1. Introduction: From Taiji to the Weave

Taiji Evolution Cosmology [1] holds that Taiji is the origin of the universe, giving rise to the two aspects, yin and yang. Countless pairs of yin and yang interweave into a network: we call each entangled pair a knot, and the way in which knots connect is the weave. The weave determines physical law—gravity, spacetime, and matter all emerge from the network. The task of this paper is to turn the "weave" from metaphor into mathematics.

Structure of the paper: Section 2 defines the weave; Section 3 gives the Weaving Formula; Section 4 lists the assumptions; Section 5 proves the fundamental theorems (area law, region intrinsicality, thermodynamic-limit saturation, universal mass gap); Section 6 presents the numerical verification; Section 7 discusses the boundaries of the Weaving Formula; Section 8 concludes.

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2. Definition of the Weave

Definition 1 (Weave): A weave is a charged network: W = (V, E, T)—V is the vertex set, E the edge set, and T the vertex-tensor family. Each edge e ∈ E carries a charge a_e ∈ G (G the charge space, a finite group). For a vertex v ∈ V, let ∂v := {e ∈ E : v ∈ e} denote the set of edges incident to v, let the valence be d_v := |∂v| (trivalent means d_v ≡ 3), and let G^{∂v} := {b : ∂v → G} be the space of charge assignments on the edges incident to v. The vertex tensor T_v : G^{∂v} → ℂ takes as its input the charge configuration on the edges incident to v; legs correspond one-to-one with incident edges and are not numbered.

The vertex tensor satisfies two constraints (for each v ∈ V):

1. Charge conservation: Tv ≠ 0 only if the group sum of the charges on all edges incident to v vanishes, ⊕{e∈∂v} a_e = 0;
2. Normalization: Σ_{(ae){e∈∂v}} |T_v((ae){e∈∂v})|² = 1.

The definition holds for arbitrary charge space G and arbitrary valence (different vertices may have different valences), independent of the particular group and valence chosen.

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3. The Weaving Formula

Definition 2 (Weaving Formula): The Weaving Formula defines the quantum state of the network—a superposition over all charge configurations whose amplitudes are products of vertex tensors. A charge configuration is an assignment of charges to edges. The set of all edge-charge configurations is the function space G^E := {a : E → G} (a assigns to each edge e ∈ E a charge a_e ∈ G); |a⟩ is the quantum basis vector corresponding to the configuration a; a|∂v ∈ G^{∂v} is the restriction of the configuration a to the edges ∂v incident to v, i.e., the input of T_v:

|Ψ[W]⟩ = Σ_{a ∈ G^E} ( Π_{v∈V} T_v(a|∂v) ) |a⟩
Z[W] = ⟨Ψ[W]|Ψ[W]⟩ = Σ_{a ∈ G^E} | Π_{v∈V} T_v(a|∂v) |²

The sum runs over all edge-charge configurations (charge conservation is enforced automatically by the support of the T_v); the product runs over all vertices v. Z[W] is the partition function. The expression holds for arbitrary charge space G and arbitrary valence—the Weaving Formula is abstract: legs are labelled by incident edges and not numbered, the valence d_v = |∂v| is encoded in the dimension of the domain of T_v, and the specific group is merely a parameter inserted into it.

Relationship to the ground state: The Weaving-Formula state |Ψ[W]⟩ is the string-net ground state (the TV state): the allowed configurations are jointly selected by vertex charge conservation (the δ term of the tensor) and by the flat-face constraints (Hamiltonian realization of the constraints and numerical confirmation in §5 and §6; on K4 the overlap between |Ψ[W]⟩ and the ground state is 1.00000000 exactly).

Construction process: Given the weave data W = (V, E, T)—

1. assign a charge a_e ∈ G to each edge, obtaining an edge-charge configuration a ∈ G^E;
2. the charge-conservation constraint (the ⊕ of the charges on the edges incident to each vertex vanishes) selects the allowed configurations;
3. the amplitude of each allowed configuration is the product of the vertex tensors Π_{v∈V} T_v(a|∂v);
4. the network quantum state is the superposition of all allowed configurations |Ψ[W]⟩ = Σ{a∈G^E} (Π{v∈V} T_v(a|∂v)) |a⟩;
5. the partition function Z[W] = ⟨Ψ[W]|Ψ[W]⟩ is the sum of squared amplitudes.

