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Pauli Exclusion from Geometry: A Non-Invertibility Route to Fermionic Antisymmetry

J. P. Charlton

Published Jan 1, 2026• 0 citations
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Abstract

We derive Pauli exclusion as a geometric theorem inside Charlton Geometry, without assuming quantum antisymmetry as a postulate. The core primitive is the Charlton non-invertibility $\omega \sim -\omega$: the Lorentzian data induced by the Charlton Operator forgets the global polarity of the distinguished field $\omega$, producing a $\mathbb{Z}_2$ ambiguity when lifting macroscopic symmetry loops into the spinorial representation.For two indistinguishable localized defects on a spatial slice, the configuration space $C_2$ has $\pi_1(C_2)\cong \mathbb{Z}_2$, generated by the exchange loop. Non-invertibility induces a polarity bundle over $C_2$ with two inequivalent holonomy branches: a trivial (``short'') lift with exchange phase $+1$ (bosonic symmetry), and a nontrivial (``long'') lift with exchange phase $-1$ (fermionic antisymmetry). On the long-lift branch, exchange realizes the nontrivial element of $\pi_1(SO(3))$, whose lift to $Spin(3)\cong SU(2)$ terminates at the central element $-1$, forcing $\Psi(x_1,x_2)=-\Psi(x_2,x_1)$ and yielding exclusion in the coincidence limit. We emphasize classification, not compulsion: Pauli exclusion is the geometric property of a stable topological branch, with defect-species placement addressed conditionally in later work. Original DOI: 10.5281/zenodo.18214243 Version: 0.9.0

DOI
10.17605/osf.io/tgp74

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Pauli Exclusion from Geometry: A Non-Invertibility Route to Fermionic Antisymmetry

J.P. Charlton Jr.

We derive Pauli exclusion as a geometric theorem inside Charlton Geometry, without assuming quantum antisymmetry as a postulate. The core primitive is the Charlton non-invertibility $\omega \sim -\omega$: the Lorentzian data induced by the Charlton Operator forgets the global polarity of the distinguished field $\omega$, producing a $\mathbb{Z}_2$ ambiguity when lifting macroscopic symmetry loops into the spinorial representation.For two indistinguishable localized defects on a spatial slice, the configuration space $C_2$ has $\pi_1(C_2)\cong \mathbb{Z}_2$, generated by the exchange loop. Non-invertibility induces a polarity bundle over $C_2$ with two inequivalent holonomy branches: a trivial (``short'') lift with exchange phase $+1$ (bosonic symmetry), and a nontrivial (``long'') lift with exchange phase $-1$ (fermionic antisymmetry). On the long-lift branch, exchange realizes the nontrivial element of $\pi_1(SO(3))$, whose lift to $Spin(3)\cong SU(2)$ terminates at the central element $-1$, forcing $\Psi(x_1,x_2)=-\Psi(x_2,x_1)$ and yielding exclusion in the coincidence limit. We emphasize classification, not compulsion: Pauli exclusion is the geometric property of a stable topological branch, with defect-species placement addressed conditionally in later work.

•Jan 11, 2026•0 citations

Born's Rule from Geometry: A Non-Invertibility Route to Quantum Probabilities

J. P. Charlton

We derive Born’s rule as a geometric theorem inside Charlton Geometry (CG), removing probability as a primitive axiom. The core mechanism is non-invertibility: the Charlton Operator η = δ − 2 ω ⊗ ω forgets polarity, since η(ω) = η(−ω), and therefore admits no global inverse back to Euclidean parent data. This lost state is compounded by intrinsic phase: a U(1) sector whose periodicity identifies θ ∼ θ +2π, further collapsing distinguishable microstate into a single Lorentz-observable macrostate. We formalize both effects as an observational quotient q : e S ↠ S with fiber group Z2 °ø U(1), and show that any observer confined to S must represent the unresolved fiber by an invariant measure. The Born distribution then appears as the unique gauge- and rotation-invariant pushforward of Haar measure from the non-invertible fiber to measurement outcomes, yielding P(i) = ∥Πiψ∥2 without postulating ψ, Πi, or the squared modulus rule. Crucially, the multiplicative composition of these probabilities is recovered not by postulate, but as a theorem of holonomy functoriality within the lift-bundle connection. In this view, quantum probabilities are not ontic propensities but a geometric accounting ofirrecoverable information discarded by reflection and phase identification. Original DOI: 10.5281/zenodo.18214468 Version: 0.9.0

