Physical Realizability of Bordism and Homotopy Classes by Gapped Lattice Systems

1 Introduction

1.1 The realizability gap

A classification theorem for topological phases produces an abstract invariant: a bordism group, a stable-homotopy group, a value of some generalized cohomology theory. Part IV of this series constructs the target of such a classification inside the condensed-mathematics program: the invertible condensed phase spectrum $\mathbf{IP}^{\mathrm{cond}}_{d,G}$, a connective condensed spectrum whose homotopy groups are meant to enumerate the invertible gapped phases in spatial dimension $d$ with symmetry $G$. What a spectrum enumerates and what a laboratory, or a finite-dimensional local Hilbert space, can produce are not the same thing. A class $c\in\pi_0\mathbf{IP}^{\mathrm{cond}}_{d,G}$ is an equivalence class of abstract data; to say that $c$ is physically realizable is to exhibit a concrete uniformly gapped local Hamiltonian whose phase maps to $c$. The prospectus for this program names the discrepancy plainly: the gap between abstract homotopy-TQFT classifications and explicit local lattice realizations is a recognized issue, and not every abstract invertible field theory is known to arise from a microscopic gapped system.

This paper is about that gap. We take the classification of Part IV as given and study the image of the comparison map $$\rho\colon \pi_0\operatorname{Shape}\bigl(\mathfrak{Phase}_{d,G}\bigr) \longrightarrow \pi_0\mathbf{IP}^{\mathrm{cond}}_{d,G},$$ from the set of phases of genuine lattice systems (components of the stabilized condensed stack of Part III) to the abstract spectrum. Realizability is the question of which classes lie in $\operatorname{im}\rho$; more precisely, since a phase is by definition already a class of lattice systems, realizability is the essential surjectivity of $\rho$ onto a physically-meaningful subspectrum. We will argue that the honest present-day answer has three parts: a large constructive region where explicit models are known; a sharp obstructed region governed by a no-go theorem; and a residual open region.

1.2 Two notions of realizability

The single most important distinction in this paper is between two nested classes of Hamiltonians. Say that a class $c$ is

• gapped-realizable if some uniformly gapped local (or quasi-local) Hamiltonian has phase $c$; and

• commuting-projector realizable ($\mathrm{CP}$-realizable) if some Hamiltonian $H=\sum_x P_x$ with each $P_x$ a finite-range projector, $[P_x,P_y]=0$, has phase $c$.

Commuting-projector models are the exactly-solvable backbone of the field: they have exact ground states, exact gaps, string operators, and a transparent entanglement structure. Every $\mathrm{CP}$-realizable class is gapped-realizable, but (and this is the crux) the converse fails, and it fails for a reason that is a rigorous theorem rather than a modeling inconvenience. The Kapustin–Fidkowski theorem [1] states that a local commuting-projector Hamiltonian has vanishing Hall conductance. A Chern insulator, an integer quantum Hall state, the $E_8$ state (all manifestly gapped-realizable, all with nonzero Hall response) are therefore not $\mathrm{CP}$-realizable. Conflating the two notions is the characteristic error this subject invites, and much of what follows is an effort to keep them apart.

1.3 What is proved, what is conjectured

In keeping with the stance of the whole series, we label a statement a Theorem or Proposition only when it follows from cited present-day results, and everything else is a numbered Conjecture. The rigorous content of this paper is:

• Theorem V-A (Theorem 3.1): every class in $H^{d+1}(G,U(1))$ is $\mathrm{CP}$-realizable, by the Chen–Gu–Liu–Wen fixed-point construction [2] with the Else–Nayak edge characterization [3].

• Theorem V-B (Theorem 5.1): in $d=1$, for on-site finite symmetry, Ogata’s operator-algebraic index [4,5] is a complete invariant, so realized $=$ invariant there; in $d=2$ the $H^3$ index exists [6] and every in-cohomology class is realized, but completeness is open (Theorem 5.2).

• Theorem V-C (Theorem 6.1): the commuting-projector no-go, zero electric Hall conductance under $U(1)$ (Kapustin–Fidkowski [1]) and zero chiral central charge (Kapustin–Spodyneiko [7]), exactly bounds the $\mathrm{CP}$-image.

• Two elementary Propositions with complete proofs: the Kitaev-chain winding invariant (Theorem 4.3) and the cluster-state string order (Theorem 3.4), both of which are also checked by the accompanying Haskell.

