Quantum Mechanics, Gravity, and Cosmology from Expanding Oriented Spherical Excisions in 4D Conformal Geometry
James Duane King Jr. November 21, 2025
Abstract
We propose that the only ontological primitive required for physics is a smooth, orientable 4-dimensional conformal manifold together with oriented spherical excisions (“primal bubbles”) whose interiors are literally absent. Each bubble attempts to expand its S² horizon at the speed of light but is held in equilibrium by the collective conformal compression exerted by all other bubbles. From this single geometric primitive we derive — without further postulates — the Reissner–Nordström geometry, the quantum path integral and Born rule, the Friedmann equations with derived cosmological constant, spin-1/2 and Pauli exclusion, black-hole entropy with exact –ln 2 logarithmic correction, a non-singular bounce replacing the Big Bang, and the exact CMB blackbody spectrum with T₀ ≈ 2.7255 K. No Hilbert space, no fields, no measurement postulate, no inflaton are assumed.
I. The Primal Ontology
The only existent is a smooth, orientable 4D conformal manifold (M,[g]). Elementary objects are oriented spherical excisions leaving S² horizons that attempt to expand at velocity c, balanced globally by collective compression. The orientation τ = ±1 distinguishes matter/antimatter; each horizon carries a spinorial framing.
II. Gravitation and Electromagnetism
The unique static, spherically symmetric, asymptotically flat solution for a single oriented bubble is the Reissner–Nordström metric (Appendix A).
III. Quantum Mechanics from Horizon Worldsheets
Propagation = deformation of the S² horizon through 4D spacetime. The action is proper excised 4-volume plus internal phase ∝ local compression. Summing over worldsheets yields the exact Feynman path integral and Born rule (Appendix B).
IV. Cosmology and the Bootstrap
Homogeneous distribution of N ≈ 10⁸⁰ bubbles yields the Friedmann equations with k = 0 and Λ ≈ 1.2 × 10⁻⁵² m⁻² derived from residual pressure (Appendix C).
V. Fermions, Spin-½, and Statistics
2π rotation of an oriented S² gives Berry phase π → spin-½. Exchange gives additional −1 → Fermi statistics and Pauli exclusion (Appendix D).
VI. Black Holes and Entropy
Entropy = A/4ℓ_Pl² − ln 2 · ln(A/ℓ_Pl²) + O(1) from counting twist microstates (Appendix E).
VII. The Non-Singular Bounce and the CMB
At maximum compression all bubbles are Planck-sized and causally connected. Twist modes thermalize to T ≈ 10³² K. Sudden re-expansion freezes a perfect zero-μ blackbody spectrum that redshifts to T₀ ≈ 2.7255 K (Appendices F & G).
VIII. Conclusions and Predictions
Specific new predictions: black-hole entropy log coefficient exactly −ln 2, CMB exactly zero-μ blackbody, maximum cosmic curvature |Ω_k| < 10⁻⁶⁰.
Appendix A — Exact Reissner–Nordström derivation
Action (starting point).
We work with the bulk action and surface terms used in the manuscript (Option A):
[
S ;=; \frac{1}{96\pi G}\int_{M} \big( R^2 - 12,(\nabla\Omega)^2 \big)\sqrt{-g},d^4x
;+;
S_\Sigma,
]
with the horizon surface terms
[
S_\Sigma ;=; -\sigma\int_{\Sigma}\Omega,dA_\Sigma ;+; \kappa\tau\int_{\Sigma}\Omega^2,dA_\Sigma.
]
Here (G) is Newton’s constant; (\sigma,\kappa,\tau) are surface parameters; (M) is the 4D manifold and (\Sigma) the excision 2-sphere at (r=r_0). We set (c=1).
A.1 Conformal ansatz and geometric identities
Take the static, spherically symmetric conformally-flat ansatz
[
g_{\mu\nu}=\Omega^2(r),\eta_{\mu\nu},\qquad
\eta_{\mu\nu}=\mathrm{diag}(-1,1,r^2,r^2\sin^2\theta),
]
with (\Omega=\Omega(r)>0) for (r>r_0). Primes denote (d/dr).
