Twistor Configuration Geometry: The FPA Model

Abstract

We define a candidate geometric framework, Twistor Configuration Geometry (TCG), built from a chosen linear flag in projective twistor space together with chamberwise compactified labeled configuration spaces over an oriented interval. The present paper develops a specific realization of TCG denoted the FPA model (Framed Permuto–Associahedral), comprising five elementary theorem-level statements that yield four combinatorial quantities — the rank $r_n = 2n-2$, the chamber count $r_n!$, the matching count $F_{r_n+1}$, and the chamber-weighted Fubini–Study sum $\pi + \pi^2 + 4\pi^3$.

The TCG framework is the stratified space

where $\mathcal{K}_r(I)$ is the chamberwise Fulton–MacPherson/Axelrod–Singer compactification of $r$ labeled points on an oriented interval $I$. Two distinguished observables are the chamber-weighted Fubini–Study functional and the matching-graph count restricted to a fundamental ordered chamber.

The construction is offered as one candidate framework, not as a derivation; seven active structural postulates (P0–P4, P5’, P6) remain, with the original P5 retired in v3, supplemented in applications by additional phenomenological identifications. The paper presents the construction’s formal definition, the five theorem-level derivations (D1–D5) with proofs, the boundary stratification of the compactification with the precise identification of which strata yield the Fibonacci count, the explicit postulate ledger, several sharpened limitations including a non-normal-crossing observation on overlapping-but-not-nested collision blocks, and well-defined computational tasks that would promote one or more postulates from axiom to derived. The construction is positioned against existing physics-of-amplitudes uses of associahedra and configuration-space compactifications. This paper is the framework reference for a companion review of empirical content.

In v3 (May 2026), the dimensionful P5 contact-scale postulate is retired and replaced by the dimensionless weak-sector boundary condition P5’: $g_{2,W}^2 = 4/(3\pi)$, equivalently $M_W/v = 1/\sqrt{3\pi}$ (see the companion electroweak-boundary note, Zenodo DOI:10.5281/zenodo.20075926). A new appendix records the fiber-triviality of the FPA bundle over real twistor-line moduli.

DOI

https://doi.org/10.5281/zenodo.20076012