The conjectural content is three numbered statements about the global shape of $\operatorname{im}\rho$ (Section 8): that it is the short-range-entangled subspectrum (Conjecture V-1); that chiral classes are gapped-realizable but never $\mathrm{CP}$-realizable (Conjecture V-2); and that realizability descends over profinite disorder hulls under a condensed-cohomological obstruction (Conjecture V-3).

1.4 Relation to companion papers

This is Part V of a six-part program that recasts topological phases of matter in the condensed-mathematics paradigm of Clausen–Scholze [8]. The parts compose as a modular hierarchy, each taking the previous as input.

Part I, Condensed Locality [9], makes the interaction space a Banach space $\mathcal{B}_{F}$ and promotes the Heisenberg dynamics to a morphism of condensed objects; it supplies the Lieb–Robinson control that gives meaning to a “uniformly gapped family” over a profinite base. Part II, Positivity and Condensed State Spaces [10], attaches the quasi-local $C^*$-algebra and, through Aoki’s solidification bridge [11], a solid-$K$-theory invariant; the operator-$K$ invariants that appear in our status table (Hall conductance, Chern number) are values of that functor. Part III, The Uniformly Gapped Substack [12], cuts out the substack $\mathfrak{Gap}_{d,G}$ whose stabilized $\pi_0$ is the set $\operatorname{Phases}_{d,G}$ that our map $\rho$ starts from; the robustness of that $\pi_0$ under perturbation is what makes “the phase of a lattice model” well defined. Part IV, From Lattice Models to Effective Field Theories [13], group-completes the invertible sector into the target spectrum $\mathbf{IP}^{\mathrm{cond}}_{d,G}$; realizability is precisely the surjectivity question for the comparison map that Part IV constructs. Part VI, the synthesis [14], assembles the five modules into a single program and records the Master Conjecture; our obstruction subgroup is one of the two hard walls (the other being the undecidability of the gap in Part III) that bound the synthesis.

The reader needs from Parts III–IV only the following interface, restated in Section 2: a phase is a component of a stabilized condensed stack, stacking gives it an abelian-group structure, and $\mathbf{IP}^{\mathrm{cond}}_{d,G}$ is the invertible part group-completed. No result of this paper depends on the conjectural global stack structure; we use the module-level theorems of the companions and cite them where needed.

1.5 Outline

Section 2 sets up the comparison map and the two realizability notions inside the condensed program. Sections 3 and 4 survey positive constructions: the group-cohomology models and the cluster state, then beyond-cohomology and topological-order models together with the Kitaev Majorana chain. Section 5 states the low-dimension completeness theorem and is careful about its scope. Section 6 states the Kapustin–Fidkowski obstruction and draws the $\mathrm{CP}$-versus-gapped line. Section 7 gathers everything into a status table. Section 8 states the conjectures on the realizability image, and Section 9 treats disorder-robust realizability over profinite hulls. Section 10 describes the computational checks. Sections 11 and 12 discuss limitations and conclude.

2 The realizability question in the condensed program

2.1 The interface from Parts III and IV

We recall only what we need. Fix a spatial dimension $d$, a lattice $L$, on-site Hilbert spaces, an $F$-function locality class, and a symmetry group $G$ with condensation $\underline{G}$. Part I organizes the admissible $G$-symmetric interactions into the condensed moduli object $\mathfrak{Ham}_{d,G}$, whose value on a profinite probe $S$ is the set of $S$-continuous families of interactions. Part III singles out the uniformly gapped substack $$\mathfrak{Gap}_{d,G}(S) \;=\; \bigcup_{\Delta>0}\Bigl\{\,H\in\mathfrak{Ham}_{d,G}(S) : \inf_{s\in S}\operatorname{gap}(H_s)\ge\Delta\,\Bigr\},$$ where the word uniformly (the infimum over the probe) is essential: a family with $\inf_s\operatorname{gap}(H_s)=0$ is not a member. Inverting the class $\mathcal{W}$ of gapped adiabatic / finite-depth equivalences [15] and stabilizing by product-state ancillas gives the phase $∞$-groupoid $\mathfrak{Phase}_{d,G}=(\mathfrak{Gap}_{d,G}[\mathcal{W}^{-1}])^{\mathrm{st}}$, and the set of phases is $$\operatorname{Phases}_{d,G} \;=\; \pi_0\operatorname{Shape}\bigl(\mathfrak{Phase}_{d,G}\bigr).$$ Stacking $\boxtimes$ makes $\mathfrak{Phase}_{d,G}$ symmetric monoidal; its invertible objects form a Picard $∞$-groupoid, and Part IV group-completes them into the connective condensed spectrum $\mathbf{IP}^{\mathrm{cond}}_{d,G}$. Writing $\operatorname{Phases}^{×}_{d,G}\subseteq\operatorname{Phases}_{d,G}$ for the invertible (short-range-entangled) phases, group completion is a map $$\gamma \colon \operatorname{Phases}^{×}_{d,G} \longrightarrow \pi_0\mathbf{IP}^{\mathrm{cond}}_{d,G}.$$

