompatibility of overlapping but non-nested adjacent collision blocks $\{i, i+1\}$ and $\{i+1, i+2\}$, which share the vertex $i+1$.

The remaining physical statement is formulated as a logarithmic BV–BFV Bulk–Boundary Sector Localization Conjecture: there exists a logarithmic BV–BFV theory on $K_r(I)$ such that (i) marginal dimensionless coupling deformations are $S_r$-equivariant interior BV classes; (ii) chirality-changing relevant mass/Yukawa deformations are logarithmic BFV residue classes supported on adjacent collision divisors; (iii) the induced hard-core residue algebra on the primitive adjacent-collision sector is $\mathcal{A}_\partial^{\rm hc}(r)$, not the full FM/AS boundary incidence algebra. Under the conjecture, $r_n!$ and $F_{r_n + 1}$ are no longer independent assignments but joint consequences of interior versus residue localization.

Five explicit failure modes are listed (F1: full-boundary dominance — structurally decisive; F2: interior mass representatives; F3: normal-circle ambiguity; F4: Standard-Model embedding; F5: complex-normal enhancement). A side application connects the same boundary-residue picture to the electron prefactor $\langle B_e \rangle = 1 - 1/(2\pi)$ via the normal-circle phase factor of a complexified collision divisor, conditional on a complex-normal enhancement of the FPA boundary theory.

The note does not derive the Standard Model sector assignment. It proves that the two required counts arise naturally from two sharply defined algebras on the same compactified interval-configuration geometry: the bulk chamber algebra and the hard-core adjacent-residue algebra. The remaining conjecture is that marginal gauge couplings localize to the former and relevant mass/Yukawa deformations to the latter. The active TCG/FPA postulate ledger is unchanged: $P_0$–$P_4$, $P_{5'}$, $P_6$, $P_7$, $P_{H'}$, $P_{SO(10)}$. The conjecture specifically targets $P_2$ and $P_3$, the coupling/mass sector-localization postulates; it does not by itself derive $P_1$, $P_4$, $P_{5'}$, $P_6$, $P_7$, $P_{H'}$, or $P_{SO(10)}$.

DOI

https://doi.org/10.5281/zenodo.20102027