•Jan 1, 2026•0 citations

Spin/Phase/Holonomy from Defect Microstructure in Charlton Geometry: A Single Theorem Closing the 2026a–2026c Obligations

J. P. Charlton

We unify the Pauli, Born, and Bell “brick” arguments of the Charlton Geometry (CG) series into a single Spinc–holonomy framework in which (i) the Clifford/Spin lift is canonical, (ii) the Z2 exchange sign is determined from defect microstructure rather than selected as an allowed option, and (iii) the U(1) phase sector arises as intrinsic monodromy of the CG connection. The central result is a defect Spinc–holonomy determination theorem: localized ω-defects canonically induce a Spinc structure on indistinguishable configuration spaces, together with defect-determined invariants (ν, θ) controlling exchange holonomy and phase monodromy. From this we derive (a) Pauli exclusion on the ν = 1 branch, (b) Born uniqueness as the invariant pushforward of Haar measure on Z2 °ø U(1) under rotational covariance and the spinorial half-angle constraint, and (c) the precise location of Bell–CHSH as a theorem of the factorizable subclass of quotients, with violation corresponding to shared-lift coupling. We include an explicit obligation-closure map showing which lemmas discharge the open obligations stated in Charlton 2026a–2026c. Original DOI: 10.5281/zenodo.18216845 Version: 0.9.0

•Jan 1, 2026•0 citations

Born's Rule from Geometry: A Non-Invertibility Route to Quantum Probabilities

J.P. Charlton Jr.

We derive Born’s rule as a geometric theorem inside Charlton Geometry (CG), removing probability as a primitive axiom. The core mechanism is non-invertibility: the Charlton Operator η = δ − 2 ω ⊗ ω forgets polarity, since η(ω) = η(−ω), and therefore admits no global inverse back to Euclidean parent data. This lost state is compounded by intrinsic phase: a U(1) sector whose periodicity identifies θ ∼ θ +2π, further collapsing distinguishable microstate into a single Lorentz-observable macrostate. We formalize both effects as an observational quotient q : e S ↠ S with fiber group Z2 °ø U(1), and show that any observer confined to S must represent the unresolved fiber by an invariant measure. The Born distribution then appears as the unique gauge- and rotation-invariant pushforward of Haar measure from the non-invertible fiber to measurement outcomes, yielding P(i) = ∥Πiψ∥2 without postulating ψ, Πi, or the squared modulus rule. Crucially, the multiplicative composition of these probabilities is recovered not by postulate, but as a theorem of holonomy functoriality within the lift-bundle connection. In this view, quantum probabilities are not ontic propensities but a geometric accounting ofirrecoverable information discarded by reflection and phase identification.

•Jan 11, 2026•0 citations

Spin/Phase/Holonomy from Defect Microstructure in Charlton Geometry: A Single Theorem Closing the 2026a–2026c Obligations

J.P. Charlton Jr.

We unify the Pauli, Born, and Bell “brick” arguments of the Charlton Geometry (CG) series into a single Spinc–holonomy framework in which (i) the Clifford/Spin lift is canonical, (ii) the Z2 exchange sign is determined from defect microstructure rather than selected as an allowed option, and (iii) the U(1) phase sector arises as intrinsic monodromy of the CG connection. The central result is a defect Spinc–holonomy determination theorem: localized ω-defects canonically induce a Spinc structure on indistinguishable configuration spaces, together with defect-determined invariants (ν, θ) controlling exchange holonomy and phase monodromy. From this we derive (a) Pauli exclusion on the ν = 1 branch, (b) Born uniqueness as the invariant pushforward of Haar measure on Z2 °ø U(1) under rotational covariance and the spinorial half-angle constraint, and (c) the precise location of Bell–CHSH as a theorem of the factorizable subclass of quotients, with violation corresponding to shared-lift coupling. We include an explicit obligation-closure map showing which lemmas discharge the open obligations stated in Charlton 2026a–2026c.