2.2 The comparison map and realizability

Part IV’s central comparison is between the microscopic invariant computed on the lattice and the deformation class of the associated effective field theory. Composing the lattice phase with that comparison gives a map to the abstract bordism/homotopy classification; denote its value on $\pi_0$ by $$\rho\colon \operatorname{Phases}_{d,G} \longrightarrow \mathcal{C}_{d,G},$$ where $\mathcal{C}_{d,G}$ is the abstract classification group of the relevant EFT: for invertible phases, $\pi_0\mathbf{IP}^{\mathrm{cond}}_{d,G}$ after realification, conjecturally a connective cover of Kubota’s $\Omega$-spectrum [16] or the Freed–Hopkins bordism target [17]. The direction of $\rho$ matters. Constructing a lattice model and reading off its abstract class is the “forward” direction and is always available. Realizability is the “backward” question: given $c\in\mathcal{C}_{d,G}$, is there a phase $P\in\operatorname{Phases}_{d,G}$ with $\rho(P)=c$?

The whole paper is the study of the two subsets $\operatorname{im}\rho^{\mathrm{CP}}\subseteq\operatorname{im}\rho\subseteq \mathcal{C}_{d,G}$. We record the reformulation that ties realizability to the language of the program.

2.3 Short-range entanglement

The invertible phases are the short-range-entangled ones [18]: a phase $P$ is invertible for $\boxtimes$ iff there is $P'$ with $P\boxtimes P'$ trivial (a product state) after stabilization. We write $\mathrm{SRE}_{d,G}\subseteq\operatorname{Phases}_{d,G}$ for this subset; group completion $\gamma$ restricts to it. Non-invertible topological order (nonzero total quantum dimension, anyons that cannot be condensed away) lies outside $\mathrm{SRE}$ and outside the spectrum-level classification altogether; it needs the higher-categorical machinery flagged in Part VI and is not classified by $\mathbf{IP}^{\mathrm{cond}}_{d,G}$. Our realizability question for the spectrum is therefore a question about $\mathrm{SRE}$; topologically ordered models such as the toric code enter only as $\mathrm{CP}$-realizations of trivial invertible class (they are invertible-trivial but intrinsically ordered), and the chiral topological orders (e.g. Kitaev’s non-abelian phase) enter through their invertible edge data, the chiral central charge.

3 Positive realizability I:
group-cohomology SPT phases

3.1 The Chen–Gu–Liu–Wen construction

The first and largest constructive region is the group-cohomology part of the symmetry-protected topological (SPT) classification. Let $G$ be a finite group acting on-site and unitarily; work in spatial dimension $d$, so the relevant cocycle degree is $d+1$.

3.2 The cluster state as a worked example

The one-dimensional cluster state is the smallest nontrivial instance of Theorem 3.1 and the model our Haskell computes with, so we give it explicitly. Take a chain of qubits with the symmetry $G=\mathbb{Z}_2×\mathbb{Z}_2$ generated by $X_{\mathrm{even}}=\prod_{j\ \mathrm{even}}X_j$ and $X_{\mathrm{odd}}=\prod_{j\ \mathrm{odd}}X_j$. The relevant cohomology is $H^2(\mathbb{Z}_2×\mathbb{Z}_2,U(1))\cong\mathbb{Z}_2$, so there is exactly one nontrivial SPT class, and it is realized by the cluster state.

The SPT order of $\lvert C\rangle$ is invisible to any local order parameter but visible to a nonlocal string operator. The following is elementary and is verified exactly by the Haskell stabilizer engine.

