A Lean-Verified Conditional Derivation of the Born Density from Two-Cell Structural Mixing

This record contains a technical note on a conditional derivation of the quadratic Born density from a minimal two-cell structural mixing condition.

The article starts from a Hilbert-space representation of states and assumes that event weights admit a local gauge-invariant pointwise modulus density. It then adds one structural invariance condition: the total weight of two equal cells is unchanged under a real rotation mixing their amplitudes. Under these assumptions, the local density is forced to be quadratic, f(r) = cr², yielding the usual Born weight after normalization.

The result is not presented as a replacement for Gleason-type or decision-theoretic derivations of the Born rule. Its purpose is narrower: to isolate a simple local route by which the square law follows once the density form and a minimal two-cell basis-independence principle are granted.

The functional-equation core and the two-cell reduction have been formalized in Lean 4 using Mathlib. The accompanying Lean development is archived separately at Zenodo: https://doi.org/10.5281/zenodo.20528721. The reported axiom footprint for every formalized theorem in the development, including the measure-theoretic born_density and born_rule_weight, is the standard classical Lean base, with no admitted steps (sorryAx) in any of the formalized theorems.

The manuscript and the accompanying Lean development were prepared with extensive AI assistance. The author reviewed, selected, corrected, and revised the outputs and remains responsible for the final text, the mathematical claims, the Lean formalization, and the interpretive framing.