11.  Spin and Statistics from Boolean Exclusion

Matter particles obey a rule that keeps them out of each other’s way: no two can be in exactly the same state. That rule is why solid things are solid. This chapter gets that rule out of the substrate rather than assuming it.

11.1.  The fermionic sector: structure and the open problem

The excitations of the scalar sector (the order parameter, identified with the Higgs doublet $H\in\mathbb{C}^{2}$ in Chapter 13) are integer-spin fields: the Higgs scalar, the gauge bosons, and the graviton. Fermions require spinor representations of the local Lorentz group $\mathrm{SO}(1,3)$.

The spinor bundle exists. Because $(M,g)$ is orientable and time-orientable (both guaranteed by the PST derivation of the Lorentzian signature $(+,-,-,-)$), its second Stiefel–Whitney class $w_{2}(M)=0$, and a spin structure exists. The vierbein $e^{a}{}_{\mu}$ satisfying $g_{\mu\nu}=e^{a}{}_{\mu}e^{b}{}_{\nu}\eta_{ab}$ is a derived quantity (the square root of the projected metric), not an independent primitive. The spinor bundle $\mathcal{S}=\mathcal{S}^{+}\oplus\mathcal{S}^{-}$ on $(M,g)$ is therefore structurally available. Quarks are sections of $\mathcal{S}\otimes\mathbf{3}$ and leptons of $\mathcal{S}\otimes\mathbf{1}$, with $\mathbf{3}$ and $\mathbf{1}$ the colour representations. The Dirac operator on $\mathcal{S}$ formed from the projected metric generates the Dirac equation [23] at the kinematic level (the equation’s operator structure follows from the spinor bundle and the projected metric); the dynamical question of whether the substrate produces fermionic excitations that populate this equation is answered in the next paragraph and §11.2.

The mechanism for fermionic excitations from the substrate. The order parameter $H$ is bosonic; its excitations obey Bose statistics. Grassmann (anti-commuting) integration variables in $Z_{\mathrm{QFT}}$ are required for half-integer spin. The doublet’s threshold vacuum manifold $\mathcal{V}_{\mathrm{EW}}\cong S^{3}\cong\mathrm{SU}(2)$ (§13.4) admits half-integer representations; a spin structure on $S^{3}$ exists without topological obstruction ($\pi_{1}(S^{3})=0$). The question of whether the Bernoulli measure gives weight to fermionic sectors is answered in §11.2 below: the Boolean $\{0,1\}$ distinction structure is isomorphic to a fermionic Fock space via the canonical anticommutation relations (CAR) algebra, so fermionic excitations are not an additional postulate but are already present in the substrate.

11.2.  Fermionic excitations: the CAR algebra identification

The question left open in the preceding section is answered by an exact algebraic identification.

The CAR algebra. For each distinction element $a\in D$, define operators $c_{a},\,c_{a}^{\dagger}$ on the substrate Hilbert space by

$$c_{a}^{\dagger}c_{a}\;=\;n_{a},\qquad\{c_{a},\,c_{b}^{\dagger}\}\;=\;\delta_{ab},\qquad\{c_{a},\,c_{b}\}\;=\;0,$$ (110)

where $n_{a}\in\{0,1\}$ is the Boolean occupation number of element $a$. These are the Canonical Anticommutation Relations (CAR). An explicit realisation is the Jordan–Wigner transformation

$$c_{a}\;=\;\Bigl{(}\prod_{b<a}(1-2n_{b})\Bigr{)}\,\sigma^{-}_{a},\qquad\sigma^{-}_{a}\;=\;|0\rangle_{a}\langle 1|_{a},$$ (111)

which maps the Boolean algebra on $\{0,1\}^{D}$ to the CAR algebra exactly, without approximation. The string of phase factors $(1-2n_{b})$ encodes the fermionic sign picked up when two occupation numbers are exchanged.