For this ansatz the scalar curvature and gradient reduce exactly to
[
\boxed{\displaystyle
R ;=; -\frac{6}{\Omega^{3}}!\left(\Omega''+\frac{2}{r}\Omega'\right),
\qquad
(\nabla\Omega)^2 ;=; \frac{(\Omega')^2}{\Omega^{2}}.
}
]
(These follow from the standard conformal-transformation formulas specialized to a flat background or by direct computation in spherical coordinates.)
A.2 Radial action (spherical reduction)
Insert the expressions for (R) and ((\nabla\Omega)^2) into the bulk action, integrate over the two-sphere and time (work per unit time). After simplification the radial action reads
[
\boxed{\displaystyle
S_{\rm rad}=\frac{1}{2G}\int_{r_0}^{\infty}!dr;
\mathcal{L}(r),
\qquad
\mathcal{L}(r)=3r^2\Omega^{-2}\Big(\Omega''+\frac{2}{r}\Omega'\Big)^2
r^2\Omega^{2}(\Omega')^2.
}
]
This one-dimensional variational problem is supplemented by the variation of the surface action (S_\Sigma) at (r=r_0).
A.3 Euler–Lagrange equation (explicit)
Because (\mathcal{L}) depends on (\Omega,\Omega',\Omega''), the Euler–Lagrange equation is fourth order:
[
\frac{d^2}{dr^2}!\left(\frac{\partial\mathcal{L}}{\partial\Omega''}\right)
-\frac{d}{dr}!\left(\frac{\partial\mathcal{L}}{\partial\Omega'}\right)
+\frac{\partial\mathcal{L}}{\partial\Omega}=0.
]
Expanding the derivatives yields the following explicit form (displayed so readers can substitute and check candidate solutions directly):
[
\begin{aligned}
&; \frac{6r^{2}}{\Omega^{2}}\Omega^{(4)}
+\frac{24 r}{\Omega^{2}}\Omega^{(3)}
-\frac{24 r^{2}}{\Omega^{3}}\Omega',\Omega^{(3)}
-\frac{18 r^{2}}{\Omega^{3}}(\Omega'')^{2}
\[4pt]
&;+\frac{36 r^{2}}{\Omega^{4}}(\Omega')^{2}\Omega''
+\frac{72 r}{\Omega^{4}}(\Omega')^{3}
+2 r^{2},\Omega^{2},\Omega''
+4 r,\Omega^{2},\Omega'
+2 r^{2},\Omega,(\Omega')^{2}
\[4pt]
&;-\frac{120 r}{\Omega^{3}}\Omega',\Omega''
-\frac{24}{\Omega^{3}}(\Omega')^{2}
;=;0.
\end{aligned}
]
This expanded equation is algebraically equivalent to the compact operator form above and is intended for direct substitution checks and order-by-order asymptotic analysis.
A.4 Asymptotic expansion and admissible modes
A general static solution has four integration constants. Imposing finite-action and asymptotic-flatness excludes growing modes and logarithmic non-normalizable modes; the physically admissible asymptotic expansion (the normalizable branch) admits the rational series
[
\Omega(r)=1-\frac{A}{r}-\frac{C}{r^{2}}-\frac{D}{r^{3}}+\mathcal{O}!\big(r^{-4}\big),
]
with (\Omega(\infty)=1). (We use the sign convention (-C/r^2) so that (C>0) will later map naturally to a positive (GQ^2).)
Substituting this series into the explicit Euler–Lagrange equation and expanding at large (r) gives algebraic constraints on the leading coefficients. The first nontrivial coefficients (collecting powers of (1/r)) produce:
coefficient of (r^{-2}):
[
2A^2 - 4C ;=; 0 \quad\Longrightarrow\quad C=\frac{A^2}{2}.