4 Positive realizability II:
beyond cohomology and topological order

4.1 Beyond group cohomology: Walker–Wang models

Not every invertible or topologically ordered phase is captured by group cohomology. In $3+1$ dimensions there are SPT and symmetry-enriched phases whose invariants are cobordism classes outside the image of group cohomology [19]. A general commuting-projector construction for a large class of these is available.

4.2 Chiral invertible phases: the $E_8$ state

The cleanest bosonic example of the obstruction to come is the $E_8$ state: a $2+1$d bosonic short-range-entangled phase with no symmetry, nontrivial because its edge carries eight chiral bosons with chiral central charge $c_-=8$ and a thermal Hall response. It generates the free part $\mathbb{Z}$ of the $2+1$d bosonic invertible classification. The $E_8$ state is gapped-realizable (it is a genuine gapped phase, described by a $K$-matrix Chern–Simons theory and realizable in coupled-layer and network constructions), but because $c_-\ne0$ it has a nonzero thermal Hall conductance, and Theorem 6.1(ii) (Kapustin–Spodyneiko [7]) forbids any local commuting-projector realization. This obstruction rests on the chiral central charge, not on the electric Kapustin–Fidkowski no-go: the bosonic $E_8$ state carries no $U(1)$ charge and has $\sigma_H=0$, so part (i) of Theorem 6.1 does not apply to it and part (ii) is what does the work. It is the bosonic analogue, at the level of gravitational response, of the fermionic integer quantum Hall state [21]; its nonzero chiral central charge is the edge datum that also governs chiral topological order more broadly [22].

4.3 Fermionic $\mathbb{Z}_2$: the Kitaev Majorana chain

The canonical fermionic example is the Kitaev chain [23], the $1+1$d class-$D$/BDI superconductor whose two phases are distinguished by a $\mathbb{Z}_2$ invariant and whose topological phase binds an unpaired Majorana zero mode at each end. Its $\mathbb{Z}_2$ label is the class-$D$ entry of the periodic table of free-fermion phases [24], whose systematic form is the operator-$K$-theoretic classification of topological insulators and superconductors [25,26] together with its bulk-edge correspondence [27,28]; the closely related Su–Schrieffer–Heeger chain [29] is its chiral-symmetric cousin. It is the fermionic counterpart of the cluster state, and our Haskell computes its invariant across the parameter plane.

Write the chain with spinless fermions $c_j$, hopping $t$, chemical potential $\mu$, and $p$-wave pairing amplitude $\Delta_{\mathrm{p}}$ (written with a subscript to keep it visibly distinct from the uniform gap bound $\Delta=\Delta$ of Section 2): $$H = \sum_j\Bigl[-t\,(c_j^\dagger c_{j+1}+\mathrm{h.c.}) -\mu\bigl(c_j^\dagger c_j-\tfrac12\bigr) +\Delta_{\mathrm{p}}\,(c_j c_{j+1}+\mathrm{h.c.})\Bigr].$$ In momentum space the Bogoliubov–de Gennes Bloch Hamiltonian is $h(k)=d_z(k)\,\tau_z+d_y(k)\,\tau_y$ in the Nambu basis, with $$d_z(k) = -2t\cos k-\mu, \qquad d_y(k) = 2\Delta_{\mathrm{p}}\sin k, \qquad d_x(k)\equiv 0,$$ where $\tau_{x,y,z}$ are Pauli matrices in particle–hole space and the $\tau_x$ component $d_x$ vanishes identically because the pairing is real. The single-particle spectrum is $\pm\sqrt{d_y^2+d_z^2}$, so the bulk gap closes exactly when $d_y=d_z=0$, i.e. at $\sin k=0$ and $\mu=\mp2t$, i.e. on the discriminant $|\mu|=2|t|$ (for $\Delta_{\mathrm{p}}\ne0$).

In two dimensions the fermionic story is richer and, importantly, also constructive: the group-supercohomology models of Gu–Wen [30] realize a large class of interacting fermionic SPTs, the discrete-spin-structure construction of Tarantino–Fidkowski [31] gives explicit commuting-projector models for all $2+1$d fermionic SPTs with symmetry $G_f=G×\mathbb{Z}_2^f$ (including all eight members of the $\mathbb{Z}_8$ classification for $G=\mathbb{Z}_2$), and the general construction-and-classification programme of Wang–Gu [32] extends this. These are torsion classes and are $\mathrm{CP}$-realizable; they populate the fermionic rows of the status table (Table 1).

