Schrödinger from McGehee

Derived — 80% confidence

Claim: The Schrödinger equation arises as the constraint that fixes the existence measure on shape space, once McGehee’s regularisation separates scale from shape. No quantisation axioms are imposed. No dynamics are presupposed.

Existence measure |ψ|² on the shape sphere. Bright regions = high probability shapes. Red dots = collision punctures. The measure lives on shape space — shapes with higher weight are more probable, not “later.”

The Construction

McGehee coordinates split every N-body configuration into three pieces: a scale factor r (overall size), a shape s (the point on the Kendall sphere), and fibre angles θ (rotational orientation). At collision, r → 0, but s stays well-defined on the shape sphere.

(q₁, …, q_N) → (r, s, θ)
r → 0 at collision, s ∈ S² remains regular

The existence measure on shapes satisfies:
Ĥ_shape ψ(s) = E ψ(s)
(stationary constraint, not evolution — ψ assigns existence weights to shapes)

What This Means

Scale is not a physical observable — it is gauged away by the shape-space quotient. The Schrödinger equation does not need to be postulated on an external Hilbert space. It arises as the stationary constraint that determines which shapes have high existence weight and which have low, on the compact shape manifold. The Laplacian is the shape-sphere Laplacian; the potential is the shape potential restricted to the collision-regularised surface. The shapes sit there with their probabilities — |ψ(s)|² is an existence measure, not a prediction of what will happen next.

Why this matters for Barbour’s programme: Julian has always maintained that scale is unphysical and that shape space is the true arena. This derivation says: if you take that seriously and follow McGehee’s regularisation to its conclusion, the Schrödinger constraint is what you get. Not as dynamics imposed from outside — as the unique condition that fixes the existence measure on the space you already have. The shapes just sit there. The equation determines their probabilities.

Open: Extension to N > 3 requires the higher-dimensional Kendall sphere. The N=3 case is complete on S².