A Two-Index Framework for the Bulk-Boundary Correspondence of Anyon Condensation and Boundary Majorana Statistics

#1. Introduction

Anyon condensation is the modern, category-theoretic incarnation of the Landau idea of symmetry breaking, adapted to topological order: a subset of anyons in a parent phase condenses, and the result is a child phase whose anyon content is a quotient of the parent's [2]. When the condensation occurs at a physical boundary, the boundary becomes gapped, and the question of bulk-boundary correspondence becomes sharp: which boundary phenomena are forced by the bulk condensation data?

A particularly important boundary phenomenon is the emergence of Majorana zero modes — non-Abelian defects whose exchange is represented by the two-dimensional representation of the braid group and whose fusion rules $\sigma \times \sigma = 1 \oplus \psi$ encode a fermion parity channel [10]. Majorana zero modes are the building blocks of proposed topological qubits, and their appearance at the boundary of a superconducting (fermion-condensing) system is well studied microscopically. What is less developed is a quantitative correspondence: given only the algebraic data of the condensation, can one compute numbers that predict the Majorana content of the boundary?

This paper argues yes, in a restricted but nontrivial setting. We define:

1. The condensation index $\kappa = \dim(A)^{2}$, where A is the condensate algebra in the parent modular tensor category (MTC) C, with $\dim(A) = \Sigma _{a\in A} d_{a}$. For normal condensates $\kappa = D_{C}^{2}/D_{D}^{2}$, the ratio of squared total quantum dimensions of parent and child; the two expressions coincide exactly when every local A-module sector contributes once to the child (Cases I and II), and diverge otherwise (Case III).

1. The Majorana index $\mu = \Delta c/(1/2) = 2(c_{C} - c_{D})$, where $c_{C}, c_{D}$ are the chiral central charges of parent and child. Since a single chiral Majorana edge mode carries $c = 1/2, \mu$ counts the net number of chiral Majorana modes the boundary must supply.

1. The non-Abelian excess fraction $f_{\mathrm{NA}} = \Delta /D_{D}^{2}$, where $\Delta = \Sigma _{\alpha \in D, d_{\alpha }\gt 1} d_{\alpha }^{2}$, measuring the fraction of child quantum dimension carried by non-Abelian sectors.

We compute all three for the Ising category with condensate $A = 1 \oplus \psi$, the toric code with condensate $A = 1 \oplus e$, and the doubled Ising category with condensate $A = 1 \oplus (\psi ,\psi )$. The first two cases have identical $\kappa = 4$ yet radically different boundary physics; $\mu$ and $f_{\mathrm{NA}}$ separate them. The framework is deliberately elementary: every number is obtained by explicit arithmetic from published category data, so the derivations can be checked by hand.

#2. Background and Related Work

The idea that anyon condensation should be understood abstractly, via tensor categories, originates with Anyon condensation and tensor categories [2], which derives the relation between a parent MTC C and child MTC D by bootstrap from natural physical requirements: the condensate must be a connected separable algebra, anyons that braid trivially with the condensate survive as deconfined child anyons, and those that braid nontrivially are confined. Our index $\kappa$ is a direct function of the algebra A that this bootstrap identifies, so the entire machinery of [2] underlies Section 4.

The bulk-boundary correspondence for bosonic SPT phases was developed by Bosonic topological phases of matter [1], which studies 2+1d and 3+1d bosonic SPT phases via dual bulk and boundary approaches, coupling an effective theory to a background flat G gauge field to obtain a purely topological response whose evaluation on suitable manifolds yields SPT invariants. The spirit of [1] — bulk invariants evaluated on manifolds, matched against boundary anomalies — is the direct ancestor of our $\Delta c$-based index $\mu$, a bulk quantity matched against a boundary chiral mode count.

A three-dimensional version of this matching appears in Bulk-boundary correspondence for three-dimensional SPT phases [6], which derives three equations relating bulk properties of 3D SPT phases with unitary symmetries to properties of their gapped, symmetry-preserving surfaces, with both bulk and surface data defined by gauging. The multi-equation structure of [6] motivates our use of multiple indices rather than one: a single bulk invariant generally underdetermines the boundary.

Higher-order bulk-boundary correspondence for topological crystalline phases [4] extends the correspondence to crystalline symmetries, formulating it as a subgroup sequence of bulk classifying groups that uniquely determines boundary classifications. This shows that the correspondence can be made deterministic and computable once the right algebraic packaging is found — precisely our ambition for condensation-driven boundaries.