4.4 Homotopy of the space of models

Realizability is not only a $\pi_0$ question. The higher homotopy of the space of gapped Hamiltonians records adiabatic pumps ($\pi_1$) and higher families ($\pi_n$); realizing a class in $\pi_n$ means building an $n$-parameter family of gapped models with the prescribed higher invariant; the $\pi_1$ generator is the adiabatic charge pump of Thouless [33], whose rigorous lattice avatar is the higher Berry class of Kapustin–Sopenko [34], and the homotopical foundations of such parametrized families are set up by Beaudry et al. [35]; a complementary classification by the homotopy of the space of states of a $C^*$-algebra appears in [36,37]. For $2+1$d topological order, Aasen–Wang–Hastings [38] propose explicit generators of $\pi_1$, $\pi_2$, $\pi_3$ of the space of Hamiltonians via automorphisms of topological order and honeycomb/automorphism codes. We stress that these homotopy groups are conjectural: [38] give constructions and strong evidence but not a proof that these are the full homotopy groups, and we cite them as such. The moduli-space perspective of Hsin–Wang [39] is the closest ordinary-topology precedent for the object Part IV condenses.

5 Completeness in low dimension

In dimension one, realizability is completely settled for on-site finite symmetry, because the operator-algebraic invariant is a complete invariant: the realized set is forced to equal the invariant set. In dimension two the same operator-algebraic index is known to exist and every in-cohomology class is realized, but its completeness is open, so only the surjectivity half survives there. The abstract target is the group-cohomology classification, itself a special case of the generalized-cohomology description of SPT phases [40,41].

6 The obstruction: the Kapustin–Fidkowski no-go

We now state the hard wall. It is the single result that most sharply bounds realizability, and it is about the commuting-projector class specifically.

7 A status table for realizability

We collect the state of knowledge into Table 1, organized by spatial dimension and symmetry class, separating the torsion part of the classification from the free part, and recording for each the best available construction, the applicable obstruction, and whether realizability is settled, constructive, obstructed-for-$\mathrm{CP}$, or open. The entries are the theorems and constructions cited above; nothing in the table is claimed beyond them.

Two patterns organize the table. First, the torsion part of the classification is, wherever a construction is known, realized by commuting-projector models, and in $d=1,2$ with on-site finite symmetry this is complete. Second, the free (chiral) part is gapped-realizable but never $\mathrm{CP}$-realizable, uniformly, by Theorem 6.1. The next section turns these two patterns into conjectures about the whole image.

8 The realizability image: conjectures

The theorems above describe $\operatorname{im}\rho$ and $\operatorname{im}\rho^{\mathrm{CP}}$ in the regions where constructions and completeness results exist. The global shape of the two images is not a theorem today; it is the conjectural content of this paper. We state three numbered conjectures, in decreasing order of confidence, and phrase them in the condensed language so that they interlock with the rest of the program.

8.1 The image is the short-range-entangled subspectrum

8.2 Chiral classes: gapped yes, commuting-projector no

8.3 Realizability over profinite disorder hulls

The third conjecture is where the condensed formalism does work that ordinary topology does not: it concerns realizability not over a point but over a profinite disorder hull, as a descent statement.

8.4 The comparison square

The three conjectures fit into one diagram, which is the realizability face of the program. Writing $\rho^{\mathrm{CP}}$ for the restriction to commuting-projector models and $\gamma$ for group completion, the square

\begin{tikzcd}[column sep=large, row sep=large]
\{\text{$\mathrm{CP}$ lattice models}\}/\mathcal{W}\arrow[r, "\rho^{\mathrm{CP}}"] \arrow[d, hook]
  & \operatorname{im}\rho^{\mathrm{CP}} \arrow[d, hook] \\
\{\text{gapped lattice models}\}/\mathcal{W}\arrow[r, "\rho"']
  & \mathcal{C}_{d,G}
\end{tikzcd}

9 Disorder-robust realizability over profinite hulls

We expand the profinite thread, since it is the part of the realizability story that the condensed formalism is built for. Ordinary topology treats a disordered family as a section of a bundle over a compact metric space; condensed mathematics treats it as an $\Omega$-point, where $\Omega=Q^{\mathbb{Z}^d}$ is genuinely profinite and its finite quotients are finite-resolution disorder data. The realizability question then acquires a resolution-by-resolution structure.

