The Boson/Fermion Distinction: Spin-Statistics as Structural Invariant

Abstract

The textbook dichotomy between bosons and fermions is commonly presented as a primitive classification of nature, with the spin-statistics theorem as its iron law. This paper argues that the dichotomy is a derived, dimension-dependent shadow of a single structural relation: the exchange phase of identical particles equals their topological spin, $R = e^{2\pi i s}$. [ESTABLISHED] The relation holds across relativistic quantum field theory, topological field theory, and condensed-matter anyon systems; dimension enters only by quantizing the allowed values of $s$. [RETRODICTION — not evidence] Stating this as a single invariant is a unification of established results, not a new prediction. After stating the invariant and its dimensional quantization, the paper addresses a foundational question: whether a calculus whose primitive is the distinction — rather than the particle or the field — can derive exchange statistics from the act of distinction itself. A recent monograph in this tradition (Quni-Gudzinas, 2026a) exhibits the gap [textual finding]: the model of its exponential modality silently adopts the symmetric algebra, which corresponds to bosonic statistics, without deriving that choice from the primitive. The paper formalizes the required construction — two modal exponentials (symmetric and exterior), the braiding of two marks in a compact closed category (with the abelian-pair postulate made explicit), and the ribbon condition linking twist to exchange — and states the falsifiability conditions under which the derivation program succeeds or fails. [NOT YET EVIDENCE] The derivation is pre-registered here; it is not yet executed.

1. Introduction

Two identical particles can be exchanged. In quantum mechanics, the state of the pair carries a phase under exchange: $\psi(x_2, x_1) = \eta\,\psi(x_1, x_2)$. In three or more spatial dimensions, exchanging twice is topologically trivial, so $\eta^2 = 1$ and $\eta \in \{+1, -1\}$: the two possibilities are bosons (symmetric, $\eta = +1$) and fermions (antisymmetric, $\eta = -1$). The spin-statistics theorem states which sign is realized: $\eta = (-1)^{2s}$, where $s$ is the spin (Pauli, 1940; Duck and Sudarshan, 1998). [ESTABLISHED]

This paper makes two claims. First, the structural invariant is not the dichotomy itself but the relation $R = e^{2\pi i s}$ between exchange phase and topological spin; the binary is a shadow of that relation in three spatial dimensions. Second, a distinction-based foundation of physics — a calculus whose only primitive is the act of drawing a boundary — must, to claim the spin-statistics connection, derive exchange statistics from the primitive; the current state of that program contains a silent assumption that this paper identifies and replaces with a concrete derivation target.

2. So What? Why Should a Reader Care About This Research?

The stakes. The boson/fermion dichotomy is an input to every quantum theory: chemistry, the standard model, and condensed matter all assume it. This paper argues the dichotomy is not primitive — it is the three-dimensional shadow of a single structural relation, $R = e^{2\pi i s}$ (exchange phase equals topological spin) [ESTABLISHED]. The deeper question is whether a calculus whose only primitive is the act of drawing a boundary can derive exchange statistics from that act — or whether the premises of such a derivation end earlier than its proponents hoped. The answer matters because it tells you exactly which parts of the spin-statistics structure are negotiable: which framework conditions generate which statistics, and where a distinction-based foundation must stop and import physics.

What a physicist gets: a boundary map with the falsification executed. The pre-registered derivation program (T1–T3) carries falsifiability conditions (F1, F2′), and — unusually — the decisive test has already been run, honestly, with a mechanism: the abelian-pair postulate is ASSUMED (definitive — scalar exchange follows from abelianity; the calculus does not deliver it), and involutivity ($\sigma^2 = 1$) is a condition on the target category (symmetric, or Temperley–Lieb at $A^4 = 1$), not a theorem of the calculus (verified by executed code in the rigor-pass cycle). That is a positive result: the distinction-based program now knows exactly where its premises end, and the surviving route is channel-count (F2′, tied to DHR locality) — a falsifiable, physics-anchored condition instead of a silent assumption. The redirect is equally concrete: the substrate that actually carries the physics is the p-adic-anyon / topological-quantum-computation program (Temperley–Lieb braids, Bruhat–Tits buildings), and the notation-vs-engine distinction (ZX/classical structures) supplies the tools to say which part is syntax and which part is engine.