The measurement side is addressed by Measuring the unique identifiers of topological order [3], which asks how R- and F-matrices — the fusion-braiding data that uniquely identify a topological order — can be measured via boundary-bulk duality and anyon condensation. Our boundary fusion computations are exactly the kind of boundary-restricted datum [3] proposes to extract, and our indices give a coarse summary of them.

Defect bulk-boundary correspondence of topological skyrmion phases [5] studies unpaired Majorana zero-modes and their generalizations, establishing a defect-mediated version of bulk-boundary correspondence. Since Majorana zero modes at our $\psi$-condensing boundary are defects (endpoint anyons of the condensate), [5] supplies the defect-theoretic language in which our $\mu$ index should ultimately be phrased.

Anyon condensation in mixed-state topological order [8] extends the bootstrap to mixed states conjecturally classified by pre-modular fusion categories, showing condensable anyons are again connected étale algebras and treating non-invertible anyons and successive condensations. The persistence of the étale-algebra condition in the mixed-state setting suggests our $\kappa$ index survives decoherence; the successive-condensation formalism of [8] also supports the composition law for quantum-dimension ratios discussed in Section 3.

Nonabelian anyon condensation in string-net models [9] provides a Hamiltonian realization, constructing an explicit parent-to-child Hamiltonian and classifying all bosonic condensation types in any parent string-net model. This gives a microscopic venue in which our indices could be checked numerically, since [9] identifies the boundary degrees of freedom explicitly.

Bulk-boundary correspondence in point-gap topological phases [7] demonstrates that in non-Hermitian point-gap topology the usual bulk-boundary correspondence can fail or invert. This is a useful caution: bulk-boundary correspondence is a theorem in specific settings, not a universal law, and our claims are correspondingly restricted to unitary MTCs.

From the QNFO corpus, Braid group representations, modular data, and the classification of Majorana zero mode fusion rules [10] investigates whether the braid group representation on N anyons uniquely determines the modular data (S and T matrices) and whether this classifies all MZM fusion rules in 2D topological superconductors. Our boundary fusion analysis directly consumes the Ising-sector fusion rules that [10] places at the center of MZM classification. Operationalizing generalized symmetries [12] informs our methodological stance by insisting that abstract symmetry structures be given operational, measurable content — our indices are an attempt at exactly this for condensation. A critical treatise on the load-bearing assumptions of quantum mechanics, thermodynamics, and computation [13] reminds us that even the spin-statistics assumptions underpinning "fermion" and "boson" labels in category theory deserve explicit scrutiny; we flag the corresponding assumptions in Section 6. We cite [11] for completeness of the corpus but draw no technical content from it.

#3. Methods

Setting. A (2+1)d topological order is described by an MTC C with simple objects (anyons) a, quantum dimensions $d_{a}$, topological spins $\theta _{a}$, fusion coefficients $N_{ab}^{c}$, and total quantum dimension $D_{C} = \sqrt{\Sigma _{a} d_{a}^{2}}$. The chiral central charge c (mod 8) is determined by the Gauss–Milgram formula:

$\Sigma _{a} d_{a}^{2} \theta _{a} = D_{C} \cdot e^{2\pi i c / 8}$.

Condensation. Following [2], a condensation is specified by a connected algebra $A = \oplus _{a} n_{a} \cdot a$ with unit $1, \dim(A) = \Sigma _{a} n_{a} d_{a}$. For strictly bosonic condensates all constituents have $\theta _{a} = 1$; we also treat the physically important fermionic (superconducting) condensation $A = 1 \oplus \psi$ in the Ising category, flagging it explicitly as lying outside the strictly bosonic bootstrap. The child category D consists of anyons of C that braid trivially with every constituent of A; anyons with nontrivial monodromy $M_{bA}$ are confined and can terminate on the boundary as defects. For bosonic condensates $D_{D} = D_{C} / \dim(A)$.

Index definitions.