•Jan 11, 2026•0 citations

A Geometric Origin for Fermi--Dirac Statistics: The Spin-Statistics Connection from $Z_2$ Normal Holonomy Without Lorentz Invariance

G Y Li

The Pauli exclusion principle---the reason atoms have shell structure and matter is stable against collapse---is conventionally explained through the spin-statistics theorem, whose every standard proof relies on the machinery of relativistic quantum field theory (Lorentz invariance and microcausality). Yet the principle governs entirely non-relativistic physics. This paper asks whether its non-relativistic content admits a correspondingly non-relativistic explanation, and argues that it does: the Fermi--Dirac statistics of a half-integer-spin particle can be obtained from the topology of three-dimensional space together with the M\"obius geometry of the particle, with no appeal to Lorentz invariance or microcausality. The relativistic input of the standard proof is replaced by a geometric one. Concretely, the proof rests on two algebraic-topological lemmas about three-dimensional space (the configuration space of two identical particles is homotopy equivalent to ^2; real line bundles over ^2 are classified by H^1(^2;_2) _2), together with an explicit equivariant bundle map identifying the relevant exchange line bundle with the normal bundle of a non-orientable surface in ^3. The construction realises the spin-statistics correspondence as a topological identity in the spirit of Kuckert--Mund~KuckertMund2004 and Anastopoulos~Anastopoulos2004, but using only standard algebraic topology without recourse to QFT axioms or geometric quantisation. If the argument stands, the Pauli exclusion principle emerges as a statement about the topology of three-dimensional space and the geometry of the particle, rather than about the causal structure of spacetime. The required geometric input on the particle is encapsulated as a single postulate (Section~sec:setup): identical fermionic particles are modelled as compact embedded surfaces in ^3 equipped with light-speed-constrained flow fields generating _2 normal holonomy () = -1. In this setting we prove: enumerate[label=(*)] The two-particle configuration space is homotopy equivalent to the real projective plane: C_2 ^2, so _1(C_2) _2. Real line bundles over ^2 are classified by H^1(^2;_2) _2: there are exactly two, the trivial bundle (holonomy +1) and the tautological Möbius bundle (holonomy -1). The exchange holonomy of the fermionic vortex-knot wavefunction equals -1 (a unit normal section traversing undergoes a sign reversal). An explicit equivariant bundle map : N|_L_{ex}|_, (s,) = (s,), is constructed and proved to be a well-defined bundle isomorphism if and only if () = -1 (Section~sec:equivariant, Proposition~prop:chi). The argument rests on one geometric axiom---the Normal Transport Axiom (NTA), that is connection-preserving---which selects =-1 from the two phases the configuration-space topology admits, playing the role microcausality plays in the standard relativistic proof (Section~sec:equivariant). The contribution is to show this selecting input can be geometric rather than relativistic. Therefore the two-particle wavefunction is antisymmetric under exchange: Fermi--Dirac statistics follows. enumerate Points (i)--(iii) are rigorous theorems. Point~(iv) is also rigorous (the bundle map is explicit and proved); the NTA is the one remaining axiom---a precisely stated geometric assumption that supplies the selection input required by the Leinaas--Myrheim classification, with the same logical status as microcausality in the standard proof. Point~(v) follows under the NTA. At the heart of the result is a striking geometric correspondence: the non-orientability of the Möbius particle (in ^3) and the non-orientability of the two-particle configuration space ^2 are the same _2 at two different scales---the scale of one particle and the scale of a pair---connected by the explicit bundle map . : spin-statistics theorem; holonomy; Möbius surface; real projective plane; configuration space; line bundle classification; exchange statistics; normal holonomy; Fermi--Dirac statistics; equivariant b