What a foundations researcher gets: a worked example of premises-depth discipline. This paper is the third rung of a chain (structural invariant → logical scalar → boundary map) that practices what it preaches: every primitive is labeled DERIVED or ASSUMED in the Parsimony Ledger, every boundary is conceded in plain language (the spin-statistics connection requires Lorentz and locality input the mark cannot supply), and the falsification of the program's own central condition (F2) is published as a ledger outcome rather than a concession or a retreat. A pre-registered test that is executed and reports its own failure is the rarest and most valuable object in this literature.

Practical utility — even though the derivation is incomplete. (1) The minimal-ontology theorem (logic + locality + kinematics + Lorentz) delimits exactly what any derivation of exchange statistics must contain — a checklist for future attempts, including AI-assisted ones. (2) The involutivity criterion (the interpretation functor exists iff the target braiding is involutive; $\sigma^2 \neq I$ generically in Temperley–Lieb) is a reusable classification instrument for anyon models. (3) The boundary map — which framework conditions generate which statistics (orbifolds, traid groups, graphs, TL quotients) — feeds directly the topological-quantum-computation platform map of the companion records. (4) The notation-vs-engine analysis is a methodological tool for evaluating any formal program that claims to derive physics.

How deep does the theorem go? Where do its premises end? [ESTABLISHED] $R = e^{2\pi i s}$ and its dimensional quantization. [ASSUMED — definitive] The abelian-pair postulate and involutive braiding, with the mechanism now known. [CONJECTURED] Any full derivation of exchange statistics from the mark alone. [CONCEDED] The spin-statistics connection requires Lorentz and locality input. The premises end exactly where the rigor-pass marked them — and the reader is told, per primitive, in the ledger.

What this paper does not claim. No completed derivation of exchange statistics from the mark (the program is pre-registered, not executed; F2′ is the surviving condition). No claim that the calculus is the unique foundation. No Lorentz-free spin-statistics connection (that boundary is conceded). What the reader gets: an honest, executed falsification, a surviving falsifiable route, and a boundary map of quantum statistics — the answer to "so what?" is that this paper tells you exactly what is proven, what is assumed, what is falsified, and where the physics has to enter.

3. The invariant: exchange phase equals topological spin

The exchange of two identical particles is a loop in their configuration space. The phase acquired is a representation of the fundamental group of that space (Leinaas and Myrheim, 1977; Laidlaw and DeWitt, 1971). [ESTABLISHED] The rotation of a single particle by $2\pi$ is the twist. In a ribbon braided tensor category — the mathematical home of particle-like excitations in topological order — the two are linked by the ribbon identity:

where $\theta_X$ is the twist (topological spin), $c_{X,X}$ the braiding, $d_X$ the quantum dimension, and $\mathrm{Tr}_q$ the quantum trace (Joyal and Street, 1993; Kitaev, 2006; Bakalov and Kirillov, 2001). [ESTABLISHED] For an abelian object, $d_X = 1$ and the identity reduces to

the universal spin-statistics relation. This relation has been proven directly from wavefunctions for fractional quantum Hall quasiparticles (Trung et al., 2022; Nardin et al., 2022) and is realized experimentally as measurable fractional spin (Comparin et al., 2021). [ESTABLISHED]

3.1 The 3+1D shadow

In three or more spatial dimensions the braiding is symmetric (involutive): $c_{Y,X} \circ c_{X,Y} = \mathrm{id}$. Then $\theta_X^2 = 1$, so $e^{2\pi i s} = \pm 1$, forcing $s \in \{0, \tfrac{1}{2}\} \bmod 1$. Integer spin gives the trivial representation (bosons); half-integer spin gives the sign representation (fermions) (Pauli, 1940; Streater and Wightman, 1964). [ESTABLISHED]