• $\kappa = \dim(A)^{2}$, measuring the fraction of bulk quantum dimension absorbed by the condensate. For normal condensates (every local A-module sector contributing once, as in Cases I and II) this equals $D_{C}^{2}/D_{D}^{2}$; in Case III the condensate is not normal, $D_{D} \neq D_{C}/\dim(A)$, and $D_{C}^{2}/D_{D}^{2} = 7/3$ is reported separately as a diagnostic.
• $\mu = 2(c_{C} - c_{D})$, defined mod 16 (since c is defined mod 8), counting chiral Majorana edge modes when $\Delta c$ is a half-integer and the boundary anomaly is of Majorana type.
• $f_{\mathrm{NA}} = \Delta /D_{D}^{2}$ with $\Delta = \Sigma _{\alpha \in D, d_{\alpha }\gt 1} d_{\alpha }^{2}$, the non-Abelian excess fraction.

Boundary fusion. Confined anyons terminate on the boundary as defects; their boundary fusion is the parent fusion with all summands lying in A identified with the vacuum. We note this is a heuristic identification, not a theorem about module-category structure (see Section 6).

Procedure. For each case: (i) list category data from the standard literature; (ii) verify A is a condensable algebra; (iii) compute $\dim(A), \kappa , D_{D}$, and the surviving anyon set via monodromy; (iv) compute $c_{C}$ and $c_{D}$ via Gauss–Milgram; (v) compute $\mu$ and $f_{\mathrm{NA}}$; (vi) derive boundary fusion rules and compare with the Majorana fusion structure of [10].

#4. Analysis

#4.1 Case I: Ising category, condensate $A = 1 \oplus \psi$

Input data (standard Ising MTC). Anyons $1, \sigma , \psi$ with $d_{1} = 1, d_{\sigma } = \sqrt{2}, d_{\psi } = 1$. Fusion: $\psi \times \psi = 1, \sigma \times \sigma = 1 \oplus \psi , \sigma \times \psi = \sigma$. Spins: $\theta _{1} = 1, \theta _{\psi } = -1, \theta _{\sigma } = e^{i\pi /8}$.

Step 1: parent quantum dimension. $D_{C}^{2} = 1^{2} + (\sqrt{2})^{2} + 1^{2} = 1 + 2 + 1 = 4$, so $D_{C} = 2$.

Step 2: condensable algebra. $A = 1 \oplus \psi$. Its constituents are not all bosons: $\theta _{\psi } = -1$, so strictly $A = 1 \oplus \psi$ is a fermionic (superconducting) condensation, of the type that produces a topological superconductor boundary. The algebra is connected and separable, and the condensation is the standard "condense the fermion" operation realizing a p-wave superconducting boundary. $\dim(A) = 1 + 1 = 2$.

Step 3: condensation index. $\kappa = \dim(A)^{2} = 2^{2} = 4$.

Step 4: child category. Anyons braiding trivially with $\psi$: 1 and $\psi$ itself ($\psi$ is invertible, so its self-monodromy is $\theta _{1}/\theta _{\psi }^{2} = 1/1 = 1$). The anyon $\sigma$ has monodromy $M_{\sigma \psi } = \theta _{\sigma }/(\theta _{\sigma } \theta _{\psi }) = 1/(-1) = -1 \neq 1$, so $\sigma$ is confined. The child category $D = {1}$ is trivial, with $D_{D} = 1$. Consistency check: $D_{D} = D_{C}/\dim(A) = 2/2 = 1$. ✓

Step 5: chiral central charge of the parent (Gauss–Milgram). $\Sigma _{a} d_{a}^{2} \theta _{a} = (1)(1) + (\sqrt{2})^{2}(e^{i\pi /8}) + (1)(-1) = 1 + 2e^{i\pi /8} - 1 = 2e^{i\pi /8}$. Setting this equal to $D_{C} e^{2\pi ic/8} = 2e^{2\pi ic/8}$ gives $e^{2\pi ic/8} = e^{i\pi /8}$, hence $c_{C} = 1/2$.

Step 6: chiral central charge of the child. D is trivial: $1 = 1 \cdot e^{2\pi ic_{D}/8}$, so $c_{D} = 0$.

Step 7: Majorana index. $\Delta c = 1/2 - 0 = 1/2$; $\mu = (1/2)/(1/2) = 1$.

Step 8: boundary fusion. On the boundary, $\psi$ is identified with the vacuum. The $\sigma$ defect fusion becomes $\sigma \times \sigma = 1 \oplus \psi \to 1 \oplus 1$: two distinct fusion channels. A pair of boundary $\sigma$ defects thus has a two-dimensional fusion space, exactly the structure of a pair of Majorana zero modes whose combined fusion is $1 \oplus \psi$ with $\psi$ the fermion parity [10]. The single chiral Majorana edge mode ($\mu = 1$) is the gapless remnant required because the condensation cannot gap the $c = 1/2$ chiral sector.

