Physical Law from Polynomial Truncation on the Product Flag Manifold Fl₁,₂(ℂ³) × ℂP¹

This note clarifies the physical interpretation of the product flag manifold
M = Fl₁,₂(ℂ³) × ℂP¹ within a unified geometric framework for fundamental physics.

The manifold is not a background or configuration space, but the structured space of unrealized possibility. Physical process is identified with the irreversible contraction of accessible Kähler volume Ω, governed locally by the bridge law
d/dτ lnΩ = −πΓ(τ).

Different physical regimes arise as successive Kähler quotients of M, forming a nested hierarchy of effective manifolds. Each quotient inherits the same geometric structures (volume form, curvature, and contraction dynamics), but with reduced dimensionality and accessible volume.

The central result is that physical laws emerge as geometric scars of polynomial truncation of the Kähler potential under contraction. At each depth, only a characteristic polynomial degree survives:

• Quartic (degree 4): particle physics and Higgs/gauge self-interactions
• Quadratic (degree 2): relativity and metric structure
• Linear (degree 1): classical mechanics
• Constant (degree 0): black hole limit

This establishes a structural origin for the form of physical laws across scales: dynamics at each regime are uniquely constrained by the surviving curvature invariants of the underlying geometry.

Measurement is interpreted as a projection between contraction depths: an observer, defined by its own polynomial degree, can only resolve structure of equal or lower order. This provides a unified geometric account of quantum measurement, classical determinism, and horizon phenomena.

The framework recasts physics as a single contraction hierarchy on a uniquely selected compact Kähler manifold, where time, scale, and law all emerge from the progressive reduction of possibility.