Bosonic and fermionic sectors. The full substrate Hilbert space $\mathcal{H}_{D}=\bigoplus_{k=0}^{|D|}\bigwedge^{k}\mathbb{C}^{|D|}$ decomposes under the parity operator $(-1)^{N}$ (where $N=\sum_{a}n_{a}$ is the total occupation number) into

$$\mathcal{H}_{D}^{+}\;=\;\bigoplus_{k\,\text{even}}\bigwedge^{k}\mathbb{C}^{|D|},\qquad\mathcal{H}_{D}^{-}\;=\;\bigoplus_{k\,\text{odd}}\bigwedge^{k}\mathbb{C}^{|D|}.$$ (112)

Every state of $\mathcal{H}_{D}$ is antisymmetric under exchange of its constituents, even and odd $k$ alike, $\mathcal{H}_{D}$ being an exterior algebra. What distinguishes the two sectors is the fermion-parity grading: exchanging two composites of occupation $k_{1}$ and $k_{2}$ produces $(-1)^{k_{1}k_{2}}$, which is $+1$ when both are even, and even-parity observables with disjoint support commute. So states in $\mathcal{H}_{D}^{+}$ project to bosonic fields under $\Pi$ and states in $\mathcal{H}_{D}^{-}$ to fermionic fields, by composite exchange sign rather than by constituent symmetry.

The Bernoulli measure weights both sectors equally. Under $\mu=\bigotimes_{a}\mathrm{Bern}(1/2)$, the probability that $N$ is even equals the probability that $N$ is odd: each is exactly $1/2$. The measure therefore assigns non-zero weight to every fermionic sector. The question of whether the Bernoulli measure populates fermionic path-integral sectors is answered affirmatively and exactly.

Spin-statistics connection. The antisymmetry of $\mathcal{H}_{D}^{-}$ under exchange is not imposed by hand. It is a structural consequence of the CAR algebra: equations (110) require $c_{a}c_{b}=-c_{b}c_{a}$ for $a\neq b$, and the Jordan–Wigner string (111) implements this sign automatically. When $\Pi$ projects $\mathcal{H}_{D}^{-}$ to 4D fields, those fields inherit the anticommuting character and satisfy the Dirac equation on $(M,g)$. The statistics half of the spin-statistics connection, that the projected fields anticommute, is therefore a structural consequence of the Boolean distinction primitive, not a separate postulate; the spin half is attached by the choice of spinor bundle rather than derived (§11.3). This is the identification; §11.3 promotes it to a formal theorem (CAR forced by Boolean exclusion, with the converse boson case and the Wightman-axiom development) and need not be re-read as the same point.

The CAR identification establishes that fermionic excitations exist and carry the correct statistics. The remaining fermion-sector results each need the gauge structure of Chapter 13 first, and are therefore taken up in Chapter 14 once it is available: the colour multiplicity $N_{c}=3$ (§13.5, equation (163)); the generation count $N_{\mathrm{gen}}=3$ (§14.2); the representation content of $\mathcal{H}_{D}^{-}$ under $\Pi$ (§14.1); and the Yukawa hierarchy and Cabibbo–Kobayashi–Maskawa (CKM) mixing (§14.3). Nothing in the present chapter depends on any of them: spin and statistics come out of P1’s Boolean exclusion alone.

11.3.  Spin-statistics from Boolean exclusion (structural theorem)

The CAR identification of §11.2 already establishes that substrate excitations carry fermion statistics structurally. This section discharges the remaining formal work: it states that statement as a rigorous theorem (CAR algebra forced by Boolean exclusion, with the integer-spin boson case obtained by composition), contrasts it with the standard quantum field theory (QFT) spin-statistics theorem, and completes the functional-analytic development (substrate-to-continuum CAR convergence, the Wightman axioms, and the Dirac equation as substrate-derived). Recall the standard QFT theorem: half-integer-spin fields anticommute and integer-spin fields commute, derived on a relativistic field theory from Lorentz invariance, positive energy spectrum, locality (microcausality), and cluster decomposition. PST delivers the statistics half of that dichotomy from the Boolean distinction primitive of P1, without invoking the QFT axioms separately. It does not deliver the spin half: what is derived is that exclusion forces the CAR and that composition parity fixes statistics, and the spin label is attached by the choice of spinor bundle rather than derived, the book taking no step from the spinor module to a two-valued spin projection (§10.4.3).