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4. Assumptions

The Weaving Formula is abstract (Definition 2: arbitrary charge space, arbitrary valence); making it concrete requires choosing the charge space and the valence, and giving the explicit form of the vertex tensor at the chosen valence. The assumptions of our universe are:

• （G1）Charge space: G = Z₂³ (an eight-element finite abelian group). This is the sole assumption-type input; replacing the group leaves the form of the formula unchanged, only the value of lnD changes (verified by multi-weave universality scans).
• （G2）Valence and the explicit form of the vertex tensor: valence n = 3 (trivalent vertices). A point in parameter space, not a derived result; quartic valence and others are equally viable (verified numerically, with the tensor form changed accordingly). Under the trivalent setting the vertex tensor takes the explicit form:

T⁽ᵛ⁾_abc = δ_{a⊕b⊕c, 0} · t_ab · ω(a,b,c)

where the δ term is the charge-conservation constraint (T nonzero only if a⊕b⊕c = 0); t_ab is the vertex amplitude matrix (normalized, see G3); ω is the phase data (a 3-cocycle, see G4).

• （G3）Vertex amplitude: tab (the amplitude matrix of the trivalent vertex tensor, depending on the edge charges a, b; vertex-wise normalization Σ{a,b}|t_ab|² = 1, see Constraint 2 of §2). In the principal case (TV state / string-net ground state) the amplitude is taken to be uniform, t_ab ≡ 1/8, which automatically satisfies the normalization.
• （G4）Phase: ω—a 3-cocycle satisfying the pentagon equation, with ω ∈ H³(Z₂³, U(1)) ≅ (Z₂)⁷, 128 independent choices in total. The pentagon equation is the explicit form of the 3-cocycle condition δω = 1 for abelian groups (a five-term identity, for arbitrary a, b, c, d ∈ Z₂³):

ω(b,c,d) · ω(a,b⊕c,d) · ω(a,b,c⊕d) = ω(a⊕b,c,d) · ω(a,b,c)

Substituting the assumptions G1–G4 into Definition 2, the Weaving Formula of this paper takes the following complete form. Under the trivalent setting each vertex v has exactly three incident edges; fix their numbering ∂v = {e{v,1}, e{v,2}, e{v,3}}, so that the configuration a|∂v corresponds one-to-one with the ordered triple (a{e{v,1}}, a{e{v,2}}, a{e_{v,3}})—the a, b, c in the G2 tensor element are precisely the charges of the three edges incident to that vertex. Substituting into Definition 2, the network quantum state (keeping the tensor layer):

|Ψ[W]⟩ = Σ_{a ∈ (Z₂³)^E} ( Π_{v ∈ V} T⁽ᵛ⁾_{a_{e_{v,1}}, a_{e_{v,2}}, a_{e_{v,3}}} ) |a⟩

Partition function:

Z[W] = ⟨Ψ[W]|Ψ[W]⟩ = Σ_{a ∈ (Z₂³)^E} Π_{v ∈ V} |T⁽ᵛ⁾_{a_{e_{v,1}}, a_{e_{v,2}}, a_{e_{v,3}}}|²

Substituting further the explicit form of G2 (with a, b, c replaced by the corresponding edge charges) into the tensor layer, we obtain the fully expanded form:

|Ψ[W]⟩ = Σ_{a ∈ (Z₂³)^E} ( Π_{v ∈ V} δ_{a_{e_{v,1}} ⊕ a_{e_{v,2}} ⊕ a_{e_{v,3}}, 0} · t_{a_{e_{v,1}}, a_{e_{v,2}}} · ω(a_{e_{v,1}}, a_{e_{v,2}}, a_{e_{v,3}}) ) |a⟩

Fully expanded partition function (|ω| = 1, so the phase does not contribute to the weight):