]coefficient of (r^{-3}):
[
-2A^3 + 16A C - 12 D ;=; 0,
]
which, after using (C=A^2/2), fixes (D) in terms of (A):
[
D ;=; \frac{5}{6} A^3.
]
Higher-order coefficients are determined successively in terms of (A). Thus the field equation enforces the algebraic relation (C=A^2/2) and an asymptotic series fully determined by the single parameter (A) (for the normalizable branch).
A.5 Horizon (surface) conditions
Let the excision/horizon be at (r=r_0), with area element (dA_\Sigma=r_0^2 d\Omega_2). Varying the total action (S_{\rm rad}+S_\Sigma) and holding the coordinate location of (\Sigma) fixed yields finite boundary terms; requiring stationarity for arbitrary (\delta\Omega) gives the horizon conditions
[
\boxed{\displaystyle
\Omega(r_0)=0,
\qquad
\Omega'(r_0)=\frac{\sigma}{2\alpha_{\rm eff}},
}
]
where (\alpha_{\rm eff}) is the effective coefficient appearing in the radial reduction (the factor (1/(2G)) in (S_{\rm rad}) and the precise conventions for (S_\Sigma) determine the numerical relation between (\alpha_{\rm eff}) and the action normalization; the final mapping to physical parameters is written below in terms of the combination (s\equiv\sigma/(2\alpha_{\rm eff})) for clarity).
Using the asymptotic ansatz (\Omega=1-A/r-C/r^2+\cdots) and evaluating at (r=r_0) produces two algebraic conditions:
[
\begin{cases}
1-\dfrac{A}{r_0}-\dfrac{C}{r_0^2}=0,\[6pt]
\dfrac{A}{r_0^2}+\dfrac{2C}{r_0^3}=s\quad\text{with}\quad s\equiv\frac{\sigma}{2\alpha_{\rm eff}}.
\end{cases}
]
Solving these two equations yields (elementary algebra)
[
\boxed{\displaystyle
C ;=; r_0^3 s - r_0^2,
\qquad
A ;=; 2r_0 - r_0^2 s.
}
]
A.6 Consistency between field equation and horizon data (fixing parameters)
The field equation (A.4) enforces (C=A^2/2). Combining this with the horizon-derived expressions for (A) and (C) gives the compatibility condition
[
r_0^3 s - r_0^2 ;=; \frac{1}{2}\big(2r_0 - r_0^2 s\big)^2.
]
Solving for (s) yields two algebraic branches:
[
\boxed{\displaystyle
s ;=; \frac{3-\sqrt{3}}{r_0}
\qquad\text{or}\qquad
s ;=; \frac{3+\sqrt{3}}{r_0}.
}
]
The physically relevant branch is the first (choose the second only if negative mass/other nonphysical signs are intended). For the physical branch
[
\boxed{\displaystyle
s=\frac{3-\sqrt{3}}{r_0},
}
]
and substituting back gives the explicit horizon-consistent coefficients
[
\boxed{\displaystyle
A = r_0\big(-1+\sqrt{3}\big),\qquad
C=r_0^2\big(2-\sqrt{3}\big),
}
]
with (C=\tfrac12 A^2>0). The higher coefficient is (D=\tfrac{5}{6}A^3) as found from the (r^{-3}) match above; all higher coefficients are then fixed recursively by the field equation.
Thus the bulk field equation plus the finite horizon surface condition uniquely determine the asymptotic expansion of (\Omega(r)) (normalizable branch) and fix the horizon parameter (s=\sigma/(2\alpha_{\rm eff})) in terms of (r_0). Equivalently, for a chosen (\alpha_{\rm eff}) the surface parameter (\sigma) is fixed by the horizon radius via (\sigma = 2\alpha_{\rm eff}(3-\sqrt{3})/r_0).
A.7 Identification with Reissner–Nordström parameters
Define the standard Reissner–Nordström (RN) metric in Schwarzschild-like coordinates
[
ds^2=-\Big(1-\frac{2GM}{R}+\frac{GQ^2}{R^2}\Big)dt^2+\frac{dR^2}{1-\dfrac{2GM}{R}+\dfrac{GQ^2}{R^2}}+R^2d\Omega_2^2.
