The Thales partition equation of motion, based on the single Poincaré metric ds2 =
dχ2/(1−χ2)2 on the asymmetry variable χ∈(0,1), passes five of six numerical stress tests
but fails to reproduce the Schwarzschild radial geodesic equation quantitatively. We show
that the failure is resolved by a compound partition metric of the form
gχχ = χ2p(1−χ2)−2q
,
which introduces a temporal factor (χ2p, vanishing at the horizon as gtt → 0 does in
Schwarzschild) alongside the spatial factor ((1−χ2)−2q, divergent at spatial infinity). The
Christoffel symbol of this family is
Γp,q(χ) = p+ (2q+ p)χ2
χ(1−χ2).
Setting (p,q) = (1,1) reproduces the Schwarzschild connection
ΓS(χ) = 1 + 3χ2
χ(1−χ2)
exactly, with no residual mismatch. The corresponding metric is gχχ = χ2(1−χ2)−2, and
the Schwarzschild gravitational potential appears as V(χ) =−1/(8r2
sχ2).
The connection decomposes as ΓS = 1/(χ(1−χ2)) + 3χ/(1−χ2): one temporal unit
plus three spatial units, reflecting the 3 + 1 dimensionality of spacetime at the connection
level. At the metric level, the factorization is gχχ = gtime·gspace with gtime = χ2 and
gspace = (1−χ2)−2, where gtime vanishes at the horizon (χ = 0) just as the Schwarzschild
temporal component gtt = 1−rs/r→0.
Numerical integration confirms the exact match: trajectory error∼10−11 over the full
infall, compared to 73% mean mismatch for the single Poincaré metric (n = 2) and 47%
for the symmetric compound (n = 4). The compound partition framework provides the
geometric arena; Newtonian gravity provides the potential. Together they exactly reproduce
Schwarzschild radial dynamics from the constraint a+ b= 1.
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