•Jul 13, 2026•0 citations
We derive Pauli exclusion as a geometric theorem inside Charlton Geometry, without assuming quantum antisymmetry as a postulate. The core primitive is the Charlton non-invertibility ω∼−ω\omega \sim -\omegaω∼−ω: the Lorentzian data induced by the Charlton Operator forgets the global polarity of the distinguished field ω\omegaω, producing a Z2\mathbb{Z}_2Z2​ ambiguity when lifting macroscopic symmetry loops into the spinorial representation.For two indistinguishable localized defects on a spatial slice, the configuration space C2C_2C2​ has π1(C2)≅Z2\pi_1(C_2)\cong \mathbb{Z}_2π1​(C2​)≅Z2​, generated by the exchange loop. Non-invertibility induces a polarity bundle over C2C_2C2​ with two inequivalent holonomy branches: a trivial (short'') lift with exchange phase $+1$ (bosonic symmetry), and a nontrivial (long'') lift with exchange phase −1-1−1 (fermionic antisymmetry). On the long-lift branch, exchange realizes the nontrivial element of π1(SO(3))\pi_1(SO(3))π1​(SO(3)), whose lift to Spin(3)≅SU(2)Spin(3)\cong SU(2)Spin(3)≅SU(2) terminates at the central element −1-1−1, forcing Ψ(x1,x2)=−Ψ(x2,x1)\Psi(x_1,x_2)=-\Psi(x_2,x_1)Ψ(x1​,x2​)=−Ψ(x2​,x1​) and yielding exclusion in the coincidence limit. We emphasize classification, not compulsion: Pauli exclusion is the geometric property of a stable topological branch, with defect-species placement addressed conditionally in later work. Original DOI: 10.5281/zenodo.18214243 Version: 0.9.0
We derive Pauli exclusion as a geometric theorem inside Charlton Geometry, without assuming quantum antisymmetry as a postulate. The core primitive is the Charlton non-invertibility ω∼−ω\omega \sim -\omegaω∼−ω: the Lorentzian data induced by the Charlton Operator forgets the global polarity of the distinguished field ω\omegaω, producing a Z2\mathbb{Z}_2Z2​ ambiguity when lifting macroscopic symmetry loops into the spinorial representation.For two indistinguishable localized defects on a spatial slice, the configuration space C2C_2C2​ has π1(C2)≅Z2\pi_1(C_2)\cong \mathbb{Z}_2π1​(C2​)≅Z2​, generated by the exchange loop. Non-invertibility induces a polarity bundle over C2C_2C2​ with two inequivalent holonomy branches: a trivial (``short'') lift with exchange phase +1+1+1 (bosonic symmetry), and a nontrivial (``long'') lift with exchange phase −1-1−1 (fermionic antisymmetry). On the long-lift branch, exchange realizes the nontrivial element of π1(SO(3))\pi_1(SO(3))π1​(SO(3)), whose lift to Spin(3)≅SU(2)Spin(3)\cong SU(2)Spin(3)≅SU(2) terminates at the central element −1-1−1, forcing Ψ(x1,x2)=−Ψ(x2,x1)\Psi(x_1,x_2)=-\Psi(x_2,x_1)Ψ(x1​,x2​)=−Ψ(x2​,x1​) and yielding exclusion in the coincidence limit. We emphasize classification, not compulsion: Pauli exclusion is the geometric property of a stable topological branch, with defect-species placement addressed conditionally in later work.
We derive Born’s rule as a geometric theorem inside Charlton Geometry (CG), removing probability as a primitive axiom. The core mechanism is non-invertibility: the Charlton Operator η = δ − 2 ω ⊗ ω forgets polarity, since η(ω) = η(−ω), and therefore admits no global inverse back to Euclidean parent data. This lost state is compounded by intrinsic phase: a U(1) sector whose periodicity identifies θ ∼ θ +2π, further collapsing distinguishable microstate into a single Lorentz-observable macrostate. We formalize both effects as an observational quotient q : e S ↠ S with fiber group Z2 °ø U(1), and show that any observer confined to S must represent the unresolved fiber by an invariant measure. The Born distribution then appears as the unique gauge- and rotation-invariant pushforward of Haar measure from the non-invertible fiber to measurement outcomes, yielding P(i) = ∥Πiψ∥2 without postulating ψ, Πi, or the squared modulus rule. Crucially, the multiplicative composition of these probabilities is recovered not by postulate, but as a theorem of holonomy functoriality within the lift-bundle connection. In this view, quantum probabilities are not ontic propensities but a geometric accounting ofirrecoverable information discarded by reflection and phase identification. Original DOI: 10.5281/zenodo.18214468 Version: 0.9.0