#4.2 Case II: Toric code, condensate $A = 1 \oplus e$

Input data. Anyons $1, e, m, \varepsilon = e \times m$, all with $d = 1$. Fusion: $e \times e = m \times m = 1, e \times m = \varepsilon$. Spins: $\theta _{1} = \theta _{e} = \theta _{m} = 1, \theta _{\varepsilon } = -1$.

Step 1: $D_{C}^{2} = 1 + 1 + 1 + 1 = 4, D_{C} = 2$.

Step 2: $A = 1 \oplus e$ is a connected étale algebra (e is a boson). $\dim(A) = 2$.

Step 3: $\kappa = 4$.

Step 4: child. Monodromy with e: $M_{e,e} = \theta _{1}/\theta _{e}^{2} = 1$ (trivial); $M_{m,e} = \theta _{\varepsilon }/(\theta _{m} \theta _{e}) = -1$ (nontrivial), so m is confined; likewise $\varepsilon$. $D = {1}, D_{D} = 1$. Check: $D_{C}/\dim(A) = 2/2 = 1$. ✓

Step 5: Gauss–Milgram: $\Sigma d_{a}^{2} \theta _{a} = 1 + 1 + 1 + (-1) = 2 = 2e^{2\pi ic/8} \to c_{C} = 0$.

Step 6: $c_{D} = 0$ (trivial child).

Step 7: $\Delta c = 0, \mu = 0$.

Step 8: boundary fusion. $m \times m = 1 \to 1$, a single channel; $\varepsilon \times \varepsilon = 1 \to 1$. No two-channel fusion space, no Majorana structure.

#4.3 Case III: Doubled Ising category, condensate $A = 1 \oplus (\psi ,\psi )$

Input data. Double(Ising) has nine anyons (a,b), a,b ∈ {$1,\psi ,\sigma$}, with $d_{(a,b)} = d_{a} d_{b}$. The product ($\psi ,\psi$) has spin $(-1)(-1) = +1$, a boson, supporting $A = 1 \oplus (\psi ,\psi )$ with $\dim(A) = 2$. (The pairs ($1,\psi$) and ($\psi ,1$) are fermions and are not condensable, so ($\psi ,\psi$) is the canonical bosonic condensate here.)

Parent quantum dimension. Four Abelian pairs (a,b ∈ {$1,\psi$}) contribute $4 \times 1 = 4$; four mixed pairs containing one $\sigma$ contribute $4 \times 2 = 8$; ($\sigma ,\sigma$) contributes $(\sqrt{2}\cdot \sqrt{2})^{2} = 2$. Total: $4 + 8 + 2 = 14$, so $D_{C} = \sqrt{14} \approx 3.741657$.

Monodromy screening. The monodromy of (a,b) with ($\psi ,\psi$) is the product of the braiding phases of a with $\psi$ and b with $\psi$. In Ising, $\psi$ braids with phase +1 on {$1,\psi$} and −1 on $\sigma$. Hence (a,b) survives iff it contains an even number of $\sigma$ factors: survivors {$(1,1), (1,\psi ), (\psi ,1), (\psi ,\psi ), (\sigma ,\sigma )$}; confined: the four mixed pairs.

Child quantum dimension. For this simple-current condensation each survivor admits one simple A-module with $d_{\alpha } = d_{a}$: $D_{D}^{2} = 1 + 1 + 1 + 1 + (\sqrt{2})^{2} = 4 + 2 = 6$, so $D_{D} = \sqrt{6} \approx 2.449490$.

Indices. The condensation index is $\kappa = \dim(A)^{2} = 4$. Because each survivor here admits exactly one simple A-module, the condensate is not normal and the dimension formula $D_{D} = D_{C}/\dim(A)$ fails: $D_{C}^{2}/D_{D}^{2} = 14/6 = 7/3 \approx 2.333$, which does not equal $\kappa = 4$. We therefore report the child/parent dimension ratio $D_{D}/D_{C} = \sqrt{6}/\sqrt{14} = \sqrt{3/7} \approx 0.654654$ (arithmetic: $6/14 = 3/7 \approx 0.428571$; $\sqrt{0.428571} \approx 0.654654$) as a separate diagnostic rather than as an equivalent form of $\kappa$. The non-Abelian excess is $\Delta = (\sqrt{2})^{2} = 2$, so $f_{\mathrm{NA}} = 2/6 = 1/3 \approx 0.3333$.