Z[W] = Σ_{a ∈ (Z₂³)^E} Π_{v ∈ V} δ_{a_{e_{v,1}} ⊕ a_{e_{v,2}} ⊕ a_{e_{v,3}}, 0} · |t_{a_{e_{v,1}}, a_{e_{v,2}}}|²

where each edge charge a_e ∈ G = Z₂³ (G1); the tensor element of a vertex is the explicit form of G2 (G2–G4):

T⁽ᵛ⁾_abc = δ_{a⊕b⊕c, 0} · t_ab · ω(a,b,c),  a, b, c ∈ Z₂³

The three factors each play their own role:

• δ_{a⊕b⊕c, 0} (charge-conservation projector): the tensor is nonzero only if the group sum of the three legs vanishes; the allowed edge-charge configurations are selected automatically by the δ of each vertex (Constraint 1 of §2);
• t_ab (vertex amplitude, G3): normalized amplitude matrix, vertex-wise normalization Σ_{a,b}|t_ab|² = 1 (Constraint 2 of §2); in the principal case (TV state / string-net ground state) the amplitude is uniform, t_ab ≡ 1/8 (equal-weight superposition; the normalization is then automatic);
• ω(a,b,c) (phase data, G4): a 3-cocycle satisfying the pentagon equation; modulo coboundaries ω ∈ H³(Z₂³,U(1)) ≅ (Z₂)⁷, 128 independent choices.

This formula is the common starting point of all the fundamental theorems of §5, and the object of the item-by-item numerical evaluation in §6.

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5. Fundamental Theorems

The theorems of this section are all derived directly from the Weaving Formula (Definition 2).

5.1 Theorem 1 (Area law, f = L−1)

Theorem: For the Turaev–Viro (TV) state [3] (flat, equal-weight superposition—"ground state" refers to the ground state of the Levin–Wen string-net Hamiltonian [2] H(W) = −Σ_v A_v − Σ_p B_p, where A_v is the vertex charge-conservation projector and B_p the flat-face projector; numerical confirmation in §6), under a simple cut (the boundary of the region is a single closed curve), the bipartite entanglement entropy is:

S(A) = (L−1)·lnD

where L is the number of cut edges and D = |G| = 8 is the bond quantum dimension.

Proof (four steps):

1. The boundary of region A consists of L cut edges, each carrying a charge ∈ Z₂³;
2. Total-flux conservation: summing the flat-face constraints of all faces inside region A, interior edge charges cancel pairwise, leaving a single global constraint Σ⊕(boundary charges) = 0 on the boundary charges (guaranteed by the spherical topology and the single closed boundary curve); hence the number of independent boundary charges is L−1;
3. Property of the TV state: given the independent boundary charges, the interior of the region is determined uniquely by the face constraints—the Schmidt vectors are classified by the boundary charges and are orthogonal;
4. Hence the Schmidt rank is 8^(L−1) (8 choices per boundary charge), and the entropy is S = ln(rank) = (L−1)·ln8. ∎

Numerical verification (all three graphs green):

Graph	Cut	L	f measured
K4	single vertex {0}	3	2
triangular prism	upper triangle {0,1,2}	3	2
cube	half {0,1,2,3}	4	3

5.2 Theorem 2 (Region-intrinsic entropy, f = E_B − F_B)

Theorem: For an arbitrary cut, the number of degrees of freedom of the bipartite entropy of the TV state is:

f = E_B − F_B = V_B − c₀(B)

where B is the complementary region of the cut: E_B = number of edges interior to B, F_B = number of faces interior to B, V_B = number of vertices in B, and c₀(B) = number of connected components of the induced subgraph of B.

Sketch of proof:

1. Decompose the TV state by Schmidt decomposition encoded on the B side; the number of nonzero eigenvalues of ρ_A equals the number of reachable B-side configurations;
2. The independent degrees of freedom on the B side are the interior edge charges of B subject to the face constraints inside B (one independent constraint per face);
3. The number of independent degrees of freedom is E_B − F_B (for connected B the face constraints are independent; the Euler characteristic χ = 1 guarantees no redundancy);
4. The Euler formula V_B − E_B + F_B = c₀(B) gives f = E_B − F_B = V_B − c₀(B). ∎

Numerical verification: all 18 data points (every cut of K4 / triangular prism / cube) match; an independent cross-check by linear algebra (rank computation over GF(2)) is fully green. f = L−1 is the special case (for the cut-complementary region E_B = L, F_B = 1).