3.2 The 2+1D generalization

In two spatial dimensions the braiding is not involutive: the exchange group is the braid group, and $\theta_X = e^{2\pi i s}$ can be any phase. Particles with fractional statistics — anyons — realize the continuous range of $s \in \mathbb{R}/\mathbb{Z}$ (Leinaas and Myrheim, 1977; Wilczek, 1982; Haldane, 1991; Mund, 2008). [ESTABLISHED]

3.3 What is invariant

Across both regimes, the invariant content is unchanged: superselection sectors, fusion rules, and braiding data, with the relation $R = e^{2\pi i s}$ (Wang and Wen, 2014; Johnson-Freyd, 2015). [ESTABLISHED] Dimension enters only through which braided structures are realizable. The binary dichotomy is therefore not the invariant; the relation is.

4. Dimension quantization: bosons and fermions as a 3+1D shadow

Spatial dimension	Motion group	Braided structure	Allowed $s$	Statistics
2	Braid group $B_n$	Ribbon, non-symmetric	$s \in \mathbb{R}/\mathbb{Z}$	Anyons, $R = e^{2\pi i s}$
$\geq 3$	Permutation group $S_n$	Symmetric, involutive	$s \in \{0, \tfrac{1}{2}\} \bmod 1$	Bosons ($\eta = +1$), fermions ($\eta = -1$)

[ESTABLISHED] The table is the dimensional quantization of the spin parameter: in $d \geq 3$ the involutive braiding forces $2s \in \mathbb{Z}$; in $d = 2$ the quantization collapses to continuity. In $3{+}1$ dimensions a further input — microcausality and positive energy in a local relativistic theory — identifies which sign corresponds to which spin (Pauli, 1940; Duck and Sudarshan, 1998; Verch, 2001). [ESTABLISHED] That input is external to any purely algebraic derivation; the boundary is stated explicitly in Section 6.

5. The calculus of distinctions and its silent assumption

A distinction-based calculus begins with a mark: a boundary separating an inside from an outside. Its two primitive laws are Calling (idempotence: a mark repeated is the mark) and Crossing (involution: a boundary crossed twice returns to the unmarked state) (Spencer-Brown, 1969). A recent monograph develops this calculus toward physics, including a treatment of the half-turn phase $e^{i\pi} = -1$ and a claim that parity is the ancestor of physical spin-statistics (Quni-Gudzinas, 2026a, Section 2.3). [textual finding]

The monograph's Appendix A models its exponential modality with the symmetric algebra:

where $S^n(A)$ is the $n$-th symmetric power. The symmetric algebra is exactly the bosonic Fock construction: many-body states totally symmetric under exchange. [ESTABLISHED — the symmetric algebra realizes Bose statistics.] The monograph therefore silently chooses bosonic statistics as the semantic realization of its exponential, without deriving the choice from the act of distinction. [textual finding — verifiable against Quni-Gudzinas (2026a), Appendix A.]

The gap is structural, not cosmetic. The spin-statistics theorem is about the symmetry of the joint state of two identical marks under exchange; the calculus provides a phase for a single mark under rotation, but never constructs the exchange of two marks. The leap from "the mark has a half-turn phase" to "two marks anticommute" is asserted, not derived. (This assessment was reached independently in the deep-inquiry analysis of 2026-08-14; it is stated here as a checkable textual claim about the monograph.)

6. A derivation program

The program has three tasks, pre-registered with falsifiability conditions (Section 7).

T1 — Two modal exponentials. In a $\mathbb{Z}/2$-graded symmetric monoidal category with the graded braiding $\sigma_{A,B}(a \otimes b) = (-1)^{\lvert a \rvert \lvert b \rvert} b \otimes a$, the exchange operator $P = \sigma_{A,A}$ on $A \otimes A$ splits into two idempotent projectors,

whose eigenspaces are the symmetric and antisymmetric subspaces. [ESTABLISHED — elementary super-algebra.] For a mark of odd parity, the graded symmetric algebra coincides with the exterior algebra, so the symmetric and exterior exponentials,

are the two grading components of one construction. The two statistics are the two eigenvalues of exchange. [DERIVATION SKETCH — P4 notebook T1.]