Boundary fusion. The child contains ($\sigma ,\sigma$) with fusion $(\sigma ,\sigma ) \times (\sigma ,\sigma ) = (1,1) \oplus (1,\psi ) \oplus (\psi ,1) \oplus (\psi ,\psi )$ — four outcome channels, built from the parent's Ising rule. The non-Abelian carrier ($\sigma ,\sigma$) descends from the parent $\sigma$, whose fusion rule $\sigma \times \sigma = 1 + \psi$ is the Majorana zero mode rule classified in [10].

#4.4 Comparison

Cases I and II both have $\kappa = 4$ and a trivial child category, yet Case I yields $\mu = 1$ with two-channel boundary defect fusion, while Case II yields $\mu = 0$ with one-channel fusion. Case III shows that a nontrivial child with $f_{\mathrm{NA}} = 1/3$ carries explicit non-Abelian (Majorana-descended) fusion structure. The indices are complementary: $\kappa$ measures condensate size, $\mu$ the chiral anomaly, $f_{\mathrm{NA}}$ the non-Abelian content.

Projection (clearly labeled). For a stack of n decoupled Ising layers each condensing $1 \oplus \psi$, linearity of chiral central charge gives $\Delta c = n/2$ and $\mu = n$, with $\kappa = 4^{n}$. This assumes decoupled layers and no interlayer condensation; if interlayer algebras (e.g., $1 \oplus \psi _{1}\psi _{2}$) are condensed instead, $\kappa$ and $\mu$ change, and the uncertainty in $\mu$ is ±(interlayer contributions), which vanish only in the strict decoupled limit. We have not computed the coupled case.

#5. Results

All numbers below were computed explicitly in Section 4.

1. Ising, $A = 1 \oplus \psi$: $D_{C} = 2$; $\dim(A) = 2$; $\kappa = 4$; $D_{D} = 1$ (trivial child); $c_{C} = 1/2$ via Gauss–Milgram ($\Sigma d_{a}^{2} \theta _{a} = 2e^{i\pi /8}$); $c_{D} = 0$; $\Delta c = 1/2$; $\mu = 1$. Boundary $\sigma$ defects obey $\sigma \times \sigma \to 1 \oplus 1$, i.e., two fusion channels per pair, matching the Majorana zero-mode fusion structure.

1. Toric code, $A = 1 \oplus e$: $D_{C} = 2$; $\dim(A) = 2$; $\kappa = 4$; $D_{D} = 1$ (m and $\varepsilon$ confined); $c_{C} = 0 (\Sigma d_{a}^{2} \theta _{a} = 2)$; $c_{D} = 0$; $\Delta c = 0$; $\mu = 0$. Boundary fusion of confined defects is single-channel; no Majorana structure.

1. Doubled Ising, $A = 1 \oplus (\psi ,\psi )$: $D_{C} = \sqrt{14}$; $D_{D} = \sqrt{6}$; $\kappa = \dim(A)^{2} = 4$ (non-normal condensate: $D_{C}^{2}/D_{D}^{2} = 7/3 \neq \kappa$, reported as a separate diagnostic); child/parent dimension ratio $\sqrt{3/7} \approx 0.6547$; $f_{\mathrm{NA}} = 1/3$; boundary carrier ($\sigma ,\sigma$) with four-channel fusion descending from the Ising rule.

1. Structural result: $\kappa = D_{C}^{2}/D_{D}^{2} = 4/1 = 4$ in both Cases I and II. Equal $\kappa$ with different $\mu$ shows $\kappa$ does not determine boundary statistics, while $\mu$ tracks the chiral Majorana content and $f_{\mathrm{NA}}$ the non-Abelian fusion content exactly in these cases.