11.3.1.  Structural theorem (Boolean spin-statistics)

Let $D$ be a finite distinction set and $\mathcal{P}(D)$ the substrate configuration space with Bernoulli measure $\mu=\bigotimes_{a}\mathrm{Bern}(\tfrac{1}{2})$. The Boolean structure on $D$, each site $a\in D$ being either occupied ($1$) or unoccupied ($0$) in a configuration $C\in\mathcal{P}(D)$, carries a canonical anticommutation-relations (CAR) algebra structure under the creation/annihilation operators

$$(c_{a}^{\dagger}\psi)(C) \;=\;\mathbf{1}[a\notin C]\cdot(-1)^{\#\{b<a:b\in C\}}\,\psi(C\cup\{a\}),$$ (113)

$$(c_{a}\psi)(C) \;=\;\mathbf{1}[a\in C]\cdot(-1)^{\#\{b<a:b\in C\}}\,\psi(C\setminus\{a\}),$$ (114)

satisfying $\{c_{a},c_{b}\}=\{c_{a}^{\dagger},c_{b}^{\dagger}\}=0$ and $\{c_{a},c_{b}^{\dagger}\}=\delta_{ab}I$ (CAR), not the commutator relations $[c_{a},c_{b}^{\dagger}]=\delta_{ab}I$ (CCR). Therefore the substrate-emergent excitations carry fermion statistics structurally from P1.

Proof.

The Jordan–Wigner construction (114) realises the Boolean $\{0,1\}$-exclusion principle as the constraint $(c_{a}^{\dagger})^{2}=0$ (no double occupancy at a site): a single application of $c_{a}^{\dagger}$ to a configuration $C$ with $a\in C$ already returns zero, and to $C$ with $a\notin C$ the first application gives $C\cup\{a\}$ where $a$ is then occupied, so the second application again returns zero. This is exactly the $c_{a}^{\dagger}c_{a}^{\dagger}=0$ relation, equivalent under polarisation to $\{c_{a}^{\dagger},c_{a}^{\dagger}\}=0$. The cross-relation $\{c_{a},c_{b}^{\dagger}\}=\delta_{ab}I$ follows from the Jordan–Wigner sign factor; the calculation is standard [105, 107]. The corresponding commutator relations $[c_{a},c_{b}^{\dagger}]=\delta_{ab}I$ would require $c_{a}^{\dagger}c_{a}^{\dagger}\neq 0$ (multiple bosons at a site allowed), which is incompatible with the Boolean exclusion of P1. ∎

11.3.2.  Consequence: half-integer excitations are fermionic

Half-integer-spin elementary excitations on the substrate are fermionic by structure. Integer-spin excitations (photons from inner fluctuations of the gauge sector, composite bound states of even numbers of fermions) carry bosonic statistics by composition: an even-order composite has even fermion parity, so two of them exchange with sign $(-1)^{k_{1}k_{2}}=+1$ and commute at disjoint support. This is not the CCR. For a pair operator $b=c_{1}c_{2}$ one finds $[b,b^{\dagger}]=\mathbb{1}-n_{1}-n_{2}$, so the canonical commutation relation is recovered only in the dilute limit where the constituent modes are empty; fermion bilinears close on a Lie algebra, not a Heisenberg algebra. The structural statement is therefore: statistics of an excitation are fixed by its CAR-composition order; odd-order composites are fermionic, even-order composites are bosonic. Note what that statement is indexed by: composition parity, not spin.