The content of Conjecture V-3 is that the passage from resolutionwise realizability to profinite realizability is governed by a single condensed-cohomological obstruction. Two structural facts, both consequences of the companion papers, make this plausible and are theorems modulo their stated hypotheses.

10 Computational verification

The accompanying Haskell package src/bordism-realizability/ makes the two elementary propositions executable and self-checking. It is not a proof assistant; it is a battery of exact finite computations that would fail loudly if the propositions were misstated, together with QuickCheck properties that sample the parameter space.

10.1 Kitaev-chain invariant sweep

The module Kitaev.hs implements the Bogoliubov–de Gennes $d$-vector $(d_y(k),d_z(k))$ of Theorem 4.3 and computes two invariants directly from it: the winding number $w$, by summing signed angle increments of $(d_y,d_z)$ around $k\in[0,2\pi)$ on a fine grid and rounding; and the Majorana number $\mathcal{M}=\mathop{\mathrm{sgn}}(d_z(0))\mathop{\mathrm{sgn}}(d_z(\pi))$. Main.hs prints an invariant table as $\mu$ sweeps across the transition at fixed $t=\Delta_{\mathrm{p}}=1$:

for $|\mu|<2$ the winding is $1$ and $\mathcal{M}=-1$ (topological); for $|\mu|>2$ the winding is $0$ and $\mathcal{M}=+1$ (trivial); the two invariants agree (mod $2$) on the gapped locus and jump together at $|\mu|=2$.

A second routine sweeps the full $(\mu,t)$ plane and reports the phase boundary as the locus where the winding changes, recovering $|\mu|=2|t|$. The transition is detected, not hard-coded: the code locates the grid cell where the invariant changes and reports the midpoint.

10.2 Cluster-state string order

The module Stabilizer.hs implements an exact stabilizer-formalism engine over the binary symplectic (check-matrix) representation of Pauli operators with $i$-power phase tracking. It builds the cluster-state stabilizer group $\langle K_j\rangle$ of Theorem 3.3, computes $\langle P\rangle\in\{+1,-1,0\}$ for any Pauli string $P$ by testing membership in the group via Gaussian elimination over $\mathbb{F}_2$, and evaluates the string order parameter $S_{a,b}$ of Theorem 3.4. It confirms $\langle S_{a,b}\rangle=+1$, $\langle Z_j\rangle=\langle X_j\rangle=0$, and computes the symmetry-breaking degradation $\langle U(\theta)^\ast S_{a,b}U(\theta)\rangle$ by expanding the rotated string into its $2^m$ Pauli terms and summing exact stabilizer expectations — recovering $(\cos\theta)^m$ without assuming the closed form.

10.3 QuickCheck properties

Properties.hs states one property per claim of the two propositions:

• Invariant quantization: for random $(\mu,t,\Delta_{\mathrm{p}})$ off the discriminant, the computed winding is exactly $0$ or $1$ and equals $\tfrac12(1-\mathcal{M})$.

• Phase-boundary detection: for random $t,\Delta_{\mathrm{p}}\ne0$, the winding is $1$ iff $|\mu|<2|t|$, and the detected boundary matches $|\mu|=2|t|$ within grid resolution.

• String-order quantization: for random even-length windows, $\langle S_{a,b}\rangle=+1$ in the cluster state and $\langle Z_j\rangle=0$.

• String-order degradation: for random $\theta\in[0,\pi/2]$ and window length, the computed degraded string order equals $(\cos\theta)^m$ within tolerance and is monotone decreasing in $\theta$.

Main.hs runs the demonstrations and all properties and exits nonzero on any failure, so the package doubles as a regression test of Theorems 3.4 and 4.3.

11 Discussion

11.1 What the condensed viewpoint adds here

Realizability is, at bottom, a question about explicit Hamiltonians, and one might ask what the condensed formalism contributes beyond bookkeeping. Three things. First, it gives the right home for the disorder thread: Conjecture V-3 and Section 9 are statements about $\Omega$-points and descent that are awkward to phrase with ordinary bundles and natural with profinite probes. Second, it makes the comparison map $\rho$ a morphism in a category with exact derived operations, so that “obstruction subgroup” $\mathrm{Obs}_{d,G}=\mathcal{C}_{d,G}/\operatorname{im}\rho$ is a well-behaved condensed object rather than a set-theoretic quotient. Third, it aligns realizability with the rest of the tower: the same $\mathcal{W}$, the same stabilization, the same spectrum appear in Parts III–V, so realizability is literally the surjectivity of a map the program already constructs, not a separate theory.