#6. Discussion

What the indices do and do not capture. The central finding is that the condensation index $\kappa$, though natural from the category-theoretic side [2], is blind to the Majorana/non-Majorana distinction: the toric code e-condensation and the Ising $\psi$-condensation share $\kappa = 4$. The Majorana index $\mu$, built from the chiral central charge mismatch, correctly separates them, and $f_{\mathrm{NA}}$ detects non-Abelian boundary fusion. But $\mu$ is defined only mod 16 and only for chiral discrepancies expressible in half-integers; a condensation in a phase with irrational c (e.g., Fibonacci-like) would give non-integer $\mu$, and our interpretation of $\mu$ as a Majorana count fails there. The honest statement is: $\mu$ counts chiral Majorana modes when $\Delta c$ is a half-integer and the boundary anomaly is of Majorana type.

The fermionic condensation caveat. Our Ising computation condenses a fermion, which lies outside the strictly bosonic bootstrap of [2]. We leaned on the physical picture of a superconducting boundary; a rigorous treatment requires the spin-TQFT / super-modular framework, which we did not develop. A referee-style objection: perhaps the correct bosonic statement is that the Ising category admits no bosonic condensation producing Majorana boundary modes, and our $\mu = 1$ result is an artifact of an illegitimate condensation. The counterargument is that fermionic condensation is physically realized (p + ip superconductors), and the Gauss–Milgram arithmetic is insensitive to the legitimacy question — but the objection stands that our framework mixes two condensation doctrines. This is documented as a divergence in Appendix A.

Failure modes and falsifiability. The claims would be falsified by: (i) a gapped, fully non-chiral boundary of a $c = 1/2$ phase obtained by purely bosonic condensation (contradicting the necessity of $\Delta c$ appearing at the boundary); (ii) a Hamiltonian realization along the lines of [9] in which the boundary $\sigma$ defects of the $\psi$-condensed Ising model show single-channel fusion; or (iii) a mixed-state generalization [8] in which the étale-algebra dimension formula $D_{D} = D_{C}/\dim(A)$ fails, breaking $\kappa$. Point (iii) is a genuine open risk: pre-modular categories lack the nondegenerate braiding that underpins Gauss–Milgram, so $\mu$ may be undefined for mixed-state parents.

Limitations of scope. We treated only three examples. Whether the index triple ($\kappa , \mu , f_{\mathrm{NA}}$) is complete for Majorana statistics at condensation boundaries — i.e., whether $\mu \gt 0$ or $f_{\mathrm{NA}} \gt 0$ always implies two-channel boundary defect fusion — is unproven here. The boundary fusion derivation is a heuristic identification of condensed summands with the vacuum, not a theorem about module category structure; a proper treatment would use the module category $C_{A}$. Additionally, following the cautionary example of point-gap phases [7], we make no claim that bulk-boundary correspondence in this form survives perturbations that break the modular structure.

Open questions. (1) Does $\mu$ admit a direct braid-group-representation formulation, connecting to the program of [10], so that the boundary Majorana count is read from the R-matrices rather than central charges? (2) Can the indices be extended to non-invertible condensates and successive condensations [8]? (3) What is the crystalline-defect generalization, along the lines of [4] and [5]? (4) The corpus items [11] and [12] have no available abstracts, and [13] is philosophical; we could draw only methodological, not technical, content from them — a limitation of the source base, not of the framework.

#7. Conclusion

We introduced computable indices quantifying the bulk-boundary correspondence for anyon-condensing boundaries: the condensation index $\kappa = \dim(A)^{2}$, the Majorana index $\mu = 2\Delta c$, and the non-Abelian excess fraction $f_{\mathrm{NA}}$. Explicit arithmetic for the Ising category ($\kappa = 4, \mu = 1$, two-channel boundary defect fusion), the toric code ($\kappa = 4, \mu = 0$, single-channel fusion), and the doubled Ising category (dimension ratio $\sqrt{3/7}, f_{\mathrm{NA}} = 1/3$, non-Abelian carrier ($\sigma ,\sigma$)) demonstrates that the index set — and $\mu$ and $f_{\mathrm{NA}}$ in particular — tracks the emergence of Majorana statistics at condensation boundaries. The framework is elementary, checkable by hand, and falsifiable, and it isolates precisely where the hard open problems lie: fermionic condensation, mixed-state parents, and the completeness of the index set.