11.3.3.  Relation to the standard QFT spin-statistics theorem

The QFT theorem assumes Lorentz invariance, positive energy, locality, and cluster decomposition, and concludes the spin-statistics rule. PST inverts the logic: starting from the Boolean exclusion of P1, the CAR structure is forced, and those same relativistic properties follow downstream from the substrate projection $\Pi$ and the Mosco-convergence emergence of $M=\mathbb{R}\times S^{3}$ (§7.1). The two arguments run from opposite ends and do not deliver the same statement: QFT derives a dichotomy indexed by spin from the relativistic arena, requiring its full apparatus; PST derives a dichotomy indexed by composition parity from the Boolean exclusion of P1 alone, with no further input from the spacetime emergence chain. The second is a real result and it is not the first.

11.3.4.  Closure (1): CAR convergence

Extend the Mosco-convergence framework of §18.1 from the Boolean Dirichlet form to the substrate CAR algebra by identifying the substrate-side spinor degrees of freedom with the discrete creation/annihilation operators of equation (114). Concretely: choose a spinor bundle $\mathcal{S}\to M$ with structure group $\mathrm{Spin}(1,3)$ and trivialised over the Delaunay triangulation $\mathcal{T}_{n}$ of $\rho_{n}(D_{n})\subset M$. Define the continuum spinor field operator

$$\psi(x)\;:=\;\lim_{n\to\infty}\sum_{a\in D_{n}}c_{a}\cdot\phi_{a}(x)\cdot s_{a},$$ (115)

where $\phi_{a}(x)$ is the hat function on $\mathcal{T}_{n}$ centred at $\rho_{n}(a)$ (the same basis as in Closure (a) of §18.1.6) and $s_{a}\in\mathcal{S}_{\rho_{n}(a)}$ is a choice of spinor frame at site $a$, determined by the spin structure on $S^{3}$ (Computation 4 confirms uniqueness up to gauge). The limit (115) is taken in the strong-operator topology on the Gelfand–Naimark–Segal (GNS) Hilbert space of the substrate CAR.

11.3.5.  Theorem (CAR Mosco convergence)

For $f,g\in L^{2}(M,d^{4}x)\otimes\mathcal{S}$ and any state $|\Phi\rangle$ in the GNS Hilbert space:

$$\sum_{a\in D_{n}}f(\rho_{n}(a))\,g(\rho_{n}(a))\,\bigl{\langle}\Phi|c_{a}c_{a}^{\dagger}|\Phi\bigr{\rangle} \xrightarrow{n\to\infty}\;$$ (116)

$$\int_{M}\langle f(x),g(x)\rangle_{\mathcal{S}} \bigl{\langle}\Phi|\psi(x)\psi^{\dagger}(x)|\Phi\bigr{\rangle}\,d^{4}x,$$

and the substrate CAR relations $\{c_{a},c_{b}^{\dagger}\}=\delta_{ab}I$ converge, on a Cauchy slice of the sublimation-derived timelike direction, to the equal-time continuum CAR $\{\psi_{\alpha}(\mathbf{x},t),\psi^{\dagger}_{\beta}(\mathbf{y},t)\}=\delta_{\alpha\beta}\,\delta^{(3)}(\mathbf{x}-\mathbf{y})\,I$ in the distributional sense, with the rate matching the Berry–Esseen rate of §18.1.6. The slice matters, and it is supplied (Computations 186, 191, 192). Writing $\delta^{(4)}(x-y)$ instead, summing over sites realised into the four-dimensional $M$ rather than into a slice, would be dimensionally wrong by one power of mass, since $\{\psi,\psi^{\dagger}\}$ carries $[\text{mass}]^{3}$ in four dimensions; it contradicts the free Dirac field, whose unequal-time anticommutator is $iS(x-y)$ rather than a delta; and it is incompatible with the spectral condition W2 stated below, since a constant Fourier transform cannot have support in the closed forward cone. Posing the relations slice-wise is what the substrate’s lack of a preferred time makes non-trivial, and it is done in three steps. No site is realised on a measure-zero slice, so the relations are posed on a slab of thickness $h_{n}$ whose site density per unit $3$-volume carries exactly the missing power of mass (Computation 186). The slab sum converges to the slice integral for every cloud whose discrepancy vanishes faster than $h_{n}$, by Koksma–Hlawka, with no independence assumed (Computation 191). And the time function the slab is cut in is cosmic time along $\Phi(\hat{\tau})$: isotropy makes that field hypersurface-orthogonal, its level sets are the compact Cauchy surfaces, the slab is normalised by the proper $3$-volume of the slice, and the result is invariant under relabelling the leaves (Computation 192). What the limit rests on is A1 to A4 with A3’s rate supplied by P3, the standing of equicoercivity A7 (§18.1.6), and it is recorded on those terms in Appendix C.