5.3 Theorem 3 (Thermodynamic-limit saturation, c_∞ = 1)

The area-law coefficient c(L) = (L−1)/L: K4 (L=3 → c=2/3), triangular prism (3 → 2/3), cube (4 → 3/4) → thermodynamic limit:

c_∞ = 1

Twofold confirmation: (1) extrapolation of f = L−1; (2) the SVD entropy of the TRG fixed-point tensor is ln8 exactly.

Significance of the theorem: the area-law entropy S = c(L)·L·lnD (c(L) = (L−1)/L is the area-law coefficient, see above), divided by the horizon area A = L·a_P², gives the entropy density η = S/A = c(L)·lnD/aP²; in the thermodynamic limit c∞ = 1 and the entropy density saturates at η_∞ = lnD/a_P² (a_P is the physical scale of a bond, see Corollary 3).

5.4 Theorem 4 (Universal mass-gap law)

Theorem: The mass gap of the TRG fixed point of the group-constrained tensor network (valid for |G| ≥ 3):

m* = −ln((|G|−2)/(|G|−1))

depends only on the group order |G|, independent of the group structure (abelian/nonabelian). Domain of validity |G| ≥ 3: for |G| = 2 the argument (|G|−2)/(|G|−1) = 0, so m* = −ln 0 → +∞ diverges, outside the domain of this law (Z₂/toric-code-type string nets do have a gap, but it is not described by this formula); the principal group Z₂³ of this paper, with |G| = 8, is unaffected.

Numerical verification:

Group	Order	m* measured	m* predicted	Deviation
Z₃	3	0.693147	0.693147	0.00%
Z₅	5	0.287682	0.287682	−0.00%
Z₆	6	0.223144	0.223144	−0.00%
Z₇	7	0.182322	0.182322	+0.00%
Z₈	8	0.154151	0.154151	−0.00%
S₃ (nonabelian)	6	0.223144	0.223144	−0.00%

The nonabelian group S₃ gives exactly the same spectrum as the abelian group of the same order—the mass gap depends only on |G|, independent of the group type.

The corollaries below are all direct consequences of the theorems above: Corollaries 1 and 2 follow from Theorem 1 (horizon and discreteness applications of the area law); Corollary 3 combines Corollary 1 and Theorem 3 (saturation coefficient).

5.5 Corollary 1 (Logarithmic correction to black hole entropy)

Corollary 1: Taking the horizon to be a simple closed surface (L bonds), Theorem 1 gives the horizon entropy

S_BH = (L−1)·ln8 = A/(4G) − ln8

—an exact identity with no higher-order corrections: taking the horizon entropy in the Bekenstein–Hawking form S = A/(4G) (here G denotes Newton's gravitational constant, to be distinguished from the notation for the charge space) is the external physical junction (belonging, together with the l_P input of Corollary 3, to the boundary of the single-weave zero-parameter principle); A/(4G) = L·ln8 is guaranteed exactly by the bond-scale relation a_P² = 4G·ln8 (Corollary 3). The term −ln8 is the logarithmic correction—a testable prediction.

5.6 Corollary 2 (Entropy quantization)

Corollary 2: The bipartite entropy of the TV state takes only the discrete values (L−1)·ln8—the "charge-counting" structure of the entanglement entropy.

5.7 Corollary 3 (Physical scale of the bond)

Corollary 3: The external Bekenstein–Hawking junction S = A/(4G) of Corollary 1 equals the horizon entropy S_BH = (L−1)·ln8 of Theorem 1 exactly; under thermodynamic-limit saturation (Theorem 3) the leading term gives A/(4G) = L·ln8. Taking the horizon area A = L·a_P² (L bonds, each of area a_P²), substitution gives a_P² = 4G·ln8; in natural units (ħ = c = 1), Newton's constant G = l_P², so:

a_P = 2√(ln8)·l_P ≈ 2.884·l_P ≈ 4.66×10⁻³⁵ m

i.e., a_P² = 4·ln8·l_P². The dimensionless number 2√ln8 is determined entirely by the weave structure—ln8 comes from the bond quantum dimension D = 8, and the coefficient 4 from the horizon entropy formula—with no fitting parameters whatsoever. The weave outputs the ratio a_P/l_P; the absolute scale l_P is supplied by the external universe (the boundary of the single-weave zero-parameter principle).