We unify the Pauli, Born, and Bell “brick” arguments of the Charlton Geometry (CG) series into a single Spinc–holonomy framework in which (i) the Clifford/Spin lift is canonical, (ii) the Z2 exchange sign is determined from defect microstructure rather than selected as an allowed option, and (iii) the U(1) phase sector arises as intrinsic monodromy of the CG connection. The central result is a defect Spinc–holonomy determination theorem: localized ω-defects canonically induce a Spinc structure on indistinguishable configuration spaces, together with defect-determined invariants (ν, θ) controlling exchange holonomy and phase monodromy. From this we derive (a) Pauli exclusion on the ν = 1 branch, (b) Born uniqueness as the invariant pushforward of Haar measure on Z2 °ø U(1) under rotational covariance and the spinorial half-angle constraint, and (c) the precise location of Bell–CHSH as a theorem of the factorizable subclass of quotients, with violation corresponding to shared-lift coupling. We include an explicit obligation-closure map showing which lemmas discharge the open obligations stated in Charlton 2026a–2026c. Original DOI: 10.5281/zenodo.18216845 Version: 0.9.0
We derive Born’s rule as a geometric theorem inside Charlton Geometry (CG), removing probability as a primitive axiom. The core mechanism is non-invertibility: the Charlton Operator η = δ − 2 ω ⊗ ω forgets polarity, since η(ω) = η(−ω), and therefore admits no global inverse back to Euclidean parent data. This lost state is compounded by intrinsic phase: a U(1) sector whose periodicity identifies θ ∼ θ +2π, further collapsing distinguishable microstate into a single Lorentz-observable macrostate. We formalize both effects as an observational quotient q : e S ↠ S with fiber group Z2 °ø U(1), and show that any observer confined to S must represent the unresolved fiber by an invariant measure. The Born distribution then appears as the unique gauge- and rotation-invariant pushforward of Haar measure from the non-invertible fiber to measurement outcomes, yielding P(i) = ∥Πiψ∥2 without postulating ψ, Πi, or the squared modulus rule. Crucially, the multiplicative composition of these probabilities is recovered not by postulate, but as a theorem of holonomy functoriality within the lift-bundle connection. In this view, quantum probabilities are not ontic propensities but a geometric accounting ofirrecoverable information discarded by reflection and phase identification.
We unify the Pauli, Born, and Bell “brick” arguments of the Charlton Geometry (CG) series into a single Spinc–holonomy framework in which (i) the Clifford/Spin lift is canonical, (ii) the Z2 exchange sign is determined from defect microstructure rather than selected as an allowed option, and (iii) the U(1) phase sector arises as intrinsic monodromy of the CG connection. The central result is a defect Spinc–holonomy determination theorem: localized ω-defects canonically induce a Spinc structure on indistinguishable configuration spaces, together with defect-determined invariants (ν, θ) controlling exchange holonomy and phase monodromy. From this we derive (a) Pauli exclusion on the ν = 1 branch, (b) Born uniqueness as the invariant pushforward of Haar measure on Z2 °ø U(1) under rotational covariance and the spinorial half-angle constraint, and (c) the precise location of Bell–CHSH as a theorem of the factorizable subclass of quotients, with violation corresponding to shared-lift coupling. We include an explicit obligation-closure map showing which lemmas discharge the open obligations stated in Charlton 2026a–2026c.
A Geometric Origin for Fermi--Dirac Statistics: The Spin-Statistics Connection from Z2Z_2Z2​ Normal Holonomy Without Lorentz Invariance