Proof sketch.

Step 1: the Delaunay basis $\{\phi_{a}\}_{a\in D_{n}}$ is asymptotically complete in $L^{2}(M,d^{4}x)$ by Sobolev embedding on compact $K\subset M$ (matched scaling A1, Closure (a) of §18.1.6). Step 2: the substrate CAR generates a finite-dimensional CAR sub-algebra $C^{*}(D_{n})$ on $\mathcal{P}(D_{n})$. As $n\to\infty$, the inductive limit $C^{*}(D):=\varinjlim_{n}C^{*}(D_{n})$ is a UHF algebra of type $2^{\infty}$, the standard fermionic Fock representation. Step 3: the GNS representation of $C^{*}(D)$ on the Bernoulli measure $\mu$ converges under the matched-scaling limit to the continuum CAR representation on $L^{2}(M,d^{4}x)\otimes\mathcal{S}$, with the convergence rate the same $O(|D_{n}|^{-1/2})$ Berry–Esseen rate as the Boolean Dirichlet form. ∎

11.3.6.  Closure (2): Wightman axioms

Verify each Wightman axiom (W1)-(W5) [108, 109] on the substrate-projected relativistic field theory delivered by $\psi(x)$.

(W1) Hilbert space and Poincaré covariance. The projected Hilbert space $\mathcal{H}_{\mathrm{proj}}=L^{2}(M,d^{4}x)\otimes\mathcal{S}$ carries the unitary representation of the Poincaré group inherited from the matched-scaling emergence of $M=\mathbb{R}\times S^{3}$ (§7.1). The Lorentz subgroup acts on $\mathcal{S}$ by the standard spinor representation; the translation subgroup acts as $\hat{U}(a)=\exp(ia^{\mu}\hat{P}_{\mu}/\hbar)$ where $\hat{P}_{\mu}$ is the energy-momentum operator from §10.6. The continuity and strong continuity of the representation follow from the substrate-side $2^{D}$-dimensional Hilbert space being separable and the substrate Hamiltonian ($H_{\mathrm{Higgs}}$ of Computation 89, the modal potential of P3 on configurations) being bounded below by P2.

(W2) Spectral condition. The energy-momentum operator $\hat{P}^{\mu}$ has spectrum in the closed forward light cone $\bar{V}_{+}$. Positivity of $\hat{P}^{0}$: by the substrate’s directed P2–P3 tension structure, that Hamiltonian has spectrum bounded below by $0$; the projected Hamiltonian inherits this lower bound through $\Pi$. Lorentz invariance of $\bar{V}_{+}$ is inherited from the Lorentz covariance of (W1).

(W3) Vacuum cyclicity. The vacuum state $|\Omega\rangle$ corresponds to the empty substrate configuration $C=\emptyset\in\mathcal{P}(D)$, a tension-minimising configuration (every distinction-free configuration has $|T|=0$ by (14); the empty one is taken by convention). Cyclicity of $|\Omega\rangle$ under the polynomial algebra of $\psi(f),\psi^{\dagger}(f)$ for $f\in C_{c}^{\infty}(M)$ follows from the cyclicity of the empty configuration under the substrate CAR ladder $c_{a}^{\dagger}$: any configuration $C\in\mathcal{P}(D)$ is reached from $\emptyset$ by $\prod_{a\in C}c_{a}^{\dagger}$, and the continuum analogue follows from Closure (1).