6. Numerical Verification

All theorems of this paper are verified numerically to exact precision (to machine precision or finite-difference precision):

1. TV state and the area law: all cuts of K4 / triangular prism / cube, 18 data points in total, f = E_B − F_B matches in every case (Theorems 1 and 2);
2. TRG fixed points: the mass-gap law for 6 groups (Theorem 4) matches exactly;
3. Ground state = Weaving-Formula state: exact diagonalization of the string-net Hamiltonian H(W) = −Σ_v A_v − Σ_p B_p on the trivalent K4 graph (4 vertices, 6 edges, 4 triangular faces, sphere) (conserved subspace of dimension 512 = 8³, with 6 edge charges subject to 4 face constraints − 1 global redundancy = 3 independent constraints): the ground-state energy E0 = −8.000000 (= −(4 vertices + 4 faces)) is exact; the ground state is unique (sphere degeneracy 1); the ground state = TV state = Weaving-Formula state |Ψ[W]⟩ (Definition 2), with overlap 1.00000000 exact; the energy gap = 2.0 (a hallmark of topological order).

Details of the numerical methods:

6.1 Weave tensor and TRG: under the trivalent setting (§4 G2) the vertex tensor Tabc (3-leg) satisfies charge conservation δ{a⊕b⊕c,0}; coarse-graining two vertex tensors yields the 4-leg tensor A[i,j,k,l], and charge conservation is preserved—nonzero only if i⊕j⊕k⊕l = 0, i.e., the fourth leg charge is determined by the first three (l = i⊕j⊕k; assigned item by item in the numerics, see the mass-fixed-point verification script); TRG coarse-graining uses a corrected einsum (truncation-dimension pairing a↔c, b↔d; a dummy-index bug in an early version was fixed and cross-verified).

6.2 Exact diagonalization: on K4 (4 vertices, 6 edges, 4 triangular faces, sphere), the conserved subspace has dimension 512 = 8³ (6 edge charges subject to 4 face constraints − 1 global redundancy = 3 independent constraints); dense diagonalization (numpy.linalg.eigh); conserved configurations are generated by Gaussian elimination over the face constraints.

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7. Boundaries

1. The minimality of the assumptions G1–G4 means "no redundancy inside the formula"; it does not answer "why Z₂³"—that choice is supplied by the upstream philosophical framework [1];
2. The continuum limit (bond scale → continuum field theory) lies outside the scope of this paper—it is an open problem of weave dynamics.

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8. Conclusion

This paper turns the "weave" from a philosophical claim into mathematical theorems: a weave is a charged network W = (V, E, T); the Weaving Formula defines its quantum state; the assumptions G1–G4 are minimal (G = Z₂³ is the sole structural choice); the formula directly yields four fundamental theorems—the area law (f = L−1), region-intrinsic entropy (f = E_B − FB), thermodynamic-limit saturation (c∞ = 1), and the universal mass-gap law (m* = −ln((|G|−2)/(|G|−1)), |G| ≥ 3)—all verified to exact numerical precision. Starting from the Weaving Formula, variant Weaving Formulas for gauge fields can in principle be derived, together with a unified derivation of the gravitational, Maxwell, Schrödinger, and Yang–Mills equations, and the geometric origin of the cosmological constant.

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References

[1] Liu, Xinkuang. Taiji Evolution Cosmology: A Philosophical Framework of Monistic Cosmology. Zenodo, DOI: 10.5281/zenodo.22165784.

[2] Levin, M. A., Wen, X.-G. String-net condensation: A physical mechanism for topological phases. Phys. Rev. B 71, 045110 (2005).

[3] Turaev, V. G., Viro, O. Y. State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31, 865–902 (1992).

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