The Pauli exclusion principle---the reason atoms have shell structure and matter is stable against collapse---is conventionally explained through the spin-statistics theorem, whose every standard proof relies on the machinery of relativistic quantum field theory (Lorentz invariance and microcausality). Yet the principle governs entirely non-relativistic physics. This paper asks whether its non-relativistic content admits a correspondingly non-relativistic explanation, and argues that it does: the Fermi--Dirac statistics of a half-integer-spin particle can be obtained from the topology of three-dimensional space together with the M\"obius geometry of the particle, with no appeal to Lorentz invariance or microcausality. The relativistic input of the standard proof is replaced by a geometric one. Concretely, the proof rests on two algebraic-topological lemmas about three-dimensional space (the configuration space of two identical particles is homotopy equivalent to ^2; real line bundles over ^2 are classified by H^1(^2;_2) _2), together with an explicit equivariant bundle map identifying the relevant exchange line bundle with the normal bundle of a non-orientable surface in ^3. The construction realises the spin-statistics correspondence as a topological identity in the spirit of Kuckert--Mund~KuckertMund2004 and Anastopoulos~Anastopoulos2004, but using only standard algebraic topology without recourse to QFT axioms or geometric quantisation. If the argument stands, the Pauli exclusion principle emerges as a statement about the topology of three-dimensional space and the geometry of the particle, rather than about the causal structure of spacetime. The required geometric input on the particle is encapsulated as a single postulate (Section~sec:setup): identical fermionic particles are modelled as compact embedded surfaces in ^3 equipped with light-speed-constrained flow fields generating _2 normal holonomy () = -1. In this setting we prove: enumerate[label=(*)] The two-particle configuration space is homotopy equivalent to the real projective plane: C_2 ^2, so _1(C_2) _2. Real line bundles over ^2 are classified by H^1(^2;_2) _2: there are exactly two, the trivial bundle (holonomy +1) and the tautological Möbius bundle (holonomy -1). The exchange holonomy of the fermionic vortex-knot wavefunction equals -1 (a unit normal section traversing undergoes a sign reversal). An explicit equivariant bundle map : N|_L_{ex}|_, (s,) = (s,), is constructed and proved to be a well-defined bundle isomorphism if and only if () = -1 (Section~sec:equivariant, Proposition~prop:chi). The argument rests on one geometric axiom---the Normal Transport Axiom (NTA), that is connection-preserving---which selects =-1 from the two phases the configuration-space topology admits, playing the role microcausality plays in the standard relativistic proof (Section~sec:equivariant). The contribution is to show this selecting input can be geometric rather than relativistic. Therefore the two-particle wavefunction is antisymmetric under exchange: Fermi--Dirac statistics follows. enumerate Points (i)--(iii) are rigorous theorems. Point~(iv) is also rigorous (the bundle map is explicit and proved); the NTA is the one remaining axiom---a precisely stated geometric assumption that supplies the selection input required by the Leinaas--Myrheim classification, with the same logical status as microcausality in the standard proof. Point~(v) follows under the NTA. At the heart of the result is a striking geometric correspondence: the non-orientability of the Möbius particle (in ^3) and the non-orientability of the two-particle configuration space ^2 are the same _2 at two different scales---the scale of one particle and the scale of a pair---connected by the explicit bundle map . : spin-statistics theorem; holonomy; Möbius surface; real projective plane; configuration space; line bundle classification; exchange statistics; normal holonomy; Fermi--Dirac statistics; equivariant b