(W4) Locality (microcausality). For $x,y\in M$ spacelike-separated, the spinor field operators $\psi(x),\psi(y)$ anticommute. This axiom holds on the terms the Mosco chain sets, as the end of this paragraph states, and the obvious argument for it is vacuous. That argument runs: the Delaunay neighbourhoods of $\rho_{n}^{-1}(\{x\})$ and $\rho_{n}^{-1}(\{y\})$ are disjoint, so $\{c_{a},c_{b}\}=0$, so $\{\psi(x),\psi(y)\}=0$ for spacelike $(x-y)$. But $\{c_{a},c_{b}\}=0$ holds for every pair of substrate modes, $a=b$ included, since $c_{a}^{2}=0$; the disjointness premise is never used, and the same conclusion would follow for timelike and for coincident points. A premise that does no work establishes nothing. The non-trivial microcausality statement for a spinor field is in any case about $\{\psi_{\alpha}(x),\bar{\psi}_{\beta}(y)\}=iS(x-y)$, which vanishes outside the causal cone precisely because the Pauli–Jordan function does, and is nonzero at timelike separation; the argument above never touches it. And the route that would establish W4 is not merely unattempted; it is unavailable (Computation 181). It would need the substrate two-point structure to have Fourier support in the closed causal cone. But the substrate is pre-causal in the sense that carries the weight here: it has no time parameter, time being emergent, so that structure is an equal-time object and its Fourier transform does not depend on $k^{0}$, measured to machine zero along every $k^{0}$ line. (Not that the substrate is energy-free: Computation 89’s $H_{\mathrm{Higgs}}$ is a substrate Hamiltonian and §12.5 uses it. An energy on configurations weights a static ensemble; it does not generate evolution in a time the substrate does not have, and only the time-freeness is load-bearing.) Ask a $k^{0}$-independent transform to be supported in the closed cone $\{|k^{0}|\geq|\mathbf{k}|\}$ and fix $\mathbf{k}\neq 0$: the points with $|k^{0}|<|\mathbf{k}|$ lie outside the cone, so the transform vanishes there, and being $k^{0}$-independent it vanishes on the whole line. It is supported at $\mathbf{k}=0$ alone, which is to say it is trivial. A static object cannot encode a light cone, and no work on the substrate correlator will change that. Where W4 has to come from instead is the emergent dynamics: $iS(x-y)$ is built from the projected Dirac operator of Closure (3), and the Pauli–Jordan function vanishes outside the cone because of the mass shell, which is a statement about that operator rather than about $\mu$. And that statement is not an independent gap (Computation 185). $S$ is the causal propagator $G^{+}-G^{-}$, and on a globally hyperbolic spacetime the Dirac operator has unique advanced and retarded Green’s operators supported in $J^{\mp}$, so $S$ is supported in the closed causal cone and CAR quantisation on a Cauchy slice gives microcausality outright [213, 214]. This book supplies three of that theorem’s hypotheses: $M=\mathbb{R}\times S^{3}$ is globally hyperbolic with compact Cauchy surfaces (Computation 4), its Cauchy problem is well posed (§4.12), and the projected field satisfies (120). The fourth, the Cauchy-slice CAR limit, was the other residual listed beside W4, and it is supplied above on the slab construction (Computations 186, 191, 192), given A1 to A4 with A3 supplied by P3. So W4 holds on the same terms, as the corollary of a theorem with the standing of equicoercivity A7, and the two, which were one residue, have left the structural front together (Appendix C).

(W5) Cluster decomposition. The substrate Bernoulli measure $\mu=\bigotimes_{a\in D}\mathrm{Bern}(\tfrac{1}{2})$ factorises across independent sites: for disjoint $A,B\subset D_{n}$, $\mu_{n}(\{C:C\cap A\in S_{A},C\cap B\in S_{B}\})=\mu_{A}(S_{A})\,\mu_{B}(S_{B})$. Continuum cluster decomposition follows by taking the matched-scaling limit of the disjoint-region factorisation. Explicitly: for $f,g$ supported in disjoint compact $K_{1},K_{2}\subset M$ with $\mathrm{dist}(K_{1},K_{2})\to\infty$:

$$\langle\Omega|\psi(f)\psi(g)|\Omega\rangle\;\xrightarrow{\mathrm{dist}(K_{1},K_{2})\to\infty}\;\langle\Omega|\psi(f)|\Omega\rangle\cdot\langle\Omega|\psi(g)|\Omega\rangle.$$ (117)

The convergence rate is again the matched-scaling Berry–Esseen rate.

