LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Finite generation by admissible handlebody-spine states

Statement

Let $(H, I_{Σ}, a)$ be a marked handlebody with prescribed boundary bands $I_{Σ}$ and boundary labels $a$ . Fix a finite banded relative spine $G \subset H$ , meaning a banded graph whose degree-one boundary vertices give exactly $I_{Σ}$ , whose interior vertices have degree two or three, and whose regular neighborhood is $H$ ; for the unmarked $3$ -ball, the empty graph is allowed. Let each edge carry a label in ${0, \dots, r - 2}$ , with every boundary half-edge on a band $I_{j}$ labeled by its prescribed color $a_{j}$ . Call a labeling admissible when labels agree across every interior $2$ -valent vertex and the three labels at each interior trivalent vertex satisfy the root fusion rule. In the radical-quotient state space $V_{p} (Σ)$ , the states represented by admissibly colored relative spines span, and the set of such labelings is finite. This statement asserts spanning, not linear independence.

Statement references

Handlebody state space

Root fusion rule

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§4–5

Kauffman and Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 4–7

Chen, The Temperley-Lieb Categories and Skein Modules, Chapter 6

Facts & Assumptions

Given: A finite marked handlebody spine and the root state-space construction.

[F1]

Skein TQFT state space and handlebody pairing defines the state space as a radical quotient of the specialized colored relative handlebody skein module with its prescribed boundary bands.

[F2]

Root fusion rules and completeness of the simple objects proves that the finite label set is ${0, \dots, r - 2}$ and gives the exact local fusion rule; it does not itself prove that arbitrary handlebody skeins are spanned by spine states.

[F3]

External input — actual relative handlebody quotient and spine basis. BHMV, Proposition 1.9 p. 891 and Theorem 4.11 p. 909, identify the radical-quotient relative handlebody model with the universal state space and give an orthogonal basis represented by the $p$ -admissibly colored fixed banded spine whose boundary is the prescribed colored banded link. For even $p = 2 r$ , the source's $q = (p - 2) / 2 = r - 1$ and colors $< q$ are exactly $0, \dots, r - 2$ ; $p$ -admissibility is the registered truncated fusion rule. Coefficients are specialized through the given complex primitive root, with no vanishing quantum denominator inverted. This imports the actual spanning theorem, rather than inferring it from abstract semisimplicity alone.

Proof

Given: The fixed finite relative spine $G$ , its prescribed boundary labels, and the finite label set ${0, \dots, r - 2}$ .

1.1givenF2construct

The edge set of $G$ is finite and each edge has only $r - 1$ candidate labels by [F2]. The prescribed boundary colors are fixed, so there are at most $(r - 1)^{∣ E (G) ∣}$ edge labelings; the admissible ones form a finite subset. If $G$ is empty, its unique labeling is the empty labeling.

2.1F1F2F3step 1.1∎

Apply [F3] to the supplied relative spine, using [F1] for the matching colored relative radical-quotient model and [F2] for its root labels and vertex rule. Every vector is a linear combination of the admissible relative-spine states. Together with the finite count in step 1.1 this proves the full Statement. Linear independence, although present in the imported theorem, is not needed for this spanning conclusion. No infinite skein reduction or additional choice campaign is performed.

Depends on

Used by

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Sources

C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§4–5 (standard reference, not scraped)
Louis H. Kauffman and Sostenes L. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 4–7 (standard reference, not scraped)
Joshua Chen, The Temperley-Lieb Categories and Skein Modules, Chapter 6 (standard reference, not scraped)
BHMV, Topological Quantum Field Theories Derived from the Kauffman Bracket, Topology 34 (1995), exact external inputs (standard reference, not scraped)
Gilmer–Masbaum, Maslov index, Lagrangians, Mapping Class Groups and TQFT, arXiv:0912.4706v3 (standard reference, not scraped)