T2 — The braiding of two marks. In a compact closed category with a self-dual mark $M$, the exchange map $\sigma_{M,M}$ is a scalar $\eta \cdot \mathrm{id}$, and the ribbon identity forces $\eta = \theta_M$. In a symmetric category $\theta_M^2 = 1$, so $\eta = \pm 1$: the two eigenvalues of exchange are the boson and fermion signs. The sign $\eta = -1$ is the same $-1$ as the treatise's half-turn phase $e^{i\pi} = -1$. [DERIVATION SKETCH — P4 notebook T2.] The identification of $\eta = +1$ with Calling and $\eta = -1$ with Crossing is the mark-calculus reading of the two one-dimensional representations of the symmetric group.

T3 — Dimension quantization. The table in Section 4, with dimension entering only through the allowed braided structures. [DERIVATION SKETCH — P4 notebook T3.]

The boundary of the program. The program can show that statistics is forced by distinction, compact closure, an involutive braiding, and the abelian-pair postulate (the pair of marks has a unique joint state up to phase, so that the braiding acts by a scalar). The last postulate is the categorical counterpart of the exclusion of parastatistics: without it, mixed-symmetry (parastatistics-class) sectors are not excluded by the algebraic machinery alone; in algebraic quantum field theory the exclusion follows from locality (Doplicher, Haag, and Roberts, 1971, 1974; Doplicher and Roberts, 1990; Greenberg and Messiah, 1965). The program therefore must either adopt the postulate or derive a DHR-style exclusion — which again lands on locality. [2026 note — two external developments qualify this boundary. First, a model-independent, quantum-information-theoretic exclusion of parastatistics has been given that does not route through locality at all: complete invariance under quantum permutations forces Bosons or Fermions (Mekonnen, Galley, and Mueller, 2025). Second, the classical equivalence theorems (Greenberg and Messiah, 1965; Doplicher, Haag, and Roberts) do not exhaust the possibilities: R-parastatistics — parastatistics inequivalent to bosons or fermions, consistently defined in any dimension — has been shown to emerge as observable quasiparticle statistics in condensed-matter systems (Wang and Hazzard, 2023, 2024, 2026). The abelian-pair postulate's status is thereby sharpened: its exclusion of parastatistics-class sectors is neither a purely algebraic consequence nor exclusively a locality theorem — it is a substantive physical assumption whose justification the new literature can either strengthen (quantum-permutation invariance) or qualify (emergent paraparticles).] It cannot, from the mark alone, show which eigenvalue corresponds to which spin: the spin–statistics connection requires the additional postulate that the twist equals the $2\pi$ rotation of a Lorentz representation, with microcausality and positive energy (Pauli, 1940; Duck and Sudarshan, 1998). The paper states this boundary explicitly; it does not claim a full derivation of the spin-statistics theorem from distinction. [CONTESTED — the sufficiency of the minimal postulates is open.]

7. Falsifiability conditions

F1 (empirical). If a stable, local, relativistic excitation in $3{+}1$ dimensions is observed with exchange phase $\eta \neq e^{2\pi i s}$ — for example, a spin-$\tfrac{1}{2}$ particle obeying Bose-Einstein statistics, or a spin-$0$ particle obeying Fermi-Dirac statistics — the claim that $R = e^{2\pi i s}$ is the universal invariant is disconfirmed. No such particle is known in the Standard Model. [ESTABLISHED — the absence of violations is a strong constraint, not a proof.]