11.3.7.  Closure (3): the Dirac equation

The substrate spectral triple $(\mathcal{A}_{\mathrm{sub}},\mathcal{H}_{\mathrm{sub}},D_{\mathrm{sub}})$ with $\mathcal{A}_{\mathrm{sub}}$ the Boolean $*$-algebra on $D$ and $D_{\mathrm{sub}}$ the canonical Dirac operator from Computation 3 (verifying the real K-theory [KO] dimension-two structure) delivers the continuum Dirac operator via Mosco-convergence of spectral triples (cf. Chamseddine–Connes–Marcolli [50]). Explicitly, the substrate Dirac operator

$$D_{\mathrm{sub}}\;=\;\sum_{a\in D}(c_{a}+c_{a}^{\dagger})\otimes\gamma^{(a)},$$ (118)

where $\gamma^{(a)}$ are gamma matrices on the substrate-side spinor sub-bundle, converges under the matched scaling to the continuum Dirac operator

$$D_{M}\;=\;\gamma^{0}\,\partial_{t}+\sum_{k=1}^{3}\gamma^{k}\,e_{k},$$ (119)

where $\{e_{k}\}$ are the orthonormal frame fields on $S^{3}$ in the Friedrich–Bär formulation [113, 114]. The Dirac equation

$$D_{M}\,\psi\;=\;m\,\psi,$$ (120)

where $m$ is the mass eigenvalue, is then the structural consequence of the substrate spectral-triple equation $D_{\mathrm{sub}}\,\psi_{\mathrm{sub}}=m\,\psi_{\mathrm{sub}}$ projected through $\Pi$. Equation (120) is therefore not a separate dynamical postulate but a structural identity: it is the spectral-triple equation in the matched-scaling limit.

The substrate-to-continuum fermion sector is complete on the terms the Mosco chain sets: Closure (1) delivers the substrate-to-continuum CAR convergence with Berry–Esseen rate, posed slice-wise on the slab of Computations 186, 191 and 192, given A1 to A4 with A3 supplied by P3; and within Closure (2) the microcausality axiom W4 follows from it (Computation 185), the argument previously offered for it having been withdrawn as vacuous and replaced. Closure (2) verifies the Wightman axioms (W1)-(W5) for the projected spinor field theory, and Closure (3) identifies the Dirac equation as the matched-scaling limit of the substrate spectral-triple equation, structural rather than postulated. With the structural spin-statistics theorem above, these constitute the full fermion sector. What remains are explicit constant computations (spinor-bundle Sobolev embedding constants, Berry–Esseen rates), which are standard finite-element analysis with no fresh structural content. Alongside them sits the all-$(k,N)$ analytical uniform bound of the Dirac-convergence step (§7.1), the sector’s one named quantitative refinement.

11.4.  Summary

This chapter gets the exclusion rule out of the substrate. The spinor bundle exists on the projected geometry because it is orientable and time-orientable, so the Dirac operator is available at the kinematic level; the question is whether the substrate produces fermionic excitations at all. It does: the Boolean algebra on $\{0,1\}^{D}$ maps exactly onto the canonical anticommutation relation (CAR) algebra, so substrate excitations carry fermion statistics structurally (§11.2). The spin-statistics connection is then a structural theorem, the integer-spin boson case obtained by composition, and the projected Dirac equation is the substrate spectral-triple equation in the matched-scaling limit, not a separate postulate (§11.3). The fermion sector is complete on the terms the Mosco chain sets: CAR convergence with a Berry–Esseen rate given A1 to A4, and microcausality following from it.