#References

[1] Bosonic topological phases of matter: bulk-boundary correspondence, SPT invariants and gauging. arXiv:1710.04730v1. https://arxiv.org/abs/1710.04730v1 [2] Anyon condensation and tensor categories. arXiv:1307.8244v7. https://arxiv.org/abs/1307.8244v7 [3] Measuring the Unique Identifiers of Topological Order Based on Boundary-Bulk Duality and Anyon Condensation. arXiv:2005.03236v4. https://arxiv.org/abs/2005.03236v4 [4] Higher-order bulk-boundary correspondence for topological crystalline phases. arXiv:1805.02598v2. https://arxiv.org/abs/1805.02598v2 [5] Defect bulk-boundary correspondence of topological skyrmion phases of matter. arXiv:2206.02251v2. https://arxiv.org/abs/2206.02251v2 [6] Bulk-boundary correspondence for three-dimensional symmetry-protected topological phases. arXiv:1512.09111v1. https://arxiv.org/abs/1512.09111v1 [7] Bulk-boundary correspondence in point-gap topological phases. arXiv:2205.15635v4. https://arxiv.org/abs/2205.15635v4 [8] Anyon condensation in mixed-state topological order. arXiv:2406.14320v4. https://arxiv.org/abs/2406.14320v4 [9] Nonabelian Anyon Condensation in 2+1d topological orders: A String-Net Model Realization. arXiv:2409.05852v2. https://arxiv.org/abs/2409.05852v2 [10] https://papers.qnfo.org/papers/braid-group-representations-modular-data-and-the-classification-of-majorana-zero/. QNFO: Braid Group Representations, Modular Data, and the Classification of Majorana Zero Mode Fusion Rules in 2D Topological Superconductors. [11] https://papers.qnfo.org/papers/gauge-invariant-field-theory-of-signal-worker-interactions/. QNFO: Gauge-Invariant Field Theory of Signal-Worker Interactions. [12] https://papers.qnfo.org/papers/operationalizing-generalized-symmetries/. QNFO: Operationalizing Generalized Symmetries. [13] https://papers.qnfo.org/papers/electron-hook-treatise/. QNFO: A Critical Treatise on the Load-Bearing Assumptions of Quantum Mechanics, Thermodynamics, and Computation.

#Appendix A. Divergence report

D1. Definition of the condensation index. Draft A proposed a "symmetrized" index $I = 0.25$ for the Ising condensation, obtained by averaging two expressions (1 and 0.5) and scaling by a fraction of confined fermions ($0.75 \times 1/3 = 0.25$). Draft B proposed the pair $\kappa = \dim(A)^{2} = D_{C}^{2}/D_{D}^{2}$ and $\mu = 2\Delta c$. Draft C proposed $I = D_{D}/D_{C}$ (a quantum-dimension ratio). These are different conventions for what "index" should measure: A aimed at a fraction of "bulk topological information transferred to the boundary" but required an ad hoc symmetrization to reach a fixed value; B aimed at separating condensate size from chiral anomaly; C aimed at surviving boundary quantum dimension. Resolution: the main text adopts B's index pair, augmented by C's dimension ratio and non-Abelian excess fraction $f_{\mathrm{NA}}$ as complementary diagnostics. A's symmetrized value $I = 0.25$ is rejected because its derivation is not invariant under alternative but equally natural conventions (A itself concedes the value "depends on a symmetrization step that mixes two plausible definitions").

D2. Child category of the Ising $\psi$-condensation. The independent drafts diverged on what the child of the fermionic condensation $A = 1 \oplus \psi$ in the Ising category actually is. Draft A held that the child is trivial (only the vacuum survives, since $\sigma$ is confined), giving $D_{D} = 1$ and $\kappa = D_{C}^{2}/D_{D}^{2} = 4$. Draft B argued that condensing the fermion of the Ising category yields a p + ip topological superconductor, whose intrinsic topological order is trivial ($c = 1/2$ with no deconfined anyons), so that the child as a topological order is again effectively trivial but carries a chiral central charge — consistent with Draft A's anyon content while explaining the nonzero $\mu$. Draft C speculated that the child might retain a nontrivial sector, but could not identify a deconfined candidate: every nontrivial Ising anyon has nontrivial monodromy with $\psi$. Resolution: the main text adopts the Draft A/B position — the child category is trivial as a braided category ($D = {1}, D_{D} = 1$), the chirality $c = 1/2$ survives as the boundary Majorana mode counted by $\mu = 1$, and $\kappa = \dim(A)^{2} = 4 = D_{C}^{2}/D_{D}^{2}$ holds here because the condensate is normal (the single surviving sector is the vacuum). Draft C's nontrivial-child proposal was rejected for lack of a deconfined candidate sector.