F2 (formal). If the mark calculus cannot reproduce the two one-dimensional representations of $S_n$ (trivial and sign) from the primitive distinction, compact closure, an involutive braiding, and the abelian-pair postulate alone — without importing microcausality, Lorentz structure, or any other physical postulate — the derivation program is disconfirmed, and the monograph's spin-statistics claim stands as an asserted correspondence. (The abelian-pair postulate is a structural condition on the joint state of the pair, not a physical input; it is the minimal admission needed to exclude parastatistics-class sectors. Yang–Baxter alone forces the exchange phase to be uniform across pairs — see artifacts/notebooks/t1-t2-dill-full-check.md §3.)

F2′ (2026-08-16, updated after the T2 rigor pass). F2 as written above is disconfirmed in its strong form and is restated as follows. The category-theoretic rigor pass (companion notes res009-gap6-t2-rigor-pass-2026-08-16.md and res009-gap6-t2-f-construction-2026-08-16.md, live-verified 2026-08-16) established three results. (i) The exchange σ{M,M} is scalar iff M⊗M is simple: abelianity (dM = 1) is presupposed, not delivered, by the calculus. (ii) The interpretation functor F exists iff the target braiding is involutive (symmetric categories, or the Temperley–Lieb regime at A⁴ = 1): involutivity is a condition on the target, not a theorem of the calculus. (iii) The calculus's syntactic signature is the Lawvere theory of Boolean algebras: crossing is unary, and the binary exchange is added target structure. Consequently the mark calculus supplies the syntax of the involutive quotient (Sn = Bn/⟨σᵢ² = 1⟩, verified quantitatively in the Temperley–Lieb algebra: σᵢ² = A²I + (1 − A⁻⁴)Uᵢ), while the selection of the involutive quotient remains external kinematical input (π₁ of the configuration space: SN for d ≥ 3, BN for d = 2). F2′ (surviving condition): the derivability claim is restricted to the syntax of the two exchange channels — the two one-dimensional characters of the involutive quotient are realized as the idempotent projectors P± = (id ± σ)/2 (Calling's idempotence law), with the abelian-pair postulate retained as an explicit channel-count postulate tied to the locality-based exclusion of parastatistics (DHR 1971/1974). F2′ is not disconfirmed by the rigor pass; F2 as originally written is. [NOT YET EVIDENCE — the surviving claim is the pre-registered channel-count program]

Surprise accounting (KIF-60 discipline). The existence of anyons in $2{+}1$ dimensions is established and does not count as predictive evidence for this paper: anyonic statistics is expected under the null hypothesis of braid-group representations. Only F1's precision constraint and F2's derivability constraint carry evidential weight. The invariant formulation itself is [RETRODICTION — not evidence]: it restates established results in a unified language. The paper claims credit only for the derivation program (F2) and for the identification of the monograph's silent assumption (a textual finding).

8. Relation to existing programs

The derivation program sits between three established lines of work. First, the algebraic quantum field theory tradition proves the spin-statistics connection from locality and positivity (Streater and Wightman, 1964; Verch, 2001), and extends it to anyons in $2{+}1$ dimensions (Mund, 2008; Kuckert, 2002; Kuckert and Mund, 2004). Second, the topological and categorical tradition states the connection as a theorem about braided tensor categories (Joyal and Street, 1993; Bakalov and Kirillov, 2001; Johnson-Freyd, 2015; Oeckl, 2000), and condensed-matter physics realizes it for fractional quantum Hall quasiparticles (Comparin et al., 2021; Nardin et al., 2022; Trung et al., 2022). Third, the distinction-based tradition derives physical structure from the mark (Spencer-Brown, 1969; Quni-Gudzinas, 2026a, 2026b, 2026c). The present paper is the first, to the author's knowledge, to state the derivation target explicitly for the third tradition: the exchange of two marks, constructed in a compact closed category, whose eigenvalues are the two statistics. Adjacent internal work on p-adic anyon braiding (Quni-Gudzinas, 2026d) and on the topological distinction between Dirac and Majorana fermions (Quni-Gudzinas, 2026e) provides a compatible categorical language.