11.2 Limitations

We are candid about the boundaries of the rigorous content. The three Theorems are repackagings of cited results into the realizability language; their mathematical substance is due to Chen–Gu–Liu–Wen, Else–Nayak, Ogata, Kapustin–Fidkowski, and Kapustin–Spodyneiko, and we claim only the translation and the clean $\mathrm{CP}$-versus-gapped framing. The two Propositions are elementary and self-contained but concern the simplest ($1$d, torsion) corner. Everything about the global image (Conjectures V-1, V-2, V-3) is open and depends on Part IV’s still-conjectural identification of $\mathcal{C}_{d,G}$ with $\pi_0\mathbf{IP}^{\mathrm{cond}}_{d,G}$. In particular we do not prove that every SRE class is realizable, nor that every non-chiral class is $\mathrm{CP}$-realizable; we prove the obstruction (chiral $⇒$ not $\mathrm{CP}$) and organize the positive evidence.

11.3 Non-invertible order

The entire spectrum-level story is about invertible (SRE) phases. Intrinsic topological order (anyons, modular tensor categories, nonzero total quantum dimension) is not classified by $\mathbf{IP}^{\mathrm{cond}}_{d,G}$ and its realizability is a different question, requiring the condensed higher stack of phases and defects that Part VI flags as out of current scope. Commuting-projector models realize a great deal of non-chiral topological order (toric code, string-net / Levin–Wen, Walker–Wang), but the chiral topological orders (e.g. the non-abelian Ising phase with $c_-=1/2$) inherit the obstruction of Theorem 6.1 through their edge and are not $\mathrm{CP}$-realizable.

11.4 Open directions

The most consequential open problem is a general construction theorem: a procedure that, given an arbitrary SRE class in $\pi_0\mathbf{IP}^{\mathrm{cond}}_{d,G}$, outputs a uniformly gapped lattice model realizing it (non-commuting where the class is chiral). Conjecture V-1 asserts such a procedure exists; the coupled-wire and network constructions are the closest existing approximations for chiral phases. A second direction is to prove or refute the descent obstruction picture of Conjecture V-3 in a tractable family (e.g. the disordered Kitaev chain over a Bernoulli hull), where the crossed-product invariant of Part II is computable and the obstruction $o(c)$ should be explicitly checkable.

12 Conclusion

We have separated the realizability question, which abstract bordism/homotopy classes are produced by explicit gapped lattice models, into a constructive region, an obstructed region, and an open region, all inside the condensed program of this series. The constructive region is governed by two theorems: every group-cohomology class is realized by a commuting-projector model (Theorem V-A), and in $d=1$ with on-site finite symmetry the operator-algebraic index is complete, so realized equals invariant (Theorem V-B); in $d=2$ the index exists and every in-cohomology class is realized, though completeness is open. The obstructed region is governed by one theorem in two parts: local commuting-projector Hamiltonians have zero electric Hall conductance under $U(1)$ (Kapustin–Fidkowski) and zero chiral central charge (Kapustin–Spodyneiko), so every chiral class lies outside the commuting-projector image (Theorem V-C), and the inclusion $\operatorname{im}\rho^{\mathrm{CP}}\subsetneq\operatorname{im}\rho$ is strict wherever chiral phases exist. The open region is the global shape of the realizability image, stated as three numbered conjectures: that the image is exactly the short-range-entangled subspectrum (V-1), that chiral classes are gapped-realizable but never commuting-projector realizable (V-2), and that realizability descends over profinite disorder hulls under a condensed-cohomological obstruction (V-3). The two elementary propositions (the Kitaev chain winding invariant and the cluster-state string order) anchor the torsion corner concretely and are verified by the accompanying Haskell. Throughout, the discipline that makes the picture coherent is the distinction between realizability by some gapped local Hamiltonian and realizability by a commuting-projector model; the Kapustin–Fidkowski (electric) and Kapustin–Spodyneiko (thermal) walls are exactly the gap between them.

12.0.0.1 Code availability.

The accompanying Haskell package (the Kitaev-chain invariant sweep and the exact cluster-state stabilizer engine that verify Theorems 3.4 and 4.3) is available at github.com/YonedaAI/topological-phases-of-matter in the directory src/bordism-realizability/.

References