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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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Title The Compound Partition Metric and Exact Schwarzschild Recovery: Temporal–Spatial Factorization on the Thales Manifold
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.
Work type Technical Documentation
Tags mean gap ratio, thales altitude, lorentz factor, asymmetry parameter, geometric mean, sxs catalog, remnant spin, coherence corridor, pythagorean triples, conserved binary partition, binary black holes, rational lattice, thales partition manifold, am–gm–hm inequality, harmonic mean, contraction hierarchy, gravitational-wave diagnostics, adjacent mean gaps
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Identifier 2604045168916
Entry date Apr 4, 2026, 8:55 PM UTC
License Creative Commons Attribution 4.0
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Author. Holder Elias DeJesus. Date Apr 4, 2026.
Information available at https://www.safecreative.org/work/2604045168916-the-compound-partition-metric-and-exact-schwarzschild-recovery-temporal-spatial-factorization-on-the-thales-manifold
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Registration: 2604045168916