9. Conclusions

The boson/fermion dichotomy is not the primitive content of the spin-statistics theorem; the relation $R = e^{2\pi i s}$ between exchange phase and topological spin is. [ESTABLISHED] The dichotomy is its shadow in three spatial dimensions, where the involutive braiding quantizes $s$ to integers and half-integers. [ESTABLISHED] A distinction-based calculus that aims to ground quantum statistics must construct the exchange of two marks and derive the two eigenvalues of exchange from the primitive; the current monograph in that tradition silently assumes the symmetric (bosonic) algebra instead. [textual finding] This paper pre-registers the derivation program (T1-T3) and its falsifiability conditions (F1, F2), and states the boundary of the program: the spin-statistics connection requires Lorentz and locality input that the mark alone cannot supply. [NOT YET EVIDENCE]

10. The Parsimony Ledger (2026-08-16 update)

A ledger counts every primitive, labels each as DERIVED or ASSUMED, and states the standing debt — it prevents the Occam objection from being answered by relocation. Counting convention: one primitive = one named structural input not derived within the system.

System	Primitives	Derives	Status
Standard QFT	3D spacetime + locality, positivity, Lorentz	±1 exchange and the spin–statistics connection	COMPLETE (Streater–Wightman 1964)
Mark calculus	mark + compact closure + involutive braiding + abelian-pair postulate + external Lorentz/locality input	the two exchange eigenvalues (±1) — the syntax of the involutive quotient	PRE-REGISTERED (F2′); NOT YET EVIDENCE

Row-by-row debt (definitive outcome of the T2 rigor pass, 2026-08-16).

1. Abelian-pair postulate — ASSUMED (definitive). The demotion attempt failed honestly: scalar exchange follows from abelianity (M⊗M simple, d_M = 1); the calculus does not deliver it (rigor-pass Lemma 3; DD HARD-1 stands). The postulate is retained as an explicit channel-count postulate, tied to the locality-based parastatistics exclusion (DHR 1971/1974).
2. Involutive braiding (σ² = 1) — ASSUMED (definitive). The inheritance claim is resolved: F exists iff the target braiding is involutive — involutivity is a condition on the target (symmetric categories, TL at A⁴ = 1), not a theorem of the calculus (f-construction note). The calculus supplies the syntax of the quotient Sn = Bn/⟨σᵢ² = 1⟩; the selection remains external (d ≥ 3 kinematical input).
3. Compact closure — ASSUMED (structural input, never disputed).
4. External Lorentz/locality input — CONCEDED as boundary (paper §6). The spin–statistics connection requires the twist to equal the 2π rotation of a Lorentz representation; not derivable from the mark alone.
5. The mark itself — primitive (the program's single claimed primitive). Its primitiveness is itself a quotient-claim (UIA Q15: "what are the braid group, the mark, and SO(3) all quotients of?") and remains audited next.

Quotient-direction neutrality (reviewer DESIGN-1/3). Sn ≅ Bn / ⟨σᵢ² = 1⟩ (Artin) is neutral evidence: parsimony favors Sn as the exchange-statistics object until the mark calculus demonstrably needs Bn for 3D physics. The functorial form (f-construction): Bn is not a model of the calculus's signature; the calculus realizes the Sn reading natively.

Verdict. The mark calculus supplies the syntax of the involutive quotient; the selection of the involutive target is external kinematical input. F2 as originally written is disconfirmed; F2′ (channel-count postulate retained, tied to DHR locality) is the surviving pre-registered condition. This is the paper's §6 boundary, now stated as a ledger outcome rather than a concession.

Declarations

Funding. No external funding. Conflicts of interest. The author declares no conflicts of interest. Data availability. No experimental data were generated. All external claims are documented in the evidence files accompanying the source repository. Code availability. No code was required for the arguments presented; derivation notebooks are planned for the program's formal phase. Author contributions. Sole author. Ethics approval. Not applicable. Consent for publication. Not applicable. Acknowledgments. The author thanks the reviewers of the companion monograph for the discussion that sharpened Section 5. Correspondence. rowan.quni@outlook.com

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