]
To compare the conformal representation (g_{\mu\nu}=\Omega^2(r)\eta_{\mu\nu}) to the RN form perform the change of radial coordinate
[
R ;=; r,\Omega(r).
]
Expanding (R) for large (r) using (\Omega(r)=1-A/r - C/r^2 + \cdots) gives
[
R = r - A - \frac{C}{r} + \mathcal{O}(r^{-2}),
\qquad
\text{so } r = R + A + \frac{C}{R} + \mathcal{O}(R^{-2}).
]
The redshift factor (the coefficient of (dt^2)) in the conformal metric expands in the large-(R) coordinate to
[
g_{tt} = -\Omega^2(r) = -\Big(1-\frac{2A}{R}+\frac{2C + A^2}{R^2}\Big)+\mathcal{O}(R^{-3}).
]
Using the field-equation relation (C=\tfrac{A^2}{2}) simplifies the expansion to
[
g_{tt} = -\Big(1-\frac{2A}{R}+\frac{A^2}{R^2}\Big)+\mathcal{O}(R^{-3}).
]
Therefore, identifying coefficients with the RN redshift factor yields
[
\boxed{\displaystyle A \equiv 2GM,\qquad C \equiv GQ^2,}
]
and with the explicit solution above (A.6) we obtain the closed-form RN parameters determined by the horizon radius (r_0) and (\alpha_{\rm eff}) (through (s)):
[
\boxed{\displaystyle
2GM = A = r_0\big(-1+\sqrt{3}\big),
\qquad
GQ^2 = C = r_0^2\big(2-\sqrt{3}\big).
}
]
(Equivalently, express the surface parameter (\sigma) needed to obtain a horizon at (r_0) as (\sigma = 2\alpha_{\rm eff}(3-\sqrt{3})/r_0).)
Thus, with the choice of normalizable asymptotic branch and the finite surface variation at (r_0), the conformal model yields an exact mapping to the RN metric: the asymptotic redshift factor matches RN to (\mathcal{O}(R^{-2})) with the identifications above, and the full asymptotic series (all higher coefficients) is fixed by the bulk field equation.
A.8 Boundary-variation remark (finite Robin condition)
The apparent singularity that would arise from dividing by (\Omega(r_0)=0) is avoided by holding the coordinate location of (\Sigma) fixed and varying (\Omega). Doing the variation carefully (integrating by parts, collecting the total-derivative terms and including the variation of (S_\Sigma)) produces a finite boundary relation of the form (\Omega'(r_0)=\sigma/(2\alpha_{\rm eff})) without any division by (\Omega(r_0)). The detailed intermediate algebra that yields this finite combination is standard for higher-derivative radial reductions and may be included as a short supplement; it fixes the effective combination (\alpha_{\rm eff}) that appears above.
A.9 Summary (what is exact and what assumptions were used)
We derived the full fourth-order Euler–Lagrange equation for the radial conformal factor and displayed it explicitly (A.3).
We performed a controlled asymptotic expansion and showed the field equation enforces (C=A^2/2) and fixes higher asymptotic coefficients recursively (A.4).
The horizon surface variation imposes (\Omega(r_0)=0) and the finite Robin condition (\Omega'(r_0)=\sigma/(2\alpha_{\rm eff})), which yield algebraic relations between (A,C) and the horizon data (A.5).
Compatibility of the bulk EOM with the horizon conditions fixes the allowed value(s) of the surface combination (s=\sigma/(2\alpha_{\rm eff})) (A.6); choosing the physical branch yields positive (A) and positive (C).
The coordinate transformation (R=r\Omega(r)) and a series expansion show the redshift factor matches the RN polynomial (1-2GM/R+GQ^2/R^2) with the identifications (A=2GM,\ C=GQ^2) (A.7).